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Application of conditional filtering to simulation of turbulence, chemistry, and their interactions Hendra, Graham Ronald Rupert 2019

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Application of Conditional Filtering toSimulation of Turbulence, Chemistry, andtheir InteractionsbyGraham Ronald Rupert HendraB.A.Sc., The University of Waterloo, 2011A Thesis Submitted in Partial Fulfillmenet ofthe Requirements for the Degree ofDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)The University of British Columbia(Vancouver)April 2019c© Graham Ronald Rupert Hendra, 2019The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Application of Conditional Filtering to Simulation of Turbulence,Chemistry, and their Interactionssubmitted by Graham Ronald Rupert Hendrain partial fulfillment of theDoctor of Philosophyrequirements for the degree ofin Mechanical EngineeringExamining Committee:W. Kendal Bushe, Mechanical EngineeringCo-SupervisorCarl Ollivier-Gooch, Mechanical EngineeringCo-SupervisorPat Kirchen, Mechanical EngineeringSupervisory Committee MemberGwynn Elfring, Mechanical EngineeringUniversity ExaminerFariborz Taghipour, Chemical and Biological EngineeringUniversity ExaminerAdditional Supervisory Committee Members:Mark Martinez, Chemical and Biological EngineeringSupervisory Committee MemberiiAbstractCombustion technology has been applied in human society for millennia and, since theindustrial revolution, has become an integral part of most energy supply chains. Simulationis an important tool in modern combustor design; this thesis aims to improve the quality ofcombustion simulation tools, and thereby facilitate the design of improved combustors. Morespecifically, it aims to examine how generalizing and/or relaxing the definition of conditionalfilteringa common technique in turbulence-chemistry interaction modellingcan producenovel turbulence and combustion models.The work begins with an extension of the Conditional Source-term Estimation (CSE)model for turbulence-chemistry interaction modelling. A novel variation of the algorithm,termed CSE with Geometric Conditioning Variables (CSE-GCV) is proposed as a method ofcircumventing the theoretical and practical issues associated with traditional CSE ensembledivision. In CSE-GCV, the concept of the conditional filter is generalized by introducinggeometric (position-based) variables as conditioning variables. CSE-GCV is tested and foundto be workable; a theoretical analysis demonstrates that CSE-GCV also generally has theadvantage of reduced computational complexity compared to traditional CSE.In a separate study, the stabilization procedure employed in traditional dynamic sub-filter modelling for Large Eddy Simulation (LES) is re-interpreted as a form of conditionalfiltering based on position. This re-interpretation is used as the starting point for a con-ditional dynamic sub-filter model in which the stabilization procedure is based on filteringconditionally on scalar fields. Both the traditional and conditional models are applied to aturbulent flame; results suggest that the two models perform similarly, although performanceof both is sub-optimal in the case considered.The final, two-part, study is based around the suggestion that, with sufficient condition-ing, conditionally-filtered fields should be independent of position. It is found that assumingthis uniformity produces a novel turbulence-chemistry interaction model, termed the Uni-form Conditional State (UCS) model, in which the conditional scalar dissipation model is thekey un-closed parameter. The UCS model is applied to a series of turbulent non-premixedflames, and is found to predict their properties to good accuracy, with details showing somesensitivity to the conditional scalar dissipation model.iiiLay SummaryThe majority of the world's energy is derived from combustion. Combustion has severalnegative side-effects, chiefly associated with the production of greenhouse gases (which con-tribute to climate change) and pollutants (which are harmful to human health and theenvironment). Well-designed combustors can reduce these side-effects. This work advancessome of the simulation techniques used in the modern combustor design process. Advancingthese tools has the potential to facilitate the design of better combustors and to furthermitigate the negative side-effects of combustion.ivPrefaceThe contents of Chapter 5 have been submitted for publication, asG. R. Hendra and W. K. Bushe. An Ensemble-free Variant of the ConditionalSource-term Estimation (CSE) Method based on Geometric Conditioning Vari-ables.and have previously been presented (in work-in-progress form) asG. R. Hendra and W. K. Bushe. Application of Inter-Ensemble Regularization tothe CSE Model for Turbulence-Chemistry Interaction, paper presented to Com-bustion Institute/Canadian Section Spring Technical Meeting, Montreal, QC,2017.I performed all parts of the research and drafted both manuscripts. Dr. Kendal Bushesupervised the research and has provided feedback on the manuscripts.The contents of Chapter 6 have been submitted for publication, asG. R. Hendra and W. K. Bushe. Conditional Dynamic Subfilter Modelling.and have previously been presented (in work-in-progress form) asG. R. Hendra, M. M. Salehi, and W. K. Bushe. A Conditionally-Averaged Dy-namic Sub-Grid Model for Non-Premixed Turbulent Combustion, poster at Com-bustion Symposium, San Francisco, California, 2014.G. R. Hendra, M. M. Salehi, and W. K. Bushe. A Conditional Dynamic Sub-Grid Model for Large Eddy Simulation of Non-Premixed Turbulent Combustion,presentation to International Conference on Numerical Combustion, Avignon,France, 2015.G. R. Hendra, M. M. Salehi, and W. K. Bushe. A Conditional Dynamic Sub-GridModel for Large Eddy Simulation of Non-Premixed Turbulent Combustion, paperpresented to Combustion Institute, Canadian Section Spring Technical Meeting,Saskatoon, SK, 2015.I performed all parts of the research and drafted all manuscripts and presentations. M.Mahdi Salehi provided assistance in using the software employed in the earlier iterationsof this work. Dr. Kendal Bushe supervised the research and provided feedback on themanuscripts and presentations.The contents of Chapters 7 and 8 were combined into a manuscript which has beenaccepted for publication in Combustion and Flame, asG. R. Hendra and W. K. Bushe. The Uniform Conditional State Model forTurbulent Reacting Flows.The same work was also previously presented (in work-in-progress form) asvPrefaceG.R. Hendra and W.K. Bushe. The UCS Model: A CMC-based approach for sim-ulation of Partially-Premixed Turbulent Flames, paper presented to CombustionInstitute/Canadian Section Spring Technical Meeting, Toronto, ON, 2018.I performed all parts of the research and drafted both manuscripts. Dr. Kendal Bushesupervised the research and provided feedback on the manuscripts and how best to addresspeer reviewer comments.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvLatin Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvGreek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxivI Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Historical Origins of Combustion . . . . . . . . . . . . . . . . . . . . . 21.1.2 Combustion in the Present Day . . . . . . . . . . . . . . . . . . . . . 21.1.3 Future Trajectory of Combustion Technology . . . . . . . . . . . . . . 31.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 High-Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Intermediate-Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Structure of this Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Fundamental Physics of Reacting Systems . . . . . . . . . . . . . . . . . . . 62.1 State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Tracking State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Evaluating State Functions . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Species Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.4 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Constitutive Equations (Transport Models) . . . . . . . . . . . . . . . . . . . 102.3.1 Viscous Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 11viiTable of Contents2.3.2 Diffusive Flux Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 Heat Flux Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.4 Radiation Source Term . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.5 Species Source Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Emergent Behaviour in Combustion Systems . . . . . . . . . . . . . . . . . 183.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.1 Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.2 The Multi-Scale Chaos Problem . . . . . . . . . . . . . . . . . . . . . 183.1.3 Features of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . 203.1.4 The [Kinetic] Energy Cascade . . . . . . . . . . . . . . . . . . . . . . 203.2 Global Effective Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . 213.3 Premixedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.1 Non-Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.3 Partially-Premixed Combustion . . . . . . . . . . . . . . . . . . . . . 274 Simulation Techniques for Turbulent Combustion Systems . . . . . . . . . 314.1 Turbulence Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.1 The Filtering Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1.2 Filtered State and State Functions . . . . . . . . . . . . . . . . . . . . 334.1.3 Filtered Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 344.1.4 Filtered Constitutive Equations . . . . . . . . . . . . . . . . . . . . . 364.1.5 Sub-Filter Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.6 Specific Turbulence Modelling Paradigms . . . . . . . . . . . . . . . . 374.2 Chemistry Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.1 Reaction-Diffusion Manifolds (REDIMs) . . . . . . . . . . . . . . . . 414.2.2 Pure Reaction Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.3 Flamelet Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.4 Applicability of REDIM Methods . . . . . . . . . . . . . . . . . . . . 434.3 Turbulence-Chemistry Interaction Modelling . . . . . . . . . . . . . . . . . . 434.3.1 Degree of Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3.2 Evaluation of PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.3 Specific Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47II Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 An Ensemble-free Variant of the Conditional Source-term Estimation(CSE) Method based on Geometric Conditioning Variables . . . . . . . . 495.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.1 CSE Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.2 Research Question and Objectives . . . . . . . . . . . . . . . . . . . . 545.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2.1 The Traditional CSE Model . . . . . . . . . . . . . . . . . . . . . . . 545.2.2 CSE with Geometric Conditioning Variables (CSE-GCV) . . . . . . . 62viiiTable of Contents5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.5.1 Validation of CSE-GCV . . . . . . . . . . . . . . . . . . . . . . . . . . 715.5.2 Comparison of traditional CSE and CSE-GCV . . . . . . . . . . . . . 715.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Conditional Dynamic Subfilter Modelling . . . . . . . . . . . . . . . . . . . 746.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.1.1 Ensembles in Dynamic Subfilter Modelling . . . . . . . . . . . . . . . 746.1.2 Research Question and Objectives . . . . . . . . . . . . . . . . . . . . 756.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2.1 The Static Smagorinsky-Yoshizawa Model . . . . . . . . . . . . . . . . 756.2.2 The Germano Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2.3 Dynamic Localization with CSE Techniques . . . . . . . . . . . . . . 786.2.4 Stabilization via Regularization . . . . . . . . . . . . . . . . . . . . . 796.2.5 Conditioning on Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 The Uniform Conditional State (UCS) Model for Turbulence-ChemistryInteraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.1.1 Conditional Moment Closure (CMC) . . . . . . . . . . . . . . . . . . 887.1.2 Conditional Source-term Estimation (CSE) . . . . . . . . . . . . . . . 897.1.3 Research Question and Objectives . . . . . . . . . . . . . . . . . . . . 897.2 Additional Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.2.1 Species-Like Transport Equations . . . . . . . . . . . . . . . . . . . . 897.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2.3 The Multidimensional Flamelet (MFM) Model . . . . . . . . . . . . . 917.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3.2 Generating Conditional Governing Equations . . . . . . . . . . . . . . 947.3.3 Conditionally-Filtered Divergence of Diffusive Flux . . . . . . . . . . 967.3.4 General Conditional Transport Equation . . . . . . . . . . . . . . . . 997.3.5 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.4.1 Viability of the UCS System . . . . . . . . . . . . . . . . . . . . . . . 1047.4.2 Connections to Existing Chemistry and Combustion Models . . . . . 1077.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109ixTable of Contents8 Application of the Uniform Conditional State (UCS) Model to Non- andSlightly-Premixed Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.1.1 Objectives and Research Question . . . . . . . . . . . . . . . . . . . . 1108.2 Study Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.2.1 UCS Specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.2.2 Test Flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.2.3 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.3.1 Conditional Case Regimen . . . . . . . . . . . . . . . . . . . . . . . . 1138.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.3.3 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.4.1 Grid Convergence Sub-Study . . . . . . . . . . . . . . . . . . . . . . . 1188.4.2 Chemical Mechanism Sub-Study . . . . . . . . . . . . . . . . . . . . . 1228.4.3 Scalar Dissipation Sub-Study . . . . . . . . . . . . . . . . . . . . . . . 1268.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.5.1 Assessment of Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . 1528.5.2 Grid Convergence and Chemical Mechanism Studies . . . . . . . . . . 1528.5.3 Predictions of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.5.4 Closure for the UCS Model . . . . . . . . . . . . . . . . . . . . . . . . 1538.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153III Conclusions and References . . . . . . . . . . . . . . . . . . . . . . . . . 1549 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.1 Chapter-Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.1.1 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.1.2 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.1.3 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.1.4 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2 Overall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.1 Mathematical Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.2 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.2.1 Basic Vector and Tensor Operations . . . . . . . . . . . . . . . . . . . 169A.2.2 Tensor Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 170A.2.3 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172xTable of ContentsB Filter-Related Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173B.1 Filtering Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173B.1.1 The Basic (Volume-Weighted) Filter . . . . . . . . . . . . . . . . . . . 173B.1.2 The Favre (Mass-Weighted) Filter . . . . . . . . . . . . . . . . . . . . 174B.1.3 Conditional Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175B.2 Special Filtered Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2.1 Sub-Filter Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2.2 Filtered Moments of a Single Field . . . . . . . . . . . . . . . . . . . . 177B.2.3 Filtered Mixed Moments of a Set of Fields . . . . . . . . . . . . . . . 178B.2.4 Filtered Probability Density Functions (PDFs) . . . . . . . . . . . . . 179B.3 Reynolds Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180C Filter-Related Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182C.1 Filter-Function Commutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 182C.2 Limiting Values of Sub-Filter Terms . . . . . . . . . . . . . . . . . . . . . . . 185C.2.1 Generic Sub-Filter Term . . . . . . . . . . . . . . . . . . . . . . . . . 186C.2.2 Sub-Filter Mass Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 187C.2.3 Sub-Filter Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187C.2.4 Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187D Transport Equations for Filter-Based Turbulence Models . . . . . . . . . 188D.1 Transport of Filtered Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 188D.1.1 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188D.1.2 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188D.1.3 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189D.1.4 Mass Fraction, Mixture Fraction, and Progress Variable . . . . . . . . 190D.1.5 Other Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190D.2 Transport of Filtered Variances . . . . . . . . . . . . . . . . . . . . . . . . . . 191D.2.1 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193E The Sandia Non-Premixed Flame Series . . . . . . . . . . . . . . . . . . . . 195E.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195E.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195E.1.2 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195E.1.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195E.2 Experimental Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 196F Uniform Conditional State (UCS) Identities . . . . . . . . . . . . . . . . . . 197F.1 From Assumption 3(a) to Equation 7.16 . . . . . . . . . . . . . . . . . . . . . 197F.2 From Assumption 3(b) to Equation 7.16 . . . . . . . . . . . . . . . . . . . . . 198F.3 From Assumption 3(c) to Equation 7.16 . . . . . . . . . . . . . . . . . . . . . 199F.4 Equations 7.40 and 7.41 to Equation 7.42 . . . . . . . . . . . . . . . . . . . . 200F.5 Equations 7.40 and 7.41 to Equation 7.43 . . . . . . . . . . . . . . . . . . . . 201xiList of Tables1.1 Global Primary Energy Supply in 2016 . . . . . . . . . . . . . . . . . . . . . . 34.1 RANS Eddy Viscosity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Detailed chemical mechanisms for methane-air chemistry . . . . . . . . . . . . 405.1 Boundary conditions on velocity and pressure . . . . . . . . . . . . . . . . . . 665.2 Total computational cost of solving the normal equations via Cholesky de-composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3 Total computational cost of per iteration LSQR . . . . . . . . . . . . . . . . . 728.1 Relevant chemical mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 1148.2 Reference case conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.3 Names and series styles for the combinations of scalar dissipation modelsconsidered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.1 Mathematical Object Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.2 Magnitude Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.3 Vector and Tensor Product Notation . . . . . . . . . . . . . . . . . . . . . . . 170A.4 Tensor Operation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170A.5 Tensor Decomposition Operator Notation . . . . . . . . . . . . . . . . . . . . 170A.6 Vector Calculus Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172E.1 Key geometric parameters defining the Sandia flame series . . . . . . . . . . . 195E.2 Key velocities defining the Sandia flame series . . . . . . . . . . . . . . . . . . 196xiiList of Figures3.1 Visualization of density gradients in a candle's exhaust plume, showing lam-inar, transitional, and turbulent flow . . . . . . . . . . . . . . . . . . . . . . . 193.2 The turbulent cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 The Continuum of Premixedness . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Photograph of Bunsen burner flames with varying air coflow . . . . . . . . . . 233.5 An idealized triple flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1 Relative cost and accuracy of turbulence simulation paradigms . . . . . . . . 325.1 Examples of CSE ensemble shapes in a jet flame . . . . . . . . . . . . . . . . 515.2 Example of traditional CSE ensemble overlap in a jet flame . . . . . . . . . . 525.3 Example domain, ensembles, and subdomains for CSE of a jet flame usingtraditional ensemble division . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 The core CSE algorithm for a discrete solver . . . . . . . . . . . . . . . . . . . 585.5 Time-averaged conditional profiles of scalars . . . . . . . . . . . . . . . . . . . 675.6 Radial profiles of mean axial velocity and fluctuating kinetic energy . . . . . . 685.7 Radial profiles of time-averaged scalars . . . . . . . . . . . . . . . . . . . . . . 695.8 Radial profiles of root-mean-square fluctuations of scalars . . . . . . . . . . . 706.1 Visualization of the mixing field in a turbulent non-premixed jet flame. . . . . 826.2 Time-averaged conditional profiles of scalars . . . . . . . . . . . . . . . . . . . 836.3 Radial profiles of mean axial velocity and fluctuating kinetic energy . . . . . . 846.4 Radial profiles of time-averaged scalars . . . . . . . . . . . . . . . . . . . . . . 856.5 Radial profiles of root-mean-square fluctuations of scalars . . . . . . . . . . . 868.1 Conditional domain, basis, and boundary conditions . . . . . . . . . . . . . . 1148.2 Model profiles of conditional scalar dissipation of progress variable . . . . . . 