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Direct Numerical Simulation Analysis of the Partially Stirred Reactor Model for Turbulent Reacting Flows

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To support industrial sectors in facing the challenges from the energy transition, it is essential to develop accurate Computational Fluid Dynamics (CFD) models for combustion applications. In particular, turbulence-chemistry interactions (TCI) models that can predict pollutant emissions and energy efficiency from simulated systems are sought. Among many combustion closures, reactor-based models have recently drawn interest for their ability to treat chemistry with detailed descriptions at an affordable cost. Originally derived from the intermittency theory of highly turbulent reacting flows, such models have evolved to provide a valuable solution to any type of flame, up to non conventional combustion technologies. In numerical simulations employing the Partially Stirred Reactor (PaSR) approach as combustion closure, chemical processes are assumed to take place in sub-grid flow regions of typical length scale smaller than numerical control volumes. Each computational cell is then partitioned into two locally uniform regions, namely the inert surroundings, solely driven by turbulent mixing, and the reacting fine structures. The mean reaction rates, contributing to the transport equations of the chemical species, are estimated as the reaction rates from the fine structures multiplied by the cell reacting fraction, i.e., the volume fraction of the cell occupied by the fine structures. Characteristic time scales for mixing and chemistry are used to estimate the cell reacting fraction. Many time scales formulations exist and modelling efforts are to be put in selecting the most suitable candidates. Among the available computational approaches, Large Eddy Simulation (LES) has recently gained interest for its ability to provide accurate numerical solutions up to industrial scale problems. Conversely to costly Direct Numerical Simulation (DNS) where all scales are resolved, LES solves only the scales down to the computational grid resolution level, the shorter scales and their interactions being modelled. Nevertheless, DNS of turbulent combustion can supply key information on turbulence-chemistry interactions occurring at the smallest scales. \textit{A priori} testing is an example of modelling routes making use of DNS data for the development and validation of LES combustion models. The present Thesis reports modelling advances of the Partially Stirred Reactor combustion approach by means of \textit{a priori} investigations on DNS data of turbulent reacting flows. First, a layer decomposition of the Partially Stirred Reactor combustion model was conducted. Predictions of the chemical source terms and heat release rates from several combustion models have been compared. Various formulations of the chemical time scale for the PaSR model were considered. A class of mathematical functions was constructed to provide modelling guidance. Several potential modelling improvements were identified throughout the discussion and were grouped in three categories, namely parameter selection, cell reacting fraction reformulation and deep model revision. Parameter selection aims at finding an optimal set of parameters or submodels to improve model accuracy. Such a study was conducted on the PaSR model in the context of Moderate or Intense Low-oxygen Dilution (MILD) combustion, an appealing non conventional combustion technology in terms of fuel flexibility, pollutant emissions, and thermal efficiency. An optimal set of sub models was found for the specific modelling of MILD combustion where turbulence-chemistry interactions are naturally strengthened. Also, the layer decomposition demonstrated that finding an optimal submodel can be a local concept, i.e., depending on the local flow region. In this context, a data-driven methodology employing supervised clustering algorithms has been proposed for the local estimation of the optimal chemical time scale formulation in the PaSR model. A binning operation was used to partition the data into clusters of similar thermo-chemical states. Within each cluster, the best formulation was found by means of distance minimisation. Coupling the PaSR model with clustered solutions yielded a systematic modelling error cut-off. The method was found applicable to any type of flames. Besides parameter selection, the functional form of the cell reacting fraction in the PaSR model has been investigated more carefully. A methodology was proposed to extract sub-grid quantities from DNS data matching the physical representation of the cell reacting volume fraction. Each cell from the DNS was supposed extinct or reacting depending on the local intensity of heat release. Down-sampling on coarser LES grids, the extracted cell reacting fractions