# A modal decomposition-based partially stirred reactor (mPaSR) model for turbulent combustion closure: Implementation details and a posteriori validation

Arthur Péquin, Erica Quadarella, Riccardo Malpica Galassi, Salvatore

Iavarone, Hong G Im, Alessandro Parente

► To cite this version:

Arthur Péquin, Erica Quadarella, Riccardo Malpica Galassi, Salvatore Iavarone, Hong G Im, et al. A modal decomposition-based partially stirred reactor (mPaSR) model for turbulent combustion closure: Implementation details and a posteriori validation. Combustion and Flame, 2025, 279, pp.114269. 10.1016/j.combustflame.2025.114269. hal-05386192

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Submitted on 27 Nov 2025

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# A modal decomposition-based partially stirred reactor (mPaSR) model for turbulent combustion closure: Implementation details and a posteriori validation

Arthur Péquin<sup>a,b, ID</sup>,*, Erica Quadarella<sup>c ID</sup>, Riccardo Malpica Galassi<sup>d ID</sup>, Salvatore Iavarone<sup>e ID</sup>, Hong G. Im<sup>c ID</sup>, Alessandro Parente<sup>a,b,f ID</sup>

## ARTICLE INFO

**Keywords:**

Chemical timescale

Computational Singular Perturbation (CSP)

Detailed kinetics

Multiscale processes

Partially Stirred Reactor (PaSR)

Turbulent combustion closure

## ABSTRACT

This paper investigates a turbulence-chemistry interaction model based on the Partially Stirred Reactor (PaSR) paradigm where the hypothesis of relying on an individual chemical timescale is relaxed to deal with multiscale problems. The modal Partially Stirred Reactor (mPaSR) model relies on the Computational Singular Perturbation (CSP) theory and performs an eigen-decomposition of the Jacobian matrix of the chemical source terms. The CSP manifold is then corrected by modal fractions that, similarly to the cell reacting fraction of the original PaSR model, account for the individual mode timescales. The vector of the chemical source terms, to be returned to the computational fluid dynamics solver, acts as an aggregated contribution of the corrected CSP modes. The predictive capabilities of the mPaSR model are demonstrated a posteriori through a series of Unsteady Reynolds-Averaged Navier-Stokes simulations of the well-documented Sandia flames. Promising results are observed at different turbulence levels making the mPaSR approach a valuable alternative to existing turbulence-chemistry interaction models. Particular attention is given to the formation of pollutants, and accurate predictions of nitric oxide NO are obtained.

**Novelty and significance statement** The novelty of this work lies in the code development, integration and a *posteriori* testing of an innovative combustion model accounting for several timescales of dynamical chemical systems. This represents an important step towards well-suited approaches for the modelling of multiscale processes such as pollutant formation in turbulent flames. The model shows promising prediction capabilities with desirable computational efficiency on the investigated cases, motivating follow-up investigations in a larger range of combustion scenarios.

## 1. Introduction

Transitioning to cleaner energy requires the development of efficient combustion technologies, as combustion will remain the main route for energy conversion in various sectors including steel and glass industries, and transportation [1]. Accurately characterising reactive environments in combustion is crucial for optimising efficiency and managing emissions. As Direct Numerical Simulations (DNS) of industrial combustion systems are currently out of reach, a pragmatic trade-off between computational cost and accuracy is sought through widely

adopted approaches like Large Eddy Simulations (LES) or Reynolds-Averaged Navier-Stokes (RANS) computations. Unresolved reaction source terms necessitate modelling due to the non-linear nature of the Arrhenius rate expression, which deviates significantly from standard behaviour in laminar conditions when applied to turbulent environments [2].

Closures of unresolved reaction source terms are typically achieved through two main approaches [3]: one considers few representative variables transport equations, e.g., as in flamelet models [4,5]; the

<footnote><sup>a</sup> Université Libre de Bruxelles, Aero-Thermo-Mechanics Laboratory, École Polytechnique de Bruxelles, Avenue F.D Roosevelt 50, Brussels, 1050, Belgium</footnote>

<footnote><sup>b</sup> Université Libre de Bruxelles and Vrije Universiteit Brussel, Brussels Institute for Thermal-fluid systems and clean Energy (BRITE), Brussels, Belgium</footnote>

<footnote><sup>c</sup> CCRC, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Kingdom of Saudi Arabia</footnote>

<footnote><sup>d</sup> Department of Mechanical and Aerospace Engineering, Sapienza University, Rome, Italy</footnote>

<footnote><sup>e</sup> Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C, 91190, Gif-sur-Yvette, France</footnote>

<footnote><sup>f</sup> WEL Research Institute, Avenue Pasteur 6, Wavre, 1300, Belgium</footnote>

<footnote>* Corresponding author at: Université Libre de Bruxelles, Aero-Thermo-Mechanics Laboratory, École Polytechnique de Bruxelles, Avenue F.D Roosevelt 50, Brussels, 1050, Belgium.</footnote>

<footnote>E-mail address: Arthur.Péquin@ulb.be (A. Péquin).</footnote>
other deals with solving transport equations for all scalar variables, e.g., as in Transported Probability Density Functions [6], Eulerian Stochastic Fields methods [7], Linear-Eddy Model [8] and Conditional Moment Closures [9]. The latter modelling strategy preserves detailed local information on the thermo-chemical states of the reactive mixture, although expressing a comprehensive correction factor for the source term is complex.

