# Influence of preferential diffusion on the distribution of species in lean
${\mathrm{H_{2}}}$-air laminar premixed flames at different equivalence ratios

Frederick W Young  Umair Ahmed<EMAIL_ADDRESS>Nilanjan
Chakraborty  School of Engineering, Newcastle University, Newcastle upon Tyne
NE1 7RU,
United Kingdom

###### Abstract

The influence of equivalence ratio on preferential diffusion effects and the
resulting changes in the distributions of major species and their reaction
rates have been analysed based on 2D simulations of lean
${\mathrm{H_{2}}}$-air laminar premixed flames, at $\phi=0.4$ and $0.7$. The
enhancements of burning rate, flame surface area, and stretch factor increase
with decreasing equivalence ratio with the increase in stretch factor
particularly prominent when the burning rate and flame area are evaluated
based on normalised mass fraction variation of ${\mathrm{H_{2}}}$. The
preferential diffusion effects have been demonstrated to lead to significant
deviations of mass fractions of major species and their reaction rates from
the corresponding 1D unstretched laminar premixed flame solution and local
variations of equivalence ratio. This tendency is particularly strong for
${\mathrm{H_{2}}}$ among all the major species. Moreover, mass fractions of
${\mathrm{O_{2}}}$ and ${\mathrm{H_{2}O}}$ are found to assume super-adiabatic
values at the super-adiabatic temperature zones and this trend is particularly
strong for the $\phi=0.4$ case. It has been demonstrated that the deviations
of major species mass fractions and their reaction rates from their
corresponding 1D unstretched laminar premixed flame values arise principally
due to preferential diffusion effects induced by relative
focussing/defocussing of species and heat at the positively and negatively
curved regions with the nature of the deviations being opposite to each other
depending on the sign of the curvature. The variations of normalised species
mass fractions of $\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$ are found to be
significantly affected by the local equivalence ratio within the 2D laminar
flame with $\phi=0.4$ but these effects weaken with an increase in global
equivalence ratio.

###### keywords:

Premixed laminar flame , ${\mathrm{H_{2}}}$-air premixed combustion ,
Preferential diffusion , Curvature , Equivalence ratio

##  1 Introduction

Gas turbine engine technologies relying on lean premixed combustion of
hydrogen-based fuels are promising in the pursuit of improved sustainability
in power generation and propulsion sectors. No carbon dioxide is produced from
the combustion of pure hydrogen, while limiting to lean conditions reduces the
production of NOx and reduces the flame speed, which is important due to the
propensity for flashback in premixed hydrogen flames. It is well-known that
preferential diffusion rates of heat and species due to non-unity Lewis number
can lead to thermo-diffusive instability in premixed flames under certain
conditions [1, 2]. In the case of hydrogen-air mixtures, lean flames can be
characterised by an effective Lewis number, $Le_{eff}$, much smaller than
unity. Effective Lewis number is essentially a weighted average of the Lewis
numbers of the reactants, with greater weighting given to the deficient
reactant [3]. Thermodiffusive instabilities arise in the cases with
sufficiently small values of the effective Lewis number [1, 2, 3, 4].
According to Bechtold and Matalon [4], the effective Lewis number $Le_{eff}$
of ${H_{2}}$-air is defined as:
$Le_{eff}=1+[(Le_{\mathrm{O_{2}}}-1)+(Le_{\mathrm{H_{2}}}-1)\Xi]/(1+\Xi)$
where $\Xi=1+Ze(1/\phi-1)$, with $Le_{\mathrm{O_{2}}}>1.0$ and
$Le_{\mathrm{H_{2}}}<<1.0$; the Lewis numbers of ${\mathrm{H_{2}}}$, and
${\mathrm{O_{2}}}$ and $Ze$ being the Zel’dovich number. This implies that
$Le_{eff}$ is expected to decrease with decreasing equivalence ratio ($\phi$).
Therefore, the likelihood of thermo-diffusive instability increases as $\phi$
decreases for lean ${\mathrm{H_{2}}}$-air flames.

In previous studies of laminar two-dimensional lean premixed hydrogen-air
flames [5, 6, 7, 8, 9, 10, 11], thermo-diffusive instabilities have been shown
to amplify perturbations applied to the flame front, leading to the
development of a highly wrinkled flame, with high flame surface area and
burning rate. For developed wrinkled flames a broad range of curvature values
are observed along the flame front, and for sufficiently lean flames
($\phi<<1$) a comparison of heat release statistics between positive and
negatively curved regions reveals local extinction in flame cusps
characterised by a strongly negative curvature (here, flame normal is defined
to point into the unburned gas and flame surface elements convex (concave)
towards the reactants have positive (negative) values of curvature). The
empirical evidence suggests a strong influence of curvature (and by extension
strain and stretch rates) on the local flame behaviour, as suggested by
Markstein [12].