were given by the proportion of active DNS cells within the larger LES control volumes. Questioning the true representation of the modelled cell reacting fractions, the extracted quantities may cope with modelling needs by means of algebraic fraction forms. This illustrated the complexity of remodelling the cell reacting fraction form by hand. Within this context, machine learning and sparse-promoting techniques have been used to explore broad libraries of potential functional forms. Such approaches returned the solution best balancing accuracy and modelling complexity. An original functional form of the cell reacting fraction was found and provided higher accuracy results with respect to standard approaches. The results were validated on combustion datasets operating at different regimes. Lastly, the PaSR combustion model has been revised more in-depth to cope with the fundamental limitation of relying on a unique cell reacting fraction for all chemical species. In particular, tools from the machine learning community and arguments from the Computational Singular Perturbation theory have been employed to respectively derive two modelling frameworks. In the first approach, the closure of a progress variable transported equation with a PaSR approach was developed. Such equation left an unclosed term, namely the fine structures progress variable, that required modelling. To this purpose, neural networks (NN) have been trained and tested on DNS data of different turbulent flames. Great generalisation capabilities were obtained while regressing the subgrid scale variable from information at grid level. This methodology has showed its ability to be a valuable alternative to the classic closures for the source term of a progress variable transported equation. The second framework consisted in the integration of multiple chemical time scales in a PaSR approach. Abandoning the concept of the fine structures, a modal decomposition of the Jacobian matrix of the chemical source terms was performed. Each mode contributing to the final estimation of the mean reaction rates was multiplied by its modal coefficient resembling a cell reacting fraction. This innovative framework provided promising results on a rather simple test case, requiring further modelling attention.
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Direct Numerical Simulation Analysis
of the Partially Stirred Reactor Model
for Turbulent Reacting Flows
Thesis presented by Arthur P ´
EQUIN
in fulfilment of the requirements of the PhD Degree in Engineering
Sciences and Technology (”Docteur en Sciences de l’Ing´
enieur et Techno-
logie”)
Academic year 2022-2023
Supervisor : Prof. Alessandro PARENTE
Universit´
e Libre de Bruxelles
Thesis jury :
Prof. Axel COUSSEMENT (Universit´
e Libre de Bruxelles, Chair)
Prof. Thierry MAGIN (Universit´
e Libre de Bruxelles, Secretary)
Prof. B´
en´
edicte CUENOT (CERFACS, University of Eindhoven)
Prof. Nedunchezhian SWAMINATHAN (University of Cambridge)
Dr. Daniel MIRA (Barcelona Supercomputing Center)
Direct Numerical Simulation Analysis
of the Partially Stirred Reactor Model
for Turbulent Reacting Flows
Arthur PÉQUIN
Aero-Thermo-Mechanics Laboratory
University of Brussels
This dissertation is submitted in fulfilment of the requirements of the
Doctor of Philosophy Degree in Engineering Sciences and Technology
July 2023
To my loving parents, brother and sister.
“Only the disciplined ones are free in life.
Eliud Kipchoge
Acknowledgements
In running, perseverance is key. Going through discomfort requires bravery but only then you will find the
resources to move forward. As any marathon finisher, any PhD holder deserves recognition for their resilience.
This journey has been incredible and I am extremely grateful. I have learnt a lot, about combustion for
sure, but also on myself. I have met plenty of interesting persons and visited beautiful places. As when
crossing the finish line, it is now time to thank all the people who have accompanied me during these past 4 years.
My first round of thanks goes to Prof. Alessandro Parente for giving me the opportunity to undertake
this PhD journey, for his unconditional support, for the insightful and pleasant discussions on combustion
modelling and sometimes on running. Thanks for the trust you put in me along the years. Under your
supervision, I was able to grow as a fresh academic researcher.
I would like to thank all my jury members for their time and expertise. In particular, thanks to Prof. Axel
Coussement for his advices and the technical support throughout the PhD. Thanks to Prof. Thierry Magin and
Dr. Daniel Mira for their insightful comments on the manuscript. A special thank to Prof. Bénédicte Cuenot
with whom I had the privilege of getting my first grasp on combustion and academic research. Thanks as well
to Prof. Nedunchezhian Swaminathan for hosting me as one of his student at the University of Cambridge.
