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Combustion Theory and Modelling
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Conditional space evaluation of progress variable
definitions for Cambridge/Sandia swirl flames
Nikola Sekularac, XiaoHang Fang, W. Kendal Bushe & Martin H. Davy
To cite this article: Nikola Sekularac, XiaoHang Fang, W. Kendal Bushe & Martin H. Davy (2023):
Conditional space evaluation of progress variable definitions for Cambridge/Sandia swirl
flames, Combustion Theory and Modelling, DOI: 10.1080/13647830.2023.2211537
To link to this article: https://doi.org/10.1080/13647830.2023.2211537
© 2023 The Author(s). Published by Informa
UK Limited, trading as Taylor & Francis
Group
Published online: 12 May 2023.
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Combustion Theory and Modelling, 2023
https://doi.org/10.1080/13647830.2023.2211537
Conditional space evaluation of progress variable definitions for
Cambridge/Sandia swirl flames
Nikola Sekularac a,b, XiaoHang Fang a,c∗, W. Kendal Bushedand Martin H. Davy a
aDepartment of Engineering Science, University of Oxford, Oxford, UK; bCERFACS, Toulouse,
France; cDepartment of Mechanical & Manufacturing Engineering, Schulich School of
Engineering, University of Calgary, Calgary, Canada; dDepartment of Mechanical Engineering,
University of British Columbia, Vancouver, Canada
(Received 11 July 2022; accepted 9 April 2023)
Data from all spatial locations of nine turbulent flames in the Cambridge/Sandia swirl
database are combined to study how the choice of scalar variables in conditional
moment closure (CMC) type approaches affect the conditional spatial fluctuations of
reactive scalars. In order to investigate the influence of swirl and stratification, two
additional data-sets have been constructed. Principal component analysis (PCA) is
applied to help identify the number of scalar variables and the most appropriate choices
to describe the composition space. Two PCA scaling methods have been adopted,
namely Pareto and Auto-scaling. Regardless of the data-set investigated and the scal-
ing method used, the results suggest that a single principal component correlated with
temperature accounted for the largest variance. For the first moment hypothesis, four
progress variable, c, definitions identified by PCA are selected as conditioning vari-
ables to investigate the conditional fluctuations and normalised RMS of various species
and temperature from all three databases at all axial locations. The results indicate
that two control variables based on mixture fraction, Z, and progress variable signif-
icantly reduce the conditional fluctuations of scalars compared to a single variable.
The selection of progress variables had minimal effects on the RMS of conditional
fluctuations for all tested conditions, although a slight reduction of conditional fluc-
tuations was found for the temperature-based progress variable, which can potentially
help the further extension of CMC-based models in different flame configurations. The
present study also shows that using Zand c(regardless of its definition) as two condi-
tioning scalars enables the detachment of the thermo-chemical state from space, swirl
and stratification effects. This suggests that adopting a doubly conditioned source term
estimation (DCSE) approach might successfully predict the considered set of flames,
assuming that ensembles are divided along the axial direction.
Keywords: numerical simulation; Turbulent combustion; principal component analy-
sis; conditional moment method
1. Introduction
For simulations that are not fully resolved, closure of the chemical source term is needed.
Various methodologies have been developed (e.g. the flamelet model [1], or the conditional
moment closure approach [2]) to build functional filtered chemistry models that can close
∗Corresponding author. Email: xiaohang.fang@eng.ox.ac.uk,xiaohang.fang@ucalgary.ca
© 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.
org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository
by the author(s) or with their consent.
2N. Sekularac et al.
the highly non-linear source term. Although based on different assumptions, the accuracy
of these models relies on (i) an adequate description of the composition space, which seeks
to relate the different scalar variables in the thermo-chemical scalars’ vector, and (ii) an
adequate description of the statistical distribution of these variables in the form of a joint
probability density function (PDF).
Among different models and studies, local mixture, the degree of progression of the
chemical reactions, strain rate, total enthalpy, and residence time are properties that have
been found to be closely related to the chemical rates in a reactive flow field. For non-
premixed flames, the mixture fraction, Z, is usually retained as the main controlling
variable to describe the turbulent mixing effects [3]. For premixed flames, the progress
variable, c, is often selected as the primary scalar to describe the rate of progression from
the unburnt to burnt state [4]. It has been shown that the fluctuations found in the species
mass fractions and temperature are often correlated in non-premixed flames with the fluc-
tuations of Zand with the fluctuations of cin premixed flames. However, a single scalar
variable able to accurately describe the structure of partially-premixed flames and cap-
ture regions with high probabilities of local extinction and re-ignition has not yet been
found [5,6]. This has led to different combinations of scalar variables, where often the
mixture fraction and the progress variable are simultaneously considered [7]. The inclu-
sion of two different mixture fractions as control variables was also proposed to capture
the main properties of moderate and intense low oxygen dilution (MILD) combustion for
jet-in-hot-coflow (JHC) burners [8–11]. While mixture fractions are generally incorporated
using Bilger’s definition [12], various definitions of the progress variable have been used
for different approaches.
For laminar flamelet approaches, progress variables are often chosen as the primary
controlling variables to construct a reduced dimension composition space. The tempera-
ture and the mass fraction of a major combustion product (e.g. H2OorCO
2) have been
widely used to define cfor flamelet models. While suitable for many flames, using a sin-
gle product mass fraction for larger hydrocarbons may fail to represent the progress of the
chemical reactions. Therefore, more species mass fractions in the form of linear combina-
tions are often incorporated to track the reaction progress in the flow and capture different
stages of combustion [13,14]. Sun et al. [15] have investigated the modelling capabilities
of the unsteady flamelet/progress variable (UFPV) model in predicting the ECN Spray A
cases. The progress variable was defined using a combination of major and intermediate
species to obtain results across all downstream locations of the jet. While progress variables
incorporating more than a single species mass fraction are effectively better performing,
two challenges are worth mentioning, of which (i) the diffusivity of the selected scalars
needs to be accounted for in the progress variable transport equation, and (ii) the addition
of intermediate species is often in conflict with the monotonicity of c. Studies have tack-
led the injection of the progress variable by computing weight coefficients using various
automated optimisation techniques, e.g. the well-known ∂c
∂κ >0 criterion, where κoften
denotes a time or a spatial coordinate [16–18]. However, most progress variable definitions
found in the literature are based on user expertise where the chosen definition can signif-
icantly influence the numerical predictions, particularly for fuel-rich/heavy-fuel mixtures
[19]. Recently, Gupta et al. [20] studied different progress variable definitions for tabu-
lated chemistry through the analysis of premixed methane-air (CH4/Air) laminar flames.
Compared with detailed chemistry, flamelet generated manifold (FGM) results using differ-
ent species mass fraction-based progress variable definitions were shown to give different
mass burning rates. Lipatnikov & Sabelnikov [21] also examined the effect of five different
Combustion Theory and Modelling 3
progress variable definitions on the flamelet approach predictions of the mean density and
the mean mole fractions of various species using Direct Numerical Simulation (DNS) data
of a premixed hydrogen-air flame. A complementary study was carried out using DNS data
of lean hydrogen-air turbulent premixed flames operating under various Karlovitz numbers
[22]. Similar findings were obtained, suggesting that the definition of cin flamelet-based
models indeed affects the physical modelling while simultaneously impacting numerical
errors.