1158.3 Variation of doubly-conditional profiles with grid resolution (YH2O, YCO, YOH,and YH2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.4 Variation of doubly-conditional profiles with grid resolution (YNO, YC2H2 , T ,and ω˙c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.5 Normalized error associated with various scalars and grid resolutions. . . . . . 1218.6 Variation of doubly-conditional profiles with chemical mechanism (YH2O, YCO,YOH, and YH2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238.7 Variation of doubly-conditional profiles with chemical mechanism (YNO, YC2H2 ,T , and ω˙c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.8 Normalized error associated with various scalars and chemical mechanisms. . 1258.9 Variation of doubly-conditional profiles of YH2O with scalar dissipation model 1278.10 Variation of doubly-conditional profiles of YCO with scalar dissipation model . 1288.11 Variation of doubly-conditional profiles of YOH with scalar dissipation model . 1298.12 Variation of doubly-conditional profiles of YH2 with scalar dissipation model . 1308.13 Variation of doubly-conditional profiles of YNO with scalar dissipation model . 131xiiiList of Figures8.14 Variation of doubly-conditional profiles of YC2H2 with scalar dissipation model 1328.15 Variation of doubly-conditional profiles of T with scalar dissipation model . . 1338.16 Variation of doubly-conditional profiles of ω˙c with scalar dissipation model . . 1348.17 Singly-conditioned profiles of YH2O . . . . . . . . . . . . . . . . . . . . . . . . 1358.18 Singly-conditioned profiles of YCO . . . . . . . . . . . . . . . . . . . . . . . . 1368.19 Singly-conditioned profiles of YOH . . . . . . . . . . . . . . . . . . . . . . . . 1378.20 Singly-conditioned profiles of YH2 . . . . . . . . . . . . . . . . . . . . . . . . . 1388.21 Singly-conditioned profiles of YNO . . . . . . . . . . . . . . . . . . . . . . . . 1398.22 Singly-conditioned profiles of YC2H2 . . . . . . . . . . . . . . . . . . . . . . . 1408.23 Singly-conditioned profiles of T . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.24 Singly-conditioned profiles of ω˙c . . . . . . . . . . . . . . . . . . . . . . . . . 1428.25 Radial profiles of YH2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.26 Radial profiles of YCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.27 Radial profiles of YOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.28 Radial profiles of YH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468.29 Radial profiles of YNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.30 Radial profiles of YC2H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.31 Radial profiles of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.32 Radial profiles of ω˙c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.33 Normalized error associated with various scalars and scalar dissipation models.151B.1 Unconditional and conditional ensemble averaging of experimental measure-ments of Sandia Flame D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177E.1 Visualization of the mixing field of Flame D, including measurement locations 196xivList of SymbolsThis list includes only symbols which appear in the main document (excluding appendices)and have global scope. Symbols which appear in appendices only and/or have local scopeare not included as they typically only appear at or near their definition. Entries of the formAℵ∈R are shorthand for any Aℵ for which ℵ ∈ R.Latin AlphabetSymbol Description Dimensionality IntroducedAℵ∈R pre-exponential factor for reaction ℵ in mo-dified Arrhenius equationsee Eq. 2.44 Eq. 2.44Bη spectral (per wavenumber, η) blackbody in-tensitym/(L2t3) Eq. 2.37CFIYoshizawa coefficient at scale of filter F 1 Eq. 6.1CFSSmagorinsky coefficient at scale of filter F 1 Eq. 6.1c progress variable 1 Eq. 3.12D set of fields whose gradients drive diffusion − Eq. 2.17D overall diffusivity (only well-defined if diffu-sion is assumed to be Fickian with a singleglobal coefficient)L2/t Eq. 3.26DF turbulent or eddy diffusivity associatedwith filter F (typically assumed to apply toall scalars φ ∈ Z)L2/t Eq. 4.21Dψ∈DYα∈Y diffusivity describing dependence of flux ofmass fraction Yα on gradients of drivingforce ψL2/(tψ) Eq. 2.17DY ∈Y+ diffusivity of scalar Y (only well-defined ifdiffusion is assumed to be Fickian)L2/t Eq. 7.39Davα∈S average diffusivity for species α diffusinginto mixtureL2/t Eq. 2.28Dmultα∈Sβ∈S multi-component mass diffusivity for spe-cies combination α, βL2/t Eq. 2.20xvLatin AlphabetSymbol Description Dimensionality IntroducedDTα∈S thermodiffusivity (kinematic thermodiffus-ion coefficient) of species αL2/t Eq. 2.23d differential symbol − −E set of CSE ensembles (a set of sets of points) − Sec. 5.2.1Eψ normalized error in predictions of scalar ψ ψ Eq. 8.9E set of discrete CSE ensembles (a set of setsof cells)− Sec. 5.2.1E CSE ensemble core − Sec. 5.2.1Eaℵ∈R activation energy of reaction ℵ mL2/(t2n) Eq. 2.44e mass-averaged specific total (internal pluskinetic) energy of mixtureL2/t2 Sec. 2.2.4e discrete CSE ensemble core − Sec. 5.2.1fYcα∈S coefficient defining contribution of species αto progress variable-defining scalar Yc1 Eq. 3.9fZα∈S coefficient defining contribution of species αto mixture fraction Z1 Eq. 3.2G set of geometric variables expected to ac-count for variation in conditionally-filteredfields− Sec. 5.2.1G geometric domain (set of all possible valuesof G)− Eq. 5.9G discrete geometric domain (set sets of points(cells) in geometric space)− Sec. 5.2.1Gη spectral (per wavenumber, η) incident radi-ation functionm/(L2t3) Eq. 2.37H total enthalpy of mixture mL2/t2 Sec. 2.1.2h mass-averaged specific enthalpy of mixture L2/t2 Sec. 2.1.1I◦ identity matrix 1 −hα∈S specific enthalpy of species α L2/t2 Eq. 2.35xviLatin AlphabetSymbol Description Dimensionality IntroducedK∗JY ∈Y+κ∈K conditional diffusive flux associated withscalar Y and dummy condition κ∗mY/(Ltκ) Eq. 7.35~jc diffusive flux of progress variable c m/(L2t) Eq. 3.13~j1 diffusive flux of the field 1, which is iden-tically~0m/(L2t) Eq. 7.2~jYc diffusive flux of scalar field Yc used to defineprogress variable cm/(L2t) Eq. 3.8~jZ diffusive flux of mixture fraction m/(L2t) Eq. 3.4~jα∈S diffusive flux of species α m/(L2t) Eq. 2.5~jψ∈{u,h} diffusive flux of u or h, which is equivalentto the heat flux ~qm/t3 Eq. 7.3~jordα∈S ordinary diffusive flux of species α m/(L2t) Eq. 2.18~jPα∈S pressure diffusive flux of species α m/(L2t) Eq. 2.18~j subφ∈Z sub-filter mass flux of scalar φ associatedwith filter Fm/(L2t) Eq. 4.13~jTα∈S thermal diffusive flux of species α m/(L2t) Eq. 2.18K set of properties used as conditioning vari-ables− Eq. 4.46Ktradset of traditional CSE conditioning variables − Sec. 5.2.1KM˜tradset of filtered moments of traditional CSEconditioning variables− Sec. 5.2.1K conditional domain (set of all possible valuesof conditioning variables)− Eq. 4.46K discrete conditional domain (a set sets ofpoints (cells) in conditioning space)− Sec. 5.2.1Ktrad conditional domain in the traditional CSEalgorithm (set of all possible values of tra-ditional CSE conditioning variables)− Sec. 5.2.1Ktrad discretized version of the conditional do-main in traditional CSE (a set of setsof points (cells) in traditional conditioningspace)− Sec. 5.2.1xviiLatin AlphabetSymbol Description Dimensionality Introducedk mass-averaged specific kinetic energy ofmixture,~v·~v2L2/t2 Sec. 2.2.4k sub sub-filter (turbulent) kinetic energy asso-ciated with filter FL2/t2 Eq. 4.16kℵ∈R rate coefficient for reaction ℵ see Eq. 2.43 Eq. 2.43L set of all elements − Eq. 2.42L representative length scale L Eq. 3.1L F̂F Leonard stress tensor associated with filtersF̂ and Fm/(Lt2) Eq. 6.8L F̂‡ traceless component of Leonard stress ten-sor associated with filters F̂ and Fm/(Lt2) Eq. 6.20M mass-averaged molar mass of mixture m/n Eq. 2.20Mα∈S molar mass of species α m/n Eq. 2.20m total mass of mixture m Eq. 2.3nensnumber of CSE ensembles used in simula-tion1 Sec. 5.1.1nprocsnumber of processors used in parallel execu-tion1 Sec. 5.1.1nℵ∈R modified Arrhenius temperature exponentfor reaction ℵ1 Eq. 2.44P total thermodynamic pressure of mixture m/(Lt2) Sec. 2.1.1PK∗s˜ubSub-filter probability density function ofconditions K associated with the Favre ver-sion of filter F (i.e., F˜)1 Eq. 4.46PK∗tradsub˜Sub-filter probability density function ofconditions Ktradassociated with the Favreversion of filter F (i.e., F˜)1 Sec. 5.2.1PrF turbulent Prandtl number associated withfilter F1 Eq. 4.24Q local energy source due to radiation m/(Lt3) Eq. 2.11~q heat flux vector m/t3 Eq. 2.11xviiiLatin AlphabetSymbol Description Dimensionality Introduced~q cond conductive heat flux vector m/t3 Eq. 2.32~q Duf Dufour heat flux vector m/t3 Eq. 2.32~q spec heat flux vector associated with advectionby species diffusionm/t3 Eq. 2.32~q subφ∈{e,u,h} sub-filter heat flux of specific energy field φassociated with filter Fm/t3 Eq. 4.31R set of all chemical reactions − Eq. 2.41R universal gas constant, mL2/(t2nT ) Eq. 2.348.314 598 J/(molK)Re Reynolds number 1 Eq. 3.1Retturbulent Reynolds number 1 Eq. 4.1S set of species names − Sec. 2.1∣∣∣SF∣∣∣ resolved overall strain rate at scale of filterF1/t Eq. 6.4SFF resolved strain rate at scale of filter F 1/t Eq. 6.6SF‡ resolved shear rate at scale of filter F 1/t Eq. 6.5ScF turbulent Schmidt number associated withfilter F1 Eq. 4.24T absolute temperature T Sec. 2.1.1T time domain (set of all possible times) − Sec. 2.2t time t Sec. 2.2U total internal energy of mixture mL2/t2 Sec. 2.1.2u mass-averaged specific internal energy ofmixtureL2/t2 Sec. 2.1.1V total volume of mixture L3 Sec. 2.1.2~v mass-averaged velocity of mixture L/t Eq. 2.4∆~vφ∈Z drift velocity (velocity relative to mixtureaverage) of scalar φL/t Eq. 3.25X set of all extensive properties − Eq. 2.3xixGreek AlphabetSymbol Description Dimensionality IntroducedX spatial domain (set of all possible positions) − Sec. 2.2X discrete spatial domain (set of sets of points(cells))− Eq. 4.37Xα∈S mole fraction of species α 1 Eq. 2.21~x position L Sec. 2.2Y set of species mass fractions − Sec. 2.1.2Y∗ set of scalars with mass fraction-like trans-port equations− Eq. 7.1Yc scalar field used to define progress variablec1 Eq. 3.8Y prodc Yc of pure reactants 1 Eq. 3.12Y reactc Yc of pure products 1 Eq. 3.12Y ccCO2 carbon dioxide mass fraction which wouldbe observed at complete combustion1 Eq. 8.3Yα∈S mass fraction of species α 1 Sec. 2.1.1Z set composed of species mass fractions, mix-ture fraction, and progress variable− Eq. 3.24Z mixture fraction 1 Eq. 3.2Zstoichmixture fraction at stoichiometric air/fuelratio1 Eq. 3.6Z∈Lα∈S number of atoms of element  per moleculeof species α1 Eq. 2.42Greek AlphabetSymbol Description Dimensionality IntroducedαF turbulent or eddy thermal diffusivity as-sociated with filter FL2/t Eq. 4.22∆F filter width associated with filter F L Eq. 4.37ζ overall dynamic dilatational viscosity ofmixturem/(Lt) Eq. 2.14xxGreek AlphabetSymbol Description Dimensionality Introducedη wavenumber 1/L Eq. 2.37κPoverall Planck mean absorption coefficientof mixture1 Eq. 2.40κη overall spectral (at wavenumber, η) absorp-tion coefficient of mixture1 Eq. 2.37λ overall thermal conductivity of mixture mL/(t3T ) Eq. 2.33λAFair-fuel equivalence ratio 1 Eq. 3.7µ overall dynamic shear viscosity of mixture m/(Lt) Eq. 2.14µF dynamic turbulent or eddy viscosity as-sociated with filter FL2/t Eq. 4.20ν overall kinematic shear viscosity of mixture L2/t Eq. 3.1νF kinematic turbulent or eddy viscosity as-sociated with filter FL2/t Eq. 4.20νprodα∈Sℵ∈R stoichiometric coefficient representing num-ber of atoms of α which participate in eachinstance of reaction ℵ as a product1 Eq. 2.41νreactα∈Sℵ∈R stoichiometric coefficient representing num-ber of atoms of α which participate in eachinstance of reaction ℵ as a reactant1 Eq. 2.41ξ˙ℵ∈R extent rate of reaction ℵ n/(L3t) Eq. 2.43ρ total density of mixture m/L3 Sec. 2.1.1∑sum operator − −σ Stefan-Boltzmann constant, m/(t3T 4) Eq. 2.405.670 367× 10−8 W/(m2K4)σF Cauchy stress tensor m/(Lt2) Eq. 2.9τF viscous stress tensor m/(Lt2) Eq. 2.9τ subF sub-filter stress tensor associated with filterFm/(Lt2) Sec. 4.1.3φFAfuel-air equivalence ratio 1 Eq. 3.6χφ∈Zψ∈Z scalar dissipation associated with scalars φand ψ1/t Sec. 3.3.3xxiGreek AlphabetSymbol Description Dimensionality IntroducedΨ generic state − Sec. 2.1ΨˆCSE CSE state basis − Eq. 5.7Ψˆchem chemical state basis − Sec. 2.1.1Ψkin kinematic state − Sec. 2.1Ψˆkin kinematic state basis − Sec. 2.1.1Ψˆphys physical state basis − Sec. 2.1.1Ψˆrchem reduced chemical state basis − Sec. 4.2Ψˆrtherm reduced thermodynamic state basis − Eq. 4.40Ψthermo thermodynamic state − Sec. 2.1Ψˆthermo thermodynamic state basis − Sec. 2.1.1Ψtot total state − Sec. 2.1Ψˆtot total state basis − Sec. 2.1.1Ω solid angle 1 Eq. 2.38Ωˆ direction vector normal to solid angle ele-ment dΩ1 Eq. 2.38ω˙c rate of production of progress variable c 1/t Eq. 3.14ω˙Ktradrate of production of traditional CSE condi-tioning variables due to chemical reactions(a set of fields)− Eq. 5.14ω˙KM˜tradrate of production of variance of traditionalCSE conditioning variables due to reactions(a set of fields)− Eq. 5.15ω˙1 rate of production of the field 1, which isidentically 01/t Eq. 7.2ω˙Yc rate of production of scalar Yc used to defineprogress variable c1/t Eq. 3.8ω˙α∈S rate of production of species α due to chem-ical reactions1/t Eq. 2.5ω˙ΨˆCSE rate of production of CSE state basis due toreactions (a set of fields)− Eq. 5.12xxiiGreek AlphabetSymbol Description Dimensionality Introducedω˙Ψˆrchem rate of production of reduced chemical statedue to reactions (a set of fields)− Eq. 5.16ω˙ψ∈KM˜ rate of production of filtered moment ψ 1/t Eq. 5.31ω˙ψ∈{u,h} source of [absolute] internal energy or en-thalpy, which is identically zero1/t Eq. 7.3xxiiiList of AbbreviationsCFD Computational Fluid DynamicsCMC Conditional Moment ClosureCPU Central Processing UnitCSE Conditional Source-term EstimationCSE-GCV Conditional Source-term Estimation with GeometricConditioning VariablesDNS Direct Numerical SimulationFGM Flamelet Generated ManifoldFMFM Filtered Multidimensional Flamelet ManifoldLES Large Eddy SimulationLSQR Least Squares QR DecompositionMILD Moderate Inert Low DilutionMFM Multidimensional Flamelet ManifoldPDF Probability Density FunctionRANS Reynolds Averaged Navier StokesREDIM Reaction-Diffusion ManifoldSFS Sub-Filter StressTGLDM Trajectory-Generated Low Dimensional ManifoldUCS Uniform Conditional StateWALE Wall-Adapting Local Eddy-viscosityxxivPart IIntroduction and Literature Review1Chapter 1IntroductionThis chapter introduces the context which motivated this work, the objectives of the work,and the overall structure of this document.1.1 ContextCombustion technology has been a part of human society for millennia, holds an integralrole in the present day, and is likely to retain this prominent role for the foreseeable future.1.1.1 Historical Origins of CombustionArchaeological evidence suggests that human ancestors in Africa used fire approximately onemillion years ago [1]. In the ensuing millennia, humankind has developed countless directuses for the heat and light from flames: intimidating predators, cooking food, heating homes,lighting streets, and myriad others. During the industrial revolution, newly-developed heatengines made it possible to use the heat from flames indirectly, converting it into work topower industrial machines. The descendants of these industrial machines are an integralpart of the infrastructure supporting modern developed societies, andfor the most partcontinue to be powered by combustion-driven heat engines.1.1.2 Combustion in the Present DayAs illustrated in Table 1.1, the present-day global Total Primary Energy Supply is phenom-enally large: 12 150 Mtoe (million tonnes of oil equivalent) for the year 2016. Over 90 % ofthis energy is extracted from fossil fuels and waste, typically via combustion. This extensivereliance on combustion is not without its drawbacks; many of the byproducts of combustionhave harmful impacts on human health or the environment [2]:• carbon dioxide contributes to climate change through the greenhouse effect,• carbon monoxide is poisonous even in low concentrations,• oxides of nitrogen (nitric oxide and nitrogen dioxide) are the main precursors to smogand acid rain, and• unburned hydrocarbons and soot are carcinogenic and can cause respiratory damage.Despite the potential impacts on the global climate and human health, the relatively lowcost of combustion technology and fuels has prevented alternatives from making seriousinroads on the global energy supply. Against this backdrop, governments have generallychosen to introduce legislation limiting and/or taxing emissions. These regulations havehelped incentivize the development of cleaner and more efficient combustion technology.21.2. ObjectivesTable 1.1: Global Primary Energy Supply in 2016 [3]Source Energy [Mtoe] FractionOil 4326.1 31.7 %Coal/Peat 3834.8 28.1 %Natural Gas 2947.8 21.6 %Biofuels and Waste 1323.8 9.7 %Nuclear 668.7 4.9 %Hydro 341.2 2.5 %Other 204.7 1.5 %Total 13 647.0 100.0 %1.1.3 Future Trajectory of Combustion TechnologyDevelopment of combustion technology has historically relied on physical testing and simplemathematical modelling. With the relatively recent development of computers and theirexponential growth in power, it has become possible to complement physical tests with nu-merical simulations. In applications where producing and testing prototypes is expensiveand time-consuming (such as jet engine design), simulation is now less expensive than phys-ical prototyping. Replacing a program of physical prototyping with a program of physicalprototyping and simulation makes it possible for a design team with a fixed budget to ex-plore a larger parameter space than prototyping alone would, enhancing the team's abilityto improve efficiency and reduce emissions. Further development of combustion simulationtools can be expected to improve the usefulness of simulation as a design tool.1.2 ObjectivesThe objectives of this project can be sorted according to their level of detail:• the highest level objectives are those of interest to society and industrial partners,• the intermediate-level objectives are those of interest to researchers in the field ofcombustion simulation, and• the lowest-level objectives are those of interest to those involved in carrying out orreviewing this work.The high- and intermediate-level objectives are presented here, while the low-level objectivesare presented together with the research projects they pertain to.1.2.1 High-LevelThe ultimate goal of this project is to improve the quality (applicability, computationalefficiency, and/or accuracy) of combustion simulation tools, thereby facilitating the develop-ment of improved combustion technology. This objective is beneficial to industrial partnerswhose business involves combustion development, as it helps ensure that they can develop31.3. Structure of this Documentproducts respecting exacting emissions standards. It is also beneficial to society as a whole,as the introduction of cleaner and more efficient combustion technology mitigates the nega-tive impacts of combustion byproducts on human health and the environment.1.2.2 Intermediate-LevelLooking below the high level, this project explores how generalizing and/or relaxing thedefinition of conditional filtering (a technique introduced in Section 4.3 and defined fully inSection B.1.3) can produce novel turbulence and combustion models. The research chaptersare most obviously united in this theme when described as follows:Chapter 5: what happens if position is treated as a conditioning variable?Chapter 6: what happens if the conditioning variable position is replaced with the con-ditioning variable mixture fraction?Chapters 7 and 8: what happens if one assumes that conditioned fields are independentof position?The detailed (low-level) objectives of the four research projects are presented within theirrespective chapters.1.3 Structure of this DocumentAs described above, the core of this document is four chapters (58) describing four researchprojects. These chapters are united as Part II, Research Contributions, and supported by• a preliminary part, Part I, which includes this introduction and background materialcommon to the four research chapters, and• a concluding part, Part III, which includes a brief retrospective summary of eachresearch project and a list of works cited.In the interests of making this document accessible to the widest audience possible, thebackground material which completes Part I is presented over three chapters.• Chapter 2 covers the fundamental physics of reacting systems (equations of state,governing equations, and constitutive equations). It is intended to be an accessi-ble starting point for academically-minded mechanical engineers without expertise inthermofluids.• Chapter 3 introduces the dynamic phenomena which emerge from the fundamentalgoverning equations in real reacting systems (turbulence, global reactions, and depen-dence on premixedness). It is intended to be an accessible starting point for thosewith expertise in thermofluids, but not in turbulence and combustion.