This research stay was a true life experience and a dream came true.
Thanks to all my colleagues from the ULB for making my journey an enriching experience on both
professional and personal viewpoints. Thanks to Ruggero for being a close friend since the beginning. I
really enjoyed working, attending conferences and summer schools in your company. You made the daily lab
funnier and more Roma-oriented (that’s still ok). Thanks to my post-doc mentors, Salvatore and Riccardo
M.G., your venerable wisdom heavily contributed to make me a better student. Thanks to my dear friends
Andrea, Giuseppe, Magnus, Marianna, Riccardo L. and Simone. Thanks Alberto and Sara for the bouldering
sessions. Thanks Himanshu for the good laughs. Thanks Rodolfo for the good work together. Thanks to
the always-in-good-mood Spanish crew Eva and Patricia. Thanks as well to Kamila, Marco, Maryam and
Matteo. Thanks to the players of the ATM football team. I am convinced that good work comes along with
friendly working environment, so a general thanks to the past and current ATM department members, to my
combustion friends from France, Italy and Cambridge.
I would like to express my sincere gratitude to my parents, brother, sister, family and close friends for
always being my first supporters, in whatever I do. My last round of thanks obviously goes to Hélène and our
little cat Bob’ for making my daily life sweeter.
The present work has received financial support through an ASPIRANT fel lowship from the Fonds National de
Recherches Scientifiques (FNRS) and a doctoral research grant from the Foundation Wiener-Anspach (FWA).
Summary
To support industrial sectors in facing the challenges from the energy transition, it is essential
to develop accurate Computational Fluid Dynamics (CFD) models for combustion applications.
In particular, turbulence-chemistry interactions (TCI) models that can predict pollutant
emissions and energy eciency from simulated systems are sought. Among many combustion
closures, reactor-based models have recently drawn interest for their ability to treat chemistry
with detailed descriptions at an aordable cost. Originally derived from the intermittency
theory of highly turbulent reacting flows, such models have evolved to provide a valuable
solution to any type of flame, up to non conventional combustion technologies. In numerical
simulations employing the Partially Stirred Reactor (PaSR) approach as combustion closure,
chemical processes are assumed to take place in sub-grid flow regions of typical length scale
smaller than numerical control volumes. Each computational cell is then partitioned into two
locally uniform regions, namely the inert surroundings, solely driven by turbulent mixing,
and the reacting fine structures. The mean reaction rates, contributing to the transport
equations of the chemical species, are estimated as the reaction rates from the fine structures
multiplied by the cell reacting fraction, i.e., the volume fraction of the cell occupied by the
fine structures. Characteristic time scales for mixing and chemistry are used to estimate the
cell reacting fraction. Many time scales formulations exist and modelling eorts are to be
put in selecting the most suitable candidates.
Among the available computational approaches, Large Eddy Simulation (LES) has
recently gained interest for its ability to provide accurate numerical solutions up to industrial
scale problems. Conversely to costly Direct Numerical Simulation (DNS) where all scales
are resolved, LES solves only the scales down to the computational grid resolution level,
the shorter scales and their interactions being modelled. Nevertheless, DNS of turbulent
combustion can supply key information on turbulence-chemistry interactions occurring at
the smallest scales. Aprioritesting is an example of modelling routes making use of DNS
data for the development and validation of LES combustion models. The present Thesis
reports modelling advances of the Partially Stirred Reactor combustion approach by means
of a priori investigations on DNS data of turbulent reacting flows.
First, a layer decomposition of the Partially Stirred Reactor combustion model was
conducted. Predictions of the chemical source terms and heat release rates from several
combustion models have been compared. Various formulations of the chemical time scale
for the PaSR model were considered. A class of mathematical functions was constructed
to provide modelling guidance. Several potential modelling improvements were identified
throughout the discussion and were grouped in three categories, namely parameter selection,
cell reacting fraction reformulation and deep model revision.
