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Proceedings of the Combustion Institute 000 (2018) 1–8
www.elsevier.com/locate/proci

Quantifying kinetic uncertainty in turbulent combustion
simulations using active subspaces
Weiqi Ji a, Zhuyin Ren a,∗, Youssef Marzouk b, Chung K. Law a,c
a Center for Combustion Energy, Tsinghua University, Beijing 100084, China
b Massachusetts Institute of Technology, Cambridge, MA 02139, USA
c Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

Received 1 December 2017; accepted 26 June 2018
Available online xxx

Abstract
Uncertainty quantification in expensive turbulent combustion simulations usually adopts response surface
techniques to accelerate Monte Carlo sampling. However, it is computationally intractable to build response
surfaces for high-dimensional kinetic parameters. We employ the active subspaces approach to reduce the
dimension of the parameter space, such that building a response surface on the resulting low-dimensional
subspace requires many fewer runs of the expensive simulation, rendering the approach suitable for various
turbulent combustion models. We demonstrate this approach in simulations of the Cabra H2 /N2 jet flame,
propagating the uncertainties of 21 kinetic parameters to the liftoff height. We identify a one-dimensional
active subspace for the liftoff height using 84 runs of the simulations, from which a response surface with
a one-dimensional input is built; the probability distribution of the liftoff height is then characterized by
evaluating a large number of samples using the inexpensive response surface. In addition, the active subspace provides a global sensitivity metric for determining the most influential reactions. Comparison with
autoignition tests reveals that the sensitivities to the HO2 -related reactions in the Cabra flame are promoted
by the diffusion processes. The present work demonstrates the capability of active subspaces in quantifying
uncertainty in turbulent combustion simulations and provides physical insights into the flame via the active
subspace-based sensitivity metric.
© 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
Keywords: Uncertainty quantification; Active subspaces; Turbulent lifted flames; Autoignition

1. Introduction
Simulation of turbulent combustion is usually
associated with uncertainties from the turbulent
combustion model, the chemical kinetic model, the
boundary conditions, etc. For example, detailed ki∗ Corresponding author.

E-mail address: zhuyinren@tsinghua.edu.cn (Z. Ren).

netic models for hydrocarbon fuels have a large set
of elementary reactions with many rate constants
which inevitably contain uncertainties. It remains
an open question as whether the uncertainties in
kinetic models can be small enough to satisfy the
needs of combustion simulations with prescribed
accuracy requirements.
Uncertainty quantification (UQ) provides a
set of methods to address the above question by

https://doi.org/10.1016/j.proci.2018.06.206
1540-7489 © 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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quantifying the uncertainties in the models as well
as the predictions, and usually consists of two steps.
Take the kinetic uncertainty as an example. The
first step, the so-called inverse problem [1–3], aims
at quantifying the distribution of the model parameters based on experimental data under simplified
flow conditions, such as the ignition delay time. The
second step, usually referred to as the forward problem, with the combustion simulation being called
the forward model, aims at quantifying the uncertainty in the predictions by propagating the above
model uncertainties through combustion forward
models. During this step, a large number of samples are drawn from the distribution of the model
parameters characterized in the first step. Each
sample corresponds to a set of model parameters,
say the pre-factor A in the reaction mechanism, and
the forward model is evaluated once for each sample. Consequently the statistics of the predictions
from all of the samples can lead to the uncertainty
in the prediction, such as its uncertainty interval.
The forward problem has been studied for
simple combustion problems [4–7], such as the
0-D auto-ignition and the 1-D flame propagation.
Most of the cited works employ non-intrusive
methods with Monte Carlo (MC) method or its
variation. Due to the slow convergence of MC, a
large number of samples are required to reach convergence. To alleviate this computational challenge,
various response surface techniques [7,8] have been
adopted in conjunction with MC, in which one
uses a few carefully selected runs of the expensive
model to construct a cheaper response surface
that is subsequently sampled for MC. However,
building response surfaces requires a large number
of runs of the expensive model when the number
of input parameters is large. For high dimensional
kinetic parameters, various methods have been
used to reduce the dimension of the input parameter space. For example, local sensitivity analysis
and screening methods have been used in [1,9] to
identify the input parameters that contribute most
to the uncertainties in the outputs. Subsequent response surface construction will then only involve
these important parameters and requires much
fewer runs of the expensive model.
The present work focuses on the forward problem in turbulent combustion, in which the number
of affordable simulations is more severely limited
since it can take days to weeks for each sample.
Recently, the active subspace (AS) dimension
reduction method developed by Constantine et al.
[10] has attracted considerable attention for UQ
applications, and has been successfully applied to
the simulations of autoignition and laminar flame
speeds [11], and the turbulent combustion simulation of hypersonic scramjet engines [12]. Unlike
the standard sensitivity analysis that identifies the
important individual input parameters, the AS
approach identifies the important directions in

