Combustion and Flame 220 (2020) 16–22

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Combustion and Flame
journal homepage: www.elsevier.com/locate/combustflame

Dependence of kinetic sensitivity direction in premixed flames
Weiqi Ji a, Tianwei Yang b, Zhuyin Ren b, Sili Deng a,∗
a
b

Department of Mechanical Engineering, Massachusetts Institute of Technology, MA 02139, USA
Center for Combustion Energy, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history:
Received 12 April 2020
Revised 17 June 2020
Accepted 17 June 2020

Keywords:
Sensitivity Analysis
Uncertainty quantification
Sensitivity direction
Flamelet
Extinction strain rate

a b s t r a c t
The sensitivities of turbulent combustion simulations to chemical kinetic parameters can be analyzed to
understand the controlling reactions in turbulent flames and to quantify the uncertainties in simulations.
However, computing the sensitivity of turbulent combustion simulations to a large number of kinetic
parameters is still challenging. A promising approach is to estimate the sensitivity from laminar flames,
especially for cases where the flamelet model is applicable. Under these conditions, the underlying hypothesis is that the sensitivity direction of the flamelet profiles is independent of the strain rate and
the flame coordinate, which is the progress variable for premixed flames. In the present work, this hypothesis was tested in laminar premixed counterflow flames. We first studied the sensitivity directions
of two extreme cases, the near-extinction strained flames and the freely propagating unstretched flames.
It was found that the sensitivity directions of the extinction strain rate and the laminar flame speed are
aligned with each other for various fuels, equivalence ratios, and pressures. We then studied the dependence of the sensitivity direction of the maximum flame temperature on the strain rate as well as the
dependence of the sensitivity direction of the species profiles on the progress variable. It was found that
the sensitivity direction of maximum temperature was largely independent of the strain rate. Moreover,
the sensitivity directions of the temperature and species profiles were independent of the progress variable, and they were all similar to the sensitivity direction of the extinction strain rate. These findings
suggest that there is a universal sensitivity direction for turbulent premixed flames and the direction can
be estimated by the sensitivity direction of extinction strain rate. These conclusions will enable efficient
sensitivity analysis of turbulent combustion simulations when the hypothesis is valid.
© 2020 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction
In order to accurately capture the pollution formation processes, modern turbulent combustion simulations usually adopt
detailed chemical models which contain thousands of species and
reactions. Since reaction rates are inevitably associated with certain uncertainties, it is often desirable to quantify the influence of
the uncertainties in the chemical kinetic parameters on the simulation and to understand what reactions dominant the uncertainty
[1–4].
A common approach to elucidate the kinetic uncertainty is to
conduct sensitivity analysis, such that the linear response of the
flame profiles to small perturbations of the kinetic parameters is
quantified [5–7]. The uncertainty can also be estimated by firstorder Taylor expansion with the sensitivity as the derivative [8].
Due to the high computational cost of turbulent combustion sim-

∗

Corresponding author.
E-mail address: silideng@mit.edu (S. Deng).

ulations, conducting sensitivity analysis for a large number of kinetic parameters is impractical.
A potential approach to estimate the kinetic sensitivity of the
expensive turbulent combustion simulations is to relate the sensitivity to that of the less expensive laminar flames. For example, Mueller et al. [1] proposed to estimate the uncertainties of
flamelet-based large eddy simulations based on the uncertainties
of steady flamelets. Although not explicitly mentioned by Mueller
et al. [1], the underlying hypothesis for the estimation is that
the sensitivity directions of the flamelet profiles under different
strain rates and flame coordinates (the mixture fraction for nonpremixed flames and progress variable for premixed flames) are
the same. In other words, flamelet profiles under different strain
rates and flame coordinates will change simultaneously with the
perturbation of the kinetic parameters. For example, the changes
of the temperature at two different strain rates but same flame coordinates upon kinetic perturbation will be proportional to each
other. Therefore, the uncertainty of the flamelet can be parameterized by a single variable along the unified sensitivity direction, such that the high-dimensional kinetic uncertainty is reduced

https://doi.org/10.1016/j.combustflame.2020.06.027
0010-2180/© 2020 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