Parameter selection aims at finding an optimal set of parameters or submodels to improve
model accuracy. Such a study was conducted on the PaSR model in the context of Moderate or
Intense Low-oxygen Dilution (MILD) combustion, an appealing non conventional combustion
technology in terms of fuel flexibility, pollutant emissions, and thermal eciency. An
optimal set of sub models was found for the specific modelling of MILD combustion where
turbulence-chemistry interactions are naturally strengthened. Also, the layer decomposition
demonstrated that finding an optimal submodel can be a local concept, i.e., depending on the
local flow region. In this context, a data-driven methodology employing supervised clustering
algorithms has been proposed for the local estimation of the optimal chemical time scale
formulation in the PaSR model. A binning operation was used to partition the data into
clusters of similar thermo-chemical states. Within each cluster, the best formulation was
found by means of distance minimisation. Coupling the PaSR model with clustered solutions
yielded a systematic modelling error cut-o. The method was found applicable to any type
of flames.
Besides parameter selection, the functional form of the cell reacting fraction in the PaSR
model has been investigated more carefully. A methodology was proposed to extract sub-grid
quantities from DNS data matching the physical representation of the cell reacting volume
fraction. Each cell from the DNS was supposed extinct or reacting depending on the local
intensity of heat release. Down-sampling on coarser LES grids, the extracted cell reacting
fractions were given by the proportion of active DNS cells within the larger LES control
volumes. Questioning the true representation of the modelled cell reacting fractions, the
extracted quantities may cope with modelling needs by means of algebraic fraction forms.
This illustrated the complexity of remodelling the cell reacting fraction form by hand. Within
this context, machine learning and sparse-promoting techniques have been used to explore
broad libraries of potential functional forms. Such approaches returned the solution best
balancing accuracy and modelling complexity. An original functional form of the cell reacting
fraction was found and provided higher accuracy results with respect to standard approaches.
The results were validated on combustion datasets operating at dierent regimes.
Lastly, the PaSR combustion model has been revised more in-depth to cope with the
fundamental limitation of relying on a unique cell reacting fraction for all chemical species. In
particular, tools from the machine learning community and arguments from the Computational
Singular Perturbation theory have been employed to respectively derive two modelling
frameworks. In the first approach, the closure of a progress variable transported equation
with a PaSR approach was developed. Such equation left an unclosed term, namely the fine
structures progress variable, that required modelling. To this purpose, neural networks (NN)
viii
have been trained and tested on DNS data of dierent turbulent flames. Great generalisation
capabilities were obtained while regressing the subgrid scale variable from information at
grid level. This methodology has showed its ability to be a valuable alternative to the
classic closures for the source term of a progress variable transported equation. The second
framework consisted in the integration of multiple chemical time scales in a PaSR approach.
Abandoning the concept of the fine structures, a modal decomposition of the Jacobian matrix
of the chemical source terms was performed. Each mode contributing to the final estimation
of the mean reaction rates was multiplied by its modal coecient resembling a cell reacting
fraction. This innovative framework provided promising results on a rather simple test case,
requiring further modelling attention.
ix
Résumé
Pour soutenir le secteur industriel face aux problématiques liées au changement climatique, il
est primordial de développer des modèles, dits de CFD (Computational Fluid Dynamics),
performants pour la combustion. En particulier, les modèles d’intéractions entre la chimie
et la turbulence servent à prédire les émissions de polluants et le rendement énergétique
d’un système. Que ces modèles soient précis est un des objectifs majeurs de la modélisation
numérique assistée par ordinateur. Parmi plusieurs solutions, les modèles basés sur réacteurs
orent la possibilité de décrire les phénomènes chimiques avec une grande précision, moyennant
un coup de calcul raisonné. Ce type de modèle est dérivé d’études sur l’intermittence
statistique des écoulements réactifs turbulents. Un exemple, le modèle PaSR (Partially
Stirred Reactor) suppose que la combustion prend place dans des zones précises du fluide en
écoulement, les conditions de mélange de gaz sont optimales pour le déclenchement de
réactions chimiques. Les taux moyens de réactions chimiques, donnés au solveur numérique
pour continuer la simulation, sont calculés selon la réactivité et le volume de ces dites zones
propices. Des temps caractéristiques liés au mélange turbulent et aux réactions chimiques
sont utilisés pour estimer les propriétés physiques des zones réactives. Une multitude de
définitions existent dans la littérature pour définir ces temps caractéristiques et il est alors
question de sélectionner un couple optimal de paramètres.