For conditional moment closure (CMC) models, filtered chemistry is modelled through
the separation of model elements which give descriptions for the moments of reactive
parameters and model of the distribution function. CMC-based approaches are centred
around the hypothesis that fluctuations in reactive scalars are closely correlated with the
fluctuations around values of conditioning variables (e.g. mixture fraction and progress
variable) [23,24]. Originated from CMC, conditional source term estimation (CSE) avoids
solving conditionally-averaged transport equations by inversion of the integral functions
[25]. In a recent work on the methane-air non-premixed piloted Sandia flames, CSE was
found to provide a more accurate solution while being less time-consuming compared to
CMC [26]. CSE models have also been used to simulate large hydrocarbon and spray
flames. While using CSE-based approaches with a single conditioning variable has been
found suitable for non-premixed and premixed flames [7,27–30], for partially-premixed
or stratified combustion, one conditioning variable is not sufficient. Subsequently, dou-
bly CSE (DCSE) with mixture fraction and progress variable as conditioning variables
have been developed to simulate partially-premixed flames, lifted flames, and spray flames
[31,32]. The CSE and DCSE concepts have also been examined in a-priori analysis in
DNS for high-pressure conditions [33]. The results highlighted that DCSE and double
conditioning are likely to be needed for cases closer to real practical combustion appli-
cations. Similar to the flamelet approaches, when choosing the conditioning variables for
CMC approaches, the mixture fraction definition from Bilger is well accepted by the com-
munity, where the choice of progress variable and particularly its effects on conditional
variable fluctuations needs more studies. Recently, Bushe [34] and Mousemi & Bushe
[35] examined the conditional moment closure hypothesis over the Sandia/TUD, the Syd-
ney Swirl burner, and Cambridge/Sandia stratified swirl burner databases. Their studies
suggested two-condition (mixture fraction and progress variable) conditional averages in
the Sandia/TUD do not vary in space nor vary with the Reynolds number, whereas, for
Sydney and Cambridge/Sandia swirl burners, a third conditional variable (total enthalpy)
might be needed to further reduce the spatial gradients of conditional averages attributed
to the heat transfer. For these studies, progress variables based on temperature and mass
fraction of CO2were used. Perhaps more interestingly, the a-priori studies on single-step
high-pressure DNS data from Devaud et al. [33] and Bushe et al. [36]usingCO
2and
temperature-based progress variables suggested the success of DCSE does not depend on
a particular choice of the second conditioning variable which significantly increased the
capabilities of CSE models. However, further extension of these two studies on the effect
of progress variable selections is not possible due to the single-step chemistry nature of the
DNS data.
Consequently, two questions around control variables in the context of CMC-based
models are worthy of note here – the questions central to this paper:
(1) What is the minimum number of scalar variables needed to adequately characterise
conditional space for certain flames?
4N. Sekularac et al.
(2) Which scalar variables represent the best choices for this, and how do they influence
the predictions, in particular for the definition of the progress variable?
Depending on the studied case, Bushe et al. [34] also pointed out that the selection and
number of control variables must be carefully undertaken to adequately address the effects
of turbulence and chemistry. Additionally, the modelling of the joint-PDF gives a sub-
stantial challenge to the turbulent combustion modelling community as its shape strongly
depends on the selected scalar variables. The validity of the statistical independence
assumption used for modelling the presumed joint-PDF was shown to be erroneous, as
correlations between conditional variables are important [36].
The present work is motivated by the study of three different groups: Sutherland & Par-
ente’s work on principal component analysis-based models [37], Bushe’s study on spatial
gradients of conditional averages [34], and Gupta et al.’s analysis on the impact of the
progress variable definition on flamelet generated manifolds [20]. We investigate the two
questions posed above with a particular focus on the effect of progress variable selection on
the conditional fluctuations obtained with one-condition conditional averages and doubly
conditional averages of the Cambridge/Sandia swirl burner data-set. We will first intro-
duce the properties and working conditions of the burner and the methodology employed
for determining the appropriate progress variable definitions. The chosen progress variable
definitions are then used to study conditional fluctuations where all one-point, one-time
measurements are included. The results are discussed, and physical insights are provided
based on the observations.
2. Methodology
2.1. Experimental setup
Experimental measurements of premixed and stratified CH4/Air flames from the Cam-
bridge/Sandia swirl burner (referred to as SwB) are used in this study. Multiscalar data
of nine turbulent flames are examined under: (i) various ratios of stratification, and (ii)
various swirl intensity conditions, depicted in Table 1.
The burner, shown schematically in Figure 1, features a large co-flow of pure filtered air
preventing ambient air from entering the reaction zone, and two concentric outer (subscript
Table 1. Operating conditions for Cambridge/Sandia swirl burner. φiand φodenote equivalence
ratio of the flow in the inner and outer annuli, respectively. In all cases φg=0.75, Ui=8.31 m/s,
Uo=18.7 m/s and Uco−flow =0.4 m/s. Conditions highlighted in bold font denote flames under
swirling flows.
Flame Swirl ratio
Stratification
factor φiφo
SwB1 0 1 0.75 0.75
SwB2 0.25 1 0.75 0.75
SwB3 0.33 1 0.75 0.75
SwB5 0 2 1 0.5
SwB6 0.25 2 1 0.5
SwB7 0.33 2 1 0.5
SwB9 0 3 1.125 0.375
SwB10 0.25 3 1.125 0.375
SwB11 0.33 3 1.125 0.375
Combustion Theory and Modelling 5
Figure 1. Plan view schematic of the exit geometry in the Cambridge/Sandia swirl burner, showing
a plan view and a cross section through the burner axis. The curved-dashed arrows in the plan view
indicate the direction of swirling flows in the outer annulus. φiand φoin the cross section denote the
equivalence ratio of the flow in the inner and outer annuli, respectively. Adapted from [38]
o) and inner (subscript i) annuli supplying the fuel/air mixture. The annuli’s velocities
were chosen to maximise the Reynolds numbers in the flows with Rei=5,960 and Reo=
11,500. A variable degree of swirl allows the burner to mimic the flow conditions found
in many practical systems. The swirl assists flame stabilisation allowing more extreme
stratified conditions to be investigated than would otherwise be possible. The stratification
factor is defined by the ratio of the equivalence ratio in the inner and outer annuli. The
scalar measurements recorded include temperature and the mole fractions of CO2,CO,
H2,CH
4,N
2,O
2and H2O at different axial and radial positions. A minimum of 300 sam-
ples were taken at 60 different radial locations per axial position via Rayleigh and Raman
scattering to capture temperature and major species, respectively. Further information on
the measurement techniques, the experimental setup and the burner’s characteristics can
be found in [38,39]. The existence of turbulent regions within the flames, operating under
premixed and/or stratified mixture conditions, with or without swirl, makes this burner an
ideal test case for attempting to answer the two questions central to this paper.