• Chapter 4 introduces the simulation techniques which are applied to the simulation ofturbulent combustion (including turbulence models, chemistry models, and turbulence-chemistry interaction models). It is intended to be an accessible starting point to41.3. Structure of this Documentthose with expertise in turbulence and combustion, but not in turbulent combustionmodelling.Additional, project-specific, background information is also presented at the outset of eachresearch chapter.5Chapter 2Fundamental Physics of ReactingSystemsCombustion is a chemical process in which a hydrocarbon fuel reacts with an oxygen-containing oxidizer1. Combustion systems can be viewed as a special case of the moregeneral reacting systems; it is therefore instructive to examine the physics of reacting sys-tems. The properties of a reacting system are conceptually grouped as the state, and theevolution this state is described by the combination of governing and constitutive equations.2.1 StateStates, Ψ, are abstract objects which provide complete information about instantaneous,local properties. Within this work, states are specifically defined as sets of properties. Threevarieties of state are defined:1. the total state, Ψtot, is defined as the set of all properties (observable attributes ofthe system),2. the thermodynamic state, Ψthermo, is defined as the set of all thermodynamic prop-erties (properties independent of the reference frame),3. the kinematic state, Ψkin, is defined as the set of all kinematic properties (propertiesassociated with motion).Although the three varieties of state are all uncountably infinite sets, they are also redundant:for each, there exists some subset of properties which, together, define the state, such thatall other properties can be evaluated from this subset by applying so-called state functions.2.1.1 Tracking StateThis work is concerned with single-phase reacting continua which1. contain no non-thermal radiation2,2. are small relative to astronomical length scales2, and3. contain a mixture of species from the set of species names S.1Reactions between metal powders and oxidizers can also be considered combustion, but are outsidethe scope of this work.2Items 1 and 2 together imply that the radiation field is uniquely determined by the temperature field,and therefore need not be tracked.62.1. StateFor such systems, any set of (|S| + 7) independently variable fields is a basis, Ψˆtot, whichdefines the total state Ψtot. This total state basis can be decomposed into1. the kinematic state basis, Ψˆkin, with 6 independent components (typically chosenas the 3 independent components of the position, ~x, and velocity, ~v), which completelydefines the kinematic state; and2. the thermodynamic state basis, Ψˆthermo, with (|S|+ 1) independent components,which completely defines the thermodynamic state and can be further decomposedinto:(a) the chemical state basis, Ψˆchem, with (|S| − 1) independent components (typ-ically chosen as any (|S| − 1) species mass fractions, Yα∈S 3), and(b) the physical state basis, Ψˆphys, with 2 independent components, typically cho-sen as• pressure, P , or density, ρ, and• temperature, T , internal energy, u, or enthalpy, h.This decomposition can be summarized asΨˆtot = Ψˆkin⋃Ψˆchem⋃Ψˆphys´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Ψˆthermo. (2.1)2.1.2 Evaluating State FunctionsIf the values of all properties in some state basis are known, any property ψ in the cor-responding state can be evaluated using a so-called state function. Many state functionsare defined in terms of other properties, and are therefore straightforward to evaluate giventhose properties, but several thermodynamic state functions can only be evaluated usingmodels (such as the ideal gas law or empirical fits), which can become quite involved.For single-phase systems, thermodynamic state functions are typically formulated withthe inherently-intensive propertiestemperature, T ; pressure, P ; and the set of all speciesmass fractions Y ≡ {Yα : α ∈ S}as their arguments:∀ψ ∈ Ψthermo : ψ = ψ( T, P®ˆΨphys, Y®Ψˆchem) (2.2)where the leading statement ∀ψ ∈ Ψthermo : indicates the domain of discourse of the main(centred) statement by defining the free variable ψ as any element of Ψthermo 4. This makesit extremely convenient to track the thermodynamic state using (|S| − 1) mass fractions,temperature, and pressure (as suggested in Section 2.1.1). Property models with the formof 2.2 can be used with different independent variables; property evaluation then becomesa two-stage process:3Throughout this work, terms of the form Yα∈S are taken as shorthand for Yα where α ∈ S4Symbol-by-symbol interpretation: ∀®for allψ ∈®inΨthermo :®it is true that.72.2. Governing Equations1. Invert a system of equations of state (in the form of Equation 2.2) for the naturally-intensive properties T , P , Y.2. Apply other equations of state (in the form of Equation 2.2) with T , P , Y as argumentsto evaluate any other desired properties.For the special case of mass-normalized extensive properties and ideal mixtures inwhich there are no inter-species interactions (assumed throughout this work), Equation 2.2can be simplified to∀Ψ ∈(X ∩Ψthermo):∃ψ : ψ ≡ Ψm=∑α∈SYαψα(T, P ) (2.3)where X is the set of all extensive properties (total volume V , total internal energy U , totalenthalpy H. . . ), m is the mass, and ψα∈S(T, P ) represents the ψ of pure species α at thesame T and P as the mixture, implying that it is most straightforward to track the physicalstate using T and P .2.2 Governing EquationsIn the Eulerian approach to continuum mechanics, the state of the entire system at sometime t in the time domain T can be represented using (|S|+ 7) fields ψ (~x, t)one for each ψin the total state basis Ψˆtotwhich are each defined at all positions ~x in the spatial domainX 5. Because the position is a field argument, 3 of these fieldsx (~x, t), y (~x, t), and z (~x, t)are trivial. The evolution of the remaining (|S|+4) fields can be deduced by solving (|S|+4)transport equations for rates of change. The relevant equations are the Mass Balance, theSpecies Balances, the Momentum Balance, and the Energy Balance.2.2.1 Mass BalanceThe mass balance is a single scalar equation [4]:∂∂tρ®LocalChange+ ~∇ · (ρ~v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= 0. (2.4)where ρ is the density (dimensionalitym/L3) and ~v is the the velocity vector (dimensionalityL/t).5In the interests of brevity, neither functional dependence on (~x, t) nor the constraint (~x, t) ∈ (X , T ) willbe explicitly annotated except where it is necessary for clarity.82.2. Governing Equations2.2.2 Species BalancesThe species balance equations are written as:∀α ∈ S :ρDDtYαucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρYα)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vYα)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ ·~jYα´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Diffusion+ ρω˙Yα ,®Production(2.5)where• ~jYα∈Y are the diffusive flux vectors (with dimensionality m/(L2t)), and• ω˙Yα∈Y are the rates of production (with dimensionality 1/(L3t)).Only (|S| − 1) of the |S| scalar equations are independent; the last mass fraction, diffusiveflux vector, and rate of production are defined by the constraints∑α∈SYα = 1 (2.6)∑α∈S~jYα = ~0 (2.7)∑α∈Sω˙Yα = 0. (2.8)2.2.3 Momentum BalanceThe momentum balance is a vector equation with 3 independent components [4]:ρDDt~vucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρ~v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~v~v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection=~∇ · σFucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright~∇ · τF®ViscousForces−~∇P®Pressure´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces+ ρ~g.®BodyForce(2.9)The symbols σF and τF represent the Cauchy and viscous stress tensors, respectively (di-mensionality m/(Lt2)); the annotation F indicates that these tensors are symmetric butnot (in general) traceless6. The two are related by [5]σF®TotalStress= τF®ViscousStress+ −P I◦,®ThermodynamicPressure(2.10)6The notation of tensor decomposition is described more fully in Section A.2.2.92.3. Constitutive Equations (Transport Models)where I◦ represents the rank-two identity tensor, and the annotation ◦ indicates that thistensor is isotropic6.2.2.4 Energy BalanceThe Energy Balance is a single scalar equation. It can be written in terms of the specifictotal (internal plus kinetic) energy e = u+ k (dimensionality L2/t2) [4],ρDDteucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρe)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~ve)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ · ~q´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶HeatFlux+ Q®Radiation+~∇ · (σF · ~v)ucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright~∇ · (τF · ~v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ViscousForces−~∇ · (P~v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Pressure´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces+ ρ~g · ~v®BodyForce, (2.11)the specific internal energy u (dimensionality L2/t2),ρDDtuucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρu)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vu)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ · ~q´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶HeatFlux+ Q®Radiation+σF : ~∇~vucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyrightτF : ~∇~v´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ViscousForces−P ~∇ · ~v´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Pressure´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces, (2.12)or the specific enthalpy h (dimensionality L2/t2),ρDDthucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρh)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vh)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ · ~q´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶HeatFlux+ Q®Radiation+σF : ~∇~v + P ~∇ · ~vucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyrightτF : ~∇~v+DPDt´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces(2.13)where• ~q is the heat flux vector (dimensionality m/t3), and• Q is the local source due to radiation (dimensionality m/(Lt3)).2.3 Constitutive Equations (Transport Models)The transport equations above feature several transport terms which require closure, specif-ically:102.3. Constitutive Equations (Transport Models)• τF, the viscous stress tensor,• ~jYα∈Y, the diffusive flux vectors (of which there are |S|, although only (|S| − 1) areindependent),• ~q, the heat flux vector,• Q, the radiation long-range transfer term, and• ω˙Yα∈Y, the species source terms (of which there are |S|, although only (|S| − 1) areindependent).These terms can be closed by any of a variety of transport models, depending on the detailrequired.2.3.1 Viscous Stress TensorThe viscous stress is typically modelled using the Newtonian model [5]τF = 2µ(~∇~v)‡+ 3ζ(~∇~v)◦(2.14)= 2µ(~∇~v)‡+ ζ(~∇ · ~v)I◦ (2.15)where•(~∇~v)‡is the rate-of-shear tensor, in which the symbol ‡ denotes the operation ofextracting the symmetric traceless component of a tensor6,•(~∇~v)◦is the rate-of-dilatation tensor, in which the symbol ◦ denotes the operationof extracting the isotropic component of a tensor6,• µ is the dynamic shear viscosity (often shortened to viscosity), with dimensionalitym/(Lt), which describes the fluid's resistance to pure shear, must be non-negative,and can be evaluated as a function of the thermodynamic state basis; and• ζ is the dynamic dilatational viscosity, with dimensionality m/(Lt), which describesthe fluid's resistance to pure dilatation, must be non-negative, and can be evaluatedas a function of the thermodynamic state basis.The Newtonian model can also be formulated asτF = 2µ(~∇~v)F+[ζ − 23µ](~∇ · ~v)I◦ (2.16)The coefficient(ζ − 23µ)is sometimes taken as a root coefficient from which ζ is defined;sources which adopt this formulation may use different naming conventions (in conflict withthose herein) for the dilatational viscosity.112.3. Constitutive Equations (Transport Models)2.3.2 Diffusive Flux VectorsIn general, the diffusive flux can be written as∀α ∈ S : ~jYα = −∑ψ∈DρDψYα ~∇ψ, (2.17)where D is the set of fields whose gradients drive diffusion (with elements of arbitrarydimensionality) and Dψ∈DYα∈Y are kinematic coefficients (with dimensionality L2/(tψ) 7) whichcan be evaluated as functions of the state basis. The full, multicomponent diffusion modelprovides the most detailed results, but in many systems it is appropriate to adopt a simplifiedmodel.Full, Multicomponent DiffusionIn the most detailed model for the diffusive flux vectors, each~jYα∈Y is the sum of contribu-tions due to three effects [6]:∀α ∈ S : ~jYα = ~jordYα®OrdinaryDiffusion+ ~jTYα®ThermalDiffusion+ ~jPYα®PressureDiffusion(2.18)These effects each capture dependence on different gradients.1. Ordinary diffusion captures dependence on mass fraction gradients:∀α ∈ S : ~jordYα = −ρ∑β∈SDYβYα ~∇Yβ (2.19)∀α, β ∈ S : DYβYα ≡∑γ∈SDmultαγMMγ(δβγ − MYγMβ)(2.20). . . where• DYβ∈YYα∈Y are modified multicomponent mass diffusivities, with dimensionality L2/t,• Dmultα∈Sβ∈S are (raw) multicomponent mass diffusivities, with dimensionality L2/t,which can be evaluated as functions of the thermodynamic state basis,• Mα∈S is the molar mass of species α,• M is the mixture-average molar mass, defined asM ≡∑α∈SXαMα, (2.21)(where Xα∈S is the mole fraction of species α) and• δβ∈S γ∈S is the Kronecker delta.7The dimensionality of the driving force ψ ∈ D is not constrained, and therefore cannot be expressed inbase dimensions122.3. Constitutive Equations (Transport Models)When the chemical state basis is represented using mole fractions rather than massfractions, it is convenient to instead view ordinary diffusion as a response to molefraction gradients, and to evaluate it as∀α ∈ S : ~jordYα = −ρ∑β∈SDmultαβ ~∇Xβ, (2.22)which is equivalent to the combination of Equations 2.19 and 2.20.2. Thermal diffusion (the Soret effect) captures dependence on temperature gradients:∀α ∈ S : ~jTYα = −ραDTα~∇TT(2.23). . . where DTα∈S is the thermodiffusivity (kinematic thermodiffusion coefficient), withdimensionality L2/t, which can be evaluated as a function of the thermodynamic statebasis.3. Pressure diffusion captures dependence on pressure gradients:∀α ∈ S : ~jPYα = −ρ∑β∈SDmultαβ (Xβ − Yβ)~∇PP(2.24). . . where Dmultα∈Sβ∈S has the same meaning as in Equation 2.19. The term (Xβ∈S − Yβ∈S)is positive when species β is lighter than the mixture average, and negative when itis heavier; pressure diffusion can therefore be leveraged to separate species based ontheir molar masses, for example in centrifuges.This model can be cast in the general form of Equation 2.17 by defining the set of all fieldswhose gradients drive diffusion asD ≡ Y⋃{T, P} (2.25)and the corresponding coefficients as∀α, β ∈ S : DXβYα ≡∑γ∈SDmultαγMMγ(δβγ − MYγMβ)(2.20 repeated)∀α ∈ S : DTYα ≡DTαT(2.26)∀α ∈ S : DPYα ≡∑β∈SDmultαβXβ − YβP. (2.27)Simplified DiffusionIn combustion processes, it is often acceptable to simplify the ordinary diffusion and neglectthe pressure diffusion, giving the simpler model∀α ∈ S : ~jYα = −ρDavα ~∇Yα´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶OrdinaryDiffusion−ρDTα~∇TT´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ThermalDiffusion(2.28). . . where132.3. Constitutive Equations (Transport Models)• Davα∈S is an average diffusion coefficient for species α diffusing into the mixture, andcan be evaluated as a function of the thermodynamic state basis, and• DTα∈S has the same meaning as in Equation 2.23.This model can be cast in the general form of Equation 2.17 by defining the set of all fieldswhose gradients drive diffusion asD ≡ Y⋃{T} (2.29)and the corresponding coefficients as∀α, β ∈ S : DYβYα ≡ δαβDavα (2.30)∀α ∈ S : DTYα ≡DTαT. (2.31)2.3.3 Heat Flux VectorThe heat flux vector ~q is the sum of contributions due to three effects [4]:~q = ~q cond®Conduction+ ~q Duf®DufourEffect+ ~q spec®Advection byDiffusion of Species(2.32)These effects can be evaluated as follows:1. Conduction (associated with temperature gradients)~q cond = −λ~∇T (2.33). . . where λ is the thermal conductivity, with dimensionality mL/(t3T ), which can beevaluated as a function of the thermodynamic state basis,2. The Dufour effect (associated with concentration gradients)~q Duf =RTM∑α∈S∑β∈S(DTαDmultαβ[YβYα~jYα −~jYβ])(2.34). . . where• R is the universal gas constant, 8.314 598 J/(molK),• M is the mixture average molar mass, defined by Equation 2.21,• Dmultα∈Sβ∈S and DTα∈S have the same meanings as in Equations 2.20 and 2.23 re-spectively,and142.3. Constitutive Equations (Transport Models)3. Advection by diffusion of species~q spec =∑α∈Shα~jYα (2.35)where hα∈S is the specific enthalpy of species α.In combustion processes, it is often acceptable to neglect the Dufour Effect, giving~q = −λ~∇T +∑α∈Shα~jYα . (2.36)2.3.4 Radiation Source TermFor a medium which absorbs and emits radiation but does not scatter it (such as a singlegas phase), the radiation source term can be expressed as [7]Q =∫ ∞0κη(η) · (Gη(η)− 4piBη(η)) dη (2.37)where• the integration is over all wavenumbers, η• κη is the spectral absorption coefficient, a dimensionless scalar which can be evaluatedas a function of the thermodynamic state basis8,• Bη is the spectral blackbody intensity, with dimensionality m/(L2t3), which can beevaluated as a function of the thermodynamic state basis8• Gη is the spectral incident radiation function, with dimensionality m/(L2t3), definedbyGη(η) =∫4piIη(η, Ωˆ) dΩ (2.38)where Iη is the spectral radiative intensity, with dimensionalitym/(L2t3), representingthe radiative state, Ω is a solid angle, and Ωˆ is the direction vector normal to dΩ.Solving this system requires knowledge of Iη, which can be deduced by solving the quasi-steady-state radiative transfer equation for a non-scattering medium [8]Ωˆ · ~∇Iη(η, Ωˆ) = κη(η) ·(Bη(η)− Iη(η, Ωˆ))(2.39)In practical combustion simulation, the dramatically simpler optically thin approximationis generally preferred:Q = −κPσT 4 (2.40)where8The associated models require consideration of species' atomic transitions, and are thus outside of thisthe scope of this document; see [8] for further information.152.3. Constitutive Equations (Transport Models)• κPis the Planck-mean absorption coefficient, which can be evaluated as a function ofthe thermodynamic state basis, and• σ is the Stefan-Boltzmann constant, 5.670 367× 10−8 W/(m2K4).The simplification assumes that the system emits but does not absorb or scatter radiation.This approximation performs well for laboratory flames, but leads to an over-prediction ofradiative losses (and thus an under-prediction of flame temperature) in larger flames [7].2.3.5 Species Source TermThe species source terms ω˙α correspond to the effects of chemical reactions from the set Rof all reactions ℵ. The chemical equations which make up the full mechanism can be writtenas∀ℵ ∈ R :∑α∈S(νreactαℵ)α→∑α∈S(νprodαℵ)α (2.41). . . where νreactα∈Sℵ∈R and νprodα∈Sℵ∈R are (dimensionless) coefficients representing the numberof atoms of species α which participate in each instance of reaction ℵ (as reactants andproducts, respectively)9. Chemical reactions conserve each element from the set L of allelements ; this constraint (Conservation of Elements) can be written as:∀ ∈ L.∀ℵ ∈ R : ∑α∈SZα(νprodαℵ − νreactαℵ)= 0 (2.42)where Z∈Lα∈S is a dimensionless coefficient representing the number of atoms of element per molecule of species α 10.Each basis reaction can be assigned an extent rate ξ˙ℵ∈R, with dimensionality n/(L3t). For elementary reactions (effectively, reactions which could occur as a result of a singlemolecular collision), these rates can be evaluated from the Law of Mass Action as∀ℵ ∈ R : ξ˙ℵ = kℵ(T ) ·∏α[ρYαMα]νreactαℵ(2.43)where kℵ∈R(T ) is the reaction rate coefficient, with dimensionalityn(1−∑α∈S νreactαℵ )L(3−3∑α∈S νreactαℵ )t.The reaction rate coefficient can be evaluated from the modified Arrhenius equation [9]∀ℵ ∈ R : kℵ(T ) = Aℵ(T to power nℵ)ucurlyT (nℵ) exp(−EaℵRT)(2.44)where9For example, if reaction ℵ corresponds to the chemical equation (C + 2O→ CO2) then(νreactC,ℵ = 1),(νreactO,ℵ = 2),(νprodCO2,ℵ = 1), and all other νreactα∈Sℵ∈R and νprodα∈Sℵ∈R are zero.10For example, CO2has (ZC,CO2 = 1), (ZO,CO2 = 2) and (Z,CO2 = 0) for all other  ∈ L.162.3. Constitutive Equations (Transport Models)• Aℵ∈R is the pre-exponential factor, a constant with dimensionalityn(1−∑ℵ∈R νreactαℵ )L(3−3∑ℵ∈R νreactαℵ )tT (nℵ),• nℵ∈R is the (dimensionless) temperature exponent,• Eaℵ∈R is the activation energy of reaction ℵ, with dimensionality mL2/(t2n) (whichmust be non-negative), and• R is the universal gas constant, 8.314 598 J/(mol ·K).The extent rates ξ˙ℵ∈R can therefore be calculated with knowledge of the state basis and ofconstants νreactα∈Sℵ∈R, νprodα∈Sℵ∈R, Eaℵ∈R, nℵ∈R, and Aℵ∈R, which are tabulated for any completechemical mechanism. With the individual reaction rates known, the species source termscan be evaluated as:∀α ∈ S : ω˙Yα =∑ℵ∈Rξ˙ℵ(νprodαℵ − νreactαℵ)Mα. (2.45)17Chapter 3Emergent Behaviour in CombustionSystemsIn real-world combustion processes, the mathematical system laid out in Chapter 2 can giverise to dynamic phenomena which are more properly attributed to the interactions betweenand within the underlying equations than to individual terms in equations. The phenomenarelevant in this work are turbulence, global effective chemical reactions, and a dependenceon the degree to which fuel and oxidizer are mixed as they enter the combustion system(premixedness), and are collectively labelled as emergent behaviour.3.1 Turbulence3.1.1 Flow RegimesThe momentum balance (Equation 2.9) can be viewed as a balance between inertial forces,pressure forces, viscous forces, and gravitational forces. The character of the solutions tothis equation varies starkly depending on the relative strengths of the inertial and viscousforces, which can be quantified using the (dimensionless) Reynolds NumberRe ≡ Inertial ForcesViscous Forces=ρ |~v|Lµ=|~v|Lν(3.1)where L is some representative length scale, ν is the kinematic viscosity, and all propertiesare evaluated at some representative condition. When viscous forces dominate over inertialforces (low Re), the resulting velocity profiles are smooth in space, time, and response toinitial conditions; such flow fields are called laminar. At the other extreme (inertial forcesdominate over viscous; high Re), the flow has a completely different character: velocityprofiles exhibit strong dependence on space, time, and initial conditions, and are calledturbulent. In between these regimes there also exists a small transitional regime whichexhibits some characteristics of both laminar and turbulent behaviour. Figure 3.1 illustrateshow all three flow regimes appear within the exhaust plume of an ordinary candle: laminarflow in and immediately above the flame, turbulent flow far from the flame, and transitionalflow in between. Many industrially-interesting combustion systems exhibit turbulent flow.3.1.2 The Multi-Scale Chaos ProblemAlthough, in principle, the momentum equation presents a complete theoretical descriptionof turbulence, in practice this description is intractable. The complication which ultimatelythwarts the momentum equation is that turbulence is a chaotic, multi-scale phenomenon [10]:flow details spanning an enormous range of length and time scales interact with one another,183.1. TurbulenceFigure 3.1: Visualization of density gradients in a candle's exhaust plume, showing laminar,transitional, and turbulent flow.c© 2009 Gary Settles (https://upload.wikimedia.org/wikipedia/commons/0/03/Laminar-turbulent_transition.jpg), used under CreativeCommons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0).193.1. Turbulenceand this interaction is such that small variations at any scale quickly give rise to dramaticvariations across all scales. For human-scale flows, the smallest scales of turbulence are oftensmaller than those an experimenter can measure or control; a set of initial and boundaryconditions which appears completely defined at the human scale is, in fact, incomplete. Asa result of this incompleteness, and the multi-scale chaotic dependence on the un-resolveddetail,• the momentum equation is rendered useless (its inputs cannot be fully resolved, andguessing at the un-resolved detail will almost certainly give predictions which differdramatically from reality), and• turbulent flows appear (at the human scale) to exhibit random behaviour, even thoughthe underlying physics are completely deterministic when all scales are considered.The apparent randomness of turbulence makes it amenable to statistical analysis. Whileindividual realizations of a turbulent flow are not predictable (given human-scale measure-ments only), statistical properties of an ensemble of realizations typically are. Experimentsand simulation of turbulent flows therefore generally aim to predict statistical properties ofturbulence, such as the mean and root-mean-square fluctuation of the velocity field.3.1.3 Features of Turbulent FlowsIn addition to strong dependence on space, time, and initial conditions, turbulent velocityprofiles exhibit [10]:• Enhanced mixing (relative to laminar flows): the wild variations in a turbulent velocityfield give rise to significant advective mixing. This advective effect enhances andcomplements all molecular diffusion phenomena (ordinary diffusion, thermodiffusion,conduction. . . ).