the input parameter space determined in terms
of linear combinations of the input parameters.
Since the direction is not constrained to be aligned
with the canonical basis, the AS approach can lead
to further dimension reduction. In [11], the AS
method identified one-dimensional structure in
the map from rate constants to the ignition delay
time for hydrogen/air, methane/air mixtures and
five-dimensional structure for dimethyl ether/air
mixture under test conditions. In [12], the AS
method identified one-dimensional structure in
the map from seven input parameters of the operating conditions, such as the angle of attack and
turbulence intensity, to the scramjet performance;
thus the response surface of a single active variable
derived from the seven inputs can be constructed.
The total cost is much less than directly building
a response surface of seven inputs. It also helps to
understand and improve models by better visualizing the map from the high-dimensional inputs to
outputs. As a byproduct, the active subspace also
reveals the most sensitive parameters for the quantity of interest (QoI). Note that other dimension
reduction approaches have also been exploited for
turbulent combustion simulations. For example,
Mueller et al. [13] developed a dimension reduction
algorithm for flamelet-like models by decoupling
the uncertainties in the flamelet table induced by
kinetic uncertainty from the flow field. However,
the algorithm is limited to flamelet-like models and
the uncertainties induced by kinetic models. The
AS approach is more general and it is suitable for
propagating kinetic uncertainty as well as other
types of uncertainty sources in the context of
various turbulent combustion models.
In this work, the active subspace method is
for the first time demonstrated for propagating
high dimensional kinetic uncertainty in turbulent
combustion simulations; specifically, it is used to
propagate the uncertainties in 21 rate constants to
the liftoff height in the Cabra H2 /N2 jet flame [14].
In the following, we shall first introduce the active
subspace method, and then apply it to the Cabra
flame simulations.
2. Kinetic uncertainty and active subspace
Following previous works [7,11], the rate constants are presumed to be independent lognormal
distributions, and are centered and normalized as
xj , which follows the standard normal distribution,
ln k j /k j0
xj = 1
∼ N (0, 1 ), j = 1, . . . , d,
ln UF j
3

(1)

where kj is the jth reaction rate coefficient, kj0 and
UFj are its nominal value and uncertainty factor.
The number of elemental reactions is denoted as
d, which is also the number of input parameters.
The simulation inputs are then represented as a

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d-dimensional random vector x = [x1 , . . . , xd ]T .
We denote the distribution of x as π x . The deterministic forward combustion model map the
inputs to the QoIs, and the map to a specific QoI
is denoted as a function f(x).
The active subspace methodology is detailed
in [10] with algorithms, a rigorous error analysis,
and demonstrations. Here we simply state the basic
concepts. The AS method seeks an r-dimensional
subspace in the d-dimensional parameter space
that describes most of the variation of f. The idea
is to find a low-dimensional approximation of f as
f (x ) ≈ g(xr ), xr = S T x,

(2)

where g is a function of the r-dimensional input xr
with r < d, and S is an orthogonal matrix of size
d × r. The active subspace is defined as span{S}.
One way to identify the active subspace is to perform an eigenvalue decomposition of the matrix
C, defined as the expectation of the outer product
of the gradient ∇f with itself, i.e.,
C = ∫ ∇ f (x )∇ f (x )T πx (x )dx = W W T .