Recently, it has been shown based on three-dimensional Direct Numerical
Simulations (DNS) of turbulent hydrogen-air flames that the choice of the
species used to define the flame surface can significantly influence the
evaluation of the flame surface area [13]. In the context of Reynolds Averaged
Navier-Stokes (RANS) and Large Eddy Simulations (LES) turbulent combustion
models often make use of a reaction progress variable to quantify the
conversion of fresh reactants to burned products. Commonly used flamelet-based
closures [14, 15, 16] need the knowledge of a reaction progress variable in
terms of normalised mass fractions of major species so that the reaction
progress variable increases monotonically from $0.0$ in the unburned gases to
$1.0$ in fully burned products. The choice of the reaction progress variable
depends on the distributions of major species and their responses to thermo-
diffusive instability in lean ${\mathrm{H_{2}}}$-air flames which is yet to be
fully understood. Therefore, it is necessary to analyse the behaviour of
normalised major species mass fractions in lean ${\mathrm{H_{2}}}$-air flames
for different equivalence ratios to understand the influence of thermo-
diffusive instability on species distributions within the flame front. In this
respect, the main objectives for this work are: (i) to investigate the
variation of major species across lean hydrogen-air laminar premixed flames to
ascertain the choice of the species relevant for defining an appropriate
progress variable under fuel-lean conditions, (ii) to investigate the
influence of preferential diffusion on the species distribution and (iii) to
investigate the influence of the choice of the species on the evaluations of
flame surface area and the stretch factor, $I_{0}$ [17]. To achieve the
aforementioned objectives, numerical simulations for two-dimensional laminar
premixed hydrogen-air flames at two equivalence ratios, $\phi=0.4$ and $0.7$,
have been performed with a simulation setup similar to that of Berger et al.
[6]. It was demonstrated by Berger et al. [9] using dispersion relations that
${\mathrm{H_{2}}}$-air flames with $\phi=0.4$ are thermo-diffusively unstable
(with $Le_{eff}\approx 0.34$ [9]), whereas ${\mathrm{H_{2}}}$-air flames with
$\phi=0.7$ (with $Le_{eff}\approx 0.5$ [9]) are nominally thermo-diffusively
neutral to stretch effects [18], and preferential diffusion effects are
expected to be weaker in the $\phi=0.7$ case than in the $\phi=0.4$ case [9].
Thus, these two equivalence ratios have been chosen to compare and contrast
the effects of $\phi$ variation in lean ${\mathrm{H_{2}}}$-air premixed
combustion.

The rest of the paper is organised as follows. In the next section details of
the numerical simulations are provided, which is followed by the results and
the corresponding discussion. Finally, the findings are summarised and the
conclusions are drawn in the final section of the paper.

##  2 Numerical simulations

A fully compressible direct numerical simulation code, SENGA2 [19] is used to
perform the simulations. The code solves conservation equations for mass,
momentum, energy and species in time and space. A 10th-order central finite
difference scheme (gradually reducing to fourth-order at non-periodic
boundaries) is used to calculate spatial derivatives. Time advancement is
performed using a low-storage fourth-order explicit Runge-Kutta scheme. In the
simulations, the thermophysical properties such as specific heat, are assumed
to be temperature-dependent and are evaluated using CHEMKIN polynomials.
Molecular transport is implemented using a mixture-averaged approach, which
includes the Soret and Dufour effects, and is used for the calculation of
viscosity, species diffusivity, thermal conductivity and thermal diffusion
ratio. The skeletal chemical mechanism for hydrogen-air combustion proposed by
Burke et al. [20], which has 9 species and 23 reversible reactions, has been
employed for the simulations considered here.