• Eddies, defined by Pope [10] as localized regions with a characteristic size over whichthe velocity is at least moderately coherent. The overall turbulent flow field can beviewed as the superposition of many eddies of different sizes.• Vorticity (tendency for fluid elements to rotate): the vorticity field ~∇× ~v is non-zeroin a turbulent flow.• Vortex stretching, a three-dimensional phenomenon by which large eddies axially strainand accelerate smaller ones, transferring kinetic energy. In general, the reverse processmay also occur, but the forward process (energy transfer from larger to smaller scales)dominates.Detailed experimental studies of turbulent flows have also resolved specific coherent struc-tures such as hairpin vortices [10].3.1.4 The [Kinetic] Energy CascadeBased on work initiated by Kolmogorov [11, 12], it is common to divide the scales of turbu-lence into three ranges [10]:203.2. Global Effective Chemical Reactions1. The [Kinetic] Energy-Containing Range (at large length scale), in which viscous forcesare negligible and statistical properties are flow-dependent and (in general) anisotropic,2. The Inertial Subrange (at intermediate length scale), in which the viscous forces arenegligible and statistical properties are universal and isotropic, and3. The Viscous Subrange (at small length scale), in which the viscous forces are significantand statistical properties are universal and isotropic.The two smallest scales are collectively referred to as the Universal Equilibrium Range. Ascale-wise kinetic energy balance shows that kinetic energy is injected at the largest scales,transferred through the intermediate scales by vortex stretching, and dissipated (by viscousforces) at the smallest scales. This energy flow is termed the turbulent cascade, and isillustrated in Figure 3.2. As shown in the figure, the smallest scales of turbulence do notextend all the way to a length scale of zero; the smallest eddies are still orders of magnitudelarger than molecular and atomic sizes11.0LengthScaleViscousSubrangeInertialSubrangeEnergy-ContainingRangeUniversal Equilibrium RangeInjectionVortexStretchingKinetic EnergyDissipationFigure 3.2: The turbulent cascadeThe succession of turbulence scales is referred to as an energy cascade because kineticenergy flows down it from large scale to small scale: kinetic energy is injected at thelargest scales (by external driving forces), transferred to smaller and smaller scales via vor-tex stretching, and ultimately dissipated (converted into thermal energy) by viscous forcesacting at the smallest scales.3.2 Global Effective Chemical ReactionsAlthough the chemical mechanisms underlying combustion systems are generally complexand varied, they typically exhibit a strong tendency to proceed towards a state in whichhydrocarbon fuels have been converted into carbon dioxide and water. Thermodynamicsprohibits a complete conversion of fuel, but in practice a near-complete conversion is quite11If this were not the case, the continuum governing equations would no longer apply.213.3. Premixednesscommon. This tendency motivates definition of the overall or complete combustion chem-ical equation for some hydrocarbon fuel CaHb, which describes (as a first approximation)the chemical process observed. When air is the oxidizer, this is typically written asCa Hb +(a+b4)(O2 + 3.76 N2)→ aCO2 + b2H2O + 3.76(a+b4)N2.When fuel and oxidizer are present in the same ratio as they appear in the overall reaction,the composition is said to be stoichiometric; mixtures with excess oxidizer are called lean(and react to produce a mixture of combustion products and oxidizer), while those withexcess fuel are called rich (and react to produce a mixture of combustion products andunburnt fuel).3.3 PremixednessIn most industrially-relevant combustion processes, the reaction is localized within a rela-tively small region where chemical reaction rates are dramatically higher than rates of diffu-sion. As a consequence of this scale separation, the behaviour of combustion systems showsstrong dependence on whether or not the fuel and oxidizer are initially well-mixedi.e., onwhether or not diffusive mixing is a necessary step in the incoming material's relaxationtowards an equilibrium composition. For classification purposes, it is convenient to considera continuum between the two extremes of initially-mixed and initially-unmixed reactants.As illustrated in Figure 3.3, this continuum can be divided into three regimes:1. non-premixed combustion, where the fuel and oxidizer enter the combustor separately;2. premixed combustion, where the fuel and oxidizer enter the combustor as a homoge-neous mixture; and3. partially premixed combustion, a catch-all label for any case between the two ex-tremes (the slightly premixed and near-fully premixed subdivisions will be discussedin Section 3.3.3).Although this division is fairly universal, some authors prefer to speak of different varietiesof flame rather than different varieties of combustion; the nomenclature is then:1. non-premixed or diffusion flames,2. premixed flames, and3. partially premixed or stratified flames.As illustrated in Figure 3.4, a simple Bunsen burner can exhibit any of the three com-bustion regimes depending on the amount of air allowed into the burner:• on the far left, the air valve is completely closed and the flame is non-premixed,• on the far right, the air valve is completely open, and the flame is premixed, and• in the two central images, the air valve is partially open (more so at centre right thancentre left), and the flame is partially premixed.223.3. PremixednessPartially PremixedNon-Premixed[Fully]PremixedNear-FullyPremixedSlightlyPremixedFigure 3.3: The Continuum of PremixednessFigure 3.4: Photograph of Bunsen burner flames with varying air coflow. Adapted fromhttps://commons.wikimedia.org/wiki/File:Bunsen_burner_flame_types.jpg byArtur Jana Fija lkowskiego under Creative Commons Attribution-ShareAlike 3.0 Unported(CC BY-SA 3.0).233.3. Premixedness3.3.1 Non-Premixed CombustionIn non-premixed combustion, fuel and oxidizer enter the system separately and must mixbefore reacting. Since mixing is typically slow relative to chemical reactions, the overallrate of fuel consumption is limited by the rate of mixing (i.e., molecular diffusion and/orturbulent advection) [13]. The reaction is generally localized to a somewhat thin regionwhich is labelled as a non-premixed flame12in Figure 3.4, this would be located inside theluminous yellow/orange surface of the leftmost plume.When examining non-premixed combustion systems, it is convenient to define the mix-ture fraction, Z, a dimensionless variable which indicates what fraction of the local matteroriginated in each inlet stream. Mixture fraction is typically defined following Bilger [14],as a normalized linear combination of species mass fractions which is conserved in chemicalreactions (i.e., a function of the chemical state), to wit,Z ≡∑α∈SfZα Yα (3.2)where fZα∈S are dimensionless coefficients subject to the requirement that∑α∈SfZα ω˙Yα = 0 (3.3)for all physically-possible sets of production rates ω˙Yα∈Y, and the normalization is definedto give Z = 1 in pure fuel and Z = 0 in pure oxidizer. The mixture fraction then evolvesaccording to the equationρDZDtucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρZ)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vZ)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ ·~jZ´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Diffusion(3.4)where the diffusive flux~jZ (with dimensionality m/(L2t)) can be evaluated as a linearcombination of species diffusions,~jZ ≡∑α∈SfZα~jYα . (3.5)The mixture fraction is a useful parameter to define because it is a highly determinativevariable in non-premixed combustion: knowledge of the local mixture fraction typicallyprovides significant insight into the local chemical state.The rate of reaction in non-premixed combustion shows a non-linear dependence on therate at which fuel and oxidizer are forced together at the flame (the strain rate): the rateof fuel consumption generally increases with strain rate, but ultimately reaches a maximumbeyond which the reaction rate drops to zero (extinction).12also called a diffusion flame, in other works243.3. Premixedness3.3.2 Premixed CombustionIn premixed combustion, fuel and oxidizer enter the system as a homogeneous mixture.The overall ratio of fuel and oxidizer fluxes to the system can be used to define a globalvalue of the mixture fraction, Z, which is typically re-scaled into one of two (dimensionless)equivalence ratios:1. the fuel-air equivalence ratio,φFA≡ Z/(1− Z)Zstoich/(1− Zstoich)(3.6)(where the subscript stoich denotes properties evaluated at stoichiometric composi-tion), or2. the air-fuel equivalence ratio,λAF≡ (1− Z)/Z(1− Zstoich)/Zstoich=1φFA. (3.7)Both the mixture fraction and the equivalence ratios can also be assigned local values ac-cording to Equations 3.2, 3.6, and eq:lambdaAF. In premixed flames which do not featurepreferential species transport, every local Z, φ, and λ is equal to the corresponding globalquantity; preferential transport can introduce local variations, but these are typically small.Premixed systems generally exhibit flame frontsthin reaction zones which separateunburnt fuel-oxidizer mixture on one side from combustion products on the other side andspontaneously propagate from burnt to unburnt. In Figure 3.4, this would correspondwith the luminous blue surface of the rightmost plume. The overall rate of reaction isdictated by the speed with which heat and radicals are transported to and from the reactionzone (by molecular conduction/diffusion and/or turbulent advection) [13], which is typicallyfaster than the rate of mixing in non-premixed combustion since the reaction zone hasextremely steep concentration gradients. In the absence of turbulence, premixed flamefronts propagate with a characteristic speed (the laminar flame speed) which is a functiononly of the composition and thermodynamic state.When examining premixed flames, it is convenient to define the progress variable, c, adimensionless variable which varies between extremes of 0 in pure reactants and 1 in pureproducts and thereby indicates (in an rough overall sense) the degree to which the localmaterial has reacted. The first step in defining c is to identify some scalar field Yc whichincreases monotonically between reactants and products and is transported according to anequation of the same form as the species transport equation, to wit,ρDDtYcucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρYc)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vYc)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ ·~jYc´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Diffusion+ ρω˙Yc .®Production(3.8)253.3. Premixednesswhere~jYc is the diffusive flux of Yc, with dimensionality m/(L2t), and ω˙Yc is the rate ofproduction of Yc due to chemical reactions, with dimensionality 1/t. Defining Yc as a linearcombination of species mass fractions, i.e.,Yc =∑α∈SfYcα Yα (3.9)~jYc =∑α∈SfYcα ~jYα (3.10)ω˙Yc =∑α∈SfYcα ω˙Yα (3.11)(where fYcα∈S are dimensionless coefficients) is one way of guaranteeing that Yc satisfies Equa-tion 3.8 exactly; Yc can also be chosen as the sensible internal energy, sensible enthalpy, ortemperature, under appropriate simplifying assumptions (discussed further in Section 7.2.1).Regardless of the definition of Yc, the progress variable can be defined by normalizing it:c ≡ c(Z, Yc) = Yc − Yreactc (Z)Y prodc (Z)− Y reactc (Z)(3.12)where Y reactc (Z) denotes the Yc of unreacted mixture at the local mixture fraction13andY prodc (Z) denotes the Yc of fully reacted products14at the local mixture fraction. Thediffusive flux~jc of progress variable (with dimensionalitym/(L2t)) and the rate of productionof progress variable ω˙c (with dimensionality 1/t) can then be defined as~jc ≡(∂Yc∂c)−1Z[~jY c −(∂Yc∂Z)c~jZ](3.13)ω˙c ≡(∂Yc∂c)−1Zω˙Yc . (3.14)where• in premixed combustion, ~jZ = ~0, simplifying the equation for ~jc (the general casehas been displayed above as it will be referred to when discussing partially premixedcombustion), and• the partial derivatives are most convenient to evaluate if Equation 3.12 is rearrangedasYc(Z, c) = c[Y prodc (Z)− Y reactc (Z)]+ Y reactc (Z) (3.15)= c · Y prodc (Z) + (1− c)Y reactc (Z). (3.16)13In premixed combustion, the local mixture fraction is generally taken as the global mixture fraction;this is, technically speaking, only exact in the absence of preferential diffusion.14The definition of fully reacted products is somewhat arbitrary: it is often chosen as chemical equilibrium(evaluated by considering a closed, adiabatic, isobaric reactor), but could alternatively be chosen as theproducts of complete combustion or of some other model reaction.263.3. Premixednessgiving (∂Yc∂c)−1Z=1Y prodc (Z)− Y reactc (Z)(3.17)(∂Yc∂Z)c= cdY prodcdZ+ (1− c)dYreactcdZ. (3.18)With these definitions, the progress variable in premixed combustion evolves according tothe transport equationρDDtcucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρc)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vc)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ ·~jc´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Diffusion+ ρω˙c®Production(3.19)which takes the same form as the original species balance and the balance for Yc. Theprogress variable is a useful parameter to define because it is a highly determinative vari-able in premixed combustion: knowledge of the local progress variable typically providessignificant insight into the local chemical state.Premixed flames exhibit flammability limits: for a given fuel-oxidizer combination ata given physical state, only a limited range of equivalence ratios (mixture fractions) issufficiently reactive to sustain premixed flame fronts [13]. These limits are often referredto as the lean and rich premixed flammability limits. Much like non-premixed combustion,the rate of reaction in premixed combustion shows a non-linear dependence on the rate atwhich fuel and oxidizer are forced together at the flame front (the strain rate): the rate offuel consumption generally increases with strain rate, but ultimately reaches a maximumbeyond which the reaction rate drops to zero (extinction).3.3.3 Partially-Premixed CombustionAs illustrated in Figure 3.3, the continuum between premixed and non-premixed combustionis referred to as the partially-premixed combustion regime. This regime can also be furthersub-divided.• The near-fully premixed side of the continuum can arise in situations where a burneris intended to produce a premixed mixture for combustion, but the mixing is imperfectand there is therefore some variety in composition at the un-reacted side of the reactionzone. In this extreme, it is typically true that, despite the imperfect mixing, everypoint within the burner has a composition between the premixed flammability limits.• The slightly premixed side can arise when a non-premixed flame is strained: sufficientlyhigh local strain triggers local extinction, which halts reactions but not diffusion, andtherefore allows the fuel and oxidizer to mix before (potentially) reigniting [15]. In thisextreme, it is typically true that there is at least one point within the burner wherethe composition is outside of the premixed flammability limits.273.3. PremixednessFuelOxidizerRichStoichLeanFigure 3.5: An idealized triple flame (based on a similar diagram in [15] and photographsin [16]).An idealized laminar partially premixed flame is illustrated in Figure 3.5. The divergingburner area ensures that the velocity gradually decreases towards the right, allowing theflame to situate itself at a location where the leftward reaction propagation and rightwardflow velocity are balanced. The fuel and oxidizer enter at a speed high enough that thereaction zone (flame) cannot propagate upstream and anchor at the inlet; instead, threereaction zones (flame branches) appear [16]:1. The upper rich branch is situated at a composition near the rich premixed flamma-bility limit, and is similar (but not identical) to a rich premixed flame. The productswhich propagate rightward from this flame still contain a significant amount of un-reacted fuel.2. The lower lean branch is situated at a composition near the lean premixed flamma-bility limit, and is similar (but not identical) to a lean premixed flame. The productswhich propagate rightward from this flame still contain a significant amount of un-reacted oxidizer.3. The central stoich (stoichiometric) branch is similar (but not identical) to a non-premixed flame in which the fuel stream is the products of the rich flame branchand the oxidizer is the products of the lean flame branch. This flame is situated ata composition near the stoichiometric composition.As this example demonstrates, partially premixed combustion is not so much a blendedregime with intermediate behaviour as it is a combination regime, where both premixed-esque and non-premixed-esque behaviours are present at different locations.When examining partially premixed flames, it is convenient to define both the mixturefraction and the progress variable. The partially premixed mixture fraction is identical tothe non-premixed mixture fraction (Equations 3.2 to 3.5); the partially premixed progressvariable is defined in the same way as the premixed progress variable (Equations 3.8 to 3.16),283.3. Premixednessbut (since Z is no longer constant) its transport equation is more complicated [17]15:ρDDtcucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρc)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vc)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection=Diffusionucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright−~∇ ·~jc +Productionucurlyρω˙c+ρ(∂Yc∂c)−1Z(∂2Yc∂Z2)c;cχZZ+2ρ(∂Yc∂c)−1Z(∂2Yc∂Z∂c)c;ZχZcInteractionwithZ field(3.20)where the partial derivatives (which can be evaluated by differentiating Equations 3.15and/or 3.16) are (∂Yc∂c)−1Z=1Y prodc (Z)− Y reactc (Z)(3.17 repeated)(∂2Yc∂Z2)c;c= cd2Y prodcdZ2+ (1− c)d2Y reactcdZ2(3.21)(∂2Yc∂Z∂c)c;Z=dY prodcdZ− dYreactcdZ(3.22)and the χZZ and χZc are generalized scalar dissipations, with dimensionality 1/t. Theseterms can be defined in as∀φ, ψ ∈ Z : χφψ ≡ −12(∆~vφ · ~∇ψ + ∆~vψ · ~∇φ), (3.23)where• Z is the set composed of species mass fractions, mixture fraction, and progress variableZ ≡ Y⋃{Z, c} , (3.24)and• ∆~vφ∈Z is the drift velocity of scalar φ (i.e., its velocity relative to the mixture massaverage), with dimensionality L/t,∀φ ∈ Z : ∆~vφ ≡~jφρ. (3.25)Self scalar dissipations (for which φ = ψ) are generally expected to be non-negative. Inthe simple case of Fickian diffusion with a single diffusivity D, i.e., the case where∀φ ∈ Z : ~jφ = −ρD~∇φ (3.26)15Note that [17] defines the progress variable based on the fuel mass fraction, which is monotonicallydecreasing ; this is tantamount to defining Yc (as used in this work) as the sum of all species mass fractionsexcept fuel, which introduces negative signs in some locations.293.3. Premixednessthe scalar dissipation can be