W. Ji, T. Yang and Z. Ren et al. / Combustion and Flame 220 (2020) 16–22

17

Fig. 1. Sensitivity analysis of LFS and ESR of the methane/air mixture with an equivalence ratio of 0.7 and at 30 atm using a detailed mechanism [14]. (a) Unnormalized
sensitivities, (b) Normalized sensitivities.

to one-dimensional uncertainty. Furthermore, the entire turbulent
flame share the same sensitivity direction everywhere, although
the flame involves a wide range of strain rates and mixture fractions/progress variables. Similarly, the universal sensitivity direction assumption is also required for the estimation of the active
subspace of turbulent flames based on laminar flames [3].
However, the hypothesis has not been validated in the literature
since the sensitivity analysis of turbulent flames is only available
recently. Therefore, this work aims to test this hypothesis in laminar premixed flames and build theoretical foundations for developing efficient algorithms for the sensitivity analysis of turbulent
combustion simulations. We shall first study the sensitivity directions of two extreme cases of near-extinction strained flames and
freely propagating unstretched flames. We will then study the dependence of the sensitivity direction of the maximum flame temperature on the strain rate and that of the species profiles on the
progress variable.

ity analysis with respect to kinetic rate constants for ESR at the
exact strain rate determined experimentally. In the reducing factor approach, the simulation of the counterflow premixed flame
starts from a low strain rate and gradually increases to the nearextinction state. The step size of the increment of the strain rate
should be gradually decreased to retain the burning solutions up
to the extinction point. Such an approach has been implemented
in the open-source software of Ember [11], and the reducing factor
approach with Ember has been reported to be more efficient than
the continuation approach implemented in Chemkin [13]. However,
the sensitivity of ESR has to be evaluated using the finite difference
method in the reducing factor approach, which could be computationally expensive. Nonetheless, in both approaches, the sensitivity
is usually reported in its dimensionless form,

2. Sensitivity directions of extinction strain rate and laminar
flame speed

where ki is the rate constant for the i th reaction, and si is
the corresponding sensitivity coefficient. The sensitivity coefficients
for all reactions can be stacked into a sensitivity vector, i.e., S =
[s1 , s2 , . . . , sn ]. Furthermore, the sensitivity vector is normalized
by its magnitude ||S||2 to obtain the sensitivity direction S˜ (
S=
S/||S||2 ). Therefore, the sensitivity direction represents the normalized sensitivities of all reactions.
Figure 1 shows the sensitivity analysis of LFS and ESR of a
methane/air mixture using the detailed mechanism in Hashemi
et al. [14]. The equivalence ratio is 0.7, and the pressure is 30 atm,
which are kept the same as Long et al. [15]. The unnormalized and
normalized sensitivities are shown in Figs. 1a and 1b, respectively.
It can be seen that the sensitivity rankings for ESR and LFS are almost the same, with the most sensitive reaction being the primary
chain-branching reaction of H + O2 <=>O+OH. The observation is
consistent with many existing studies [15–17] which pointed out
that the ESR and the LFS share the same set of controlling reactions. Overall, the magnitude of the sensitivity of ESR is larger than
that of LFS. As pointed out in [15], this result is consistent with
previous studies that showed LFS to scale with the square root of
the overall reaction rate [18] while ESR was shown to scale linearly
[19]. Therefore, normalization of the sensitivity by the sensitivity
magnitude is necessary to assess the relative sensitivity of different reactions. Surprisingly, the normalized sensitivity for ESR and
LFS are very similar to each other as shown in Fig. 1b. We consequently denote the alignment of the sensitivity directions as the
kinetic similarity.

Laminar freely propagating flames and counterflow flames near
the extinction point are two extreme cases of strained flames. They
are usually characterized by the laminar flame speed (LFS) and the
extinction strain rate (ESR), respectively. Therefore, we first study
the sensitivity directions of ESR and LFS to investigate the effect of
strain on the sensitivity directions. Since LFS is much more extensively studied than ESR both experimentally and numerically, we
herein only briefly discuss about the computation of ESR and its
sensitivity directions.
The ESR characterizes the response of the flame to strain. A
higher ESR indicates that the mixture is more likely to sustain
combustion in strained flows. ESR is usually measured and simulated in premixed counterflow twin flames, the response of which
can be described via the S-curve analysis [9]. The maximum strain
rate that the flame can sustain with fixed composition and thermodynamic states at the inlet is defined as the ESR. Specifically, the
absolute value of the maximum velocity gradient is often utilized
to evaluate ESR. Numerically, since the ESR corresponds to a singularity point, it has to be solved using either flame continuation
approach [10] or reducing factor approach [11].
The continuation approach enables an efficient sensitivity analysis of the ESR as first proposed in [12]. By realizing that ESR becomes a dependent variable when the two-point continuation approach [10] is invoked, it is possible to perform rigorous sensitiv-