Simulations of 2D flames at two different equivalence ratios have been
performed, $\phi=0.4$ and $\phi=0.7$ under atmospheric conditions with an
unburned gas temperature of $T_{u}=300$K. A domain length of
$L_{x}=450\delta_{th}$ and a domain width of $L_{y}=260\delta_{th}$ is used in
both cases. A large domain width is chosen to allow for the development of a
sufficient number of flame structures within the domain to ensure that domain
size-independent behaviour is observed, whilst a large domain length is chosen
to allow sufficient room for the flame to propagate along the domain,
considering the expected enhancement of flame propagation speed due to the
emergence of flame instabilities [11]. The domain for $\phi=0.4$ has been
discretised using 6720 $\times$ 3840 points, whilst for $\phi=0.7$ the domain
has been discretised using 6640 $\times$ 3840 points. The resolution
requirement for both cases is determined by ensuring at least 14 grid points
are used to resolve the thermal flame thickness, $\delta_{th}$. Periodic
boundary conditions have been applied in the $y$-direction, with an inflow
boundary condition imposed at $x=0$ and a non-reflecting outflow boundary
condition imposed at $x=L_{x}$. Navier-Stokes Characteristic Boundary
Conditions proposed by Lodato et al. [21] are used for specifying the
conditions at the inflow and outflow boundaries within the simulations. In
order to trigger the unstable flame response for the analysis of the developed
flame, an initial sinusoidal perturbation is applied in the $y$-direction to
the initial fields, which themselves are taken from the steady-state 1D
unstretched laminar flames. Low amplitude random numerical noise is imposed on
the wave amplitude to ensure the development of non-repeating, asymmetric
flame structures. The simulations are run until the burning rate and flame
area has reached a quasi-steady state.

Figure 1: Distributions of $c_{T}$ fields for the developed flames with (left)
$\phi$ = 0.4 and (right) $\phi$ = 0.7

##  3 Results & Discussion

Figure 1 shows the distributions of instantaneous non-dimensional temperature
$c_{T}=(T-T_{u})/(T_{ad}(\phi)-T_{u})$ fields for the two cases considered.
Figure 1 shows a strongly corrugated flame front for both cases. This flame
wrinkling gives rise to a significant increase in the flame area. In the case
of $\phi=0.4$ case, super-adiabatic flame temperatures (i.e., $c_{T}>1.0$) can
be seen where the flame is convexly curved towards the unburned mixture,
whereas super-adiabatic temperatures are rarely obtained in the $\phi=0.7$
case. In both cases, comparatively small temperature values (especially
$c_{T}<1.0$ in the burned gas region) are obtained in the wake of the
concavely curved regions. For flames with $Le_{eff}<1$ focusing of reactants
takes place faster than the defocusing of heat at the positively curved zones
[22, 23]. This gives rise to the simultaneous occurrence of high reactant
concentration and temperature, leading to high fuel consumption and heat
release rates at the positively curved zones in the flames with $Le_{eff}<1$.
This mechanism strengthens with decreasing $Le_{eff}$ [22, 23]. Thus, super-
adiabatic temperatures are more prevalent in the $\phi=0.4$ case than in the
$\phi=0.7$ case. The negatively curved regions in the $Le_{eff}<1$ flames are
associated with high defocusing of reactants and weak focusing of heat, and
this combination reduces the fuel consumption and heat release rates in these
zones [22, 23], which reduces the temperature at the negatively curved zones
in both cases but this trend strengthens with decreasing $Le_{eff}$ [22, 23].
Hence, low temperature wakes in the burned gas are more prominent in the
$\phi=0.4$ case than in the $\phi=0.7$ case. Figure 1 also shows that
positively curved flame segments are interrupted by sharp cusps with strong
negative curvatures for flames with both values of $\phi$. All of the features
of flame wrinkling mentioned above are consistent with previous findings for
$Le_{eff}<1$ flames [23, 24, 6, 11, 8].

The thermo-diffusive instability enhances the burning rate [1, 2, 6, 11, 8],
which can be quantified in terms of the burning velocity $S_{f}$, defined in
two dimensions in the following manner based on species $\alpha$:

$S_{f}=\frac{1}{\rho_{u}(Y_{\alpha,b}-Y_{\alpha,u})L_{y}}\int_{0}^{Ly}\int_{0}^{Lx}\dot{\omega}_{\alpha}\
dx\ dy,$ (1)

where $\rho_{u}$ is unburned gas density, $L_{y}$ is the span-wise domain
dimension and $\dot{\omega}_{\alpha}$ is the reacion rate of species $\alpha$.
It has been found that $S_{f}/S_{L}$ assumes greater values
($S_{f}/S_{L}\approx 3.5$ and $2.0$ in the $\phi=0.4$ and $0.7$ cases,
respectively) in the $\phi=0.4$ case than in the $\phi=0.7$ case as a result
of stronger thermo-diffusive effects induced by preferential diffusion in the
$\phi=0.4$ case, which results in a larger increase in reactivity relative to
that of the 1D unstretched laminar flame. This is consistent with previous
findings by Berger et al. [11, 9].