simplified to∀φ, ψ ∈ Z : χφψ = χψφ = D~∇φ · ~∇ψ, (3.27)and the non-negativity of self scalar dissipations is guaranteed.Whereas mixture fraction and progress variable are individually highly determinative innon-premixed and premixed combustion respectively, only the combination is highly deter-minative in partially premixed combustion: knowledge of both the local mixture fractionand the local progress variable typically provides significant insight into the local chemicalstate.30Chapter 4Simulation Techniques for TurbulentCombustion SystemsBoth turbulence and chemical reactions are extremely computationally expensive to simulateexactly. This has motivated the development of separate techniques for modelling turbulenceand chemistry. Unfortunately, these techniques are not natively compatible with one an-other. This incompatibility has motivated the development of distinct turbulence-chemistryinteraction models, which seek to allow existing turbulence and chemistry modelling tech-niques to be used together.4.1 Turbulence ModellingThe multi-scale, chaotic nature of turbulence implies that an enormous range of length andtime scales must be resolved16. Resolving all details in this way is called Direct NumericalSimulation (DNS). It is often labelled as a turbulence modelling approach, but can be morerigorously labelled as a simulation paradigm since it involves no modelling beyond thatalready present in the basic transport equations and flux laws (Chapter 2). For a discrete(mesh-based) solver, the computational cost of DNS can be estimated as a function of the(dimensionless) turbulent Reynolds number, defined as [10]Ret≡[12 (~v − 〈~v〉)]22ν2∣∣∣∣(~∇ (~v − 〈~v〉))F∣∣∣∣2(4.1)where 〈~v〉 represents the time average of the velocity. In terms of Ret, the cost of DNS scalesas follows [10]:• the number of cells/grid points required to resolve all flow length scales is proportionateto Re9/4t,• the number of time steps required to resolve all flow time scales is proportionate toRe3/4t, and• the overall computational cost of simulating the flow scales as the product of thenumber of grid points and the number of time steps, i.e., as Re3t.16Since most dissipation occurs at the smallest scales, a simulation which does not resolve the small-scaledetails will under-predict dissipation and, by extension, over-predict the kinetic energy of the flow (leadingto incorrect predictions and, typically, instability).314.1. Turbulence ModellingComputational CostAccuracyRANSLESDNSFigure 4.1: Relative cost and accuracy of turbulence simulation paradigms. Scaling of bothaxes is purely qualitative.As a result of this cubic scaling, the computational cost of simulating highly turbulent flowsusing DNS is prohibitively high. This has motivated the development of two true modellingapproaches: Reynolds Averaged Navier Stokes (RANS) and Large Eddy Simulation (LES).As illustrated in Figure 4.1, the three simulation paradigms each offer a different trade-offbetween computational cost and accuracy. Current processing technology is such that RANSof combustion is commonplace in industry, LES of combustion is typically only performedin academia, and DNS of combustion is intractable in all but the simplest cases (i.e., simplechemistry at low Reynolds numbers).RANS and LES can both be viewed as specializations of a more general filter-basedturbulence model, in which the Re3tscaling rule is relaxed by resolving some details andmodelling others. In the interests of highlighting the commonalities of the two models, thegeneric filter-based approach is introduced over the course of Sections 4.1.1 through 4.1.5,and the details of the two specific paradigms are reserved for Section 4.1.6.4.1.1 The Filtering OperationAs defined fully in Sections B.1.1 and B.1.217, each field φ (~x, t) can be assigned a corre-sponding filtered field φ (~x, t) and Favre filtered field φ˜ (~x, t), which can be related byφ˜ =ρφρ. (4.2)Both filters are linear operators and (although it is not always rigorously true) it is typicallyassumed that the basic (non-Favre) filter commutes with time and space derivatives:∀φ : ~∇φ ≈ ~∇φ ∂∂tφ ≈ ∂∂tφ. (4.3)17For an even more thorough discussion of the filter operator, see [18].324.1. Turbulence ModellingAs demonstrated in Section C.1, state functions relate un-filtered properties do not, ingeneral, hold between filtered properties:∀φ ∈ Ψthermo : [φ = φ(Ψˆthermo)]6=⇒[φ˜ = φ(Ψˆthermo˜)](4.4)The commutation may hold in special cases, for example if:1. the filter has a negligible impact on the thermodynamic state,Ψˆthermo ≈ Ψˆthermo˜, (4.5)or2. the function is locally (near Ψˆthermo) linear in all arguments,φ(Ψˆthermo) ≈ φ0 +∑ψ∈Ψˆthermomψψ. (4.6)As will be discussed further in Sections 4.1.3 and 4.3, approximate commutation is typicallyassumed for all thermodynamic properties except for reaction rates.In turbulence modelling, the objective of the filtering operation is to remove (filter away)small-scale details, leaving a simplified representation of the system in which only the large-scale characteristics are retained. This reduces the range of length and/or time scales whichmust be resolved, allowing the system to be represented with fewer cells and/or time stepsin a discrete solver. Since turbulence is, fundamentally, a multi-scale phenomenon, the un-resolved details cannot be neglected ; the resolved and un-resolved components of the systemproperties interact, and this interaction must be modelled in order to properly predict theresolved properties. Filter-based turbulence models therefore involve a resolved (filtered)state (Section 4.1.2), models for the interaction of this state with itself (Sections 4.1.3and 4.1.4), and models to estimate the impact of the un-resolved details on the resolvedstate (Section 4.1.5).4.1.2 Filtered State and State FunctionsIn the Eulerian approach to continuum mechanics, the filtered state of the entire system atsome time t can be represented using (|S|+ 7) filtered fields ψ (~x, t) or ψ˜ (~x, t)one for eachψ in the total state basis Ψˆtot. The choice is typically:• three elements of the filtered position ~x, (which are trivial since position ~x is a fieldargument),• three elements of the Favre filtered velocity, ~˜v,• any (|S| − 1) of the Favre filtered species mass fractions Yα˜,• ρ or P , and334.1. Turbulence Modelling• T˜ , u˜, or h˜.For compactness, this set of variables is labelled as Ψˆtot˜, even though some of the fieldswhich may appear are filtered rather than Favre filtered. The sets Ψˆchem˜ and Ψˆphys˜ aredefined by analogy, as the set of (|S| − 1) Favre filtered species mass fractions and the setwith elements (ρ or P ) and (T˜ , u˜, or h˜), respectively.As foreshadowed in Section 4.1.1, equations of state that relate un-filtered properties donot, in general, hold between filtered properties. Despite this lack of formal commutation,it is typical to assume that equations of state can apply to the filtered state described abovein the same way as they would to the un-filtered state, i.e.∀φ ∈ Ψtot :[φ = φ(Ψˆtot)]=⇒[φ˜ ≈ φ(Ψˆtot˜)]. (4.7)The chemical source termwhich is, formally, a state functionis one exception to this ruleof thumb; it is discussed further in Section 4.3.4.1.3 Filtered Governing EquationsApplying the basic filtering operation to the standard governing equations produces filteredgoverning equations (presented in detail in Section D.1). The filtered mass balance is essen-tially identical to its un-filtered analog:∂∂tρ+ ~∇ ·(ρ~˜v)= 0. (4.8)The remaining transport equations can be defined compactly as follows: for some field φwhich has an un-filtered transport equation of the formρDDtφucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂tρφ®LocalChange+ ~∇ · (ρ~vφ)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= Tφ®Transport(4.9)the corresponding filtered transport equation isρD˜Dtφ˜ucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂tρφ˜®LocalChange+ ~∇ ·(ρ~˜vφ˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ResolvedAdvection= − ~∇ · j subφ´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶UnresolvedAdvection+ Tφ®FilteredTransport(4.10)whereD˜Dt is the material derivative with respect to the Favre filtered velocity,D˜Dtφ ≡ ∂∂tφ+ ~˜v · ~∇φ (4.11)344.1. Turbulence Modellingand j subφ is a generic sub-filter transport term, which can be written as18j subφ = ρ(~vφ˜− ~˜vφ˜). (4.12)Sub-filter transport terms formally represent the impact of un-resolved details on resolvedonesspecifically, un-resolved advectionand are assigned more descriptive names based onthe field transported.• For species mass fractions, mixture fraction, and progress variable, j subφ is called thesub-filter mass flux, and has dimensionality m/(L2t)∀φ ∈ Z : j subφ = ~j subφ = ρ(~vφ˜− ~˜vφ˜), (4.13)which (as shown in Section C.2.2) is constrained by∀φ ∈ Z : −ρ~˜vφ˜ ≤ ~j subφ ≤ ρ~˜v(1− φ˜). (4.14)• For velocity, j subφ is called the sub-filter stress, and has dimensionality m/(Lt2)j sub~v = τsubF = ρ(~˜v~v − ~˜v~˜v), (4.15)which is guaranteed to be symmetric19(as indicated by the annotation F) andfor a non-negative filter kernel (Section B.1.1), as is assumed in this workpositivesemidefinite20(see Section C.2.3). This is typically broken up asτ subF = τsub‡ +23ρk subI◦´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶τ sub◦(4.16)wherek sub ≡ 12(~v · ~v˜ − ~˜v · ~˜v)=12ρTr(τ subF)(4.17)is the sub-filter (turbulent) kinetic energy, which has dimensionality L2/t2 and mustbe non-negative21.• For energy, internal energy, and enthalpy, j subφ is called the sub-filter heat flux, andhas dimensionality m/t3∀φ ∈ {e, u, h} : j subφ = ~q subφ = ρ(~vφ˜− ~˜vφ˜). (4.18)These terms follow the same forms as diffusive fluxes, viscous stresses, and conductive fluxesrespectively, and can therefore each be viewed as effective turbulent contributions to theirrespective phenomena.18Section B.2.1 provides an alternative formulation of j subφ . The alternative formulation is equivalent, andmore obviously related to the fluctuation about the mean field, but also more mathematically complicated.19The outer product of any vector with itself is symmetric.20A positive semidefinite matrix M has the property that ∀~x : ~xTM~x ≥ 0.21The diagonal elements (and, by extension, the trace) of a positive semidefinite matrix (such as thesub-filter stress tensor) are guaranteed to be non-negative.354.1. Turbulence Modelling4.1.4 Filtered Constitutive EquationsThe filtered transport equations feature several as-yet-unclosed filtered transport terms,specifically• ~∇ · τF ≈ ~∇ · τF, the divergence of the filtered viscous stress tensor,• ~∇ ·~jφ ≈ ~∇ ·~jφ, the divergence of the filtered diffusive flux vectors,• ~∇ · ~q ≈ ~∇ · ~q, the divergence of the filtered heat flux vector,• Q the radiation long-range transfer term, and• ρω˙φ = ρω˙Yα˜ , the filtered source terms.With the exception of the source term (which will be discussed in Section 4.3), the filteredfluxes can generally be evaluated by applying the laws of Section 2.3 using elements of thefiltered state basis in place of the corresponding un-filtered elements of the state basis. Forexample, Equation 2.14 corresponds to the filtered constitutive relationτF ≈ 2µ(~∇~˜v)‡+ 3ζ(~∇~˜v)◦(4.19)where µ and ζ are evaluated using the same expressions that define µ and ζ, but with Yα˜,T˜ , and P in place of Yα, T , and P .4.1.5 Sub-Filter TermsMany closure models for the sub-filter transport terms begin by recognizing that the act offiltering away flow details and replacing them with transport terms representing their neteffect on the resolved field is analogous to the act of neglecting the motions of individualmolecules and instead tracking the net transport of mass, momentum, and energy due tomolecular effects. This analogy between un-resolved eddies and molecular effects suggeststhat the same scaling rules which describe molecular transport might describe sub-gridterms, to witτ sub‡ ≈ −2ρνF(~∇~˜v)‡= −2µF(~∇~˜v)‡(4.20)∀φ ∈ Z : ~j subφ ≈ −ρDF~∇φ˜ (4.21)∀φ ∈ {e, u, h} : ~q subφ ≈ −ραF~∇T˜ .(4.22)Under this assumption, there are now just four un-closed scalars:• νF (with dimensionality L2/t) or µF (with dimensionality m/(Lt)), the turbulentor eddy kinematic or dynamic viscosity associated with filter F, first proposed byBoussinesq [19],364.1. Turbulence Modelling• k sub (dimensionality L2/t2), the sub-filter kinetic energy, bounded byk sub ≥√32νF∣∣∣∣√2(~∇~˜v)F∣∣∣∣´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Resolved OverallStrain Rate(4.23)to respect the constraint that τ subF is positive semidefinite [20]22,• DF (with dimensionality L2/t), the turbulent or eddy mass diffusivity associatedwith filter F [21], and• αF (with dimensionality L2/t), the turbulent or eddy thermal diffusivity associatedwith filter F.The most common closure approach for these scalars is to employ a detailed model for νFand k sub and then express the remaining constants in terms of dimensionless ratios asDF = νFScFαF =νFPrF(4.24)where ScF and PrF are the turbulent Schmidt and Prandtl numbers associated with filter F,which are dimensionless and of order 1.4.1.6 Specific Turbulence Modelling ParadigmsReynolds-Averaged Navier-Stokes (RANS)Reynolds-Averaged Navier-Stokes (RANS) is based on the idea that, since it is often thetime or ensemble average of the turbulent fields that are of interest, it is most reasonable tosolve time- or ensemble-averaged governing equations23. It can be demonstrated that, forthe time-average filter, the commutation in Equation 4.3 holds exactly, and filtered fieldsare independent of time:~∇φ = ~∇φ ∂∂tφ =∂∂tφ = 0 (4.26)22The resolved overall strain rate will be assigned its own symbol in Equation 6.4; the factor of√2 ensuresthat the expression reduces to the natural definition of strain rate in the case of one-dimensional flows.23The time average can be viewed as a filter with the kernel (Section B.1.1)∀~x, ~x∗ ∈ X .∀t, t∗ ∈ T : G (~x, ~x∗, t, t∗) = δ(~x− ~x∗)|T | (4.25)where X and T are, respectively, the time and space domains, and δ(∆~x) is the Dirac delta (an infiniteimpulse at the origin and zero elsewhere) and |T | represents the magnitude of the time domain.374.1. Turbulence ModellingTable 4.1: RANS Eddy Viscosity ModelsCategory Model ReferenceZero-Equation Prandtl Mixing Length [23]Baldwin-Lomax [24]Cebeci-Smith [25]One-Equation Prandtl One-Equation [26], cited in [27]Spalart-Allmaras [28]Two-Equation k-ε (also written as k-epsilon) [29]k-ω (also written as k-omega) [30]The time average is also a Reynolds Operator (discussed further in Section B.3), whichimplies that variances and sub-filter terms can be re-expressed as∀φ : φ var = φ fluc2 (4.27)∀φ : φ v˜ar = φ fl˜uc2 (4.28)∀φ ∈ Z : ~j subφ = ρ~v fl˜ucY fl˜ucα (4.29)τ subF = ρ~vfl˜uc~v fl˜uc (4.30)∀φ ∈ {e, u, h} : ~q subφ = ρ~v fl˜ucφ fl˜uc (4.31)where the RANS sub-filter stress tensor is more commonly referred to as the Reynolds stresstensor and φ fluc and φ fl˜uc are fluctuation fields,φ fluc ≡ φ− φ φ fl˜uc ≡ φ− φ˜. (4.32)In RANS, the sub-filter transport terms typically overwhelm their molecular counter-parts, implying that the molecular terms can be neglected:∀φ ∈ Z : ~j subφ +~jφ ≈ ~j subφ (4.33)τ subF + τF ≈ τ subF (4.34)∀φ ∈ {e, u, h} : ~q subφ + ~q ≈ ~q subφ . (4.35)RANS eddy viscosity models typically involve prescribed values for ScF and PrF (typicallyas 0.70.9 [22] and 0.85 respectively) and algebraic expressions for νF (or µF) and k sub.Table 4.1 lists several well-known RANS eddy viscosity models, categorized based on thenumber of additional transport equations that they introduce.384.2. Chemistry ModellingLarge Eddy Simulation (LES)Large Eddy Simulation is based on the idea that, since it is the large-scale motions that aretypically of interest, it is most reasonable to solve spatially filtered governing equations24.Each LES filter F has a corresponding filter kernel GF (~x, ~x∗), and can be assigned a charac-teristic width∆F (of dimensionality L). Theoretical analyses of LES typically consider filterswhose kernels have infinite extent, such as the Gaussian or spectral cutoff filter, but mostpractical LES is based on the premise that the act of evaluating control volume averagesasis necessary in finite volume approachesconstitutes an implicit top-hat filter called thegrid filter, for which∀~x ∈ X : ∆F (~x) = 3√|~x| (4.37)where X is the discretized spatial domain (a set of cells) and |~x| is the volume of cell~x. Itcan be demonstrated that, for homogeneous filters, the commutation between differentiationand filtering (Equation 4.3) is exact, while for inhomogeneous filters it is not; for implicitfiltering, this implies that the mesh becomes more amenable to Large Eddy Simulation asit becomes more uniform.LES eddy viscosity models typically involve prescribed values for ScF and PrF (typicallyas 0.5 [22] and 0.47 [31] respectively) and algebraic expressions for νF (or µF) and k sub.Popular models for the LES eddy viscosity include:• the Smagorinsky model [32],• the Dynamic Smagorinsky model [33], and• the Wall-Adapting Local Eddy-viscosity (WALE) model [34].The Smagorinsky and Dynamic Smagorinsky models are of particular interest in this work,and are presented in detail in Chapter 6. For a more thorough overview of LES, see [18].4.2 Chemistry ModellingCombustion chemistry is phenomenally complicated: in theory, covalent bonding permitsthe formation of arbitrarily large molecules, meaning that the number of intermediateswhich could be formed and the number of reactions which could occur is effectively infinite.This challenge is compounded by the experimental reality that simultaneously measuringthe concentrations of many species with the temporal resolution required to characterizecombustion dynamics is essentially impossible.Despite the fact that chemistry is theoretically infinitely complicated, it is often possibleto identify finite sets of species and reactions which capture a sufficiently complete picture24Spatial filtering can be viewed as a general filter with the kernel (Section B.1.1)∀~x, ~x∗ ∈ X .∀t, t∗ ∈ T :G (~x, ~x∗, t, t∗) = δ(t− t∗) ·G (~x, ~x∗) (4.36)where X and T are, respectively, the time and space domains, and G(~x, ~x∗) is a kernel which depends onspace only.394.2. Chemistry Modellingof the reaction dynamics to accurately predict key flame parameters such as the laminarflame speed and the ignition delay. Models of this nature are referred to as detailed orfull chemical mechanisms (or, occasionally, skeletal mechanisms, if the number of speciesand reactions is relatively small). Although they present simplified models of reality, thesemechanisms are still remarkably intricate: as illustrated in Table 4.2, mechanisms for themethane/air system, the simplest hydrocarbon reacting with the most common oxidizer,typically involve |S| ≈ 40 species and |R| ≈ 200 reactions [35]. This complexity makesfull-chemistry numerical simulations extremely computationally expensive. The expense iscompounded by the fact that using full chemistry typically introduces numerical stiffness:the range of reaction time scales in full mechanisms is so wide that (even in the absence offluid dynamics effects) numerical simulations typically diverge unless extremely small timesteps or specialized stiff solvers are employed. While these expenses are manageable insituations where the fluid dynamics are trivial (such as well-stirred reactors) or relativelysimple (such as laminar flames), the computational effort required to apply such mechanismsin turbulent combustion simulation (e.g. DNS) is prohibitive.Table 4.2: Detailed chemical mechanisms for methane-air chemistryMechanism Version Species Reactions ReferenceLu and Law - 30 184 [36]GRI-Mech 2.11 49 277 [37]3 53 325 [38]UBC-Mech 1.0 38 192 [39]2.0 54 277 [40]2.1 40 194 [41]3.0 71 379 [42]The high cost of full-chemistry simulations has motivated the development of simplifiedchemistry models, which attempt to reduce the computational cost of simulation by rep-resenting the chemical state using fewer degrees of freedom. Simplified chemistry modelsassume that there exists some reduced chemical state basis, Ψˆrchem, which• has no more elements than the original chemical state basis,∣∣∣Ψˆrchem∣∣∣ ≤ ∣∣∣Ψˆchem∣∣∣ , (4.38)• can be used to reconstruct the full chemical state (even in the presence of transporteffects), for example via table look-ups,∀ψ ∈ Ψchem : ψ = ψ(Ψˆrchem), (4.39)and• can be evolved in time by solving closed governing equations.404.2. Chemistry ModellingWhen such a Ψˆrchem exists, it is no longer necessary to track or solve transport equations forevery element of the chemical state basis; tracking and transporting the reduced chemicalstate basis is sufficient. A reduced thermodynamic state basis can also be defined as theunion of the reduced chemical and physical state bases :Ψˆrtherm ≡ Ψˆrchem⋃Ψˆphys; (4.40)this reduced basis permits reconstruction of any thermodynamic property,∀ψ ∈ Ψthermo : ψ = ψ(Ψˆrtherm). (4.41)Equation 4.38 intentionally admits the possibility∣∣∣Ψˆrchem∣∣∣ = ∣∣∣Ψˆchem∣∣∣ so that full chem-istry can be classified as a reduced chemical model (the trivial no reduction case); inpractice, the models of interest are those for which∣∣∣Ψˆrchem∣∣∣ ∣∣∣Ψˆchem∣∣∣ , (4.42)since these facilitate a significant reduction in the number of transport equations requiredto evolve the system and thereby reduce the computational cost of simulation.4.2.1 Reaction-Diffusion Manifolds (REDIMs)The relationship between the full and reduced chemical states can be visualized using theconcept of a manifold: although the chemical state basis Ψˆchem defines a∣∣∣Ψˆchem∣∣∣-dimensionalstate space, the set of state points which are accessible in a given application typically doesnot fill the spaceit instead approximately corresponds to a (d ≤∣∣∣Ψˆchem∣∣∣)-dimensional25manifold suspended in state space. The reduced chemical state basis then corresponds to aminimal set of variables that parameterize the manifold, and Equation 4.39 represents theact of looking