si =

∂ ln[ESR]
ki ∂ [ESR]
=
,
∂ ln[ki ]
ESR ∂ [ki ]

(1)

18

W. Ji, T. Yang and Z. Ren et al. / Combustion and Flame 220 (2020) 16–22
Table 1
The kinetic similarity between ESR and LFS evaluated for the sensitivities in the literature.
No.

Fuel

Phi

Cosine

Top N

p[atm]

Ref.

Alias

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32

methanol
isooctane
n-dodecane
n-dodecane
n-dodecane
dimethyl ether
dimethyl ether
dimethyl ether
n-butanol
n-butanol
n-butanol
n-butanol
n-butanol
n-butanol
n-dodecane
n-dodecane
n-dodecane
n-dodecane
n-dodecane
methyl-butanoate
methyl-butanoate
methyl-butanoate
n-propylbenzene
methane/C2 HF5
methane/C2 HF3 Cl2
methane/C3 H2 F3 Br
methane
methane
methane
methane
methane
methane

0.77
0.87
1.0
1.0
1.0
0.6
1.0
1.5
0.6
1.0
1.5
0.6
1.0
1.5
0.7
1.0
1.4
1.0
1.0
0.7
1.0
1.5
1.0
1.0
1.0
1.0
0.7
0.7
0.7
1.3
1.3
1.3

1.00
0.46
0.94
0.97
0.95
1.00
0.99
0.91
0.59
0.89
0.96
0.95
0.97
0.97
0.96
0.97
0.97
0.95
0.97
0.97
0.98
0.97
0.96
0.96
0.99
0.97
0.98
0.99
0.98
0.99
0.97
0.98

16
12
10
10
10
10
10
10
8
8
8
8
8
8
10
10
10
12
10
14
14
14
10
14
14
14
15
15
15
15
15
15

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
7
30
1
7
30

[27]
[27]
[24]
[24]
[24]
[28]
[28]
[28]
[26]
[26]
[26]
[26]
[26]
[26]
[16]
[16]
[16]
[29]
[29]
[30]
[30]
[30]
[31]
[32]
[32]
[32]
[15]
[15]
[15]
[15]
[15]
[15]

Holley2006
Holley2006
Kumar2007
Kumar2007
Kumar2007
Wang2009
Wang2009
Wang2009
Veloo2010
Veloo2010
Veloo2010
Veloo2010
Veloo2010
Veloo2010
Ji2010
Ji2010
Ji2010
Kumar2010
Kumar2010
Wang2011
Wang2011
Wang2011
Hui2012
Xu2017
Xu2017
Xu2017
Long2019
Long2019
Long2019
Long2019
Long2019
Long2019

To measure the kinetic similarity between two sensitivity directions, the inner product between two normalized sensitivity vectors can be quantified, which is also termed as the cosine similarity [20–23]. The cosine similarity of unity corresponds to a perfect
parallel, while zero indicates that two directions are orthogonal to
each other. For the normalized sensitivities in Fig. 1b, the cosine
similarity S˜ESR , S˜LF S  is 0.98, which suggests that the two sensitivity directions are almost parallel. This result implies that the ESR
and the LFS for the specified mechanism and conditions are kinetically similar, in the sense that they share not only the same set
of controlling reactions but also the same response to the coupling
among these reactions.
Next, we shall investigate the kinetic similarities between ESR
and LFS with other fuels, equivalence ratios, and pressures. Computing the sensitivity of ESR for large hydrocarbon fuels could be
time consuming since computing the ESR itself for large reaction
model is already intensive [11] and hyperparameter tuning is often needed to save computational time while retaining a stable
solution. Fortunately, the kinetic sensitivities of ESR are usually reported in the literature in addition to the experimental measurement of ESR. Therefore, we compiled the sensitivity analysis results
of ESR as well as those of LFS from literature and normalized the
sensitivities to obtain the cosine similarity.
The compiled database is summarized in Table 1. The investigated conditions cover a wide variety of fuel, equivalence ratios
(phi), and pressures (p [atm]). The “top N” in the table refers to
the number of top-ranked sensitive reactions presented in the literature. For most of the cases, the cosine similarity is higher than
0.95, which suggests that the two sensitivity directions for ESR
and LFS are very similar to each other. The consistency between
large hydrocarbon fuels and methane agrees with previous studies
[16,17,24] which suggest that the ESR is most sensitive to smallmolecule reactions, i.e., H2 /CO/C1 –C2 kinetics.