(3)

Note that C is symmetric, positive semi-definite,
and of size d × d. The unitary matrix W consists
of the d eigenvectors w1 , . . . , wd and  is a diagonal matrix whose components are the eigenvalues
λ1 . . . λd , sorted in descending order. If there is a
gap in the eigenvalues, meaning λr >> λr+1 , then
the function f varies mostly along the first r eigenvectors and is almost constant along the rest of the
eigenvectors. The first r eigenvectors are selected
as active directions, i.e., S ≡ [w1 , . . . , wr ]; and its
complement span{wr+1 , . . . , wd } is identified as the
inactive subspace. Then one can build a response
surface, RS(xr ), with xr as input, and it is chosen
as the function g, i.e., f (x ) ≈ g(xr ) = RS(xr ).
The matrix C can be approximated by MC simulations [10]. The number of runs M of the forward
model required depends on how the gradient of f
is evaluated, as discussed next.
(I) In the first scenario, f and its gradient
are evaluated simultaneously. Constantine
et al. [10] have shown that M increases
logarithmically with d, i.e., M = αβlog(d ).
The constant α is the over-sampling factor
and is recommended to be between 2 and
10, and β is the largest dimension of the
subspace allowed in the subsequent building
of the response surface.
(II) In the second scenario, since derivatives
with respect to the kinetic parameters are
often not available in legacy combustion
codes, the gradients are estimated via finite
difference methods and the evaluation of
gradients in general takes d runs of the forward model. Thus M will scale with dlog(d),
i.e., M = αβd log(d ).
(III) In the third scenario, f is approximately
linear with the inputs over the entire input

3

space, e.g., the integrated exit pressure with
respect to the seven inputs in the scramjet
simulation [12]. Then, the function has a
one-dimensional active subspace, which can
be estimated by global linear fitting through
ordinary least square (OLS) [10], i.e.,
f (x ) = bT x + b0 .

(4)

The active direction is given by w1 = b /b2 .
For such a case, M linearly increases with d, i.e.,
M = αd.
In all scenarios, the error in the estimated eigenvalues and eigenvectors due to insufficient number
of runs can be estimated with bootstrapping [12],
in which M samples are resampled with replacement for hundreds of times from the runs already
evaluated. Then eigen-decomposition is performed
for each time of the resampling. The lower and upper bounds of the active subspace provide a good
approximation to the error bounds in the computed active subspace. Once the active subspace
is identified, various response techniques, such as
polynomial fitting, Polynomial Chaos Expansion
(PCE) [7] and High Dimensional Model Representation (HDMR) [8], can be readily applied to the
low-dimensional active subspace. Depending on
the degree of nonlinearity and coupling, PCE and
HDMR are suitable for moderate dimensions of
the input parameter space.
The entire workflow for propagating the kinetic
uncertainty to the QoI in turbulent combustion
simulation will be: (1) Estimate the active subspace
based on a small number, M, of samples drawn
from π x . The function f and its gradient for each
sample are evaluated by performing expensive
simulations. (2) Build the response surface RS(xr )
to the low-dimensional variable xr . (3) Estimate the
distribution of the QoI by evaluating a large number, N, of the samples drawn from π x . The function
f is acquired via the cheap response surface RS(xr ).
It is worth noting that other dimension reduction methods, such as the basis adaptation method
[15], could also be applied, especially if the distribution π x is non-Gaussian and sampling from it
is difficult. A review of the relationship between
the active subspace method and other subspace
methods is given in [10]. Comparisons among
different methods in the context of combustion
simulations merit further studies.
As a byproduct, the active subspace also reveals
the global sensitivity of the QoI to the inputs [16],
i.e., large components of the active direction correspond to the influential parameters. In addition,
the active direction provides information on how
the kinetic parameters are coupled with each other.
T

3. TPDF simulations of turbulent lifted jet flames
The Cabra flame [14], which is a turbulent jet
flame of H2 /N2 issued into a hot co-flow of lean

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Table 1
The uncertainty factor for each elementary reaction in the Li mechanism.
No.