2.2. Data-processing
Three data-sets have been constructed to investigate the effects of spatial coordinates, swirl
flow ratio, and stratification factor on the conditionally-averaged reactive scalars, depicted
in Table 2. In the first case, the data collected from all nine flames are grouped together to
create a general conditional domain for each of the scalars (SwB|all). Here, it is assumed
that the conditional averages are independent of spatial coordinate, swirl flow ratio and
stratification factor. The second data-set (SwB|Hstratified) combines the measurements of
6N. Sekularac et al.
Table 2. Characteristics of the three data-sets used for investigating the behaviour of conditional
averages.
Case Flames Swirl ratio
Stratification
factor Comments
SwB|all SwB1-11 0–0.33 1–3 All nine flames grouped together
SwB|Hstratified SwB9-11 0–0.33 3 Fixed high stratification
SwB|Hswirl SwB3,7,11 0.33 1–3 Fixed high swirl
three flames exhibiting varying swirl flow ratio intensities with a single high stratifica-
tion factor to investigate the dependence of conditional averages on swirl. For the third
case, data from 3 flames with different stratification factors and a fixed high swirl flow
ratio are grouped to investigate the dependence of conditional averages on stratification
(SwB|Hswirl). Grouping the flames in these three distinct data-sets, each tackling a specific
characteristic of the flow, allows the exploration of the most optimal scalar or combina-
tion of scalars needed to sufficiently accurately represent the thermo-chemical state under
different conditions.
The data was first ‘cleaned’ to remove mole fractions displaying negative values asso-
ciated with experimental uncertainty. All mole fraction measurements were subsequently
converted to mass fractions values (note that mass fractions will be used throughout the
present work). An extensive analysis of the scalars profiles showed that the measurements
of YH2 are concatenated between the interval [0, 0.0002] for all three data-sets. As such, an
artificial exclusion for values of H2mass fraction above 0.00025 is adopted and carried out
to remove potential outliers. The mixture fraction Zwas calculated for every instantaneous
single-point measurement. The definition of Zproposed by Bilger is used to calculate the
mixture fraction of all nine flames, as
Z=2YC−YC,2
MC+1
2
YH−YH,2
MH−YO−YO,2
MO
2YC,1−YC,2
MC+1
2
YH,1−YH,2
MH−YO,1−YO,2
MO
(1)
Mixture fractions of zero and unity are respectively assigned to pure air and the richest
entry of the flow through all of the cases with an equivalence ratio of 1.125. For the con-
sidered sets of flames, the stoichiometric mixture fraction lies at Zstoich =0.9, with a lower
flammability limit located at Z=0.575 [40]. To investigate the role of control variables in
conditional spaces, four of the most common combustion progress variable cused by the
community are considered in this study, defined as
ck=φk−φk,min
φk,max −φk,min
(2)
where φ1denotes the temperature, while φ2,φ3and φ4are the mass fractions of CO2,
CO +CO2and CO +H2+H2O+CO2, respectively (cf. Table 3). The local maximum is
determined using a function that returns the upper peak envelopes of the scalar kselected to
define c. The envelope is computed using spline interpolation over local maxima separated
by 2,500 samples, for which a parametric study was performed to find the optimal number
of samples. The local minimum values have been fixed to zero and to 290 K for the species-
based progress variables and temperature, respectively.
A second condition based on mixture fraction and the four progress variables (i.e.
Z,ck<0 and Z,ck>1) is applied to take potential outliers from the analysis as
Combustion Theory and Modelling 7
Table 3. Summary of the four progress variables investigated in this study using Equation (2).
Label Scalar(s) φkMark
c1Temperature
c2YCO2
c3YCO2 +YCO •
c4YCO +YH2 +YH2O +YCO2 ♦
these points are considered to be unphysical. After executing all previous steps, each
database consists of 5,518,536 point-based measurements for SwB|all, 1,887,496 for
SwB|Hstratified and 1,901,113 for SwB|Hswirl.
To examine the conditional space of Cambridge/Sandia flames, the methodology pro-
posed in [34,40] is followed and applied to all three data-sets. The conditional averages are
obtained via a discrete process involving binning, dividing each progress variable dimen-
sion into 30 bins. This is justified given that more bins increase the possibility of having
intervals with an insufficient number of data points which imposes unrealistically small
fluctuations around the mean value [40]. Moreover, considering that DCSE will likely
be needed for modelling reactive flows relevant to practical combustion systems [33], if
more bins are included then more computational time is needed during the inversion pro-
cess with the matrix of the joint-PDF. This implies that a much larger matrix needs to be
inverted compared to previous implementations of CSE in premixed and non-premixed
flames. Consequently, the inversion process becomes much more challenging. For the first
moment hypothesis, the conditional fluctuations of species mass fractions and temperature
around one-condition (c1–c4) conditional average are calculated at each axial location, such
that
f
i,k=fi−f|ξ=ck(x)(3)
where idenotes a single-point measurement, f
i,kis the fluctuation of either mass fraction or
temperature around one-condition (ck), fiis the point measurement of that reactive scalar
and f|ξ(x)is the conditional average of that reactive scalar evaluated by averaging all of
the measurements of the chosen data-set at all radial locations together at each downstream
distance. Two reasons for investigating how much conditional averages vary in the axial
direction are worth mentioning. First, the conditional fluctuations are larger in the axial
positions and are spatially independent in the radial direction [34,41]. That was shown
to be particularly true for jet flames, suggesting that within a CSE framework, group of
localised cells, referred to as an ensemble, should be divided along the axial direction. Sec-
ond, Mousemi et al. [40] showed for the same burner that the global conditional averages
(equivalent to defining a single CSE/DCSE ensemble where all of the reactive control vol-
umes in the domain are included) did not exhibit a particular functional dependence on the
flow dynamics and the burner’s geometry assuming that three conditioning variables are
selected/retained. Suppose that ensembles are split across all axial positions, the number of
conditioning scalars needed to accurately represent the chemical state is reduced, where a
single control variable could perhaps be sufficient to separate the conditional averages from
spatial coordinate, swirl and/or stratification effects. This approach is of particular interest
for CMC-based models as it can be seen as a viable alternative to bypass the challenges
associated with joint-PDFs defined by a minimum of two scalars.
8N. Sekularac et al.
If the conditional average is a good representation of the local thermo-chemical state,
then it is expected that the mean of the conditional fluctuations will be zero. However, the
RMS of those fluctuations is clearly not, as shown by Bushe [34] using the Sandia/TUD
database and the Sydney swirl burner. Therefore, the square root of the average of the
square of conditional fluctuations can be computed, as
RMSi,k=f2
i,k(4)
The proposed RMS is normalised by the maximum value of the considered reactive scalar.