up a point in the manifold. The most general category of manifold is theReaction-Diffusion Manifold (REDIM) [43], which attempts to capture the impact of bothreactions and diffusion on the chemical state. In a general REDIM, both chemical properties(which drive reactions) and functions of their gradients (which drive diffusion) may serve aselements of the reduced chemical state.Most popular chemistry reduction techniques can be viewed as a specialization of theREDIM concept. Two special categories of REDIMpure reaction manifolds and flameletmanifoldsare introduced in the following two sections, and their applicability is comparedin Section 4.2.4.4.2.2 Pure Reaction ManifoldsPure reaction manifolds are based on an assumption that the full chemical mechanism canbe cleanly partitioned into two components [35]:25It is not guaranteed that a d-dimensional manifold can be represented using a single coordinate mapthat is, a single bijection between points in d-dimensional Euclidean space and points on the d-dimensionalmanifold that preserves the topology of the manifold. Defining position unambiguously (i.e., defining thereduced chemical state) may require more than d variables.414.2. Chemistry Modelling• fast chemistry, which proceeds far faster than all other effects, and is therefore in-sensitive to flow details and history (i.e., stateless); and• slow chemistry, which proceed no faster than other effects, and is therefore sensitiveto flow details and history (i.e., stateful).Under this assumption, any minimal set of variables which represents the state of the slowchemistry provides complete information about the full chemical state. This can be viewedas a specialization of the REDIM in which diffusive effects are assumed to be negligible [43]and only chemical properties (representing the state of the slow chemistry) serve as manifoldparameters (functions of gradients of properties, such as scalar dissipations, do not), andthus the reduced chemical state basis is a subset of the chemical state,Ψˆrchem ⊆ Ψchem. (4.43)The oldest approach for generating a pure chemistry manifold involves invoking quasi-steady-state and partial equilibrium assumptions [44] to simplify a full chemical mechanisminto a reduced chemical mechanism. This technique was first proposed before the develop-ment of the manifold view of chemistry reduction, and has therefore only retroactively beenclassified as a manifold method [45]. Reduced chemistry manifolds are typically identified byhand, and tend to introduce stiffness just as full chemistry does. In contrast, pure reactionmanifolds designed in light of the manifold concept are typically identified algorithmicallyusing computers; they are often able to produce extremely low-dimensional manifolds whichintroduce less stiffness than the corresponding full mechanisms (non-stiff 2-D manifolds arecommon for premixed combustion of simple fuels).4.2.3 Flamelet ManifoldsFlamelet manifolds are based on an assumption that the length and time scales associatedwith chemistry are small compared to those associated with the flowin particular, thoseassociated with any turbulence which may be present [46]. This flamelet assumption26implies that, when viewed from an appropriate reference frame, the reaction zone in anyflow27appears like that in some analogous laminar flame [47]. This implies that, if onecan. . .• construct a flamelet library consisting of all the laminar flames whose analoguesmight appear in the system of interest, each identified by distinct values of some setof controlling variables, and• represent position within each flamelet using some set of parameterizing variables,. . . then the combination of the controlling and parameterizing variables provides a completerepresentation of the local chemical state. This can be viewed as a specialization of theREDIM in which26A flamelet is defined by Peters [46] as an asymptotically thin reaction layer embedded in a turbulentflow.27(satisfying the flamelet assumption)424.3. Turbulence-Chemistry Interaction Modelling• diffusive effects are assumed to be universal functions that can be probed by simulatinglaminar flamelets [43], and• both chemical properties and functions of their gradients (typically scalar dissipations)may serve as manifold parameters.As with reduced chemistry, the oldest flamelet manifold (the steady laminar flameletmodel for non-premixed combustion [48]) predates the manifold concept, but can be retroac-tively labelled as a manifold approach for consistency.4.2.4 Applicability of REDIM MethodsPure chemistry manifolds capture only chemical effects, and effectively assume that fastreactions dominate over transport; flamelet manifolds capture chemical effects and simpletransport, and effectively assume that turbulence does not impact transport at the scale ofthe chemistry. In the limit where both assumptions are valid (reactions dominate over alltransport and turbulence is not intense enough to impact transport at the chemical scale)pure chemistry and flamelet manifolds converge to the same REDIM. There also exist regionsin which only one manifold remains applicable:• when turbulence is high and reaction rates are high (such as in the reaction zone of ahighly turbulent flame), only pure chemistry manifolds apply; whereas• when turbulence is low or moderate but reaction rates are low (such as in the preheatregion of a laminar flame), only flamelet manifolds apply.In the limit where turbulence is high and reaction rates are low (such as the preheat regionof a highly turbulent flame), neither pure chemistry nor flamelet manifolds are applicable; ageneralized REDIM which captures all relevant interactions would be required for accuratesimulation.4.3 Turbulence-Chemistry Interaction ModellingThe two main turbulence modelling paradigms, RANS and LES, are both based on filteringoperations, and are therefore not natively compatible with any of the chemistry modelspresented in Section 4.2. The root of this incompatibility is the fact that chemistry modelsrequire the evaluation of chemical source term and/or species reconstruction functions whichare designed to apply to un-filtered fields, and (due to the lack of automatic filter-functioncommutation28) cannot necessarily be applied to filtered fields:∀ψ ∈ Ψˆrchem : [ω˙ψ = ω˙ψ(Ψˆrtherm)]6=⇒[ω˙ψ˜ = ω˙ψ(Ψˆrtherm˜)](4.44)∀ψ ∈ Ψrchem :28See Sections 4.1.1 and C.1.434.3. Turbulence-Chemistry Interaction Modelling[ψ = ψ(Ψˆrtherm)]6=⇒[ψ˜ = ψ(Ψˆrtherm˜)]. (4.45)Although the implications above would hold if the filter operation had a negligible impact orthe reaction rate and species reconstruction functions were linear in all arguments28, neitherof these conditions holds in general:• the filter must have an impactif it does not, then the turbulence model reduces toDNS and the Re3tscaling law must be satisfied (Section 4.1)29, and• regardless of the chemistry model, the reaction rate and species reconstruction func-tions have nonlinear dependence on their arguments (often highly nonlinear depen-dence).Equation 4.44 is often labelled as the filtered chemical source term problem: the filtered re-duced chemical state basis cannot be transported without further modelling since its filteredsource is un-closed. The closely related Equation 4.45 implies that, even if the filtered re-duced chemical state basis were available (and was combined with the filtered physical statebasis to form the filtered reduced thermodynamic state basis), this would not be sufficientto deduce the filtered chemical state without further modelling.One approach to this closure problem is to abandon any attempt to predict the chemicalstate, and instead model only its impact on the physical state (typically captured in a filteredsource term for sensible energy, sensible enthalpy, or temperature) in an ad hoc fashion30.This is approximately comparable to the tactics employed in RANS and LES models for thesub-grid stress (which typically make no attempt to resolve the underlying fluid dynamics);models of this nature include• G Equation (for premixed combustion) [49],• Flame Surface Density (for premixed or non-premixed combustion) [50],• Eddy Break-Up (for premixed combustion) [51],• Eddy Dissipation Concept (for non-premixed combustion) [52],• Bray Moss Libby (for premixed combustion) [53].Ad hoc models are computationally inexpensive, but give only a rough approximation ofcombustion dynamics: many details (such as individual species mass fractions) cannot bepredicted, and those that can be are often predicted poorly.Models which attempt to predict the filtered chemical state are significantly more com-plex than ad hoc models. Although these models are conventionally referred to as turbulence-chemistry interaction models, they can be more accurately viewed as attempting to capturethe interaction of a filter (or turbulence model) with chemistry, and would therefore be more29The spatial resolution requirement may be even more stringent than the flow scales suggest if the smallestreaction scales are smaller than the smallest flow scales.30This modelling may involve the definition of additional state variables, which can be viewed as repre-senting an ad hoc chemical state, butin contrast to the reduced chemical state of Section 4.2the ad hocchemical state cannot be used to reconstruct all properties in the full chemical state.444.3. Turbulence-Chemistry Interaction Modellingproperly referred to as filter-chemistry interaction models. These models typically evaluatethe un-closed filtered terms using conditionally-filtered fields (defined fully in Section B.1.3)and filtered probability density functions (PDFs; defined fully in Section B.2.4), to wit,∀ψ ∈ Ψˆrchem : ω˙ψ˜ =∫KK∗ω˙ψ˜ · PK∗s˜ub dK∗ (4.46)∀ψ ∈ Ψrchem : ψ˜ =∫KK∗ψ˜ · PK∗s˜ub dK∗ (4.47)where K is the set of conditioning variables (often {Z} for non-premixed combustion or{c} for premixed combustion) and K is the set of all possible values of the conditioningvariables or conditional domain (often [0, 1]). Specific models differ in two aspects: thedegree of conditioning in the previous equations (full or partial), and the method by whichthe PDF is evaluated. These options are discussed in the following two subsections, whilespecific turbulence-chemistry interaction models (corresponding to different combinations ofoptions) are presented in Section 4.3.3.4.3.1 Degree of ConditioningThe conditioning in Equations 4.46 and 4.47 may be complete (every element of the reducedthermodynamic state basis is a conditioning variable) or partial (not all elements of thereduced thermodynamic state basis are conditioning variables).Full ConditioningWhen full conditioning is applied, the conditionally-filtered quantities in Equations 4.46and 4.47 can be evaluated by using the conditioning variables as arguments to the chemistrymodel. No further modelling is required to close conditional quantities, but the correspond-ing PDF represents the joint probability distribution of all variables in the reduced ther-modynamic state basis, which may be very complex (and, by extension, difficult to predictcorrectly) if the reduced thermodynamic state basis has many elements.Partial ConditioningThe complexity of the PDF to be evaluated can be reduced by conditioning on only someelements of the reduced thermodynamic state basisideally those which are most determi-native. This partial conditioning gives a PDF which is less complex (and, by extension,easier to predict correctly) but leaves all of the conditionally-filtered quantities in Equa-tions 4.46 and 4.47 un-closed. The typical route to closure is to assume that the selectedconditioning variables are sufficiently determinative that the conditional filter has a smallimpact, and therefore commutes with function application. This gives rise to the so-called454.3. Turbulence-Chemistry Interaction Modellingfirst moment closure approximation,∀ψ ∈ Ψˆrchem :K∗ω˙ψ˜ ≈ ω˙ψ K∗Ψˆrtherm˜(4.48)∀ψ ∈ Ψchem :K∗ψ˜ ≈ ψ K∗Ψˆrtherm˜(4.49)where the functions ω˙ψ and ψ are provided by the chemistry model, and the conditionally-filtered reduced thermodynamic state basis is defined asK∗Ψˆrtherm˜ ≡{K∗ψ˜ : ψ ∈ Ψˆrtherm}. (4.50)The first moment closure approximation implies that closure for the conditional thermody-namic stateK∗Ψˆrtherm˜ provides closure forK∗ω˙ψ˜ andK∗ψ˜. As discussed further in Section 4.3.3, theconditional state may be closed by solving conditional transport equations or by invertingEquation 4.47.4.3.2 Evaluation of PDFsThe PDF may be evaluated by presuming a functional form or by solving a transportequation.Presumed PDFIn the presumed PDF approach, the PDF is assumed to take some functional form, param-eterized by filtered mixed moments of the conditioning variables. For the PDF of a singlescalar, for example, it is typical to parameterize the PDF in terms of the filtered field (thefiltered raw moment of degree one) and filtered variance (the filtered central moment ofdegree two). The required moments may be evaluated through direct modelling (in termsof resolved variables) or by solving transport equations.As the number of conditioning variables increases:1. the PDF becomes more difficult to probe experimentally, making it harder to generatean empirical model;2. the mathematical form of the PDF can become more complicated, increasing thenumber of moments required for parameterization; and3. the number of mixed moments required to parameterize the PDF grows super-linearly:for a PDF parameterized using moments of degree d or lower, the number of mixedmoments requiring closure is |K| · d when the variables are statistically independentand(|K|+d−1d)in general.In light of these factors, the presumed PDF approach is generally only tractable when thenumber of conditioning variables is small and many or all of the variables are statisticallyindependent.464.3. Turbulence-Chemistry Interaction ModellingTransported PDFIn the transported PDF approach, a transport equation for the PDF is derived and solved.This introduces two main challenges:1. the diffusion term in the PDF transport equation is un-closed, and must be modelled;and2. the computational cost of a discrete simulation becomes phenomenally large: if thePDF is discretized using a rectangular grid with n bins for each of the |K| conditions,then transporting the PDF is tantamount to solving n|K| transport equations.The second challenge can be addressed by using Monte Carlo techniques in lieu of resolvingthe full PDF, but even with this simplification transporting the PDF is quite expensive. Inexchange for this expense, the joint PDF allows one to evaluate not only the filtered valuesof the conditioning values, but also detailed statistical information such as filtered mixedmoments.4.3.3 Specific ModelsIt is traditional to recognize three varieties of turbulence-chemistry interaction model.Flamelet Models apply flamelet manifold chemistry, may apply full or partial condition-ing, and model the PDF. In these models, the inner product operations in Equa-tions 4.46 and 4.47 are typically interpreted as constructing the turbulent (filtered)flame from an ensemble of laminar flamelets. Overall, flamelet models are computa-tionally straightforward and very successful [48, 53, 46, 15], but they are not (strictlyspeaking) applicable at high turbulence intensity, where the chemical and turbulentlength scales become comparable and the flamelet assumption breaks down.Conditional Moment models apply any chemistry, invoke partial conditioning, and modelthe PDF. The two main conditional moment approaches differ in how they close theconditional state:1. in Conditional Moment Closure (CMC) [54, 55, 56], the conditional state is eval-uated by solving conditional transport equations (which must be closed by mod-elling transport terms associated with the physical gradients of conditional quan-tities), while2. in Conditional Source-term Estimation (CSE) [57, 58], the conditional state isevaluated by inverting integral equations.PDF models [59] may apply any chemistry, invoke full conditioning, and transport thePDF.This work focusses on conditional moment models. Further background on CSE is presentedin Chapter 5 (which applies and extends the technique), while further background on CMCis presented in Chapter 7 (which presents a novel conditional moment method and, inevaluating it, contrasts it with CMC).47Part IIResearch Contributions48Chapter 5An Ensemble-free Variant of theConditional Source-term Estimation(CSE) Method based on GeometricConditioning VariablesThis chapter introduces and tests a novel variation of the Conditional Source-term Estima-tion (CSE) algorithm for turbulent combustion modelling. In contrast to traditional CSE,which accounts for spatial variation in conditional properties by dividing the domain intoensembles, the novel variation treats position as an additional conditioning variable anddispenses with the notion of ensembles altogether.5.1 IntroductionAs discussed in Section 4.3.3, the CSE model is a conditional moment method for turbulence-chemistry (filter-chemistry) interaction modelling. Like all conditional moment methods,it may theoretically be employed with any of the simplified chemistry models introducedin Section 4.2, and relies on a modelled PDF; a generic version of the CSE algorithm,applicable to any combination of chemistry and presumed PDF models, will be presented inSection 5.2.1.The CSE approach has been applied successfully to a wide variety of turbulent combus-tion systems, including non-premixed [57, 58], premixed [60], partially premixed [61], andeven Moderate Inert Low Dilution (MILD) combustion [62, 63]. The approach has also beencombined with a variety of reduced chemistry models, including two-step chemistry [57, 58],Trajectory-Generated Low Dimensional Manifold (TGLDM) [64], and Flamelet GeneratedManifold [65] (as first suggested in [66]). One aspect of the model which remains the subjectof active research is the algorithm by which spatial variation in conditionally-filtered fieldsis accounted for.5.1.1 CSE EnsemblesThe key assumption in CSE is that conditionally-filtered fields vary slowly in space, andthus• the domain can be divided into ensembles across which conditionally-filtered fields areeffectively uniform, and495.1. Introduction• each ensemble can be assigned a distinct conditionally-filtered state by solving anensemble-specific inverse problem.The details of how ensembles are identified, how inter-ensemble smoothing is applied, andhow ensemble data is managed when the domain has been decomposed into disjoint subdo-mains for parallel execution are discussed below.Ensemble Division SchemesGiven that CSE is based on the idea that conditionally-filtered fields vary slowly in space, itis natural to require that each CSE ensemble should be as localized as possible (ideally con-tiguous). Unfortunately, this constraint does not define a unique set of ensembles; Figure 5.1illustrates three of the many ways in which the domain enclosing a jet flame could be dividedinto four contiguous ensembles. For jet flames, the slabs division scheme (Figure 5.1a) isgenerally viewed as the most reasonable, and has been used with some success [58, 64, 65],but it is not clear how this scheme would generalize to the case of multiple inlets and/oroutlets.One ensemble division scheme which attempts to consider arbitrary geometries in animpartial way (i.e., without requiring physical insight from the user) is the Morton orderingscheme of Nivarti and Bushe [67]. The scheme is based on drawing a space-filling Mortoncurve [68] through the discretized domain. Since points which are close to one another in thedomain are often close to one another along the Morton curve, dividing the Morton curve intosegments and grouping the cells that each segment passes through into an ensemble tendsto give relatively localized ensembles31. Although this scheme would be both impartial andgeneral, it has yet to be applied to a full CSE calculation.Inter-Ensemble SmoothingWithout some sort of inter-ensemble smoothing, CSE ensemble inversions are distinct prob-lems; there is no constraint to prevent the results from varying significantly between neigh-bouring ensembles. This prospect is theoretically unpalatable, as it is in direct contraventionof CSE's stated assumption that conditional fields should vary slowly in space. In practice,failure to apply smoothing can produce reaction rate fields which feature non-physical stepchanges at ensemble boundaries. This issue is traditionally addressed by defining overlappingensembles via the following algorithm:1. divide the domain into disjoint ensemble cores, then2. enlarge each ensemble core into an ensemble by unifying it with an overlap regionconsisting of one or more neighbouring ensemble cores.Figure 5.2 illustrates one way in which this algorithm might define overlapping ensemblesfor a jet flame.When applying CSE with overlapping ensembles, the inverse problem defining each en-semble's conditional state is based on data from the entire ensemble, but the resulting31The Morton curve also occasionally leaps between points which are physically distant in the spatialdomain, and (as discussed in [67]) a CSE ensemble division algorithm must account for this.505.1. Introduction0123(a) Slabs11 22 33 00(b) Annuli0123(c) UnstructuredFigure 5.1: Examples of CSE ensemble shapes in a jet flameconditional state is only applied across the ensemble core. The overlap ensures that theinversions associated with any pair of neighbouring ensembles are, in part, based on thesame data, effectively applying an ad hoc form of inter-ensemble smoothing.Interaction with Parallel Execution SubdomainsEnsemble division effectively introduces a layer of structure to the domain by decomposingit into overlapping ensembles. As illustrated in Figure 5.3, this can present a challenge whenimplementing CSE in parallel computational fluid dynamics (CFD) solvers, which introducean