Figure 2: Magnified views of local normalised equivalence ratio $\phi$ in the
2D $\phi=0.4$ (left) and $\phi=0.7$ (right) laminar flame cases. Figure 3:
Mean values of $c_{\alpha 1}$ and $c_{\alpha 2}$ based on $\mathrm{H_{2}}$,
$\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$ mass fractions conditioned upon $c_{T}$
for both the original definition of $c_{\alpha}$ (denoted as $c_{\alpha 1}$),
and the local mixture fraction dependent definition of $c_{\alpha}$ (denoted
as $c_{\alpha 2}$). The error bars indicate standard deviations of each
$c_{\alpha}$ conditioned upon $c_{T}$.

In addition to the enhancement of burning velocity and the curvature
dependence of temperature, the preferential diffusion effects alter the
distribution of the species in the lean $\mathrm{H_{2}}$-air flames. This
variation of species can lead to variation in local equivalence ratio,
$\phi=(\xi(1-\xi_{st}))/((1-\xi)\xi_{st})$ (where $\xi$ is the local mixture
fraction and $\xi_{st}$ is the stoichiometric mixture fraction), as reported
by Berger et al. [11]. Here, the mixture fraction is defined as
$\xi=(\beta-\beta_{O})/(\beta_{f}-\beta_{O})$, where
$\beta=2Y_{C}/W_{C}+0.5Y_{H}/W_{H}-Y_{O}/W_{O}$,
$\beta_{f}=(2a+0.5b-c)/W_{C_{a}H_{b}O_{c}}$ (where $a=0$, $b=2$ and $c=0$),
and $\beta_{O}=-Y_{O\infty}/W_{O}$ with $Y_{O\infty}$ being the elemental mass
fraction of oxygen in the pure oxidiser stream, and $Y_{\alpha}$ and
$W_{\alpha}$ are mass fraction and molecular mass of the element or species
$\alpha$, respectively [25]. Figure 2 shows the local variation of $\phi$
across the flame for the two cases considered here, where the local $\phi$
values have been normalised by the global equivalence ratio for each case
($\phi=0.4$ and $0.7$ respectively) to aid comparison. It can be seen from
Fig. 2 that the local variation in $\phi$ across the flame is much higher in
the case of $\phi=0.4$ than in the $\phi=0.7$ flame which is consistent with
the earlier findings of Berger et al. [11]. The observed differences between
the positively and negatively curved regions in Fig. 2 for $\phi=0.4$ case are
indicative of the strong preferential diffusion effects induced by flame
curvature, which affect the local fuel consumption rate and cause significant
variations in local equivalence ratio. While a departure from the global
equivalence ratio (and corresponding mixture fraction value) is expected
across the flame due to the conversion of reactants to products via
intermediate species, significant differences in local equivalence ratio are
observed between positive and negative curvature regions for both cases, with
lower $\phi$ regions observed adjacent to negatively curved flame segments,
and higher $\phi$ regions found adjacent to positive curvature flame segments.
This is especially visible on the burned gas side of the flame for $\phi=0.4$,
where local equivalence ratios are significantly greater than the global value
of $\phi=0.4$ behind positively curved segments. At positive curvatures, a
relative increase in equivalence ratio with higher reactivity supports the
assertion that positively curved regions display enhanced burning rates.
Conversely, low equivalence ratio values seen in negatively curved regions
indicate that these regions have smaller concentrations of fuel. The
combination of strong defocusing of reactants and weak focusing of heat leads
to local flame extinction in the negatively curved cusps. This local
extinction behaviour in the negatively curved cusps for $\phi=0.4$ was
previously reported by Berger et al. [11].

Figure 4: Mean values of normalised reaction rate of $c_{\alpha 1}$ (i.e.,
$\dot{\omega}_{c}$) based on $\mathrm{H_{2}}$, $\mathrm{O_{2}}$ and
$\mathrm{H_{2}O}$ mass fractions, conditioned upon $c_{T}$. Normalisation of
$\dot{\omega}_{c}$ and $\dot{\omega}_{T}$, has been performed using the
maximum values corresponding to unstretched 1D laminar premixed flames.