This last step is justified in two ways: (i) to compare the different scalars to one another
which the relative magnitude should be comparable, and (ii) the data has been filtered
to eliminate outliers, suggesting that the maximum measured value is unlikely to be the
consequence of a major measurement error.
2.3. Principal component analysis
Over the past decade, low-dimensional manifold representations have been frequently used
to mitigate the costs associated with turbulent reacting flows and detailed kinetics [42].
Data-driven analytical tools have seen considerable success in combustion applications
for building low-dimensional manifolds while preserving an adequate representation of
the thermo-chemical state [43]. Among many others, principal component analysis (PCA)
may be employed to find new sets of conditioning variables that have the highest corre-
lations with the reactive scalars to detach the conditional averages from the real domain.
PCA parameterises the thermo-chemical state-space using a reduced number of optimal
scalars identified in the directions of maximal data variance, principal components (PCs).
Projecting the state-space on those PCs gives the PC-scores, and adopting only a subset
of those scores as conditioning variables is expected to result in a more accurate repre-
sentation of the chemical state with smaller discrepancies for the unconditional averages
[40]. However, a number of issues are yet to be addressed regarding the applicability of
PCA with CMC-based approaches. Suppose more reactive scalars are combined to define
a PC, the diffusion term for the selected principal component becomes more complex, and
evaluation of the diffusive fluxes for each component is required [44]. Similar to the diffu-
sion problem, the chemical source terms of all scalars used to define the selected PC must
be combined to appropriately describe the principal component’s source term. Moreover,
the PCs are often difficult to associate with previously presented control variables, where
physical interpretations are not always straightforward depending on the studied case. This
raises an additional complexity, in particular with the closure of the chemical source terms,
where presuming the shape of the PCs’ PDFs is not trivial. Accordingly, rather than adopt-
ing PC-scores as controlling variables, here, PCA is utilised as a data-driven technique
to identify which definition of cis needed/preferred to accurately describe the flames of
interest. As such, PCA can be used as a guideline for building an appropriate look-up table
parametrised by an optimum progress variable definition that encompasses the most rele-
vant features/effects of the flames. Previous studies [45] suggested that one of the first PCs
was often found to be highly correlated with Zfor non-premixed flames. While this has
been thoroughly validated for Sandia/TUD jet flames, to the best of the authors’ knowl-
edge, premixed flames have not been studied yet with PCA, suggesting that further research
is needed.
Combustion Theory and Modelling 9
The mathematical approach to compute the principal components of a given data-set X
(n×Q) reduces to an eigenvalue decomposition problem, where rows nrepresent indi-
vidual measurements of Qvariables. Suppose X has been appropriately standardised (i.e.
centred and scaled), PCA projects all Qvariables onto a rotated basis obtained from the
eigenvalue decomposition of the covariance matrix S(Q×Q)as
S=1
n−1XTX=ALAT(5)
where A is the (Q×Q) matrix whose columns are the eigenvectors of S, and L is a (Q×Q)
diagonal matrix containing the eigenvalues of S. Following the details of the PCA reduction
provided in [37,46], PC-scores are obtained as
=XA (6)
where is an (n×Q) matrix. Each column of A describes the weight between the Q
variables of X and the corresponding principal component. The dimensionality reduction
is undertaken by truncating A, such that only the first qPCs that account for the maximum
variance are retained, with q<Q. The original data-set X is retrieved as
XXq=qAT
q(7)
where Xqis the approximation of X based on the first qeigenvectors of A, and qis the
(n×q) matrix of the principal component scores. Detailed mathematical formulation of
PCA is not elaborated here where more details can be found in the literature [47].
Principal component analysis requires high-fidelity data-sets to generate the PC-basis
and accurately describe the thermo-chemical state-space. The experimental measurements
of all three data-sets fed to PCA have been cleaned out following the steps presented pre-
viously. It should be noted that the mixture fraction and the progress variables have been
excluded from the databases before being passed to PCA.
Various studies have tackled the effects of scaling methods on PCA [48,49]. Scaling
has an important outcome on the method’s accuracy as it can change the PCA structure
by altering the relative importance of various scalars. Auto-scaling, Range scaling, VAST
(variable stability) scaling, Level scaling and Pareto scaling are among the most common
options used in conjunction with PCA for combustion studies. Range scaling divides each
variable by the difference between the minimal and the maximal value, whereas Level scal-
ing adopts the mean values of the variables as the scaling factor. VAST scaling focuses on
using the product between the standard deviation and the so-called coefficient of variation,
defined as the ratio of the standard deviation and the mean. Pareto scaling was recognised
as having a distinct advantage for major species and source terms reconstruction while
needing fewer components [50]. Level, VAST, Range and Auto-scaling options were found
to provide similar results with often more components needed to achieve the same recon-
struction accuracy obtained with Pareto [51]. Therefore, in order to study the scaling effect
on the accuracy of the method, the PCA analysis is carried out using two scaling options,
assuming that the data-sets have been previously centred:
(1) Pareto scaling, which adopts the square root of the standard deviation
(2) Auto-scaling (AS), which uses the standard deviation as the scaling factor
Previously, Parente & Sutherland [52] found that Auto-scaling is more adapted when
an exploratory analysis on the chemical manifold should be performed, whereas Pareto
10 N. Sekularac et al.
appears more suitable for capturing the principal features of the systems and the behaviour
of the main species. Parente & Sutherland also [52] showed that the square root of the
standard deviation enhances the temperature scalar in carrying most of the data variance,
and thus, forcing the first principal component to align with temperature. For this reason,
the temperature was excluded from the three databases passed on to PCA.
The real utility in PCA comes by founding correlations among the variables defining the
state-space. A new coordinate system is identified in the directions of maximal data vari-
ance, allowing less important dimensions to be eliminated while maintaining the primary
structure of the original data. In that sense, one can suppose applying PCA to a given reac-
tive flow where no prior knowledge about the physical and chemical phenomena is known,
and help identify the adequate number of control variables needed to accurately quantify
the thermo-chemical state-space within a manifold. In order to determine the amount of
information captured by each principal component and thus replace the Qelements of X
by q<Qprincipal components, the fraction of total variance accounted by each PC is
calculated as
tqi=qi
k=1lk
Q
k=1lk
(8)
where iand lkdenote a single PC and the variance located on the diagonal of the covariance
matrix S, respectively. Since outputs resulting from the two scaling methods have different
numerical ranges, their PC-scores have been scaled to the interval [−1, 1].
3. Results & discussion
3.1. Principal component analysis
The PC analysis was individually performed on all three databases with all radial and
axial locations grouped together. Figure 2illustrates the variance accounted by each PC
using Equation (8). To clarify, figures depicting PCA results do not include the temperature
scalar within the analysis. Regardless of the scaling method adopted, a single principal
component seems to account for the largest amount of variance present in all three data-
sets, with ∼0.9 using Pareto and ∼0.8 with AS.
The variance explained by PC1 is in good agreement with the threshold proposed by
Parente et al. [46]. Their study showed that by accounting for ∼0.9 of the total variance, all
main species and temperature can be recovered with satisfactory levels of approximation.