Reactor-based models [10] pertain to this modelling strategy and have found broad applicability. In particular, the Partially Stirred Reactor (PaSR) model [11] accounts for finite-rate chemistry by assuming that reactions occur at small scales and divides each computational cell into reactive and non-reactive parts. The reactive zone is known as the fine structures [12], and its volume fraction is determined by a model parameter, i.e., the cell reacting fraction $κ$, related to the characteristic chemical and mixing timescales. The PaSR model has shown promise in various combustion regimes [13–16], but accurately estimating characteristic chemical and mixing times remains a challenge due to their variability within reactive environments [17–22]. The cell reacting fraction was also found sub-optimal due to its functional form and physical bounds [23]. Recent studies have employed machine learning like neural networks or sparse symbolic regression to conceive and explore alternative expressions [24,25].

In addition, the uniqueness of the cell reacting fraction, i.e., when a single chemical timescale is considered, is a fundamental limitation of reactor-based models in all combustion regimes [26,27]. This assumption is questionable from a physical standpoint, since different chemical species have different characteristic scales and multiple reacting layer thicknesses exist as a consequence of the multiscale nature of reacting flows [28]. However, associating a fraction to each chemical species may not assure closure of the mass balance.

Simulating the Adelaide Jet-in Hot-Coflow (AJHC) burner [29], Iavarone et al. [18] proposed two- or multiple-$κ$ PaSR solutions, obtained by adaptive chemical timescales, to improve the source term predictions of selected species, such as nitric oxide NO. Although resulting in small mass imbalances, the proposed solutions showed great improvement in the prediction of small chemical species for the AJHC burner. Péquin et al. [23] highlighted the difficulty of finding a family of cell reacting fractions ensuring mass conservation. While evaluating the sources of errors from each model component, the authors motivated the need to develop a more comprehensive model able to integrate multiple reacting fractions for individual or group of species.

To address this modelling issue, Quadarella et al. [30] proposed a revision of the PaSR approach using arguments from the Computational Singular Perturbation (CSP) theory [31,32]. This approach consists in the direct manipulation of the species production rates through a modal decomposition of the Jacobian matrix and seeks to include the whole set of chemical times involved in the reactive system. Following Fox [33], these characteristic chemical timescales can be approximated by the inverse of the eigenvalues of the Jacobian matrix. Although deviating from the fine structure concept of the original PaSR formulation, the *a priori* results demonstrated the benefits of accounting for multiple times and reacting fractions.

This work presents the first *a posteriori* validation of the modal decomposition-based Partially Stirred Reactor model, referred as the “mPaSR” model. Unsteady-RANS (URANS) simulations of flames exhibiting processes with distinctive timescales, i.e., configurations D and F of the well-documented piloted jet diffusion Sandia flames [34], are used to investigate three key points: (i) the framework of the mPaSR model; (ii) the assessment of its features; and (iii) its performance compared to existing solutions.

The present paper is organised in the following way. Section 2 presents the methodology employed in this study, i.e., the numerical set-up and the combustion closures. The results are presented in two parts: Section 3 details the construction of the mPaSR chemical source terms while Section 4 presents the full URANS simulation results. Conclusions and prospects are exposed in the final Section 5.

## 2. Methodology

### 2.1. Numerical test case: Sandia flames

Flames D and F are two piloted diffusion flames with an axisymmetric geometry [35]. A mixture of methane $CH_4$ and air exits the main jet at 294 K and a velocity of 49.6 m/s for flame D and 99.2 m/s for flame F, exhibiting a Reynolds number $Re$ of 22400 and 44800, respectively. The surrounding pilot jet consists of a mixture of burnt gases of temperature 1880 K and a bulk velocity of 11.4 m/s and 22.8 m/s for flames D and F, respectively. An air coflow with a velocity of 0.9 m/s and a temperature of 291 K surrounds the flame. The amount of local extinction increases from configurations D to F, with flame F representing the most challenging test case, being close to global extinction. The experimental data consist of the mean and root mean square of temperature and species mass fractions at several axial locations. More details about the experimental set-up and measurements can be found in Refs. [34,35].

The mesh quality (20 000 cells) resulted from a numerical grid selection on the domain geometry presented in Fig. 1. Both the KEE [36] and the GRI-Mech 3.0 [37] mechanisms are used in this study to describe partially premixed flames. URANS simulations are carried out using the $k-ε$ turbulence model and the FiReSMOKE solver [38]. The mesh selection, kinetic schemes and turbulence modelling are further discussed in Supplementary Material. From a cold solution, a reference reacting simulation was obtained before assessing the combustion models. Starting from the reference, all simulations were run for 500 ms which corresponds to 11.4 and 22.8 pilot jet flow-through times $τ_f$ to ensure statistical steadiness for flames D and F, respectively.