In order to examine the preferential diffusion effects on the reaction zone
structure, the variations of the mean values of the progress variable defined
using two definitions,
$c_{\alpha_{1}}=(Y_{\alpha}-Y_{\alpha,u})/(Y_{\alpha,b}-Y_{\alpha,u})$ and
$c_{\alpha_{2}}=(Y_{\alpha}-Y_{\alpha}(\xi)_{u})/(Y_{\alpha}(\xi)_{b}-Y_{\alpha}(\xi)_{u})$,
based on major species mass fractions and their normalised reaction rates
(i.e., $\dot{\omega}_{c}=\dot{\omega}_{\alpha}/(Y_{\alpha,b}-Y_{\alpha,u})$
for $\alpha=\mathrm{H_{2}},\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$) conditioned
upon $c_{T}$ are shown in Fig. 3 and Fig. 4, respectively for both $\phi=0.4$
and $0.7$. The mean values of the normalised heat release rate,
$\dot{\omega}_{T}$, conditioned upon $c_{T}$ are also shown in Fig. 4. The
profiles corresponding to 1D unstretched laminar flames are also plotted in
Fig. 3 and Fig. 4, and error bars in Fig. 3 indicate the standard deviation of
each averaged data point. Normalisation of $\dot{\omega}_{c}$ and
$\dot{\omega}_{T}$, has been performed using the maximum values obtained for
1D unstretched laminar flames. For both Fig. 3 and Fig. 4, data from across
the entire domain has been considered in the averaging process.

It can be seen from Fig. 3 that $c_{\alpha 1}$ based on $\mathrm{H_{2}}$ mass
fraction shows the highest standard deviations of the different species used
to define $c_{\alpha 1}$, for both equivalence ratios. This is a consequence
of the small Lewis number value of $\mathrm{H_{2}}$, which gives rise to a
considerable scatter of $c_{\alpha 1}$ based on $\mathrm{H_{2}}$ mass fraction
for a given value of $c_{T}$ due to preferential diffusion effects. Moreover,
the mean profile of $c_{\alpha 1}$ based on $\mathrm{H_{2}}$ mass fraction
significantly deviates from the corresponding 1D unstretched laminar flame
profile for the $\phi=0.4$ case but this trend is much weaker in the
$\phi=0.7$ case due to weakened preferential diffusion effects in the latter
case. Figure 3 shows that the profiles for $c_{\alpha 1}$ based on
$\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$ mass fraction agree reasonably well
with the corresponding 1D unstretched laminar flame profiles, apart from the
behaviour in the super-adiabatic region ($c_{T}>$ 1) for the $\phi=0.4$ flame.
In the super-adiabatic region, $c_{\alpha 1}$ based on $\mathrm{O_{2}}$ and
$\mathrm{H_{2}O}$ mass fractions assume values greater than unity, indicating
the enhanced consumption (generation) of oxygen (water) in the 2D laminar
$\phi=0.4$ premixed laminar flame in comparison to the corresponding 1D
unstretched laminar flame. This can be substantiated by the non-zero values of
$\dot{\omega}_{c}$ for $c_{T}>1.0$ in the $\phi$ = 0.4 case. For the purpose
of modelling, the reaction progress variable is usually bound between 0.0 and
1.0 and super-unity values of reaction progress variables based on
$\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$ mass fractions indicate that a standard
reaction progress variable definition based on $\mathrm{O_{2}}$ and
$\mathrm{H_{2}O}$ mass fractions may not be suitable for thermo-diffusively
unstable lean $\mathrm{H_{2}}$-air flames.

The standard deviations of all $c_{\alpha 1}$ profiles for the $\phi=0.7$ case
are smaller than those for the $\phi=0.4$ case, which is a consequence of the
reduction in the variability of the $c_{\alpha 1}$ values in comparison to the
$\phi=0.4$ case. It can further be seen from Fig. 4 that qualitative
behaviours of $\dot{\omega}_{c}$ and $\dot{\omega}_{T}$ for the 2D laminar
flames are mostly in agreement with those of the corresponding 1D unstretched
laminar flames. However, significant quantitative differences can be seen in
the profiles of $\dot{\omega}_{c}$ and $\dot{\omega}_{T}$ between the 2D case
and the corresponding 1D unstretched laminar flame case for $\phi=0.4$ in Fig.
4, indicating an enhancement of heat release rate (e.g., there is an almost
$50\%$ increase in $\dot{\omega}_{T}$) and $\dot{\omega}_{c}$ values for each
species mass fraction, highlighting the tendency of thermo-diffusive
instabilities to enhance reactivity in lean $\mathrm{H_{2}}$-air flames. This
is consistent with the previous findings by Berger et al. [11].