Reaction

UF

No.

Reaction

UF

1
2
3
4
5
6
7
8
9
10
11

H + O2 [=] O + OH
O + H2 [=] H + OH
H2 + OH [=] H2 O + H
O + H2 O [=] OH + OH
H2 + M [=] H + H + M
O + O + M [=] O2 + M
O + H + M [=] OH + M
H + OH + M [=] H2 O + M
H + O2 (+ M) [=] HO2 (+ M)
HO2 + H [=] H2 + O2
HO2 + H [=] OH + OH

1.5
1.3
2
1.5
2
2
3
2
1.2
2
3

12
13
14
15
16
17
18
19
20
21

HO2 + O [=] O2 + OH
HO2 + OH [=] H2 O + O2
HO2 + HO2 [=] H2 O2 + O2
HO2 + HO2 [=] H2 O2 + O2
H2 O2 (+ M) [=] OH + OH (+ M)
H2 O2 + H [=] H2 O + OH
H2 O2 + H [=] HO2 + H2
H2 O2 + O [=] OH + HO2
H2 O2 + OH [=] HO2 + H2 O
H2 O2 + OH [=] HO2 + H2 O

1.2
3
2.5
1.4
2.5
3
2
3
2
2

combustion products, is employed to demonstrate
the propagation of kinetic uncertainty through
active subspace. Comprehensive investigations of
this flame, such as the detailed comparison between calculations and measurement and the flame
stabilization mechanism, etc., have been conducted
in [17–19]. In addition, sensitivities to the coflow
temperature and chemistry, etc. have been reported by using either divided difference [17–19] or
particle-level sensitivity analysis [20]. However,
these studies are limited to either the sensitivity
with respect to different mechanisms or local sensitivities to individual reactions without accounting
for the coupling between the kinetic parameters.
The focus of this study is on the propagation
of the kinetic uncertainty to the predicted liftoff
height H using active subspace, and the global sensitivities with respect to the kinetic parameters. The
flame is simulated using the transported probability
density function method (TPDF) in conjunction
with the standard k-ε model and enhanced wall
treatment in Fluent. The constant C1 ε is adjusted
from 1.44 to 1.6 for better jet spreading rate
prediction. The Interaction-by-Exchange-withthe-Mean (IEM) model with Cϕ = 2.0 is employed
for micromixing. The In Situ Adaptive Tabulation
(ISAT) tolerance is εtol = 6.25 × 10−6 , which is
sufficiently small for ensuring numerical accuracy
according to the study in [17]. The coflow temperature is 1045 K if not otherwise specified. The
rest of the solver settings and the computational
domain are identical to those in [17]. A grid of 22k
cells with 50 particles per cell is employed, which
ensures grid independence as shown in Fig. 1b.
The Li mechanism consisting of 21 elementary
reactions [21] is employed with the temperature independent uncertainty factors being adopted from
Konnov [22], as listed in Table 1. The predicted OH
contour and mixture fraction profile are shown
in Fig. 1, which demonstrates good agreement in
the mixture fraction between the experimental and
numerical results. As in [14], the liftoff height H
is defined as the first axial location normalized by
the diameter of the fuel jet, z/D, at which the Favre
mean mass fraction of OH reaches a value of

Fig. 1. (a) Contour plot of YOH from the base case. (b)
Profiles of mixture fraction along the axial line from both
simulation and experiment.

2 × 10−4 . It is noted that the definition of lift-off
height does not affect the conclusions drawn herein.
Each TPDF run costs about 2000 CPU-hours using 48 Xeon-E5 processors. It is computationally
infeasible to build a 21-dimensional response
surface due to the large number of runs needed.
Previous budget analysis [18] and sensitivity analysis [23] suggest that flame stabilization is mainly
attributed to the auto-ignition process. Hence we
first study the active subspace of the ignition delay
(IDT) in 0-D autoignition processes at representative conditions of the Cabra flame. These 0-D tests
can explore the level of linearity for H in the kinetic
parameters space and thereafter the dimension of
the active subspace for H in Cabra flame since they
share the same dominant physical processes.