Il existe plusieurs types de simulation numérique, par exemple, les simulations aux grandes
échelles (LES) ont récemment gagné en intérêt auprès de la communauté industrielle pour
leur capacité à fournir des résultats numériques détaillés sur des systèmes à grandes capacités
(industrielles). Contrairement à la simulation numérique directe (DNS) toutes les échelles
de la turbulence sont résolues, les simulations LES appliquent un filtre au delà duquel les plus
petites échelles sont modélisées au lieu d’être calculées. Cela permet notamment de limiter
les coûts de calculs et de considérer des maillages numériques plus volumineux. Cependant,
les données DNS sont très riches en informations sur les intéractions chimie-turbulence et
peuvent être employées pour renforcer les capacités prédictives des modèles LES. La présente
Thèse rapporte quelques développements du modèle de combustion PaSR à partir de données
DNS.
En premier lieu, le modèle PaSR est analysé en ses diérents composants. Chaque
terme du modèle est étudié séparemment dans cette première partie. Les taux de réactions
chimiques et de production de chaleur sont calculés selon le modèle PaSR et comparés à
d’autres approches. Plusieurs définitions du temps chimique sont employées. Une classe
de fonctions mathématiques est également construite pour servir de modèle de référence.
Des améliorations du modèle PaSR sont identifiées au cours de la présentation des résultats
et peuvent être regroupées en trois catégories: la sélection de paramètres optimaux, la
reformulation du modèle pour les fractions volumiques réactives, et les révisions du modèle
en profondeur.
La sélection de paramètres cherche à trouver un jeu optimal de définitions et à améliorer la
précision du modèle pour un système donné. Une de ces études est présentée dans le cadre de
la combustion MILD (Moderate or Intense Low-oxygen Dilution), un procédé de combustion
peu polluant et à hauts rendements énergétiques. De surcroît, la décomposition du modèle a
mis en évidence qu’employer une unique définition du temps chimique est sous-optimal, la
meilleure option pouvant varier selon les conditions locales de combustion. Une méthodologie,
basée sur les données, est présentée afin de tenir compte d’une multitude de temps chimiques.
Un algorithme de classification dit supervisé est utilisé pour grouper les zones de combustion
de conditions similaires et leur attribuer la définition du temps chimique qui convient le
mieux. Cette méthode s’est trouvée convaincante sur diérents régimes de combustion.
Au-delà de la sélection de paramètres, la définition même des fractions volumiques
réactives est analysée plus en détails. Une méthode est proposée pour extraire ces quantités à
partir de données DNS. À l’échelle de la DNS, chaque cellule numérique est considérée comme
chimiquement active ou non-réactive en fonction de l’intensité locale du taux de dégagement
de chaleur. Un filtre est alors appliqué pour calculer les fractions volumiques réactives à
l’échelle LES. Des corrélations sont trouvées entre les besoins du modèles PaSR et ces données
issues de la DNS. Cela a cependant démontré la compléxité de redéfinir un tel paramètre à
partir d’arguments physiques. Certains algorithmes, orientés sur le traitement des données,
peuvent également être employés pour identifier des formes de modèles alliant précision
et coûts de calculs modérés. Une formulation originale est trouvée grâce à ce processus
améliorant les résultats existants, et ce sur une multitude de régimes de combustion.