Consequently, the physical interpretation of all other principal components is omitted in
this work, as is it believed to be out of the scope of this study.
In order to determine the underlying structure of PC1, the weights of the original vari-
ables characterising the three databases (i.e. matrix A) are presented in Figure 3for both
scaling methods. Regardless of the scaling method used, it is interesting to note that PCA
is able to automatically distinguish reactants from products, with PC1 being negatively
and positively correlated with reactants and combustion products, respectively. Regardless
of the data-set, it appears that the mass fractions of CO2and O2have the most important
contributions to PC1-Pareto, and to a larger extent YH2O, with coefficients equal to approx-
imately 0.6, 0.65 and 0.35, respectively. This trend is also apparent for PC1-AS, with the
latter having non-negligible weights on intermediate species, as opposed to Pareto, which
clearly emphasises main species. This observation agrees with the study undertaken by
Combustion Theory and Modelling 11
Figure 2. Comparison of variance explained with Pareto (triangles) and Auto-scaling (pentagons)
for each principal component of (a) SwB|all, (b) SwB|Hstratified and (c) SwB|Hswirl.
Parente et al. [52] which has shown that the variance accounted for minor species by Auto-
scalingisupto ∼20% higher than that explained by the other scaling methods investigated
in their work.
Considering the criterion proposed by Ranade & Echekki [53], only coefficients with
magnitudes ≥0.4 are kept to help identify the more prominent contributors to PC1. As
PC1-Pareto, the same three species appear to have dominant weights on PC1-AS, namely
the mass fractions of CO2,O
2and H2O, with ∼0.4. It is worth mentioning that all three
scalars are known to behave linearly with temperature, thus suggesting that PC1 is perhaps
correlated/aligned with temperature. This trend is illustrated in Figure 4, where results
of all three data-sets considered herein promote a PC1 monotonically increasing with
temperature. As expected, this behaviour is clearly accentuated by adopting the Pareto
method.
A supplementary analysis was carried out by including temperature in SwB|all and using
only Auto-scaling, as PC1-Pareto will be constrained to align with temperature. Figure 5
illustrates the dominant contributions to PC1-AS. The temperature scalar and the same
three species mass fractions have the largest weights on PC1, with ∼0.4, suggesting that
PC1, regardless of the scaling method adopted, is correlated with temperature.
12 N. Sekularac et al.
Figure 3. Comparison of weights obtained with Pareto and Auto-scaling (with strips) for the
leading principal component and scalars of (a) SwB|all, (b) SwB|Hstratified and (c) SwB|Hswirl.
After identifying the structure of the first principal component, the first-moment condi-
tional fluctuations analysis of all three databases is carried out to investigate which of the
proposed progress variables can sufficiently accurately characterise the composition space.
As suggested by PCA, particular attention is brought to the temperature-based progress
variable c1.
3.2. All flames (SwB1-11)
Conditional averages of the Qvariables describing the SwB|all thermo-chemical state
can be calculated and consequently determine the conditional fluctuation associated with
each experimental measurement. One-condition conditional averages using one of the four
progress variables are investigated and compared to one another in order to determine the
most optimal definition of ck. Figure 6illustrates the conditional fluctuations of tempera-
ture and five different species mass fractions. To clarify, throughout the entire document,
figures with axial locations account for all data at different radial locations.
Regardless of the progress variable investigated, at all eight downstream locations, con-
ditional fluctuations of the mass fraction of CH4, CO and H2around ckexhibit an important
functional dependence on the physical domain, stratification and/or swirl, visually high-
lighted by conditional fluctuation points spread far from zero. This trend is emphasised
near the burner’s tip, where the heat exchange with the bluff body might be significant,
Combustion Theory and Modelling 13
Figure 4. Comparison of correlations obtained with Pareto (triangles) and Auto-scaling (pen-
tagons) for the leading principal component with temperature for (a) SwB|all, (b) SwB|Hstratified
and (c) SwB|Hswirl. The markers illustrate 500 point-based measurements randomly selected within
the flames’ and PC1-scores’ data-sets.
Figure 5. Comparison of weights obtained with Auto-scaling (with strips) for the leading principal
component and scalars of SwB|all with temperature included. PC1-AS accounts for ∼0.8 of the
total variance, R2=0.97-AS with temperature.
14 N. Sekularac et al.
Figure 6. Conditional fluctuations of species mass fractions and temperature around the condi-
tional average f|ξ=ck(x)for SwB|all database using only ckas the single conditioning variable
and collecting all points at different radii together; are also shown the local average of these
conditional fluctuations f
i,k(golden markers).
but also where an important recirculation of the flow is encountered. Due to high swirl,
the recirculation zone is extended further downstream, leading to important conditional
fluctuations of intermediate species at axial distances corresponding to z=40, 50 mm. It
is interesting to note that all investigated ckpromote very similar results for the same three
previous species. On the contrary, the conditional fluctuations of temperature and mass
fractions of CO2and H2O vary in function of the progress variable retained. The condi-
tional fluctuations of YH2O around c1suggest that a temperature-based progress variable
can perhaps more effectively decrease the functional dependence on the physical domain
compared to the other ck. This can be attributed to the fact that the mass fraction of H2O
is closely relevant to temperature. As expected, a similar behaviour can be seen for the
conditional fluctuations of T around c1, and the conditional fluctuations of YCO2 around c2.
While local averages of all conditional fluctuations are anchored at zero, and thus, sug-
gesting that all progress variables investigated are doing a good job of characterising the
considered data-set, it is nearly impossible to find which definition of cis effectively the
Combustion Theory and Modelling 15
Figure 7. Normalised RMS of the conditional fluctuations of temperature and species mass frac-
tions for the SwB|all database around the conditional average f|ξ=ck(x)(markers) using ck
as the single conditioning variable and around the conditional average f|η=Z,ξ=c1,c2(x)
using the mixture fraction and, the temperature-based progress variable (crosses) or the YCO2-based
progress variable (pluses), and collecting all points at different radii together.
best choice to accurately describe the thermo-chemical state-space, and detach it from
spatial coordinates, but also swirl and/or stratification effects.
Figure 7enables to distinct the performances of each ckby analysing each variable’s
normalised RMS. The normalised RMS of Y
CO2 around c2provides the best results due
to the inclusion of the CO2mass fraction in c2. As expected, the same trend is observed
for the RMS of Taround c1. It is interesting to note that further downstream the axial
position, the RMS of Y
CO2 around c1are improved or off the same order of magnitude as
the YCO2-based progress variable. Additionally, the RMS of Y
CH4 obtained using any ck
are unchanged and remain close to 10%. Assuming that RMS of conditional fluctuations of
the order of 10% can be considered as ‘relatively small’ [34], one can suppose that using a
single conditioning variable for this database might still give acceptable predictions of the
considered reactive scalar for conditional moment closure models.