independent layer of structure by decomposing the domain into disjoint subdomains andassigning each subdomain to a processor. In the standard message passing approach toparallel computation, each processor tracks the data associated with its own subdomain;in the absence of some simple relationship between subdomains and ensembles, there isno straightforward algorithm for re-grouping this data by ensemble for inversion. Thisprogramming challenge is sufficiently intimidating that traditional CSE implementationsenforce the constraint that subdomains and CSE ensemble cores must coincide.Althoughthis choice circumvents a tedious data-rearrangement problem, it implies that the numberof ensembles, nens, must equal the number of processors used for execution, nproc, which hasboth theoretical and practical drawbacks.515.1. Introduction012345(a) Ensemble 0012345(b) Ensemble 1012345(c) Ensemble 2Figure 5.2: Example of traditional CSE ensemble overlap in a jet flame with six ensemblecores and an overlap of one ensemble core in each direction. Numbers identify ensemblecores; within each sub-figure, the ensemble core is heavily shaded, while the ensemble overlapregion is lightly shaded.Theoretical The predictions of the CSE approach, and thus of the entire simulation, de-pend on the choice of ensembles [69]. This means that solving the same problemwith a different numbers of processors (and thus a different number and arrangementof ensembles) will generate results which differ by more than the inevitable floatingpoint imprecision-associated error. It is problematic for the result of a computationto depend on the number of processors used to perform it.Practical The requirement that CSE ensembles contain enough diversity of data to permitmeaningful inversion places a lower bound on the number of cells per ensemble, andthus an upper bound on the number of ensembles which can be used for a given domain.Because nproc= nens, this in turn places an upper bound on the number of processorswhich can be used, and prevents CSE from being applied with a truly massive numberof processors (as would be required for the fast turnaround times which are requiredby industry).Tsui and Bushe [65] recently proposed two alternate ensemble division schemes designedto alleviate the practical issues above.525.1. Introduction(a) Domain012345(b) Ensembles01234567(c) SubdomainsFigure 5.3: Example domain, ensembles, and subdomains for CSE of a jet flame usingtraditional ensemble division. Ensemble 4 is shaded in the subdomains sub-figure to illustratehow the two levels of structure clash.Semi-Dynamic Ensemble Division: Each CSE ensemble remains associated with acore sub-domain, but the number of contiguous overlap sub-domains varies in orderto maintain the required diversity of data for inversion. This eliminates the lowerbound on the number of cells per ensemble, and thus the upper bound on the numberof processors which can be used in a CSE simulation, without removing the nproc= nensconstraint.Dynamic Ensemble Division: CSE ensembles are still constructed as the union of sub-domains, but the requirement that each sub-domain serve as the core of an ensembleis removed. This replaces the nproc= nensconstraint with the less restrictive re-quirement that nproc≥ nens, allowing for CSE with an arbitrarily large number ofprocessors.Although both of these schemes theoretically permit CSE with an arbitrarily large numberof processors, neither address the theoretical issue that the associated CSE solution canvary with the number of processors used in simulation. The schemes are also only capableof applying inter-ensemble smoothing using the ad hoc approach (inter-ensemble overlap).535.2. Theory5.1.2 Research Question and ObjectivesThis chapter aims to answer the question of whether it is possible to modify the CSEalgorithm in such a way as to alleviate some or all of the ensemble division-associated issuesdiscussed above. The associated objective is to propose and test such a CSE variant. As willbe shown presently, the technique proposed is based on a generalization of the conditionalfiltering procedure, specifically a generalization in which position is treated as an additionalconditioning variable. In the larger context of this thesis, this chapter can therefore beviewed as answering the research question what are the implications of treating position asa conditioning variable in CSE?5.2 TheoryAlthough the proposed modification to the CSE algorithm is quite straightforward, theoriginal CSE algorithm is rather complicated. In the interests of clarity (particularly ofnotation), the traditional CSE algorithm is presented in Section 5.2.1 before introducing theproposed modification in Section 5.2.2.5.2.1 The Traditional CSE ModelEach CSE calculation can be divided into three stages: pre-processing, ensemble divisionand simulation.Pre-ProcessingCSE pre-processing proceeds as follows:1. Identify an appropriate set of (traditional) conditioning variables Ktrad, and an associ-ated conditional domain Ktrad representing the set of all allowable values of Ktrad. Theconditional domain can also be viewed as a set of points in a traditional conditioningspace. Typical choices areKtrad={{c} (premixed){Z} (non-premixed) (5.1)Ktrad ={[0, 1] (premixed or non-premixed) . (5.2)2. Select a presumed PDF model for estimating the PDF, defined asPK∗trads˜ub ≡ ρ · δ (K (~x, t)−K∗)ρ. (5.3)where K∗ is a set of independent variables corresponding to K. In general, this modelwill require, as inputs, the set of filtered conditioning variablesKtrad˜={ψ˜ : ψ ∈ Ktrad}(5.4)545.2. Theoryand also some set KM˜tradof filtered moments of conditioning variables. Typical modelsgiveKM˜trad={c v˜ar}={c˜2 − c˜2}(premixed){Z v˜ar}={Z 2˜ − Z˜2}(non-premixed).(5.5)3. Select a reduced chemistry model (Section 4.2), which describes how to estimate anylocal chemical property given the local reduced chemical state basis Ψˆrchem. Formethane-air chemistry, manifold-based CSE approaches [64, 65] often use the setΨˆrchem = {Z, YCO2 , YH2O} (5.6)while full chemistry would correspond to defining Ψˆrchem = Ψˆchem.Steps 13 collectively define the CSE state basis,ΨˆCSE ≡ K˜trad⋃KM˜trad⋃Ψˆrchem˜ (5.7)which provides complete information about the filtered chemical state.4. Select minor sub-models:• one sub-filter advection model for each element of ΨˆCSE (typically of the sameform as Equation 4.21), and• one sub-filter scalar dissipation model for each element of KM˜.These auxiliary models are required to close the transport equations for the elementsof ΨˆCSE.5. Partition the spatial domain X (a set of points in real space) into a discrete spatialdomain X. In a finite volume solver, the discrete domain is a set of cells (sets ofpoints) in real space.6. Partition the conditional domain Ktrad (a set of points in traditional conditioningspace) into a discrete conditional domain Ktrad . In a finite volume solver, the discreteconditional domain is a set of cells (sets of points) in traditional conditioning space.Another aspect of pre-processing (which is not unique to CSE) is selecting a physicalstate basis Ψˆphys (i.e. a representation of the non-chemical aspects of the thermodynamicstate). Common choices areΨˆphys ={{P, h} (isobaric, adiabatic combustion){v, u} (isochoric, adiabatic combustion). (5.8)As previously illustrated in Equations 4.40 and 4.41, the combination (union) of the re-duced chemical state basis Ψˆrchem and the physical state basis Ψˆphys can be labelled as thereduced thermodynamic state basis Ψˆrtherm, and provides complete information about thethermodynamic state.555.2. TheoryEnsemble DivisionTraditional CSE ensemble division can be formalized as follows.1. Identify a set G of geometric variables g which is collectively expected to account for allvariation in conditionally-filtered fields (i.e., for which it is expected that, at any giveninstant, points with the same G (~x) always have the same conditionally-filtered fieldvalues). In three dimensions, the set G = {x, y, z} is guaranteed to be sufficient, butsmaller sets are often selected; the slabs division scheme in Figure 5.1a, for example,corresponds to an assumption that all spatial variation in conditionally-filtered fieldscan be accounted for by downstream position z, i.e., G = {z}.2. Identify the geometric domain G, defined as the set of all combinations of G valueswhich could appear in the system of interest:G ≡ {G (~x) : ~x ∈ X} . (5.9)This set can be viewed as a set of points in a geometric space, or as a projection ofthe set of all possible positions ~x (i.e., of the spatial domain X ).3. Partition the geometric domain into a discrete geometric domain G. In a finite vol-ume solver, the discrete geometric domain is a set of cells (sets of points) in geometricspace.4. Define each continuous ensemble core E and discrete ensemble core E as the set ofspatial points or cells whose G fall within a given bin G∗ , i.e.∀G∗ ∈ G : E (G∗) = {~x ∈ X : G (~x) ∈ G∗} (5.10)∀G∗ ∈ G : E (G∗) = {~x ∈ X : G (~x) ∈ G∗} . .(5.11)5. Use the ensemble cores E as building blocks to construct ensembles ε. In most CSEimplementations, the process is performed once during pre-processing, giving a staticset E or E of continuous or discrete ensembles, but in the semi-dynamic and dynamicapproaches of Tsui and Bushe [65], this is performed at each time step based oninstantaneous conditions, giving a dynamic set of continuous or discrete ensemblesE(t) or E(t).SimulationDuring simulation, the CSE state basis is evolved by solving transport equations (one perstate basis element). With sub-models available for the sub-filter advection and sub-filterscalar dissipation, the only un-closed terms in these transport equations are the reaction-associated sources,ω˙ΨˆCSE ≡{ω˙ψ : ψ ∈ ΨˆCSE}, (5.12)which can be broken up asω˙ΨˆCSE = ω˙Ktrad˜⋃ω˙KM˜trad⋃ω˙Ψˆrchem˜ (5.13)565.2. Theorywhereω˙Ktrad≡ {ω˙ψ : ψ ∈ Ktrad} (5.14)ω˙KM˜trad≡{ω˙ψ : ψ ∈ KM˜trad}(5.15)ω˙Ψˆrchem ≡{ω˙ψ : ψ ∈ Ψˆrchem}(5.16)and the reaction-associated sources of filtered moments will be formally defined in Equa-tion 5.31. The core CSE algorithm allows these sources terms and (optionally) the fullfiltered chemical state Ψchem˜, to be closed in terms of ΨˆCSE. Figure 5.4 illustrates thisalgorithm as applied in a discrete solver at some time t. The figure makes a distinctionbetween the set of traditional conditioning variables, Ktrad(with domain Ktrad or Ktrad ina discrete solver), and the set of all conditioning variables, K (with domain K or K in adiscrete solver). In the traditional CSE algorithm, these sets and their associated domainsare identical, but in the proposed modification which will be introduced in Section 5.2.2, thesets will become distinct. The steps in the core CSE algorithm are as follows.1. Set the PDF. Use the PDF model to estimate the sub-filter PDF in each cell usingthe local filtered conditioning variables Ktrad˜(~x, t) and the local filtered moments ofconditioning variables KM˜trad(~x, t). In a continuous solver, the PDF would be set ateach position ~x in the domain X ,∀~x ∈ X : PK∗s˜ub (~x, t) = f(Ktrad˜(~x, t) ,KM˜trad(~x, t)), (5.17)where the form of the function f is determined by the PDF model. In a finite volumesolver, however, the discretized PDF is set for each cell ~x in the discretized domainX,∀~x ∈ X : P K∗s˜ub (~x, t) = f (Ktrad˜ (~x, t) ,KM˜trad(~x, t)) , (5.18)where the sinuous over-arrow indicates an array in which• there is one value per bin in discretized conditional space K, and• each array element represents the integral of the PDF over the correspondingconditional bin K∗ .The discretized PDF array elements are therefore defined as∀K∗ ∈ K :P K∗s˜ubK∗≡∫K∗PK∗s˜ub dK∗. (5.19)2. Assemble ensemble-level data arrays. These arrays are annotated with a solidover-arrow, indicating that they contain one value per cell in the ensemble and thateach array element represents the value in the corresponding cell. This step can bedivided into two sub-steps.575.2. TheoryKM˜tradK˜trad Ψˆrchem˜ Ψˆphys˜ΨˆCSEPK∗s˜ub12aPK∗s˜ub2bΨˆrtherm˜3K∗Ψˆrtherm˜K∗ω˙Ktrad˜K∗ω˙Ψˆrchem˜4aK∗Ψchem˜4b5aω˙Ktrad˜6ω˙KM˜trad5bω˙Ψˆrchem˜5cΨchem˜ω˙ΨˆCSEPer Cell(∀~x ∈ X)Per Ensemble(∀ε~x ∈ E~x(t))Figure 5.4: The core CSE algorithm for a discrete solver. Boxes denote variables, circlesdenote steps, and lines illustrate the flow of information. Dotted lines pass uninterruptedbehind both the Per Ensemble panel and solid lines. Steps 4b and 5c are optional extensionswhich together allow calculation of any filtered chemical property.585.2. Theory(a) Assemble discretized PDFs (which are each arrays with one element per condi-tional bin) into an array:∀ε ∈ E(t) :∀~x ∈ ε :P K∗s˜ub (ε, t)~x≡ PK∗s˜ub (~x, t) . (5.20)The resulting array of arrays can be viewed as a 2-D array (matrix) with oneelement per ordered pair (~x ∈ E(t),K∗ ∈ K).(b) Combine the reduced chemical state basis with the physical state basis (givingthe thermodynamic state basis) and assemble the result into an array:∀ε ∈ E(t) :∀~x ∈ ε : [Ψˆrtherm˜ (ε, t)]~x≡ Ψˆrchem˜ (~x, t)⋃ Ψˆphys˜ (~x, t) . (5.21)The resulting 1-D array of sets of values can be viewed as a 2-D array of valueswith one element per ordered pair (~x ∈ E(t), ψ ∈ Ψˆrtherm).CSE implementations can be structured in such a way that this step has a negligiblecomputational cost.3. Invert for the conditional reduced thermodynamic state basis. This involvessolving one integral equation for each element of the reduced thermodynamic statebasis. In a continuous solver, the equations are∀ε ∈ E(t).∀ψ ∈ Ψˆrtherm :∀~x ∈ ε :∫KPK∗s˜ub (~x, t) ·K∗ψ˜ (ε, t) dK∗ = ψ˜ (~x, t) (5.22)while in a discrete solver they are∀ε ∈ E(t). ∀ψ ∈ Ψˆrtherm :∑K∗∈KP K∗sub (ε, t)K∗·K∗ψ˜ (ε, t)K∗= ψ˜ (ε, t) (5.23)where the dotted over-arrow on the conditionally filtered field indicates an array inwhich• there is one value per bin in discretized conditional space K, and595.2. Theory• each array element represents the average of the PDF over the correspondingconditional bin K∗ .The conditionally filtered fields are therefore defined by∀K∗ ∈ K : K∗Ψˆrtherm˜K∗≡∫K∗K∗Ψˆrtherm˜ dK∗∫K∗dK∗. (5.24)In both the continuous and discrete cases, the goal is to identify the conditionally-filtered ψ, but this is (typically32) complicated by the fact that the system is• over-constrained, meaning that the solution can only satisfy the equation insome bulk (e.g. least-squares) sense, and• ill-conditioned, meaning that the solution is sufficiently sensitive to variationsin the input that it is effectively meaningless unless the system is solved withregularization.CSE solvers therefore seek a regularized least-squares solution. This inversion processis by far the most computationally demanding step in the algorithm; one optimizationavailable to combat this is to assume that the conditional reduced thermodynamicstate basis evolves slowly in time, and update it (via inversion) relatively infrequently.4. Apply the chemistry model conditionally. For a continuous solver, this wouldamount to∀ε ∈ E(t).∀K∗ ∈ K.∀ψ ∈(ω˙K⋃ω˙Ψˆrchem⋃Ψchem):K∗ψ˜ (ε, t) = ψ K∗Ψˆrtherm˜ (ε, t)(5.26)(where the function ψ comes from the reduced chemistry model) while for a discretesolver it is instead∀ε ∈ E(t). ∀K∗ ∈ K.∀ψ ∈ (ω˙K⋃ ω˙Ψˆrchem⋃Ψchem) :K∗ψ˜ (ε, t)K∗= ψ K∗Ψˆrtherm˜ (ε, t)K∗ . (5.27)32Under certain assumptions, the conditionally-filtered values of the physical (non-chemical) propertiescan be deduced in advance: for example, if the system is assumed to be approximately isobaric and heattransfer and viscous dissipation are assumed to be insignificant, thenK∗P˜ = PK∗h˜ = h (5.25)because P and h are global constants.605.2. TheoryIn both cases, applying the relationship to ψ ∈ Ψchem (i.e., step 4b) is an optionalextension which facilitates prediction of any filtered chemical property.5. Evaluate unconditionally-filtered fields. For a continuous solver, this wouldamount to∀~x ∈ X . ∀ψ ∈(ω˙K⋃ω˙Ψˆrchem⋃Ψchem):ψ˜ (~x, t) =∫KPK∗s˜ub (~x, t) ·K∗ψ˜ (ε∗ (~x, t) , t) dK∗ (5.28)where function ε∗(~x, t) returns the main ensemble associated with point ~x at time t33, while in a discrete solver it is instead∀~x ∈ X. ∀ψ ∈ (ω˙K⋃ ω˙Ψˆrchem⋃Ψchem) :ψ˜ (~x, t) = ∑K∗∈KP K∗s˜ub (~x, t)K∗·K∗ψ˜ (ε∗ (~x, t) , t)K∗(5.29)where function ε∗(t) returns the main discrete ensemble associated with cell  at timet 34. In both cases, applying the relationship to ψ ∈ Ψchem (i.e., step 5c) is an optionalextension which facilitates prediction of any filtered chemical property.6. Evaluate unconditionally-filtered sources of filtered moments. In practicalCSE implementations, it is typically assumed that the highest-order filtered momentof interest is the variance and that the conditioning variables are independent, meaningKM˜ = K v˜ar ≡{κ v˜ar : κ ∈ K}(5.30)The associated source terms are then defined (in this work) by∀κ ∈ K : ω˙κ v˜ar ≡ 2(κω˙κ˜ − κ˜ω˙κ˜). (5.31)In a continuous solver, these can be evaluated as∀~x ∈ X . ∀κ ∈ K :ω˙κ v˜ar (~x, t) =∫K2 · (κ∗ − κ˜ (~x, t)) ·K∗ω˙κ˜ (ε∗ (~x, t) , t) dK∗ (5.32)33For overlapping ensembles, the main ensemble associated with (~x, t) is the unique ensemble whose corecontains ~x at time t; for non-overlapping ensembles, the main ensemble associated with (~x, t) is the uniqueensemble containing ~x at time t.34For overlapping ensembles, the main discrete ensemble associated with (~x, t) is the unique discreteensemble whose core contains ~x at time t; for non-overlapping ensembles, the main ensemble associatedwith (~x, t) is the unique ensemble containing ~x at time t.615.2. Theorywhile in a discrete solver they are instead∀~x ∈ X. ∀κ ∈ K :ω˙κ v˜ar (~x, t) = ∑K∗∈K2[(κ(K∗)− κ˜ (~x, t)) · K∗ω˙κ˜ (ε∗ (~x, t) , t)] (5.33)where the function κ(K∗) returns the κ∗ coordinate of the centroid (in conditionalspace) of conditional cell K∗ .5.2.2 CSE with Geometric Conditioning Variables (CSE-GCV)The proposed modification to the CSE algorithm, labelled CSE with Geometric ConditioningVariables (CSE-GCV), is based on the idea of enlarging the set of conditioning variables Kto include both the traditional conditioning variables Ktradand the geometric variables G,to witK ≡ Ktrad⋃G. (5.34)Through this modification, it is possible to eliminate CSE ensembles altogether, although itis necessary to modify the conditioning space and PDF and to add regularization conditionsto give a solution which is smooth in geometric conditioning space.Elimination of EnsemblesWhen filtered conditionally on the enlarged set of variables K, fields become functions ofthe set independent/dummy conditioning variables K∗tradassociated with traditional scalarfields and the G∗ set of independent/dummy geometric variables; this is convenient becauseit implies that all spatial variation in conditionally filtered fields is accounted for throughdependence on G∗, and thus the conditional state is independent of ~x. This independenceimplies that no additional effort is required to account for spatial variation in conditionally-filtered fields; in particular, the ensemble formalism of traditional CSE can be dispensedwith entirely. In the context of the algorithm of Section 5.2.1, removing the ensembleformalism is tantamount to defining a single, global ensemble, to wit∀t ∈ T : E(t) = {X} E(t) = {X} . (5.35)Modification of Conditioning Space and PDFSince CSE-GCV has additional conditioning variables, the conditioning space is enlargedK = Ktrad× G |K| = |Ktrad| · |G| (5.36)K = Ktrad × G |K| = ∣∣∣Ktrad ∣∣∣ · |G| (5.37)where × is the Cartesian product. This implies that the PDF to be modelled has more binsthan in traditional CSE, and represents the joint probability of a different constellation of625.2. Theoryconditions:PK∗s˜ub =PK∗trads˜ubtraditional CSEPK∗trad⋃G∗s˜ubCSE-GCV.(5.38)Fortuitously, closing the new PDF requires no additional modelling assumptions. The tra-ditional presumed PDF model makes no allowance for possible variation with positioni.e.,it implicitly assumes that Ktrad(~x, t) and G (~x, t) are statistically independentso the newPDF can invoke the same assumption to justify the simplificationPK∗s˜ub = PK∗trads˜ub ⊗ PG∗sub. (5.39)The symbol ⊗ indicates the operation by which an array with one element per K∗ (or,equivalently, per (K∗trad,G∗) combination) is generated as the outer product of an arraywith one element per K∗tradand an array with one element per G∗ . This can be writtenmore explicitly as∀K∗ ∈ K∗ : P K∗s˜ubK∗=PK∗tradsub˜K∗trad(K∗ )·P G∗subG∗ (K∗ )(5.40)where• K∗trad(K∗) returns the cell in the traditional conditioning space K∗tradcorrespondingto a cell in the overall conditioning space K∗, and• G∗ (K∗) returns the cell in the geometric conditioning space G∗ corresponding to acell in the overall conditioning space K∗.The first PDF component in Equations 5.39 and 5.40 is precisely the PDF of traditionalCSE, which is closed by the presumed PDF model; the second component can be evaluatedwithout further modelling by applying its definition,∀~x ∈ X. ∀G∗ ∈ G :P G∗sub (~x, t)G∗=∫G∗ δ (G (~x, t)−G∗) dG∗. (5.41)which is closed because the grids X and G are fully defined35. Equation 5.41 can be viewedas a parameter-free PDF model; it follows that CSE-GCV requires no additional filtered35For static grids, all t-dependence is eliminated, so this term can even be evaluated as a pre-processingstep.635.2. Theoryfields or moments to define the full PDF, despite the addition of conditioning variables. InRANS, or in LES where |X|  |G| (the spatial domain resolution is far finer than binningof geometric conditioning variable space, as is standard with traditional CSE ensembles),this model reduces to a delta function for all spatial cells except the vanishingly small setwhose range of Gs spans more than one G∗ , i.e.∀~x ∈ X.∀G∗ ∈ G :P G∗sub˜ (~x, t)G∗≈ δ (G (~x, t)−G(G∗)) . (5.42)where the function g(G∗) returns the g∗ coordinate of the centroid (in geometric space)of the geometric cell G∗ . This allows Equation 5.29 to be simplified to∀~x ∈ X.