Note that there are a few outliers that have noticeable lower cosine similarity than the others. Regarding the outliers of #9–10, the
difference is potentially because the ESR at lean condition is insensitive to the reaction of CO+OH<=>CO2 +H with the kinetic model
published in 2009 [25]. The model significantly underpredicted the
ESR for lean and stoichiometric conditions. The predictions with
the updated model employed in [26] agreed much better with the
experimental data. Correspondingly, the difference in the sensitivity direction is significantly reduced, as shown in #12–13. For the
dataset of #2, the cause of the difference is suspected to be that
the ESR is extremely sensitive to C3 H4 +O<=>CH2 O+C2 H2 based
on the kinetic model published in 1998 [12]. The model showed
large discrepancies from the experimental data of ESR. Therefore,
it can be summarized that the similarity between the kinetic sensitivity of ESR and LFS is an inherent property of fuel-air mixtures,
although such similarity might not show in computations if the
models significantly off predict the experimental measurements.
It is worth to mention that the kinetic similarity is consistent
with the historical views on the connection between ESR and LFS
based on the time scale analysis by Peters [33]. The flame residence time in flame propagation is given by the ratio between the
flame thickness and LFS, while the quench time for flame extinction is given by the inverse of ESR. The numerical simulations by
Rogg [34] for a stoichiometric methane-air flame show that the
flame residence time and the quench time are very close to each
other, such that the quench time is subsequently approximated by
the flame residence time in flame propagation for turbulent combustion modeling. Since both time scales are equal to the chemical
time scale [33], one can imply that flame extinction has similar
chemical time scale with the unstretched flame propagation. Since
the chemical time scale is determined by the controlling reactions,
the similarity of the controlling reactions and the sensitivity directions is consistent with the similarity in the time scales. It is

W. Ji, T. Yang and Z. Ren et al. / Combustion and Flame 220 (2020) 16–22

19

Fig. 2. (a) The response of the maximum flame temperature versus the strain rate. (b) The evolution of the cosine similarity and normalized magnitude with strain rates.
The sensitivity magnitude is normalized by the maximum magnitude at the strain rate of 1075 1/s. The cosine similarity refers to the inner product between the sensitivity
direction at each strain rate and the sensitivity direction of ESR.

worthy to note that the above time scale analysis is qualitative,
since the chemical time scale of LFS is not always equal to that of
ESR [19], and the flame temperature at extinction is much lower
than the unstretched flame. Thus, the classical time analysis cannot
fully explain the quantitative similarity of the sensitivity directions.
Therefore, more rigorous analysis on the dependence of sensitivity
directions on the strain rate and flame temperature will be carried
out in Section 3.
Leveraging the kinetic similarity between ESR and LFS, the sensitivity of ESR can be efficiently computed based on the sensitivity
of LFS. Specifically, the similarity implies that

∂ ln [ESR] ∂ ln [ESR] ∂ ln [LF S] ∂ ln [LF S]
/
=
/
,
∂ ln [ki ] ∂ ln [km ]
∂ ln [ki ] ∂ ln [km ]