### 2.2. Turbulent combustion closures

#### 2.2.1. Laminar Finite Rate (LFR) model

In the Laminar Finite Rate (LFR) model [39], any effect of turbulence on the filtered quantities is neglected. Hence, the LFR approach cannot be regarded as a proper Turbulence-Chemistry Interaction (TCI) model, but serves as a reference closure for detailed kinetics. The i-th species LFR chemical source term is determined by Arrhenius expressions on the filtered thermo-chemical, i.e., temperature and species mass fractions, quantities $\tilde{\Phi} = [\tilde{T}, \tilde{Y}]$, with $i = [[1, N_S]]$, $N_S$ being the number of chemical species in the kinetic mechanism, and reads as

$$ \tilde{\omega}_i(\tilde{\Phi}) = M_{w,i} \sum_{r=1}^{N_R} \hat{R}_{i,r}(\tilde{T}, \tilde{Y}), \quad (1) $$

where $M_{w,i}$ is the molecular weight of the i-th species, and $\hat{R}_{i,r}(\tilde{T}, \tilde{Y})$ is the molar net production rate of species i from reaction r. Each reaction rate $\hat{R}_{i,r}$, among $N_R$, follows an Arrhenius law.

#### 2.2.2. Partially Stirred Reactor (PaSR) model

The works of Chomiah [11] and Karlsson [40] put forth the initial description of the Partially Stirred Reactor (PaSR) model which, under turbulent conditions, partitions any computational grid cell into a non-reacting zone, solely driven by turbulent mixing and referred to as the *surroundings*, and a reacting part called the *fine structures*. The volume fraction occupied by the fine structures is defined by a factor $κ$, i.e., the cell reacting fraction, defined as

$$ \kappa = \frac{\tau_c}{\tau_c + \tau_{mix}}, \quad (2) $$

where $τ_c$ and $τ_{mix}$ denote the characteristic chemical and mixing timescales, respectively.

Assuming most chemical reactions occur within the fine structures [41], the filtered or averaged source terms are expressed as

$$ \tilde{\omega}_i = \kappa \frac{\bar{\rho}(Y_i^* - Y_i^0)}{\tau^*}, \quad (3) $$
<figure><img src="image_4.png" /><figcaption>Fig. 1. Sketch of the numerical domain. Three positions are marked as vertical black dashed lines at normalised axial distances x/D = 15, 30 and 45 from the nozzle exit, with D = 7.2 mm being the main jet diameter.</figcaption></figure>

where $Y_i^*$ and $Y_i^0$ are the i-th species mass fractions in the fine structures and in the surroundings, respectively, $i = 1, ..., N_S$, $\bar{\rho}$ is the Reynolds-averaged density, and $\tau^*$ represents the residence time in the fine structures. To estimate $Y_i^*$, the fine structures are treated as an ideal reactor by means of a Plug-Flow Reactor (PFR) [10,42] evolving from $Y_i^0$, which can be approximated by the filtered mass fraction $\bar{Y}_i$ [41], over $\tau^*$. Originally based on the mixing timescale only [41], $\tau^*$ is defined as $\tau^* = \min(\tau_{mix}, \tau_c)$ following Refs. [18,43]. This allows the reactor to be chemically, i.e., $\tau_c > \tau_{mix}$, or mixing, i.e., $\tau_c < \tau_{mix}$ controlled. The modelling of $\tau_{mix}$ and $\tau_c$ is required for the PaSR closure. Regarding the mixing time, we consider the following formulations:

### Integral-based timescale (I.):

Kärrholm and Nordin [41,44] estimated the characteristic timescale for mixing processes as a certain fraction of the integral timescale $\tau_I$,

$$\tau_{mix} = C_{mix} \tau_I = C_{mix} \frac{k}{\epsilon}, \quad (4)$$

where $k$ and $\epsilon$ represent the turbulent kinetic energy and its dissipation, respectively. $C_{mix}$ is a constant ranging from 0.001 to 0.9 [18,19,43,45,46]. In this study, a value of 0.7 is selected.

### Kolmogorov timescale (K.):

The Kolmogorov timescale [47], determining the local molecular interaction for the reactions, can be considered as a representative scale of the turbulent motions at the dissipative level and is defined as

$$\tau_{mix} = \tau_{\eta} = (\nu/\epsilon)^{1/2}, \quad (5)$$

where $\nu$ represents the kinematic viscosity.

All previous formulations can be regarded as global approaches. In particular, the wide variability of the constant $C_{mix}$ motivated the development of more comprehensive approaches:

### Local scalar timescale (S.):

Senouci et al. [48] developed a local formulation for $\tau_{mix}$ involving the scalar mixing timescale $\tau_{\phi}$ from Raman and Pitsch [49],

$$\tau_{mix} = \tau_{\phi} = \frac{\tilde{\phi}''_{2}}{\tilde{\epsilon}_{\phi}}, \quad (6)$$

where $\tilde{\phi}''_{2}$ is the filtered scalar variance and $\tilde{\epsilon}_{\phi}$ the filtered scalar dissipation rate. Transported equations for the filtered mixture fraction $\phi = \tilde{Z}$ are usually solved with model constants $C_{D1}, C_{D2}, C_{P1}$ and $C_{P2}$ of respective standard values 2.0, 1.8, 3.4 and 1.4 [50], as in this study.