The profiles shown in Fig. 3 corresponding to $c_{\alpha 2}$ are markedly
different to those observed for $c_{\alpha 1}$ (as well as the profiles for
the corresponding 1D unstretched laminar flames), which is a result of
equivalence ratio variation within the flame, as shown in Fig. 2. Profiles for
$c_{\alpha 2}$ based on H2 and O2 are very similar to each other at both
equivalence ratios, whereas significant differences are observed between the
$c_{\alpha 1}$ profiles for H2 and O2. Notably, $c_{\alpha 2}$ for O2 is
bounded between 0.0 and 1.0 as a result of mixture fraction dependence, which,
as previously mentioned, is an important feature when selecting a suitable
reaction progress variable. However, for $\phi=0.4$, relatively large standard
deviations are seen in $c_{\alpha 2}$ profiles, particularly for $c_{\alpha
2}$ based on $\mathrm{H_{2}O}$, which also displays clear non-monotonic
behaviour, reaching a value close to 1.0 at $c_{T}\approx 1$, before
decreasing at $c_{T}>1$. This implies that the inclusion of mixture fraction
dependence makes the variation of normalised $\mathrm{H_{2}O}$ mass fraction
with $c_{T}$ non-monotonic and this may not be suitable for the definition of
reaction progress variable. The standard deviations of $c_{\alpha 2}$ based on
$\mathrm{O_{2}}$ mass fraction show a moderate increase when compared with the
corresponding values for $c_{\alpha 1}$ whilst the standard deviations for
$c_{\alpha 2}$ based on $\mathrm{H_{2}}$ mass fraction remain approximately
the same as that obtained for $c_{\alpha 1}$.

Figure 5: Mean values of $c_{\alpha}$ based on $\mathrm{H_{2}}$,
$\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$ mass fractions conditioned upon $c_{T}$
for positively (i.e., $\kappa>0$) and negatively (i.e., $\kappa<0$) curved
regions for both the original $c_{\alpha}$ definition ($c_{\alpha 1}$)
($1^{st}$ row), and the local mixture fraction dependent $c_{\alpha}$
definition ($c_{\alpha 2}$) ($2^{nd}$ row). The error bars indicate standard
deviations of $c_{\alpha}$ conditioned upon $c_{T}$. Figure 6: Mean values of
normalised reaction rate of $c_{\alpha 1}$ (i.e., $\dot{\omega}_{c}$) based on
$\mathrm{H_{2}}$, $\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$ mass fractions
conditioned upon $c_{T}$ for positively (i.e., $\kappa>0$) and negatively
(i.e., $\kappa<0$) curved regions. Normalisation of $\dot{\omega}_{c}$ and
$\dot{\omega}_{T}$, has been performed using the maximum values corresponding
to unstretched 1D laminar premixed flames.

The departures of the 2D averaged profiles for $c_{\alpha 1}$ and $c_{\alpha
2}$ along with $\dot{\omega}_{c}$ from the corresponding 1D unstretched
laminar flame profiles shown in Fig. 3 and Fig. 4 have been explored further
by plotting mean values of $c_{\alpha 1}$ and $c_{\alpha 2}$ based on major
species mass fractions and their normalised reaction rates conditioned upon
$c_{T}$ for positive and negatively curved regions in Fig. 5 and Fig. 6
respectively. A comparison between Fig. 3 and Fig. 5 reveals that conditioning
on curvature leads to a significant reduction in the standard deviations of
$c_{\alpha}$ for both $\phi=0.4$ and $0.7$ cases. This reduction in standard
deviations when conditioning on curvature applies to both $c_{\alpha 1}$ and
$c_{\alpha 2}$ profiles and implies that the preferential diffusion due to
flame curvature is principally responsible for the variability of $c_{\alpha}$
values for a given value of $c_{T}$. Therefore, a much narrower distribution
of $c_{\alpha 1}$ and $c_{\alpha 2}$ about their conditional mean values on
$c_{T}$ is seen in Fig. 5 than in Fig. 3. The profiles of mean values of
$c_{\alpha 1}$ and $c_{\alpha 2}$ based on H2 mass fraction conditional on
curvature signs deviate significantly from the corresponding 1D unstretched
laminar flame profiles, reflecting that curvature induces significant
differences in the local flame structure due to preferential diffusion
effects. However, the tendency for the 2D profiles to diverge from the
corresponding 1D laminar profiles is weaker for $c_{\alpha 1}$ based on
$\mathrm{O_{2}}$ and $\mathrm{H_{2}O}$ mass fractions in comparison to that of
$c_{\alpha 1}$ based on $\mathrm{H_{2}}$ mass fraction. The profiles of mean
values of $c_{\alpha 1}$ and $c_{\alpha 2}$ based on $\mathrm{H_{2}}$ mass
fraction exhibit smaller magnitudes than the corresponding 1D unstretched
laminar values in regions of positive curvatures for both equivalence ratios.
The opposite is true for the regions with negative curvature, with mean
$c_{\alpha 1}$ and $c_{\alpha 2}$ values based on ${\mathrm{H_{2}}}$ mass
fraction being greater than the corresponding 1D unstretched laminar values
across the vast majority of the flame. These trends of mean values of
$c_{\alpha 1}$ and $c_{\alpha 2}$ based on $\mathrm{H_{2}}$ mass fraction are
consistent with high (low) equivalence ratios at positively (negatively)
curved zones, as shown in Fig. 2.