(2)

where the sensitivity of LFS to all reactions can be efficiently computed with the adjoint approach. Therefore, to evaluate the sensitivity coefficients of ESR for all reactions, we just need to compute the sensitivity of ESR to the most sensitive reaction km via
the finite difference approach, and sensitivities to other reactions
are readily available via Eq. (2). The efficient computation of the
sensitivity of ESR will benefit the gradient-based optimization and
uncertainty quantification for ESR [35].
3. Dependence of sensitivity direction on strain rate and
progress variable
Next, we shall study the evolution of sensitivity directions of
flame temperatures and species concentrations with the strain rate
and the progress variable. The sensitivities of the flame temperature and species concentrations are also computed using Ember
with the finite difference approach for the top 15 reactions preselected based on the sensitivity of LFS. The simulation is conducted for a methane/air mixture using the detailed mechanism of
GRI3.0 [36]. Details on the composition and thermodynamic states
are: equivalence ratio of 0.7, inlet temperatures of 300 K, and atmospheric pressure.
Figure 2a shows the response of the maximum temperature
with the strain rate in the upper branch of the S-curve, and the
ESR is around 1187 1/s. Since the perturbation of the kinetic rate
constants also affects the ESR, the temperature and species sensitivity are computed from the strain rate of 500 to 1075 1/s.
Figure 2b then shows the evolution of the sensitivity vector of
the maximum temperature Tmax with strain rates. The sensitivity
magnitude is normalized by the maximum magnitude at the strain

rate of 1075 1/s. In order to display the sensitivity directions at
all strain rates examined, we compute their cosine similarity with
a reference sensitivity direction, which is that of the ESR. As expected, the sensitivity magnitude increases with the strain rate
since the flame is more sensitive to chemical kinetics when approaching extinction.
Moreover, the sensitivity directions of Tmax are independent of
the strain rate, and they are similar to the sensitivity direction of
ESR with the minimum cosine similarity of 0.97 for the specified
range of strain rates. Therefore, the controlling reactions of Tmax
are independent with the strain rate. The independence is intriguing since the peak temperature, local chemical time scale, and sensitivity magnitude of the flame all vary with strain rate. The independence also explain why the ESR and LFS share the same sensitivity direction while the local chemical time scales could be different for ESR and LFS [19].
The evolution of the sensitivity directions of flame temperatures and species concentrations with the locations in the flame
coordinate is further investigated to examine if there is a universal kinetic sensitivity direction for premixed flames. Figure 3a
presents the temperature profile as a function of the distance to
the stagnation-point. The perturbed profiles correspond to the solutions with perturbed rate constants. The gap between the original and perturbed profiles indicates the magnitude of sensitivity.
It can be seen that the sensitivity magnitude is substantial near
the flame front, which corresponds to a large temperature gradient. The evolution of the sensitivity magnitudes is further detailed
in Fig. 3b. The magnitude is normalized by the maximum magnitude in the entire flame domain. The magnitude of the sensitivity
is much weakened in the cold unburned regime and the hot equilibrium regime. Figure 3b also shows the evolution of the sensitivity direction, where the cosine similarity is computed with respect
to the sensitivity direction of ESR. The evolution of the cosine similarity suggests that the sensitivity directions are almost the same
across the entire flame and are the same as that of ESR. The cosine similarity is above 0.9 for most of the locations, and the sensitivity magnitude is negligible when the cosine similarity is below
0.9. In other words, a global similarity of the sensitivity direction is
elucidated. Such global similarity has been previously observed in
the homogenous auto-ignition system and one-dimensional burner
stabilized flame [20–22].
To further elaborate the implications of the independence of
sensitivity directions with locations in the flame coordinate, Fig. 3a
presents the changes of temperature at two distinct locations x1

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W. Ji, T. Yang and Z. Ren et al. / Combustion and Flame 220 (2020) 16–22

Fig. 3. (a) The temperature profiles of flames with unperturbed and perturbed reactions in the flame coordinate. (b) The evolution of the direction and magnitude of the
temperature sensitivity with the distance to stagnation-point. The cosine similarity is computed with respect to the sensitivity direction of ESR. All results are computed
under the strain rate of 500 1/s.