4. Results
4.1. Autoignition tests
Wang and Pope [24] have shown that the composition of particles originating from the fuel
jet evolves along the mixing line between the jet
and coflow compositions before ignition, and the

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a

b

c

Fig. 2. (a) The eigenvalues, (b) summary plot using 670
samples (670 = 10 × 22 × log(d )) with gradient information, and (c) the pdf distribution of IDT[s] using
N = 50,000 samples.

ignition first occurs near mixture fraction ξ = 0.04.
Consequently, the composition of ξ = 0.04 from
the mixing line is chosen as the initial condition for
the adiabatic, isobaric autoignition tests. The corresponding temperature is 1015.4 K and the mass
fractions are YH2 = 9.9049 × 10−4 , YO2 = 0.1642,
YH2 O = 0.0620 and YN2 = 0.7728. The gradient
of the IDT to the rate constant is computed using
finite difference. The IDT in log scale, instead of
linear scale, is chosen as the QoI as it tends to be
more linearly dependent on variations of the rate
constants.
The corresponding eigenvalues estimated using 670 samples are shown in Fig. 2a. It is seen
that the first eigenvalue is larger than the second
one by two orders of magnitude, implying the
existence of a one-dimensional active subspace.
This is further confirmed by the summary plot of

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Fig. 2b, showing the distribution of IDTs along
the active variable, which is the projection of the
vector of 21-dimensional input variables onto the
one-dimensional active subspace, i.e., wT1 x. The
IDTs lie close to a one-dimensional curve and the
scattering in the direction orthogonal to the active
direction is small. A one-dimensional response
surface based on a 3rd order polynomial fitting can
be built and is shown as the solid line in Fig. 2b.
Then N = 50,000 samples are drawn and evaluated
using the response surface. The resulting probability distribution function (pdf) of IDT, termed as
MC + RS and shown in Fig. 2c, agrees very well
with the one from directly evaluating each sample
through autoignition integration with Cantera,
termed as MC. The differences in the mean and
standard deviation of IDT from the response of
the two approaches are within 2%. This demonstrates the accuracy of the one-dimensional active
subspace in propagating the kinetic uncertainty.
The components of the active direction, i.e.,
the first eigenvector, are shown in Fig. 3. The
magnitude of each component provides a global
sensitivity score for the individual reaction. As
shown, the most sensitive reaction is R1, which
is the major chain branching reaction for hydrogen/air ignition. Other important reactions are
those related to HO2 , i.e., R9 and R11. Note that
R9 is a chain termination reaction that inhibits
ignition, while R11 is a chain branching reaction
that promotes ignition. As such, the components
of R1 and R11 have the opposite sign as that of
R9. It is found that the active direction almost does
not change for the most reactive mixtures of ξ between 0.02 and 0.06, indicating that it is invariant
for the reactive mixtures of interest (not shown).
Since it is computationally intractable to evaluate the gradient of the liftoff height H in the Cabra
flame, global linear fitting with OLS is employed
to identify the active direction. The adequacy of
this approach is first investigated for the IDT. The
estimated active direction and its error by using
different numbers of samples are studied by bootstrapping. It is found that M = 4d = 84 samples
are adequate for estimating the active direction.
As shown in Fig. 3, the relative errors for the large
components in magnitude are small, even though
the fitting with OLS introduces certain errors
(small in magnitude) for insignificant reactions
due to the limited samples. Overall, the estimated
active direction is very close to the accurate one
from Eq. (3) using function gradients with an inner
product of 0.99.
4.2. The Cabra flame
Based on the autoignition tests, the lift-off
length in log-scale for the Cabra flame is expected
to have a one-dimensional active subspace. The
log(H) is assumed to be nearly linearly dependent