In Fig. 6, the profiles for $\dot{\omega}_{c}$ and $\dot{\omega}_{T}$ at
$\phi$ = 0.4 support the assertion that local extinction occurs at negative
curvature regions, with the reaction rate profiles for the negatively curved
zones roughly an order of magnitude smaller than the corresponding 1D
unstretched laminar flame values. Conversely, a significant enhancement of the
reaction rates in positive regions is evident from Fig. 5, which shows an
approximately three-fold increase in the 1D unstretched laminar flame values
in the $\phi=0.4$ case. In the $\phi=0.7$ case, the mean values of
$\dot{\omega}_{c}$ conditional upon $c_{T}$ show much smaller variation from
the corresponding 1D unstretched laminar flame values, where slightly smaller
(greater) values are observed than the 1D unstretched laminar flames values
for negative (positive) $\kappa$.

The profiles corresponding to $c_{\alpha 2}$ are significantly different from
both those seen for $c_{\alpha 1}$ as well as the 1D unstretched laminar flame
profiles conditioned upon signs of curvature. The non-monotonic behaviour of
$c_{\alpha 2}$ based on H2O is clearly explained by the behaviour in negative
curvature regions where large overshoots of $c_{\alpha 2}$ are observed, with
a maximum average value of $c_{\alpha 2}$ based on H2O mass fraction close to
$1.25$. Furthermore, as in Fig. 3, a strong agreement between $c_{\alpha 2}$
based on H2 and O2 mass fractions can be observed. This suggests that the
introduction of mixture fraction variation results in coupling between these
two normalised species mass fractions. In contrast to $c_{\alpha 1}$, which
exhibits super-unity values for the $\mathrm{O_{2}}$ mass fraction in the
positively curved regions, $c_{\alpha 2}$ based on O2 mass fraction is bounded
between $0.0$ and $1.0$ for both positively and negatively curved regions.

The differences in species distributions in response to the preferential
diffusion effects induced by flame curvature are reflected in the differences
in $c_{\alpha 1}$ and $c_{\alpha 2}$ distributions in Fig. 3 and Fig. 5. This
has implications on the evaluations of flame surface area and the stretch
factor or efficiency factor, $I_{0}$, which signifies the ratio of reaction
rate of the species in question per unit area normalised by the corresponding
1D unstretched laminar premixed flame value [17]. For the present
configuration, the stretch factor is computed using the relation:
$S_{f}=S_{L}I_{0}A^{\prime}$, where $S_{f}$ is the flame propagation speed
defined as per Eq. 1, and $A^{\prime}$ represents the flame area enhancement,
$A^{\prime}=A_{\mathrm{wrinkled}}/A_{\mathrm{flat}}$. The wrinkled flame
surface area $A_{\mathrm{wrinkled}}$ is evaluated as:
$A_{\mathrm{wrinkled}}=\int_{0}^{Ly}\int_{0}^{Lx}|\nabla c_{i}|\ dx\ dy$ where
$i=\alpha 1$ or $\alpha 2$ depending on $c_{i}$ definition. Figure 7 shows the
values of $A^{\prime}$ and $I_{0}$ calculated for both $c_{\alpha 1}$ and
$c_{\alpha 2}$ at both equivalence ratios.

Figure 7: Variations of $A^{\prime}$ and $I_{0}$ for different definitions of
$c_{\alpha 1}$ and $c_{\alpha 2}$ for $\phi=0.4$ and $0.7$.