Enfin, le modèle PaSR, et notamment l’unicité de la fraction volumique réactive, sont revus
en profondeur. Deux cadres de travail sont alors proposés. Premièrement, des algorithmes de
machine learning (ML), tels que les réseaux neuronaux (NN), sont utilisés pour la fermeture
de l’équation de transport d’une variable de progrès. Plusieurs bases de données DNS
servent à entrainer et valider les algorithmes qui s’avèrent viables pour une multitude de
cas de combustion. La deuxième révision s’appuie sur les arguments de la théorie de la
Computational Singular Perturbation (CSP) afin d’inclure une multitude de fractions réactives
au sein d’une approche PaSR fortement révisée. Sur un cas test simple, cette proposition
innovante donne des résultats satisfaisants et une attention particulière est nécessaire pour la
valider sur des cas plus complexes.
xii
Table of Contents
List of Figures xvii
List of Tables xxix
Nomenclature xxxi
1 Introduction 1
1.1 Generalcontext................................... 1
1.2 Reactor-based models for turbulent reacting flows . . . . . . . . . . . . . . . . 5
1.3 Objective and outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Turbulent combustion modelling 13
2.1 Turbulentcombustion ............................... 14
2.1.1 Turbulence ................................. 14
2.1.2 Combustionregime............................. 16
2.2 Governing equations for turbulent reacting flows . . . . . . . . . . . . . . . . 20
2.3 Computational approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Turbulent combustion models . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Fast chemistry models . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 Models for turbulent premixed flames . . . . . . . . . . . . . . . . . . 34
2.4.3 Models for turbulent non-premixed flames . . . . . . . . . . . . . . . . 35
2.5 Finite-rate reactor-based combustion models . . . . . . . . . . . . . . . . . . . 37
2.5.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.2 Eddy Dissipation Concept model (EDC) . . . . . . . . . . . . . . . . . 39
2.5.3 Ideal reactor treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.4 Partially Stirred Reactor model (PaSR) . . . . . . . . . . . . . . . . . 44
3 Methodology 51
3.1 Aprioritesting................................... 52
3.2 DNSdatacollection ................................ 52
3.2.1 Case 1: 2D turbulent non-premixed CO/H2jet flame . . . . . . . . . . 53
Table of Contents
3.2.2 Case 2: 3D turbulent non-premixed sooting flame . . . . . . . . . . . . 54
3.2.3 Case 3: 3D turbulent premixed methane-air flame . . . . . . . . . . . 55
3.2.4 Case 4: 3D turbulent non-premixed MILD flames . . . . . . . . . . . . 57
3.3 Discussion...................................... 59
3.4 DNS chemical source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 LESlters...................................... 63
3.5.1 Widths ................................... 63
3.5.2 Filtering eect ............................... 66
3.6 Concludingremarks ................................ 68
4 Layer decomposition of the Partially Stirred Reactor model 69
4.1 Subgrid scale Partially Stirred Reactor combustion model . . . . . . . . . . . 71
4.1.1 Filtered chemical source terms . . . . . . . . . . . . . . . . . . . . . . 71
4.1.2 Mixing time scale formulations . . . . . . . . . . . . . . . . . . . . . . 72
4.1.3 Chemical time scale formulations . . . . . . . . . . . . . . . . . . . . . 72
4.2 Alternative finite-rate combustion models . . . . . . . . . . . . . . . . . . . . 73
4.2.1 Quasi-Laminar Finite Rate model (QLFR) . . . . . . . . . . . . . . . 73
4.2.2 Laminar Finite Rate (LFR) model . . . . . . . . . . . . . . . . . . . . 74
4.2.3 Enhanced Laminar Finite Rate (LFR*) model . . . . . . . . . . . . . 74
4.2.4 Summary .................................. 75
4.3 Eect of modelling the fine structures quantities . . . . . . . . . . . . . . . . 76
4.4 Impact of the fine structures volume fraction . . . . . . . . . . . . . . . . . . 79
4.5 Impact of the chemical time scale formulation . . . . . . . . . . . . . . . . . . 84
4.5.1 Eects on the reacting fraction estimation . . . . . . . . . . . . . . . . 84
4.5.2 Eects on the chemical source terms estimation . . . . . . . . . . . . . 87
4.6 Error quantification from modelling layers . . . . . . . . . . . . . . . . . . . . 95