As far as the characteristic timescale for chemistry is concerned, the following formulations can be considered:

### Jacobian eigenvalues timescale (EV):

Fox [33] calculated the timescales developing in complex chemical dynamics systems as the inverse of the eigenvalues $\lambda$ of the

Jacobian matrix $J_{jk} = \partial\dot{\omega}_j/\partial Y_k$, where $\dot{\omega}_j$ is the j-th species net production rate and $Y_k$ the k-th species mass fraction. The slowest of the non-conservative timescales can be regarded as a leading scale for chemistry, i.e.,

$$\tau_c = \max \left( \frac{1}{|\lambda|} \right). \quad (7)$$

Although accurate, the method yields a higher computational cost with respect to other chemical timescale formulations due to the Jacobian matrix workload.

### Slowest Formation Rate timescale (SFR):

As a compromise between accuracy and computational cost, a widely employed approach [51–53] is to consider the slowest formation rate timescale as the leading one (after removing the $N_D$ dormant species of absolute rate $|\dot{\omega}_i| < \epsilon = 10^{-16}$ [kg.m⁻³.s⁻<fnref n="1" />]),

$$\tau_c = \max_{i=[1,N_S-N_D]} \left( \frac{Y_i}{|\dot{\omega}_i|/\rho} \right). \quad (8)$$

### 2.2.3. Modal Partially Stirred Reactor (mPaSR) model

Fig. 2 presents the graphical abstract of the modal Partially Stirred Reactor (mPaSR) model. The concept of fine structures from the PaSR model is abandoned and replaced by a manifold correction using CSP information.

As described by Fox [33], reacting systems have characteristic chemical timescales that can be approximated<fnref n="1" /> by the inverse of the eigenvalues of the isothermal Jacobian matrix $\mathbf{J}$,

$$\mathbf{J} = \begin{bmatrix} \frac{\partial \dot{\omega}_1}{\partial Y_1} & \cdots & \frac{\partial \dot{\omega}_1}{\partial Y_{N_S}} \\ \vdots & \ddots & \vdots \\ \frac{\partial \dot{\omega}_{N_S}}{\partial Y_1} & \cdots & \frac{\partial \dot{\omega}_{N_S}}{\partial Y_{N_S}} \end{bmatrix}. \quad (9)$$

The isothermal Jacobian matrix of the chemical source terms has dimensions $N_S \times N_S$ ($N_S$ being the total number of species). Each eigenvalue $\lambda_m$ of $\mathbf{J}$ is associated with a right and left eigenvector, denoted by the column vector $\mathbf{a}_m$ and the row vector $\mathbf{b}^m$, respectively. By definition, the latter two are orthonormal to each other so that $\mathbf{a}_m \cdot \mathbf{b}^m = \delta_{m,n}$, where $\delta_{m,n}$ is the Kronecker delta. The general description of the chemical source terms vector $\dot{\omega}$ can be rewritten with a Computational Singular Perturbation (CSP) modal decomposition [31] in $N$ modes ($N \le N_S$) $\mathbf{a}_m$ of amplitudes $f^m$,

$$\dot{\omega} = \sum_{m=1}^{N} \mathbf{a}_m f^m, \quad (10)$$

<footnote n="1">¹ The eigenvalues $\lambda$ of the isothermal Jacobian matrix $\mathbf{J}$ of the chemical source terms are only estimates of the characteristic scales of the dynamic system. Instead, those from the non-isothermal Jacobian matrix represent the leading order approximations of the scales. To be exact, it is necessary to account for the rotation of the eigen-directions due to the non-linearity of the problem through the CSP refinements of the projection basis, as demonstrated in Ref. [54]. In practice, however, such details are usually not implemented.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Graphical abstract of the modal Partially Stirred Reactor (mPaSR) model. The Jacobian matrix J of the laminar chemical source terms $\dot{\omega}_i$ is populated before being decomposed into sets of eigenvalues A and right/left eigenvectors $v_r/v_l$. Modal cell reacting fractions $\kappa_m$ are calculated from characteristic mixing $\tau_{mix}$ and modal $\tau_{c,m}$ timescales, following a similar procedure as in the PaSR model. The mPaSR rates $\bar{\omega}_i$ are obtained from the fractions $\kappa$, the modes a and their respective amplitudes f.</figcaption></figure>

where the amplitude $f^m$ is calculated as $b^m \cdot \dot{\omega}$. Following Ref. [30], Eq. (10) can be expressed as the sum of the modal contribution to access the filtered chemical source term of a given species i:

$$ \bar{\omega}_i = \sum_{m=1}^{N} \kappa_m \mathbf{a}_{i,m} f^m, \quad (11) $$

where

$$ \kappa_m = \frac{\tau_{c,m}}{\tau_{c,m} + \tau_{mix}}, \quad (12) $$

are the CSP cell reacting fractions, or modal fractions. This yields a decoupling of the time evolution of the mode amplitudes [55] so that each mode evolves according to its local chemical timescale calculated from the Jacobian eigenvalues: $\tau_{c,m} = 1/|\lambda_m|$. Thus, a chemical timescale is associated with a mode rather than a species. Note that a mode amplitude is contributed, in principle, by all species through the components of $b^m$. In other words, the vectors of $b$ define a new set of N system variables $\{\xi_m\}_{m=1,...,N}$, $\xi_m := b^m \cdot Y$, which may be called the CSP variables, where each $\xi_m$ is a linear combination of the chemical species. The m-th timescale $\tau_{c,m}$ is therefore associated with the CSP variable $\xi_m$. Therefore, the modal decomposition of the chemical source term, see Eq. (10), is the only projection that obtains time-decoupled intrinsic coordinates. Hence, Eq. (11) is the proper way of incorporating multiple $\kappa$'s in the PaSR model, i.e., by associating each $\kappa_m$ to the m-th intrinsic variable $\xi_m$ and its direction $\mathbf{a}_m$ in phase space. However, the CSP variables are state functions, meaning that their definition is not unique in a non-linear problem, but hold on a local basis.

The last, in a magnitude-wise ordering of the eigenvalues, $N_{elem}$ timescales are associated with zero eigenvalues. The eigenvectors associated to the zero-eigenvalues are the invariant directions corresponding to the physical atomic conservation laws involving $N_{elem}$ elements, e.g., 4 in the case of C, H, N, and O, whose existence translates into the rank deficiency of J. These $N_{elem}$ directions span the null-space of J and constitute the invariant subspace E. Note that their co-vectors are such that $f^{cons} = b^{cons} \cdot \dot{\omega} = 0$, which are the linear combinations of the system variables rates-of-change expressing the atomic mass conservation.

The existence of a conservation sub-space is a great advantage, since the inclusion of multiple fractions $\kappa_m$ in Eq. (11) still ensures mass conservation, conversely to Eq. (3) [18]. The basis is conservative in all of its first $N - N_{elem}$ directions, being then orthonormal to the conservation $N_{elem}$-dimensional dual space $b^{cons}$. Hence, any linear combination of the first $N - N_{elem}$ modes still satisfies the mass conservation $\sum_{i=1}^{N_S} \bar{\omega}_i = 0$. The proposed method efficiently accounts for the contribution of relevant chemical modes to the production rates, allowing a complete description of turbulence-chemistry interactions in all directions of chemistry evolution.

The mPaSR approach was implemented in the CFD solver FiReSMOKE<fnref n="2" /> [38], based on the OpenFOAM® framework. The modal decomposition of the Jacobian matrix was performed using the LAPACK package [56] and all CSP routines were adapted in C++ from the Python-based library PyCSP<fnref n="3" /> [57]. All definitions of the mixing timescale presented in Section 2.2.2, are applicable in the mPaSR context. The following sections present the results of the numerical test cases for the a posteriori validation of the mPaSR approach.

### 3. Construction of the mPaSR rates

#### 3.1. Chemical source terms

As the LFR approach may lead to tremendous approximations [3], the essence of the mPaSR model and its validity depend on the ability of the modal fractions to modify the resulting manifold. The LFR and mPaSR source terms are compared on two samples, namely A and B, of filtered thermo-chemical spaces from a URANS simulation of the Sandia flame D, see Fig. 3. The KEE mechanism [36] (17 species) is preferred to the GRI-Mech 3.0 [37] (53 species) to clearly present the insights into the mPaSR model with fewer species. The local scalar formulation, based on the transported equation of the mixture fraction Z, is used for the mixing timescale. Samples A and B are respectively located at the coordinates ($Z_A = 0.42, T_A = 1895$ K) and ($Z_B = 0.15, T_B = 1224$ K) in the mixture fraction - temperature space of the simulation. Stoichiometric mixture fraction is set at 0.351, making conditions for samples A and B fall in the slightly rich and lean combustion zones of the diffusion flame, respectively. A graphical representation of the samples locations is available in the Supplementary Material.

Differences on the consumption/production rates between the two methods can be observed in Fig. 3. For the mixture reactants, i.e., methane CH$_4$ and oxygen O$_2$, all chemical terms are found negative indicating a consumption of the corresponding species. The chemical rates of point A are noticeably greater than those on point B, which is expected considering the close-to-stochiometry and high temperature conditions found at point A. The mPaSR rates are lower than the LFR ones and illustrate a suppression effect from the CSP modal fractions. The two latter comments are still valid for the positive rates of the major combustion products, e.g., carbon dioxide CO$_2$ and water H$_2$O. The source terms drop by several orders of magnitude for certain faster species: HO$_2$, CH$_2$, HCO, H$_2$O$_2$ and CH. In addition to a suppressing effect, the modal fractions completely rearrange the chemical dynamics of the system. To better understand this phenomenon, indices from the CSP theory are explored in the following section.