Fig. 4. The evolution of the direction and magnitude of sensitivity of the flame
temperature with the progress variable and the strain rate. The cosine similarity is
computed with respect to the sensitivity direction of ESR. The magnitude is normalized by the maximum magnitude across the entire flame domain at each strain
rate.

and x2 . The independence implies that the ratio of the temperature
changes T(x1 )/T(x2 ) is constant no matter what and how reaction rate constants are perturbed. The constant ratio will enable us
to parameterize the uncertainties of the temperature profiles using a single random variable, which could potentially reduce the
computational cost for uncertainty quantifications.
We then present the sensitivity in the progress variable space
for clear visualization since the compositions could change sharply
in the narrow reaction zone. The progress variable c is defined in
terms of the temperature, i.e.,

c=

T − Tmin
,
Tmax (a ) − Tmin

(3)

where the maximum temperature Tmax changes with the strain
rate a. The progress variable c is within the range of [0, 1]. The
minimum temperature Tmin is the inlet temperature of 300 K.
Figure 4 shows the evolution of sensitivity directions of the flame
temperature with progress variable at two representative strain
rates. It again can be shown that the cosine similarity with ESR is
higher than 0.9 in most of the high-temperature regime. It is noted
that the cosine similarity deviates from unity when the progress
variable is close to zero or is between 0.7 and 0.9. However, it
is found that the magnitude of the sensitivity drastically reduces
in the region where the cosine similarity deviates from unity, indicating that the local temperature is not sensitive to the kinetic
parameters. In addition, the cosine similarity recovers to unity in

the reaction zone, i.e., near c = 1. Therefore, evaluating the overall
similarity across the progress variable based on the global sensitivity direction is still valid. Furthermore, the sensitivity direction
of the flame temperature is also approximately independent of the
strain rate by comparing the evolutions at two distinct strain rates.
These observations suggest that the temperature sensitivity direction keeps the same across the flame in the flow with various degrees of strain. Intuitively, the temperature sensitivity is a result of
the sensitivity of heat release rate, and the changes of the flame
temperature due to flame stretch do not alter the controlling reactions for heat release.
Next, the evolution of the sensitivity direction of the species
mass fractions with progress variable and strain rate are also studied for their relevance to emissions, as shown in Fig. 5. The sensitivity directions for major species, CH4 and CO (Figs. 5a and 5b,
respectively), are also largely independent of the progress variable and are similar to the sensitivity direction of ESR. The sharp
switching between positive and negative correlations corresponds
to the local maximum species concentration [20]. The progress
variable corresponding to the local maximum species concentration divides the flame into two kinetic regimes. Take Fig. 5b as an
example, the net production rate of CO is positive (produced) for
c < 0.75, while it is negative (consumed) for c > 0.75, and thus CO
concentration reaches maximum at c = 0.75. Thus, increasing the
total reactivity will enhance CO production when c < 0.75, while
enhance CO consumption when c > 0.75. Therefore, the sensitivity direction changes sign at local maximum concentration. The kinetic similarity is much weaker for the mass fraction of H radical
compared to the temperature, and the less pronounced similarity
could be attributed to the diffusion process [20]. While the diffusion process make the global similarity of all species less pronounced comparing with the homogenous ignition system in [20],
the effect of diffusion on H radical is more significant comparing
to CH4 and CO, since H radical is one of the most important chain
carriers and its diffusion is more pronounced due to the lighter
molecular weight.
Similar independence of the strain rate has been previously observed in the sensitivity directions of OH concentration in the simulation of a turbulent lifted flame [5]. Therefore, the independence
suggests that the sensitivity direction of ESR might be able to represent the sensitivity direction of the entire temperature field for
turbulent premixed flames although a wide range of strain rates
or scalar dissipation rates are involved in the flow field. Such representation could substantially accelerate the uncertainty quantification of turbulent combustion simulations as demonstrated
in [1].

W. Ji, T. Yang and Z. Ren et al. / Combustion and Flame 220 (2020) 16–22

21

Fig. 5. The evolution of the sensitivity direction of species concentrations with the progress variable and strain rate. The cosine similarity is computed with respect to the
sensitivity direction of ESR.