However, using ckas a single conditioning variable gives poor results for intermediate
species, i.e. CO and H2, where normalised RMS exceed 10% of the maximum value of
16 N. Sekularac et al.
that particular scalar in the regions near the burner’s tip. This suggests that the conditional
averages are different (changing in function of space, stratification and/or swirl) and that a
single controlling variable is not sufficient. The normalised RMS analysis contradicts the
PCA results, where a single principal component direction was found to have the high-
est correlation with the reactive scalars. This is possibly due to the inherent nature of the
PCA model, where PC-scores provide a rigorous mathematical formalism to reduce the
dimensionality of the original data while retaining most of the variance induced by the
turbulent fluctuations. Consequently, conditional fluctuations around two-condition condi-
tional averages using mixture fraction and a temperature-based progress variable (Z,c1)
are considered in this study. Conditional fluctuations around mixture fraction and a YCO2-
based progress variable (Z,c2) are also included to provide further insight. Each mixture
fraction dimension is divided into 50 bins. It can be deduced that: (i) doubly conditioning
is of particular interest for intermediate species, in particular those believed to be highly
correlated with Z(e.g. YCO), and (ii) regardless of the ckselected, one-condition con-
ditional averages seem to not deviate that much from two conditions (e.g. temperature,
carbon dioxide and water). Interestingly, normalised RMS of fluctuations around the one-
condition conditional averages of YCO and YH2 are nearly as efficient as Z,c1and Z,c2
further downstream the axial direction. The normalised RMS of major species and temper-
ature using the mixture fraction and the temperature-based progress variable are somewhat
more effective compared to Z,c2, excluding, again, the RMS of Y
CO2. The differences
remain minor, suggesting that the choice of a particular progress variable definition does
not seem important, as deduced in [33]. However, it is believed that adopting Zand c1
as the two controlling variables will provide a much more accurate representation of the
Cambridge/Sandia flames’ chemistry compared to mixture fraction and a species-based
progress variable. The underlying assumption here is that diffusion effects play an essen-
tial role in describing the chemical states. From this perspective, it is assumed that c(Yi)
would be a poor choice as the diffusion coefficient is often modelled using the unity Lewis
number assumption, whereas the reduced temperature progress variable includes the ther-
mal diffusivity by solving the diffusion flux term. Recently, Turkeri et al. [54] showed
through a parametric study that preferential diffusion is less relevant than heat losses for
the studied burner, as it was found that the latter are more prevalent to accurately capture
the underlying physics, particularly at the inlet of the burner.
The two-condition conditional averages of the temperature and mass fractions of sev-
eral species around mixture fraction and c1are shown in Figure 8. The obtained results
are in good agreement with [40], where minor discrepancies are attributed to the differ-
ent data-processing steps adopted in this study. The contours of conditionally-averaged
scalars around Zand a YCO2-based progress variable are illustrated in Figure 9. Despite
gathering data from all axial and radial locations, one region common to both conditional
domains can be identified in which no measurement has been found. Assuming the mixture
fraction lies within the limits of flammability, the empty region suggests that the unburnt
reactants are becoming unstable and start to react, such that methane is consumed and the
progress variable rises. This trend is highlighted in both figures, where the mass fractions
of CH4peak near the lower flammability limit (i.e. Z=0.575) of the methane-air mixture.
It appears that adopting c1as a second control variable reduces the complete filling of the
η,ξ1space, as opposed to selecting a YCO2-based progress variable. A second region with-
out measurements is found in the conditional domain using Zand the temperature progress
variable. The presence of the top-left region suggests that it is improbable to have a com-
plete reaction with local equivalence ratios well below the lower flammability limit of
Combustion Theory and Modelling 17
Figure 8. Two-condition conditionally averaged reactive scalars from SwB|all using ηand ξ1as
the sampling space variables of mixture fraction and the temperature-based progress variable c1,
respectively, and collecting data at all spatial locations (radial and axial). The temperature colourbar
is expressed in Kelvin.
methane. This behaviour is back-supported by Figure 9where all conditionally-averaged
scalars falling within this region are associated with values equal to zero. Interestingly, the
conditional domain built using the mixture fraction and c2provides a much more complete
mapping than Z,c1, in particular for regions associated with high mixture fractions and
progress variable values far from unity. The contours for conditionally-averaged temper-
ature, CO2mass fraction and H2O mass fraction exhibit, as expected, similar behaviours,
where their maximum values lie in the vicinity of Zst =0.9 and progress variable of
unity.
Intermediate species, namely CO and H2mass fractions have similar behaviour in both
conditional domains. However, it should be noted that using c2as a second control variable
promotes a conditional mapping of regions associated with maximum values to be much
more spread across η,ξ2, as opposed to the trends observed with Zand the temperature-
based progress variable.
18 N. Sekularac et al.
Figure 9. Two-condition conditionally averaged reactive scalars from SwB|all using ηand ξ2as
the sampling space variables of mixture fraction and the YCO2-based progress variable c2, respec-
tively, and collecting data at all spatial locations (radial and axial). The temperature colourbar is
expressed in Kelvin.
3.3. Fixed high stratification, swirl sweep (SwB9, SwB10, SwB11)
Within this section, the conditional fluctuations of the Qscalars describing SwB|Hstratified
around one-condition conditional averages are studied to investigate which progress
variable definition can reduce the swirl and spatial dependences, assuming high fixed
stratification mixture conditions. The conditional fluctuations of temperature and various
species mass fractions, depicted in Appendix 1 (cf. Figure A1), exhibit similar results com-
pared to SwB|all. All studied ckpromote similar behaviours for the conditional fluctuations
of YCH4,Y
CO and YH2. This suggests that the progress variable is perhaps not a good choice
for describing the variables of interest and that mixture fraction would be a better decision,
as the mass fractions of CH4and CO are strongly correlated with Z. Moreover, the height
of conditional fluctuations (in particular for intermediate species) seems to remain con-
stant throughout all axial positions, as opposed to the trends observed with SwB|all. The
local averages of the presented conditional fluctuations (cf. golden markers in Figure A1)
Combustion Theory and Modelling 19
Figure 10. Normalised RMS of the conditional fluctuations of temperature and species mass frac-
tions for the SwB|Hstratified database around the conditional average f|ξ=ck(x)(markers) using
ckas the single conditioning variable and around the conditional average f|η=Z,ξ=c1,c2(x)
using the mixture fraction and, the temperature-based progress variable (crosses) or the YCO2-based
progress variable (pluses), and collecting all points at different radii together.
are fixed at zero, suggesting that using either of the proposed ckas a conditioning vari-
able should provide an accurate approximation of the turbulent reaction rate, i.e. a closure
utilising only the first term of a Taylor expansion of the reaction rate.
Figure 10 provides further insights. The normalised RMS of all variables investigated
for the considered database around one-condition (i.e. using ck) exhibit similar results as
seen with SwB|all.