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Fig. 3. Components of the first eigenvector for the autoignition tests computed with gradient and OLS. Also shown are
the components of the active direction for the liftoff height H in the Cabra flame.

a

b

Fig. 4. (a) Summary plot and (b) pdf distribution of log(H). Blue square symbols in (a) are from three additional runs of
the simulation collected to improve the performance of the response surface at the boundary of the parameter space.

on the change of the rate constants, and the active
direction is computed using OLS with 84 samples
(M = 4d). The summary plot is shown in Fig. 4a.
The identified one-dimensional active subspace
captures most of the variations of H. Similar
to the case of IDT, the small variation in other
directions results from the error in the estimated
active subspace due to insufficient number of
samples. The response surface, based on secondorder polynomial fitting is also shown in Fig. 4a.
The R-square value of the fitting is 0.97, which
suggests that the fitting error is sufficiently small
for the present application. Then 50,000 samples
are drawn and evaluated using the second-order
polynomial fitting. The resulting pdf of log (H) is
shown in Fig. 4b. The mean of log(H) is 0.67 or
H = 4.6. The 95% confidence interval of H is [1.1,
28.3], which covers the measured 10D.

The components of the active direction in
Cabra flame are shown in Fig. 3. Compared to
the ones of IDT, the sensitivities to the reactions
R11 and R13 are promoted while R1 is relatively
weakened in the Cabra flame. Both reactions R11
and R13 are the major consumption reactions of
HO2 . According to the budget analysis in [18], the
diffusion term leads to the presence of high concentration HO2 ahead of the flame stabilization zone,
which accounts for the promoted sensitivity of the
HO2 consumption reactions. It can also explain
the promoted sensitivity of R3. However, given the
uncertainty associated, the conclusion is weak, as
the change is less significant than R11 and R13.
Nevertheless, the top sensitive reactions, e.g., R1,
R9, and R11 remain the same, and the active directions between H and IDT are quite similar to each
other, with an inner product of 0.91. This therefore

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suggests that one can first study the cheap test cases
to estimate the dimension and large components
of the active subspace for the expensive turbulent
flame simulations, if the test cases and the turbulent flame simulations share the same dominant
physics. The similarity in dominant physical processes can be analyzed via various computational
diagnostic tools such as budget analysis [18] and
Lagrangian tracking [24] referred to above. The
number of required runs of the model to accurately
estimate the active subspace can also be estimated
in the test cases. In addition, one can identify the
unimportant parameters based on the test cases,
and exclude them when identifying the active
subspace for the expensive model. This will result
in fewer samples needed for the expensive model,
as M is at least linearly increases with the number
of parameters. However, as shown in Fig. 3, the
insensitive reactions for IDT can be promoted by
the diffusion process in the Cabra flame. To be
conservative, a sufficiently small threshold should
be chosen to determine the insensitive reactions.
Since the number of parameters for the hydrogen
mechanism is relatively small, determination of the
threshold is unnecessary for the present work; and
is subject to future study for larger mechanisms.
Previous studies [19,20] have also shown that
H is very sensitive to the co-flow temperature Tc ,
such that a decrease of 10 K in Tc can double
H. The change of Tc can lead to the change of
reaction rates ahead of the flame zone. It can also
change the maximum temperature in the flame
zone as well as the relevant characteristics of the
flame, such as the flame speed. To study how Tc
affects H, a simulation under Tc = 1031 K is
performed. The Tc of 1031 K is chosen as the
predicted H = 9.75 is very close to the measured
10D. Then the effect of different Tc on the rate
constant under ξ = 0.04 is scaled to the changes
of the pre-factor A, and is further scaled to the
input parameter x. Specifically, the rate constants
are evaluated at the temperatures corresponding
to the mixture fraction of 0.04 from the mixing
line, i.e., 1015.4 K for Tc = 1045 K and 1002 K
for Tc = 1031 K, respectively. Previous Lagrangian
analysis [24] shows that the composition state is
attained to the mixing line before ignition, thus
this characteristic temperature can represent the
global influence of the co-flow temperature on the
flame. The corresponding centered and normalized
x and H, shown in Fig. 4a, demonstrate excellent
consistency with the active subspace for the rate
constants. This suggests that the effect of Tc is
mainly through changing the rate constant ahead
of the flame zone. It also questions the common
approaches to match H by adjusting Tc . To make
further judgment about the major uncertainty
source, one can conduct the inverse problem mentioned in the introduction to reduce the uncertainty
in the mechanism. As the active directions for H