<footnote n="2">² https://github.com/burn-research/FiReSMOKE.</footnote>

<footnote n="3">³ https://github.com/rmalpica/PyCSP.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 3. Absolute values of the LFR and mPaSR chemical source terms for samples A and B. The negative non-absolute values are marked by a ⊘ symbol on top of each corresponding bar.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Amplitude Participation Indices $P_k^m$ of the k-th chemical reactions to the m-th modes for samples (a) A; and (b) B. As the KEE mechanism is employed, 13 out of 17 modes are considered active, ranked from the fastest to the slowest, and $2N_R = 116$ chemical reactions contribute to the modes amplitudes (forward reactions from 1 to 58 and backward reactions from 59 to 116). For visibility purposes, only API respecting the condition $|P_k^m| > 0.1$ are shown with markers.</figcaption></figure>

## 3.2. Amplitude participation indices

The Amplitude Participation Indices (API) measure the relative contribution of each chemical reaction to the CSP modes amplitudes [58]. The k-th chemical reaction contribution to the m-th mode amplitude is expressed by the participation index $P_k^m$ defined as

$$P_k^m = \frac{(b^m \cdot S_k) R^k}{\sum_{j=1}^{2N_R} |(b^m \cdot S_j) R^j|} \quad (13)$$

It is to be noted that the forward and backward directions are considered as separate reactions, hence $k = 1, ..., 2N_R$. In Eq. (13), $S_k$ is the stoichiometric vector of the k-th reaction and $R^k$ its vector of reaction rate of progress. By definition, the sum $\sum_{k=1}^{2N_R} |P_k^m| = 1$, and a large value of $P_k^m$ indicates a significant contribution of reaction k to mode m [58].

The API for the CSP modes of the mPaSR model for samples A and B, see Figs. 4(a) and 4(b), indicate that the chemistry of intermediate and radicals species contribute to the amplitudes of the faster modes. Strong correlations are observed between HCO chemistry and the two fastest modes. Noticeable API are found for reactions 27: $CH_2O + H \rightarrow H_2 + HCO$, and 31: $HCO + M \rightarrow CO + H + M$ at point A, and from reaction 13: $CH + O_2 \rightarrow HCO + O$ at point B. Conversely, slower modes are associated to the production pathways of the major combustion products such as $CO_2$ and $H_2O$. In particular, API of modes 11 and 12 on reaction 40/97: $H_2 + OH \rightleftharpoons H + H_2O$ are significant. The production of carbon dioxide $CO_2$ is linked to the slowest mode, i.e., mode 13, with the recombination of CO through reactions 36 and 93: $CO + OH \rightleftharpoons CO_2 + H$.

Conditions differ between samples A and B and may favour different reactions and chemical pathways depending on the local temperature

and mixture fraction values [59,60]. For sample A (high temperature and rich conditions), fuel-based reactions are found to be predominant, e.g., reaction 3: $CH_4 + H \rightarrow CH_3 + H_2$ and reaction 5: $CH_4 + OH \rightarrow CH_3 + H_2O$. These reactions are no longer present for the low temperature and lean conditions at point B. Instead, $O_2$-based reactions are found such as reactions 20: $CH_2 + O_2 \rightarrow CO_2 + 2H$, 21: $CH_2 + O_2 \rightarrow CH_2O + O$ and 22: $CH_2 + O_2 \rightarrow CO_2 + H_2$. This demonstrates the method ability to respond to local combustion conditions and regimes. This flexibility appears as an appealing feature for any TCI model.

The findings of the API study support the necessity to account for a plurality of modes. An individual mode solely depicts a group of chemical species or reactions. Timescale Participation Indices (TPI) and CSP importance indices, available in the Supplementary Material, confirm this analysis and the a priori results from Ref. [30].

## 3.3. Modal fractions

Fig. 5 presents the CSP modal fractions for the mPaSR model calculated following Eq. (12). The local scalar mixing timescale values are $2.10^{-3}$ s and $5.10^{-3}$ s for samples A and B. Comparable fraction values between A and B are only obtained for the faster modes 2 and 3, and the slower modes 11 and 12. Smaller fractions are observed for sample A, well characterising systems composed of a large number of fast timescales. This is in line with an intense chemical activity at a local level where combustion occurs in thinner reacting structures.

As faster modes are associated to smaller fractions, this indicates a significant suppressing effect of the modal fractions on the faster chemical dynamics. The respective contribution of the faster modes to the overall system may appear small. This can be viewed as a limitation
<figure><img src="image_8.png" /><figcaption>Fig. 5. CSP modal fractions of the mPaSR model for samples A and B. As the KEE mechanism is employed, 13 out of 17 modes are considered active.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Centreline and radial profiles of characteristic PaSR and mPaSR timescales for Sandia flame D: (K.) Kolmogorov; (I.) integral-based; (SFR) slowest formation rate; (EV) eigenvalues.</figcaption></figure>

to capture fast chemical phenomena. Addressing the key modelling issue of relying on a single timescale to describe complex chemical dynamics, the mPaSR approach offers flexibility on the treatment of relatively slow chemical dynamics. This is an important feature to capture multi-scale phenomena such as main oxidation pathways and pollutant formation, including NOₓ.

# 4. Comparing PaSR and mPaSR models in URANS simulations

This section presents the predictive capabilities of the PaSR and mPaSR models on a series of URANS simulations. The results obtained on the Sandia flame D with a reduced number of chemical species, i.e., using the KEE mechanism [36], are first presented. Then, results for flames D and F using the GRI-Mech 3.0 [37] are shown to assess the capability of the models to simultaneously predict fuel oxidation and NOₓ formation.