As expected, the RMS obtained using a single control variable have much higher swirl
dependence at the inlet of the burner than two-condition conditional averages, in particu-
lar for methane, CO and H2mass fractions, emphasised by normalised RMS values above
10%. Further downstream the axial direction (i.e. z=60, 70 mm), the normalised RMS
are nearly of the same order of magnitude as both combinations of doubly conditioning,
suggesting that adding the mixture fraction as a second control variable does not signifi-
cantly affect the fit. For the same species, the definition attributed to the progress variable
20 N. Sekularac et al.
seems irrelevant. The differences are more straightforward for major species and tempera-
ture, where all ckprovide very good results. Based on these results, the temperature-based
progress variable seems to be the most optimal choice (as concluded from the PCA analy-
sis), followed by c4and c3, and with the YCO2-based progress variable being the worse
among the tested reaction variables. These findings are consistent with recent results
computed by analysing methane-air [55] and hydrogen-air [21] premixed flames. The nor-
malised RMS around two-condition conditional averages are unchanged compared to the
trends observed with the first database. The definition attributed to the progress variable
as a second control scalar seems irrelevant for this case. Once again, two-condition condi-
tional averages are sufficient to describe the considered database and detach it from swirl
and space. This suggests that a DCSE calculation of these flames (including SwB|all) using
both mixture fraction and progress variable as conditioning variables might be successful.
The contours of conditionally-averaged reactive scalars are illustrated in Appendix 1.
The conditional averages of temperature and several species mass fractions around the
mixture fraction and the temperature-based progress variable are shown in Figure A2.
Figure A3 presents the conditional averages of the same scalars using Zand c2. Similar
conclusions drawn for SwB|all can be applied to the considered data-set. Regions with no
available measurements are much more accentuated in both conditional domains, attributed
to the exclusion of data from 6 flames that do not exhibit the desired characteristics
of the current database. Regardless of the second control variable adopted, the condi-
tional averaged scalars vary moderately throughout the two databases assessed, suggesting
that the underlying physics and chemistry remain quasi-unchanged in the conditional
domains.
3.4. Fixed high swirl, stratification sweep (SwB3, SwB7, SwB11)
The conditional fluctuations around one-condition conditional averages of the variables
describing the SwB|Hswirl database are studied to investigate the most optimal choice of
progress variable definition to cancel stratification and spatial dependences, assuming high
fixed swirl intensity. The conditional fluctuations of the scalars previously investigated are
shown in Appendix 2-Figure A4. Similar to SwB|all and SwB|Hstratified, the conditional
fluctuations of YCH4,Y
CO and YH2 appear to be still affected by space and stratification
effects, particularly at the inlet of the burner. All ckpromote similar results, suggesting that
another choice of scalar is perhaps more suitable. Just like c2for Y
CO2, the temperature-
based progress variable provide the lowest fluctuation heights for Tand Y
H2O, with c2
and c3being the worse for the considered scalars. As in the two previous databases, the
local averages throughout all downstream locations are equal to zero, suggesting that the
definition attributed to the progress variable is perhaps less relevant in a closure context
than in an accurate representation of the chemical state.
Surprisingly, Figure 11 shows that using a single conditioning variable for intermediate
species, regardless of the definition attributed to c, equally well performs as Z,c(T) and
Z,c(YCO2), excluding the results obtained near the inlet of the burner (i.e. z=10, 20 mm)
where values deviate by a factor of ∼2, attributed to the recirculation zone and possibly
the heat exchange with the bluff body. For these axial distances, this suggests that all
investigated ckare enabling to decrease the functional dependence of conditional averages
on spatial coordinates and stratification.
Moreover, compared to the two other data-sets, the normalised RMS of CH4condi-
tional fluctuations remain below 10%, suggesting that stratification effects have perhaps
Combustion Theory and Modelling 21
Figure 11. Normalised RMS of the conditional fluctuations of temperature and species mass frac-
tions for the SwB|Hswirl database around the conditional average f|ξ=ck(x)(markers) using
ckas the single conditioning variable and around the conditional average f|η=Z,ξ=c1,c2(x)
using the mixture fraction and, the temperature-based progress variable (crosses) or the YCO2-based
progress variable (pluses), and collecting all points at different radii together.
less influence on conditional averages than spatial coordinates and/or swirl. Differences of
magnitude between the four ckare more pronounced for temperature and major species
(excluding YCH4), where c1provides the best fit for the mass fraction of H2O, but temper-
ature as well (as expected). As was foreseeable, the inclusion of carbon dioxide in c2gives
the most optimal results for decreasing the RMS of Y
CO2, with no apparent differences
compared to other ckfurther downstream. The differences between the two combinations
of doubly conditioning remain minor, with slightly better results in favour of Z,c1, exclud-
ingtheRMSofCO
2conditional fluctuations. This suggests that the definition attributed
to the progress variable might be less relevant to conditional space fluctuations. This find-
ing perhaps relates to the fundamental basis of conditional moment closure-based models
where one focuses on the separation of model elements which give descriptions for the
moment of reactive parameters concerning the scalar description in state-space. In that
sense, the main purpose of the controlling variable is to construct a functional approxi-
mation of the conditional space where the number of control variables and their ability to
22 N. Sekularac et al.
capture major physical behaviour of the system (e.g. mixing and/or reaction progress) can
be more relevant. This work, together with a previous study from Mousemi et al. [40],
has indeed demonstrated the importance of including appropriate controlling variables to
capture all physical processes and reduce conditional space fluctuations for reacting flows
where the definition of progress variable can be more flexible. This perhaps has shown
some differences compared to previous studies using one-dimensional (1D) flamelet mod-
els where a different progress variable using a simple linear combination of mass fraction
definition resulted in significant differences in mass burning rate predictions [20]. One of
the reasons identified by Gupta et al. is that when tabulated chemistry is constructed in
1D manifold approaches, a direct projection using control variables is involved, whereby
the stretch and individual species transport phenomena in turbulent reactive flows might be
neglected, resulting in significant differences in mass burning rate predictions [20]. Special
treatment is therefore needed for the construction of a 1D manifold to reduce the impact of
reaction progress variable choices via the projection of the source term and the diffusion
term. Alternatively, extending the manifold dimension to properly account for mixing and
chemical time scales can help account for variations in chemically conserved quantities
such as element mass fractions [19]. This later study perhaps corresponds better with the
results shown here in this work where, when two conditional variables are considered, the
conditional fluctuations are significantly reduced irrespective of the choices of progress
variables. However, a direct comparison with the above-mentioned studies is not available
as the current study is based on experimental data measurements whereby the mass burning
rate of the species is not available. Therefore the quantitative differences of progress vari-
able choices in the context of turbulent combustion modelling using conditional moment
closure approaches remain to be investigated.
Additionally, the contours of two-condition conditional averages have also been added in
Appendix 2. The conditional averages of temperature and mass fractions of several species
around mixture fraction and the temperature progress variable are shown in Figure A5.The
conditionally-averaged scalars around Zand c2are illustrated in Figure A6. No apparent
differences can be identified among the conditional domains computed from each of the
three databases (assuming the use of the same two control variables), suggesting that the
conditional averaged scalars behaviour is not affected by the underlying characteristics and
effects of the studied burner.