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Fig. 5. Summary plot of temperature and YH2 O against
the active direction of log(H).

and IDT are similar, a small number of experimental datasets of IDT can significantly reduce the
uncertainty in the predicted IDT as well as H.
It is also interesting to study the active subspace
of the temperature and species profiles. However,
these profiles tend to be extremely nonlinear and
the OLS is not suitable here. While it is infeasible
to compute their active subspace with the collected
samples above, we can easily check whether the
active subspace for the liftoff height can explain
the variations of these profiles. The summary plot
of temperature T and major product of H2 O from
the location z/D = 10, r/D = 1 against the active
direction of H is presented in Fig. 5. It is seen that
the active subspace of H can capture the major
trend of T and YH2 O . Compared to the variation
of H in Fig. 4a, T and YH2 O only change when
z/D = 10, r/D = 1 is close to the flame stabilization
zone, i.e., H is close to 10. Therefore, the species
profiles and liftoff height H are most likely to share
the same active subspace and be correlated in terms
of the response to the kinetic parameters. Such
similarity can be used to estimate the uncertainty
of the species profiles.

5. Conclusions
The active subspace approach is applied to
propagate 21 kinetic parameters to the liftoff height
H in the TPDF simulation of the Cabra H2 /N2
lifted flame. We first identified a one-dimensional
active subspace based on 84 runs of the simulation.
Then a one-dimension response surface is built.
Finally, the distribution of H is characterized
by evaluating 50,000 samples using the response
surface, showing that kinetic uncertainty alone
is large enough to account for the discrepancies
between the measurements and simulations.

Please cite this article as: W. Ji et al., Quantifying kinetic uncertainty in turbulent combustion simulations using active subspaces, Proceedings of the Combustion Institute (2018),
https://doi.org/10.1016/j.proci.2018.06.206

JID: PROCI

8

ARTICLE IN PRESS

[m;July 9, 2018;10:32]

W. Ji et al. / Proceedings of the Combustion Institute 000 (2018) 1–8

The present work demonstrates the capability
of active subspace in revealing the effect of flowchemistry in turbulent combustion simulations.
The identified active subspace for H is compared
with that for the ignition delay under flame conditions, and they are generally similar to each other
with the most sensitive reactions being the same.
The difference reveals that the sensitivity of the
HO2 consumption reactions is promoted due to
the diffusion process in the Cabra flame. As the
results indicate that the diffusion process alters the
sensitivities of the HO2 consumption reactions,
future work might include transport parameters
as an additional uncertainty source to investigate
the effect of flow-chemistry interaction more
conclusively.
Compared to previous work on the uncertainties of the operating conditions [12], the kinetic uncertainty has higher dimensions and is more challenging. In addition, the simple zero/one-D simulations share the same kinetic uncertainty inputs with
the turbulent combustion simulations, and as such
are used to estimate the active subspace in turbulent
flames. The identification of active subspaces via
inexpensive surrogate models is a potential effective
solution for expensive turbulent combustion simulations with large mechanisms or strongly nonlinear response. The approach is open to other types
of flame and various turbulent combustion models.
Acknowledgments
This work was supported by the National Natural Science Foundation of China Nos. 91441202
and 51476087.
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Please cite this article as: W. Ji et al., Quantifying kinetic uncertainty in turbulent combustion simulations using active subspaces, Proceedings of the Combustion Institute (2018),
https://doi.org/10.1016/j.proci.2018.06.206
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