## 4.1. Multiple scales and fractions

The mixing timescale fields for Sandia flame D using the KEE mechanism [36], see Fig. 6, are characterised by scales of order 10⁻⁴–10⁻² s within a cone departing from the inlet region and expanding downstream as free jet flows. Chemical timescales identify regions of chemical activity by taking smaller values as representative scales, the SFR timescale being higher than the EV timescale. For the PaSR-K. solutions, the resulting cell reacting fractions are chemically controlled, i.e., $\kappa > 0.5$, as the condition $\tau_{mix} < \tau_c$ is met in most points. Whereas two characteristic scales are present in the PaSR approach, prone to flame blow-off for large $\tau_{mix}$ values [23], the mPaSR model encompasses a multitude of chemical scales and has more potential for a correct synergistic prediction of a wider range of species. The modal timescales are ranked by decreasing order of eigenvalue magnitude, i.e., from the fastest, i.e., $\tau_{c,1}$, to the slowest mode, i.e., $\tau_{c,13}$, and span

across the time spectrum of the chemical dynamics. Fig. 6 presents only a few modal timescales for clarity. All modal timescales, except the slowest, are faster than the PaSR chemical scales and result in smaller modal fractions $\kappa_m$ with respect to $\kappa$. It is worth noticing that the eigenvalues-based (EV) method is equivalent to accounting for the slowest mode of the active subspace A of the mPaSR approach, i.e., $\tau_{c,13}$. The variations observed between the two scales are explained by a different converged solution and final flame positioning. Conversely to PaSR, the mPaSR approach considers a combination of chemically and mixing controlled fractions, becoming mostly mixing controlled in regions of chemical activity, characterised by a steep decrease of the timescales magnitudes. This shows the enhanced ability of the mPaSR model to cope with a large range of characteristic timescales, as in the presence of turbulence-chemistry interactions and complex chemical dynamics.

Fig. 7 presents the Sandia flame D centreline and radial profiles of temperature and species mass fractions for the PaSR and mPaSR models using the KEE mechanism [36]. Experimental measurements are marked as black squares with root-mean-square-based vertical error bars $y \pm y_{rms}/2$. Although strongly deviating from a modelling perspective, both PaSR and mPaSR approaches provide a good agreement with the Sandia flame D experimental results. Using the Kolmogorov formulation as mixing timescale, slight overpredictions are observed for temperature and mass fractions species, e.g., CO₂ and CO, whereas H₂ mass fractions profiles are well predicted. The PaSR K./EV and mPaSR K. solutions have similar dominant characteristic timescales for chemistry, respectively EV and $\tau_{c,13}$. Comparable predictions for temperature and CO₂ are obtained while slightly larger discrepancies are observed for intermediate species CO and H₂. Changing $\tau_{mix}$ definition, the mPaSR model with the integral-based formulation (mPaSR I.) greatly predicts temperature and CO₂ profiles at all positions. Accurate predictions are obtained for the CO statistics while slightly overestimating H₂ statistics. The simulation results with other $C_{mix}$ constant
<figure><img src="image_10.png" /><figcaption>Fig. 7. Centreline and radial profiles of temperature and species mass fractions of CO₂, CO and H₂ of the Sandia flame D using the PaSR and mPaSR models. The KEE mechanism [36] is employed.</figcaption></figure>

values and mixing timescale formulations for mPaSR are reported in the Supplementary Material.

## 4.2. Fuel oxidation and NO formation

The mPaSR approach has shown promising prediction capabilities for the Sandia flame D with simple chemistry. In this section, the GRI-Mech 3.0 scheme replaces the KEE mechanism to consider nitrogen chemistry along with fuel oxidation. The performance of the model is tested on flame D and the more challenging flame F, i.e., a flame with an increased turbulence level and close to global extinction.

Fig. 8 presents the Sandia flame D centreline and radial profiles of temperature and CO₂, CO and NO species mass fractions for the PaSR and mPaSR models using the GRI-Mech 3.0 [37]. The associated timescale profiles are available in the Supplementary Material. Results for the Sandia flame F are reported in Fig. 9. For both flames and both models, the Kolmogorov formulation leads to a systematic over-prediction of temperature and mass fractions. The PaSR ‘I.’ solution captures temperature, CO₂ and CO statistics well but fails at predicting NO peak, and so does the PaSR ‘S.’ solution. The mPaSR solution ‘I.’ provides the best results among the global mixing timescales with accurate peak predictions for most variables. Slight overpredictions of CO concentrations are observed for flame F. Similar observations can be drawn for the solution mPaSR ‘S.’. Temperature and CO₂ mass fraction profiles are well predicted for both flames. The mPaSR ‘S.’ solution provides the most accurate peak centreline predictions for the NO mass fraction for both flames.

To further assess the NO concentration levels across the flames, Fig. 10 presents the profiles of the mean NO mass fractions conditioned on mixture fraction Z for flames D and F using the PaSR ‘S.’ and mPaSR ‘S.’ solutions. The stoichiometric mixture fraction is marked at 0.351. For flame D, the PaSR model greatly overestimates the NO statistics around stoichiometric conditions. Underpredictions are observed for the mPaSR approach at radial positions, while the experimental data

around the stoichiometric conditions are well captured at the centre-line. For flame F, a systematic overprediction is observed when using the PaSR model. The mPaSR approach greatly captures the NO statistics at all positions.