4.7 Concludingremarks ................................ 97
5 Model parameter and data-driven approaches 99
5.1 Optimal sub modelling for MILD combustion regime . . . . . . . . . . . . . . 100
5.1.1 MILD combustion regime . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1.2 DNS of MILD combustion . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1.3 Parameter selection of reactor-based models in MILD combustion . . . 102
5.1.4 Aprioriapproach .............................105
5.1.5 Sensitivity to the time scales formulations . . . . . . . . . . . . . . . . 105
5.1.6 Assessment at additional xyplanes ..................115
5.1.7 Sensitivity to the filter width . . . . . . . . . . . . . . . . . . . . . . . 116
5.1.8 Sensitivity to conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
xiv
Table of Contents
5.2 Local sub model optimisation with clustering algorithms . . . . . . . . . . . . 122
5.2.1 PaSR model based on a single chemical time scale . . . . . . . . . . . 122
5.2.2 PaSR model based on optimal local time scales . . . . . . . . . . . . . 124
5.2.3 Supervised clustering procedure . . . . . . . . . . . . . . . . . . . . . . 126
5.2.4 Conditioning variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.5 Optimal local time scale selection . . . . . . . . . . . . . . . . . . . . . 129
5.2.6 Testing on unseen data . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2.7 Alternative error metric . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2.8 Robustness across DNS cases . . . . . . . . . . . . . . . . . . . . . . . 138
5.2.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6 Functional form of the cell reacting fraction 143
6.1 Cell reacting fraction in reactor-based models . . . . . . . . . . . . . . . . . . 144
6.1.1 Eddy Dissipation Concept . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.1.2 Partially Stirred Reactor model . . . . . . . . . . . . . . . . . . . . . . 146
6.2 Extraction of cell reacting fractions from DNS . . . . . . . . . . . . . . . . . . 148
6.2.1 Extraction methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.2 Statistics of extracted quantities . . . . . . . . . . . . . . . . . . . . . 151
6.2.3 Identifying modelling needs . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 Sparsity-promoting techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3.1 Sparse regression for TCI model identification . . . . . . . . . . . . . . 158
6.3.2 Trainingstrategy.............................. 161
6.3.3 Trainingresults...............................163
6.3.4 Extended candidates library . . . . . . . . . . . . . . . . . . . . . . . . 165
6.3.5 Testing on unseen data . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.3.6 Generalisation capabilities of the identified model . . . . . . . . . . . . 168
6.3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7 Coping with the cell reacting fraction uniqueness 171
7.1 Fine structures quantities modelling with Neural Networks . . . . . . . . . . . 172
7.1.1 A PaSR closure for a progress variable source term . . . . . . . . . . . 173
7.1.2 Direct Numerical Simulation Data . . . . . . . . . . . . . . . . . . . . 174
7.1.3 Machine learning methodology . . . . . . . . . . . . . . . . . . . . . . 175
7.1.4 Training...................................177
7.1.5 Testing ................................... 177
7.1.6 Influence of submodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.1.7 Generalisation capabilities . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.1.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
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Table of Contents
7.2 Inclusion of multiple cell reacting fractions . . . . . . . . . . . . . . . . . . . . 187
7.2.1 Inclusion of multiple chemical times . . . . . . . . . . . . . . . . . . . 188
7.2.2 Prediction capabilities of the generalised PaSR model . . . . . . . . . 190
7.2.3 Time scales participation . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8 Concluding remarks and prospects 197
References 201
List of Publications 222
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List of Figures
1.1
Observed warming of the Earth global surface temperature during the last 20
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