4. Conclusion
Within this study, the Cambridge/Sandia swirl measurements are used in conjunction with
principal component analysis (PCA) to attempt to find which set of control variables has
the highest correlation with the reactive scalars. Three databases have been constructed to
investigate the influence of swirl, stratification and spatial coordinates. Two scaling meth-
ods for the PCA model have been adopted, namely Pareto and Auto-scaling (AS). For all
three data-sets, and regardless of the scaling method adopted, it was found that: (i) the first
principal component (PC1) accounts for the largest amount of variance, and (ii) PC1 is
well-aligned with temperature.
The conditional spaces of Cambridge/Sandia flames are examined by investigating the
conditional fluctuations of temperature and various species mass fractions obtained with
single-conditional averages around four different progress variable definitions. While con-
ditional fluctuations of intermediate species and methane are unchanged using the progress
Combustion Theory and Modelling 23
variables tested, it was found that adopting a temperature-based progress variable provides
minor improvements for major reactive scalars, in particular for temperature and H2O. For
all three databases, the local averages of conditional fluctuations throughout all down-
stream locations are anchored at zero, suggesting that the definition attributed to the
progress variable is perhaps less relevant in a closure context. Regardless of the data-set,
the normalised RMS of the reactive scalars indicates that a single control variable based on
cis unable to detach the thermo-chemical state from spatial coordinate, swirl or stratifica-
tion, in particular for regions near the burner’s tip characterised by an intense recirculation
of the flow and significant heat exchanges with the bluff-body. The RMS analysis was fol-
lowed by comparing the conditional averages obtained with the progress variables against
doubly conditional averages using the mixture fraction and the progress variable c(T). Nor-
malised RMS around two-condition conditional averages adopting mixture fraction and a
YCO2-based progress variable have also been included. Here, it was shown that the condi-
tional fluctuations using both sets of two-condition conditional averages did not improve
the dependence on the physical domain compared to a single progress variable condition
further downstream the axial direction. The results are significantly improved at the burner
inlet, with values not exceeding the 10% threshold used within this study as a guideline.
The differences observed between both combinations of doubly conditioning are minor,
suggesting that the choice of a particular progress variable definition does not seem to
have an importance. Consequently, it is believed that a conditional moment closure calcu-
lation using both Zand cas two conditioning scalars might be successful, assuming that
the ensemble has been divided along the axial direction. The conditional space fluctua-
tions indicate that the success of CMC-based approaches does not depend on the definition
attributed to the progress variable, which in fine increases the applicability of such models
to different fuels and structures of flame. However, additional posterior research would be
required for these flame configurations to obtain the mass burning rates using chemistry
calculations, enabling a more rigorous/coherent comparison with previous studies given
by other modelling approaches.
Following the PCA analysis results, and given that doubly conditioning seems to
decrease the reactive scalars’ dependence more effectively, it is suggested that c(T) could
give a more accurate representation of the Cambridge/Sandia flames’ chemistry (such as
in a manifold) than a species-based progress variable, considering that species diffusivity
is often simplified by assuming unity Lewis numbers. For future studies, other criteria may
have to be tackled to evaluate the most suitable progress variable, such as the influence
of heat losses, monotonicity, species recombination in post-combustion zones, and radia-
tion effects. It should also be mentioned that the modelling of the joint probability density
function (PDF) of mixture fraction and progress variable remains an important unresolved
issue for CMC-based models, where the statistical independence assumption is anticipated
to be invalid and how the definitions of progress variable affect joint-PDF also requires
additional studies.
Acknowledgments
The authors wish to express gratitude to Professor Alessandro Parente for having provided the
impulse to pursue the work presented here. His various suggestions for improvements of the work are
greatly appreciated. Special thanks are given to Dr. M. S. Sweeney for the high-quality measurements
of the Cambridge/Sandia swirl burner and for making them available to the community. Substantial
thanks go to Professor Simone Hochgreb and Mr A. Mousemi who kindly share the full point-based
measurements with the authors. The authors would also like to acknowledge the funding support
24 N. Sekularac et al.
from EPSRC Prosperity Partnership (EP/T005327/1). Dr XiaoHang Fang gratefully acknowledges
the financial support provided by the Department of Engineering Science, University of Oxford dur-
ing the completion of this work. This publication also arises from research funded by the John Fell
Oxford University Press Research Fund and the NSERC Discovery Grant (RGPIN-2023-03309).
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
This work was supported by Engineering and Physical Sciences Research Council[EP/T005327/1]
John Fell Fund, University of Oxford[0010894] and Natural Sciences and Engineering Research
Council of Canada [RGPIN-2023-03309].
ORCID
Nikola Sekularac http://orcid.org/0000-0003-4023-7160
XiaoHang Fang http://orcid.org/0000-0001-6360-9065
Martin H. Davy http://orcid.org/0000-0001-7866-9028
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Combustion Theory and Modelling 27
Appendices
Appendix 1. SwB|Hstratified
Figure A1. Conditional fluctuations of species mass fractions and temperature around the condi-
tional average f|ξ=ck(x)for SwB|Hstratified database using only the progress variable as the
single conditioning variable and collecting all points at different radii together; are also shown the
local average of these conditional fluctuations f
i,k(golden markers).
28 N. Sekularac et al.
Figure A2. Two-condition conditionally averaged reactive scalars from SwB|Hstratified using η
and ξ1as the sampling space variables of mixture fraction and the temperature-based progress vari-
able c1, respectively, and collecting data at all spatial locations (radial and axial). The temperature
colourbar is expressed in Kelvin.
Combustion Theory and Modelling 29
Figure A3. Two-condition conditionally averaged reactive scalars from SwB|Hstratified using η
and ξ2as the sampling space variables of mixture fraction and the YCO2-based progress variable c2,
respectively, and collecting data at all spatial locations (radial and axial). The temperature colourbar
is expressed in Kelvin.
30 N. Sekularac et al.
Appendix 2. SwB|Hswirl
Figure A4. Conditional fluctuations of species mass fractions and temperature around the condi-
tional average f|ξ=ck(x)for SwB|Hswirl database using only the progress variable as the single
conditioning variable and collecting all points at different radii together; are also shown the local
average of these conditional fluctuations f
i,k(golden markers).
Combustion Theory and Modelling 31
Figure A5. Two-condition conditionally averaged reactive scalars from SwB|Hswirl using ηand
ξ1as the sampling space variables of mixture fraction and the temperature-based progress variable c1,
respectively, and collecting data at all spatial locations (radial and axial). The temperature colourbar
is expressed in Kelvin.
32 N. Sekularac et al.
Figure A6. Two-condition conditionally averaged reactive scalars from SwB|Hswirl using ηand
ξ2as the sampling space variables of mixture fraction and the YCO2-based progress variable c2,
respectively, and collecting data at all spatial locations (radial and axial). The temperature colourbar
is expressed in Kelvin.