Finally, we briefly discuss how the universal sensitivity direction enables us to parameterize the uncertainties of hundreds
of rate constants into one random variable, which is one of the
key algorithms in [1]. Without loss of generality, let us assume
that there are two rate constants, k1 and k2 , in the chemical
model. Perturbing them by 1 and 2 , respectively, we can obtain
three flamelet profiles including the unperturbed one, as shown in
Fig. 3a. By definition, the temperature changes at two distinct locations x1 and x2 can be calculated with the corresponding sensitivity coefficients [s1, 1 , s2, 1 ] and [s1, 2 , s2, 2 ], and further related
to the sensitivity directions [s
1,1 , s
2,1 ], [s
1,2 , s
2,2 ], as shown in
Eqs. (4) and (5). Since there is a universal sensitivity direction
across all locations, i.e., [s
1,1 , s
2,1 ] = [s
1,2 , s
2,2 ], we can then parameterize the high-dimensional kinetic uncertainty as a single
random variable ξ , according to Eq. (6). Therefore, the temperature
response at any location due to the perturbation is always proportional to ξ , as shown in Eqs. (7) and (8), with the difference in
the sensitivity magnitude. Similarly, since the universal sensitivity
direction is also valid for various strain rates, the temperature response at any strain rate is also always proportional to ξ .

T ( x 1 ) = s 1 , 1 1 + s 2 , 1 2 =
T ( x 2 ) = s 1 , 2 1 + s 2 , 2 2 =





s21,1 + s22,1 (s
1 , 1 1 + s
2 , 1 2 )

(4)


s21,2 + s22,2 (s
1 , 2 1 + s
2 , 2 2 )

(5)




ξ = s
1 , 1 1 + s
2 , 1 2 = s
1 , 2 1 + s
2 , 2 2

T ( x 1 ) =
T ( x 2 ) =




(6)

s21,1 + s22,1 ξ

(7)

s21,2 + s22,2 ξ

(8)

In the context of flamelet-based turbulent combustion simulations, the turbulent flames can be represented as ensembles of
laminar flames under various strain rates. Since the sensitivity directions is independent of strain rate, one can imply that the entire turbulent flame also has a universal sensitivity direction. The
magnitude of the sensitivity at different locations can be efficiently
computed by sampling along the one-dimensional random variable
ξ . It has been demonstrated that the uncertainties of the 325 reaction rate constants in the GRI 3.0 model were quantified with only
7 independent simulations with the rate constants varying along
ξ [1]. Without assuming the universal sensitivity direction, direct
Monte Carlo sampling would require at least 20 0 0 samples [1].
The universal sensitivity direction could significantly expedite
the uncertainty quantification of turbulent flame simulations if
there are low-dimensional manifolds in turbulent premixed flames,
such that we can use laminar flamelets to estimate the sensitivity

direction. While such assumption is usually valid in the wrinkled
and corrugated flamelet regimes, it is an open question whether
the analysis still holds in the thin reaction zone and the broken
reaction zone. We hope that current analysis can facilitate the research on uncertainty quantification of turbulent flames and provide an analysis framework to the community for future investigations.

4. Conclusions
We have shown that the kinetic sensitivity direction of extinction strain rate is similar to that of laminar flame speed for various kinds of fuels, equivalence ratios, and pressures. The similarity is consistent with the flame time scale analysis which shows
that flame extinction and propagation share similar chemical time
scale although the flame temperature at extinction is much lower.
Furthermore, it is shown that the sensitivity direction of flame
profiles is largely independent of the strain rates and progress
variable. Moreover, the sensitivity directions of temperature and
species concentrations are similar to that of extinction strain rate.
The behavior could be attributed to the fact that the controlling reactions that determine the sensitivities of temperature are not altered by the maximum flame temperature. Therefore, the identified
independence of sensitivity direction on strain rates and progress
variable suggest that there exists a universal sensitivity direction
in complex turbulent premixed flames involving a wide range of
strain rates and progress variables. Furthermore, the direction can
be estimated as the sensitivity directions of extinction strain rate
and laminar flame speed. While the current study focuses on laminar premixed flames, the analysis framework can be extended
to nonpremixed flames. These findings shall facilitate future algorithm development for sensitivity analysis and uncertainty quantification of turbulent combustion simulations.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Acknowledgments
SD would like to acknowledge the support from the d’Arbeloff
Career Development allowance at Massachusetts Institute of Technology. TY and ZR were supported by the National Natural Science
Foundation of China 91841302.

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