The mPaSR approach has proven its effectiveness on a series of URANS simulations of Sandia flames with accurate predictions of temperature and key species mass fraction profiles. This model can be regarded as a valuable TCI modelling alternative to existing reactor-based approaches.

## 4.3. Computational costs

Fig. 11 compares the computational costs of both models (SFR definition used for $τ_c$ in PaSR). The relative percentage of time elapsed in the combustion model functions is marked on the red bars. For the PaSR model, it consists in the time integration of the ideal reactors over their residence time $τ^*$, whereas for the mPaSR model, the loop contains the evaluation of the laminar rates, the construction of the isothermal Jacobian matrix J and its eigen-decomposition. As expected when dealing with CFD of turbulent reacting flows using detailed kinetics, most of the computational cost goes to the treatment of chemistry. With the KEE mechanism, no less than 95% and 79% of the computational time is spent on chemistry while using the PaSR and the mPaSR combustion models, respectively. When more complex kinetics are considered, these percentages increase to 98% and 96%. Additional transport equations are solved and the time spent on solving fluid dynamics grows with the number of species, illustrated by the blue zones. Employing the scalar-based definition of the mixing timescale also brings three additional transport equations influencing the fluid dynamics cost, but is not sufficient to explain the observed variations for the PaSR model. The differences lie in the time integration of the ideal reactors where longer residence times $τ^*$ induce more computationally expensive Ordinary Differential Equation (ODE) solutions. As $τ^* = τ_{mix}$ for most reactors, the mixing timescale values significantly affect the overall simulation cost. This explains the costs elevations,
<figure><img src="image_11.png" /><figcaption>Fig. 8. Centreline and radial profiles of temperature and species mass fractions of CO₂, CO and NO of the Sandia flame D using the PaSR (SFR for τ_c) and mPaSR models. The GRI-Mech 3.0 [37] is employed.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Centreline and radial profiles of temperature and species mass fractions of CO₂, CO and NO of the Sandia flame F using the PaSR (SFR for τ_c) and mPaSR models. The GRI-Mech 3.0 [37] is employed.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Centreline and radial conditional mean of NO mass fractions on mixture fraction Z for Sandia flames D (blue) and F (red) using the PaSR and mPaSR models ('S' mixing timescale). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Normalised computational costs of Sandia flame D URANS simulations employing the KEE or the GRI-Mech 3.0 mechanisms [36,37] and the PaSR or mPaSR models. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

e.g., from 1 to 1.41 for KEE, and from 5.92 to 8.35 for GRI-Mech 3.0, when considering slower mixing timescales. As no fine structures modelling is required in the mPaSR approach, the computational expense remains similar across mixing timescale definitions. On average, the costs of mPaSR simulations are respectively 4 and 2 times less than a classical PaSR approach when using the KEE and the GRI-Mech 3.0 kinetic mechanisms, respectively. This numerical cost investigation sheds light on the mPaSR model ability to be computationally efficient with respect to existing combustion models.

## 5. Concluding remarks

As a response to the limitations of existing reactor-based approaches for turbulent-chemistry interaction modelling, this paper presents the implementation details and the a posteriori validation of the modal Partially Stirred Reactor (mPaSR) model, which accounts for several timescales of dynamical chemical systems. The construction of the chemical source terms and the role of mPaSR elements are detailed on thermo-chemical samples. The predictive capabilities of the approach are demonstrated on a series of URANS simulations of the well-documented Sandia flames with multiscale processes. Satisfactory numerical predictions against experimental data are obtained along with attractive computational efficiency. Hence, the mPaSR model can be regarded as a promising alternative to existing reactor-based models.

The dimensionality of the Jacobian matrix can be seen as the method bottleneck and calls for the application of dimensionality reduction techniques. Extended validation campaigns are expected on more practical configurations and within the context of Large Eddy Simulation.

### CRediT authorship contribution statement

**Arthur Péquin:** Writing – original draft, Writing – review & editing, Methodology, Formal analysis, Conceptualization. **Erica Quadarella:**

Writing – review & editing, Formal analysis, Conceptualization. **Riccardo Malpica Galassi:** Writing – review & editing, Supervision, Formal analysis. **Salvatore Iavarone:** Writing – review & editing, Supervision, Formal analysis. **Hong G. Im:** Writing – review & editing, Supervision, Conceptualization. **Alessandro Parente:** Writing – review & editing, Supervision, Funding acquisition, Conceptualization.

### Declaration of competing interest

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

### Acknowledgements

A. Péquin acknowledges the financial support of the Fonds National de la Recherche Scientifique (FRS-FNRS) through an ASPIRANT fellowship. R. Malpica Galassi acknowledges the financial support from the Italian Ministry of University and Research in the framework of the project "National Center for HPC, Big Data and QC" funded by EU – NextGenerationEU. This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 714605.

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.combustflame.2025.114269.

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