Moderate differences are observed in the values calculated for flame area
enhancement and stretch factor for the different $c_{\alpha}$ definitions. It
should be noted here that there are differences in the values for $A^{\prime}$
depending on if $c_{\alpha 1}$ or $c_{\alpha 2}$ are used for the evaluation.
The flame area enhancement, $A^{\prime}$, based on the ${\mathrm{O_{2}}}$ mass
fraction assumes the highest value when $c_{\alpha 1}$ is considered for both
equivalence ratios, whereas the smallest $A^{\prime}$ values are obtained for
${\mathrm{H_{2}}}$ mass fraction when $c_{\alpha 1}$ is used for both
equivalence ratios. In the case when $A^{\prime}$ is based on $c_{\alpha 2}$,
the overall trends for $A^{\prime}$ across equivalence ratios remain the same,
with an increase in area can be seen as $\phi$ decreases, but a notable
reduction in $A^{\prime}$ is observed for the $A^{\prime}$ values using
${\mathrm{O_{2}}}$ based definition of $c_{\alpha 2}$, compared with
$c_{\alpha 1}$ values. For all definitions of $c_{\alpha 1}$ and $c_{\alpha
2}$, area enhancement is greater for the $\phi$ = 0.4 case than in the $\phi$
= 0.7 case, which is a consequence of enhanced flame wrinkling in the $\phi$ =
0.4 case as a result of strong thermo-diffusive instability. However, the
stretch factor $I_{0}$ exhibits an opposite trend to those seen for
$A^{\prime}$ with the ${\mathrm{H_{2}}}$ mass fraction based $c_{\alpha 1}$
yielding the highest $I_{0}$ values and the lowest values are obtained for
${\mathrm{O_{2}}}$ based $c_{\alpha 1}$. In the case of $c_{\alpha 2}$,
${\mathrm{O_{2}}}$ based definition yields significantly larger values of
$I_{0}$ for both equivalence ratios, compared with $c_{\alpha 1}$. This
implies that the influence of mixture fraction variation is significant on
$I_{0}$ and this should be accounted for in the modelling strategies.
Moreover, $I_{0}$ values for the $\phi=0.4$ case are found to be greater than
the corresponding values for $\phi=0.7$ (e.g., $I_{0}$ values range between
0.83 and 1.70 for the $\phi=0.4$ case and between 0.73 and 1.09 for the $\phi$
= 0.7 case.), indicating that the influence of flame stretch strengthens with
decreasing equivalence ratio $\phi$ in the case of lean ${\mathrm{H_{2}}}$-air
premixed flames.

##  4 Summary and Conclusions

High-fidelity 2D simulations of lean ${\mathrm{H_{2}}}$-air premixed flames
subjected to perturbations have been conducted for equivalence ratios of
$\phi=0.4$ and $0.7$ to analyse the influence of preferential diffusion
effects on the distributions of major species and their reaction rates. It has
been found that the preferential diffusion effects give rise to significant
deviations of mass fractions of major species and their reaction rates from
the corresponding 1D unstretched laminar premixed flame solution and this
tendency is particularly strong for ${\mathrm{H_{2}}}$ among all the major
species. These deviations arise principally due to preferential diffusion
effects induced by relative focussing/defocussing of species and heat at the
positively and negatively curved regions with the nature of the deviations
being opposite to each other depending on the sign of the curvature. This is
reflected in the local increases (decreases) in $\phi$ in the positively
(negatively) curved zones in the 2D $\phi=0.4$ case. These trends strengthen
and enhancements of burning rate, flame surface area, and stretch factor are
observed with a decrease in equivalence ratio as the flame becomes
increasingly susceptible to preferential diffusion and thermo-diffusive
instability. It has been found that the likelihood of super-adiabatic
temperature increases with a decrease in $\phi$, and super-adiabatic values of
${\mathrm{O_{2}}}$ and ${\mathrm{H_{2}O}}$ are obtained in the super-adiabatic
temperature zones, especially for the $\phi=0.4$ case. This suggests that
${\mathrm{O_{2}}}$ and ${\mathrm{H_{2}O}}$ mass fraction based reaction
progress variables are likely to be problematic for highly lean
${\mathrm{H_{2}}}$-air premixed flames because the reaction progress variable
values for these definitions are not going to be bounded between $0.0$ and
$1.0$. However, $\mathrm{O_{2}}$ mass fraction based reaction progress
variable remains bound between $0.0$ and $1.0$ when the equivalence ratio
variation is taken into account, but $\mathrm{H_{2}O}$ mass fraction based
reaction progress variable shows increased non-monotonic behaviour within the
flame front when the local equivalence ratio variation is accounted for. The
normalised species distributions analysed here provide the platform based on
which an optimal choice of reaction progress variable can be produced for lean
$\mathrm{H_{2}}$-air flames using automated methodologies (e.g., principal
component analysis, computational singular perturbation method, etc.) in the
future.

## Acknowledgements

The authors are grateful for the financial support from Tony-Trapp PhD
studentship (provided by Dr Tony Trapp) and the Engineering and Physical
Sciences Research Council (Grant: EP/W026686/1). The computational support for
this work was provided by ARCHER2 Pioneer projects (e691 & e817) and ROCKET
HPC facility at Newcastle University.

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