∀ψ ∈ (ω˙K⋃ ω˙Ψˆrchem⋃Ψchem) :ψ˜ (~x, t) = ∑K∗trad∈KtradPK∗tradsub˜K∗trad·K∗ψ˜ (ε∗ (~x, t) , t)K∗trad,G∗ (~x)(5.43)where G∗(~x) returns the bin in geometric space associated with cell ~x (replacing thePDF of G) and the summation is now over the traditional conditioning space rather than theextended conditioning space. This simplification significantly reduces the number of floating-point operations required (i.e., the computational cost and, by extension, simulation time)when the number of bins in geometric conditioning space is large.Smoothing in Geometric Conditional SpaceAs stated in Section 5.1, traditional CSE requires inter-ensemble smoothing to ensure thatconditional fields do, in fact, vary slowly in space. Since CSE-GCV uses one ensemble,the ad hoc smoothing procedure of traditional CSE is not applicable; some other proceduremust be invoked to ensure that conditional fieldsK∗ψ˜ vary slowly with G∗. Fortuitously, thisis straightforward to account for in step 3 of the core CSE algorithm by adding regulariza-tion conditions requiring that the conditionally-filtered state should be smooth along eachgeometric dimension g∗ ∈ G 36.SummaryOverall, the changes required to transform traditional CSE into CSE-GCV are:1. skip steps 45 of traditional CSE ensemble division, and instead define a single globalensemble,36For details on the mathematics of regularization conditions, see [70].645.3. Methods2. modify step 1 of the core CSE algorithm (PDF evaluation) to account for sub-filtervariation in G as well as Ktrad, and3. modify step 3 of the core CSE algorithm (inversion) to add additional regularizationconditions enforcing smoothness in G∗.Notably, the CSE state basis (Equation 5.7) is un-changed. The core CSE algorithm cantherefore still be represented by the diagram in Figure 5.4.5.3 MethodsThe hypothesis of this work is that CSE-GCV can simulate a turbulent flame just as tradi-tional CSE can. Given this, Sandia Flame D (described thoroughly in Section E) is selectedas a test case; it has previously been successfully simulated using CSE [64] and has extensiveexperimental data available for comparison [71, 72]. The spatial domain has a cylindricalshape, and is discretized using 2.12× 106 cells, with finer axial resolution near the inlet andfiner radial resolution near the shear layer.The conditioning variables are selected as mixture fraction Z and axial position zKtrad = {Z} G = {z} K = {Z, z} . (5.44)Mixture fraction space is discretized using 50 bins (with increased resolution between thepremixed flammability limits), while geometric conditioning space is discretized using 128bins (with increased resolution near the jet inlet); the latter aligns with how ensembleswould have been defined in a traditional CSE simulation using 128 processors. TGLDM [73]is selected as the reduced chemistry model (as in [64]), givingΨˆrchem = {Z, YCO2 , YH2O} (5.45)and the sub-filter PDF of Z is modelled using the beta distribution, givingKM˜ ={Z v˜ar = Z 2˜ − Z˜2}. (5.46)Although the TGLDM table is based on full chemistry (UBC Mech 3.0 [42]), only ninespecies (CH4, O2, N2, CO2, H2O, CO, OH, H2, NO) are resolved in the LES. Following theexample of previous CSE simulations [64], the NO mass fraction is not closed via steps 4band 5c of the CSE algorithm, but rather by solving a transport equation in which the sourceterm is closed via steps analogous to 4a and 5a.The system is evolved using a custom finite-volume solver based on OpenFOAM [74](version 3.0.1), with standard OpenFOAM boundary conditions applied for velocity andpressure as described in Table 5.1. All simulations use second-order approximations forderivatives in space and time. Fields are time-averaged over a period of 2 pilot flow-throughtimes.655.4. ResultsTable 5.1: Boundary conditions on velocity and pressure. Scheme implementations aredescribed thoroughly in the documentation and source code of OpenFOAM version 3.0.1 [74].Boundary Velocity PressureFuel Stream turbulentInlet zeroGradientPilot Stream turbulentInlet zeroGradientCoflow fixedValue zeroGradientOuter Radius pressureInletOutletVelocity totalPressureOutlet pressureInletOutletVelocity totalPressure5.4 ResultsMean conditional scalar fields are presented in Figure 5.5; conditional profiles of Z, YNO,and T are not presented, since they are not used in the simulation:• the conditional Z is trivially defined,• NO is fully transported, so CSE calculates a conditional ω˙NO rather than a conditionalYNO, and• T is deduced by solving a sensible enthalpy transport equation, so CSE calculates aconditional sensible enthalpy source rather than a conditional T .The profiles of the major scalars, CO2and H2O, offer the most direct insight into the per-formance of CSE-GCV, since they are solved for directly via inversion. As the results show,these predictions of CSE-GCV show modest variation with downstream position (consistentwith the fact that CSE-GCV allows for spatial variation in conditionally-filtered quantities,but assumes that it is small) and good agreement with experimental results and traditionalCSE predictions. Errors at z = 7.5djetare likely amplified by the fact that, near the inlet,the flow field features two mixing layers (one between fuel and pilot, the other betweenpilot and coflow) rather than a single fuel-coflow mixing layer, and the presumed β-PDF istherefore incorrect. The profiles of the minor scalars CO, OH, and H2offer insight into theperformance of the TGLDM chemistry: the fact that the predictions show worse agreementwith experiment than those of the major scalars illustrates that the TGLDM chemistryintroduces some error. The jaggedness of the conditional OH mass fraction also illustratesthat the TGLDM table does not always produces a smooth output given a smooth input;this is likely a consequence of the fact that the underlying TGLDM manifold is sometimesself-intersecting, and therefore small changes in input parameters may cause one to jumpbetween different branches of the manifold37.The mean axial velocity and fluctuating kinetic energy are presented in Figure 5.6. Theresults illustrate that the turbulence is predicted well at and near the inlet, but poorlydownstream (profiles at stations further downstream, which are not presented, suggest thatthis trend continues).37Following [64], the TGLDM manifold is projected onto the YCO2 -YH2O plane, and self-intersection ishandled by clipping; this procedure can introduce cliffs where the branch retained and (by extension) thespecies mass fractions suddenly change.665.4. Results05 · 10−20.1〈YCO2 |Z〉z = 7.5djetz = 15djetz = 30djetz = 45djet05 · 10−20.1〈YH2O|Z〉02 · 10−24 · 10−26 · 10−28 · 10−2〈YCO|Z〉02 · 10−34 · 10−3〈YOH|Z〉0 0.2 0.4 0.6 0.8 102 · 10−34 · 10−36 · 10−38 · 10−3Z〈YH2 |Z〉0 0.2 0.4 0.6 0.8 1Z0 0.2 0.4 0.6 0.8 1Z0 0.2 0.4 0.6 0.8 1ZExperiment Traditional CSE CSE-GCVFigure 5.5: Time-averaged conditional profiles of scalars. Experimental data from [71],traditional CSE results from [75]. The shaded area and dashed line mark the premixedflammability limits [76] and stoichiometric composition, respectively.675.5. Discussion0204060〈vz〉 [m/s]z = 7.5djetz = 15djetz = 30djetz = 45djet0 1 2020406080100r/djet12〈~v′ · ~v′〉[m2/s2]0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djetExperiment CSE-GCVFigure 5.6: Radial profiles of mean axial velocity and fluctuating kinetic energy. Experi-mental data from [72].The mean and root-mean-square fluctuations of the scalar fields are presented in Fig-ures 5.7 and 5.8 respectively. Given that the turbulence model has not predicted the velocityfield well downstream, it is not surprising that the mixing profile is predicted well near theinlet but less effectively downstream. CSE-GCV predictions of the major scalars CO2andH2O and temperature T are in good agreement with experiment and (for CO2and H2O)traditional CSE predictions, and tend to differ only at locations where the mixture fractionis also predicted incorrectly. This, and the fact that the conditional profiles of CO2andH2O are predicted well, suggests that a CSE-GCV simulation which predicted the mixturefraction with better accuracy would also predict the major scalars more accurately. Profilesof the minor scalars CO, OH, and H2are in better agreement with experiment than thecorresponding conditional profiles; this suggests that the PDF has a smoothing effect whichmoderates the impact of errors in the conditional state on the unconditional state. As withthe major scalars, the most pronounced errors occur at locations where the mixing fieldis predicted incorrectly. Profiles of the transported minor scalar NO are in generally goodagreement with experiment, again with the greatest errors where the mixture fraction isincorrect.5.5 DiscussionWith the theory and results of the CSE-GCV method presented, several questions remain:• Do the results validate the CSE-GCV approach?• What is the practical impact of the differing implementation details of traditional CSEand CSE-GCV?• Were the study objectives fulfilled?685.5. Discussion00.20.40.60.81〈Z〉z = 7.5djetz = 15djetz = 30djetz = 45djet05 · 10−20.1〈YCO2〉05 · 10−20.1〈YH2O〉02 · 10−54 · 10−56 · 10−58 · 10−51 · 10−4〈YNO〉02 · 10−24 · 10−26 · 10−2〈YCO〉05 · 10−41 · 10−31.5 · 10−32 · 10−3〈YOH〉01 · 10−32 · 10−33 · 10−3〈YH2〉0 1 205001,0001,5002,000r/djet〈T 〉 [K]0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djetExperiment Traditional CSE CSE-GCVFigure 5.7: Radial profiles of time-averaged scalars. Experimental data from [71], traditionalCSE results (for CO2and H2O only) from [75].695.5. Discussion05 · 10−20.10.150.2〈Z′2〉12z = 7.5djetz = 15djetz = 30djetz = 45djet02 · 10−24 · 10−2〈Y ′2CO2〉1202 · 10−24 · 10−2〈Y ′2H2O〉1202 · 10−54 · 10−5〈Y ′2NO〉1201 · 10−22 · 10−23 · 10−2〈Y ′2CO〉1205 · 10−41 · 10−31.5 · 10−3〈Y ′2OH〉1205 · 10−41 · 10−31.5 · 10−32 · 10−3〈Y ′2H2〉120 1 20200400r/djet〈T ′2〉12 [K]0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djetExperiment CSE-GCVFigure 5.8: Radial profiles of root-mean-square fluctuations of scalars. Experimental datafrom [71].705.5. DiscussionThe first two questions are each addressed in their own sub-sections below, while the thirdis addressed in Section 5.6.5.5.1 Validation of CSE-GCVThe successful predictions of conditional profiles of the major scalars CO2and H2O validatethe CSE-GCV approach. The turbulence model's performance can be explained as a con-sequence of the fact that the dynamic procedure tends to perform poorly when cells havepencil-like anisotropy (two dimensions resolved significantly better than the third) [77],as is the case in the downstream section of the present grid. Future work should thereforeconsider a grid with reduced pencil-like anisotropy.5.5.2 Comparison of traditional CSE and CSE-GCVAs previously laid out in detail in Section 5.2, the implementation details of traditionalCSE and CSE-GCV are different. At the high level, these changes can be summarized asfollows: in traditional CSE, the ensemble organization is reflected in the structure of thesolver's data, whereas in CSE-GCV the positional information in is captured through thevalues of the solver's data. The differing implementations also give the two models distinctcomputational costs.Computational CostIn both traditional CSE and CSE-GCV, the dominant cost is the integral inversion (step 3),which can be performed by generating and solving the normal equations or by using aniterative method such as LSQR [78, 79] (previously combined with CSE in [65]). Thecomputational costs of applying two CSE variants with inversion performed via the normalequations and via LSQR are summarized in Tables 5.2 and 5.3 respectively; in both tables,• f is the overlap factor in traditional CSE (the number of ensembles which each spatialcell is part of), which is typically ∼ 3,• |X| is the number of cells in the spatial domain, which is typically O (106),• |G| is the number of ensembles/subdomains/processors in traditional CSE, and thenumber of processors and bins in geometric conditional space in CSE-GCV38, whichis typically O (100),• ∣∣Ktrad ∣∣ is the number of bins in traditional conditioning space, which is typically ∼ 50,and it is assumed that the PDF of G is modelled using a delta distribution, permitting theoptimization of Equation 5.43. In both cases, the inversion is highly amenable to paralleliza-tion; while there is some overhead associated with calculating global sums, this cost scaleslogaritmically with the number of processors and is generally overwhelmed by the cost ofcell-by-cell inner products, which are trivially parallelizable.38Using the same G for both approaches provides a fair comparison in which the models both capturethe same spatial information.715.5. DiscussionTable 5.2: Total computational cost of solving the normal equations via Cholesky decom-positionGenerating CholeskyApproach Normal System DecompositionTraditional CSE O(f · |X| · ∣∣∣Ktrad ∣∣∣2) O (∣∣Ktrad ∣∣3)CSE-GCV O(|X| ∣∣∣Ktrad ∣∣∣2) O (∣∣Ktrad ∣∣3 |G|3)Ratio (GCV/Trad.) O (1/f) O(|G|3)Table 5.3: Total computational cost of per iteration LSQRMatrix-Vector LSQRApproach Multiplication BookkeepingTraditional CSE O(f · |X| · ∣∣∣Ktrad ∣∣∣) O (3 ∣∣Ktrad ∣∣+ 5 |X|)CSE-GCV O(|X| ∣∣∣Ktrad ∣∣∣) O (3 ∣∣Ktrad ∣∣ |G|+ 5 |X|)Ratio (GCV/Trad.) O (1/f) O (∼ 1)When inversion is performed by solving the normal equations, substituting CSE-GCV fortraditional CSE reduces the cost of generating the system by a small factor, but significantlyincreases the cost of solving the system via Cholesky decomposition. For the exampleparameter values specified above, the normal system would be 3 times less expensive togenerate, but 106 times more expensive to solve. In traditional CSE, the cost of generatingthe normal inversion dominates and the cost of solving it is negligible, but when this cost isscaled up by a factor of one million, it becomes prohibitive; the normal equations thereforedo not provide a tractable route to inversion in CSE-GCV.When inversion is performed via LSQR, substituting CSE-GCV for traditional CSEreduces the cost of matrix-vector multiplications by a small factor and increases the costof LSQR bookkeeping negligibly (the ratio is very close to one because |X|  |K|,i.e., the term which increases makes negligible contribution to the total cost). For theexample parameter values specified above, the cost of matrix-vector multiplications wouldbe scaled by 1/3, while that of LSQR would be scaled by 1.003. Given that the cost ofthe matrix-vector multiplication dominates, CSE-GCV is less computationally demandingthan traditional CSE (or negligibly more expensive, when there is no ensemble overlap andf = 1).Taken together, these comparisons suggest that CSE-GCV is a tractable (and, indeed,generally less expensive) substitute for traditional CSE, so long as inversion is performedusing LSQR.725.6. Concluding Remarks5.6 Concluding RemarksThe objective of this chapter was to determine whether it is possible to modify the CSEalgorithm in such a way as to alleviate some or all of the ensemble division-associatedissues articulated in Section 5.1. This objective is fulfilled, with the question answeredin the affirmative: the new CSE-GCV approach alleviates the theoretical and practicalissues associated with traditional CSE ensemble division schemes by introducing functionsof position as additional conditioning variables. Despite this success, CSE-GCV retainssome dependence on user inputs: the model's geometric conditioning variables are intendedto leverage symmetries in the flow, and identifying these symmetries requires some a prioriinsight from the user. One way of generating a truly impartial (input-free) scheme wouldbe to define a Morton curve as proposed by Nivarti and Bushe [67], then apply CSE-GCVwith coordinate along Morton curve as the single geometric conditioning variable.The conclusions of this work will be re-visited in the context of the thesis as a whole inSection 9.1.1.73Chapter 6Conditional Dynamic SubfilterModellingThis chapter introduces and tests a novel variation of the dynamic approach for modelling ofLES sub-filter terms. In contrast to the traditional dynamic closure, which stabilizes rawdynamic coefficients by averaging across ensembles of expected statistical homogeneity, thenovel variation averages conditionally on some set of scalars whose local values are expectedto correlate with the local degree of turbulence.6.1 IntroductionAs discussed in Section 4.1.5, the Large Eddy Simulation (LES) transport equations featurea sub-filter stress term representing the un-resolved advection of momentum. The simpleststatic closure models for the sub-filter stress (SFS) combine some information about theresolved flow with one or more fixed modelling coefficients. More sophisticated dynamicmodels can be derived from static ones by assuming that the values of the required modellingcoefficients can be deduced locally and instantaneously as those which apply best across arange of scales near the local filter scale [33]. This assumption of scale-invariance is generallynot viewed as controversial; it is particularly reasonable if the filter scale falls within theinertial subrange (where some features of the flow become scale-independent) [77].6.1.1 Ensembles in Dynamic Subfilter ModellingAlthough static models for the sub-filter stress tensor are tuned to ensure that resolvedkinetic energy is dissipated, raw dynamic versions of the same models (which assign in-stantaneous, local values to the sub-filter modelling coefficients) do not natively enforce thisconstraint [33]. This can be viewed as a feature, since it accurately represents the fact thatkinetic energy may be transferred from large scale to small (forward-scatter) or from smallscale to large (back-scatter), but in practice it is also a source of instability: when dissipationis negative, the sub-filter stress amplifies resolved velocity gradients rather than dampingthem, and this process is self-reinforcing. The observed instability of the raw dynamicSmagorinsky procedure [33] can be explained as a consequence of the backscatter effect andthe strong auto-correlation of the Smagorinsky coefficient (a modelling parameter which willbe defined in Section 6.2.1) in time: once it [the Smagorinsky coefficient] becomes negativein some region, it may remain negative for excessively long periods of time during whichthe exponential growth of the local velocity fields, associated with negative eddy viscosity,causes a divergence of the total energy [80, p. 231].Given their instability, raw dynamic models must be modified for use in practical LES.The traditional approach to stabilization of the raw dynamic procedure is to identify direc-746.2. Theorytions of expected statistical homogeneity and assume that the sub-filter modelling coefficientsdo not vary in these directions [33, 80]a near-perfect analog of the ensemble division intraditional CSE [57, 58] (discussed in Section 5.1.1). In flows without obvious directions ofhomogeneity, averaging may also be performed in a Lagrangian sense, using fluid-particletrajectories in place of ensembles [81].6.1.2 Research Question and ObjectivesThe parallels between dynamic sub-filter modelling and CSE suggest that there may beundiscovered synergies between the two techniques. This chapter aims to explore the con-nections between these procedures, and in particular to answer the research question ofwhether it is worthwhile to apply dynamic sub-filter modelling techniques in the contextof CSE or vice versa. As will be shown presently, a theoretical analysis suggests that theconditional filtering concept of CSE can be used as a framework for understanding dynamicsub-filter models, and suggests a novel variation in which turbulence parameters are as-sumed to vary based on scalars rather than geometric variables. In the larger context of thisthesis, this chapter can therefore be viewed as answering the research question what arethe implications of conditioning on scalars rather than position in the dynamic Smagorinskymodel?6.2 TheoryThis section begins with a review of the static models used in this work (Sections 6.2.1and 6.2.2) and a re-interpretation of existing SFS modelling approaches inspired by CSEtechniques (Section 6.2.3). Sections 6.2.4 and 6.2.5 build on this foundation by proposingnovel CSE-inspired variations to the approach presented in Section 6.2.3.6.2.1 The Static Smagorinsky-Yoshizawa ModelThe original dynamic SFS models [33, 82] were based on applying the static Smagorinskymodel [32] to isochoric flow. Moin et al [83] generalized this to the case of non-isochoric flowby incorporating the Yoshizawa model [84] for the sub-filter kinetic energy. The combinationof the Smagorinsky and Yoshizawa models gives the closureρ(~˜v~v − ~˜v~˜v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶τ subF= −2ρCFS(∆F)2 ∣∣∣SF∣∣∣´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶νFSF‡ +23ρCFI(∆F)2 ∣∣∣SF∣∣∣2´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶k subI◦ (6.1)where• τ subF is the SFS tensor, with dimensionality m/(Lt2),• k sub is the sub-filter kinetic energy, with dimensionality m2/t2,• νF is the (kinematic) turbulent eddy viscosity, with dimensionality L2/t,• ∆F is the filter width, with dimensionality L,756.2. Theory• CFSis the Smagorinsky coefficient39[32], which is dimensionless and, per Lilly [85], isprescribed valueCFS=1pi2(23cK)3/2≈ 0.0300 (6.2)where cK≈ 1.5 is the Kolmogorov constant [10],• CFIis the Yoshizawa coefficient [84], which is dimensionless and is prescribed valueC subI=0.1640.0862≈ 0.0886 (6.3)as calculated by [20] from the values of [84],•∣∣∣SF∣∣∣ is the resolved overall strain rate, with dimensionality 1/t,∣∣∣SF∣∣∣ ≡ ∣∣∣√2SFF∣∣∣ = √2SFF : SFF (6.4)(where the factor of√2 ensures that the expression reduces to the natural definitionof strain rate in the case of one-dimensional flows),• SF‡ is the resolved shear rate, with dimensionality 1/t,SF‡ ≡(SFF)‡= SFF −13Tr(SFF)I◦, (6.5)• SFF is the resolved strain rate, with dimensionality 1/t,SFF ≡(~∇~˜v)F=12[~∇~˜v +(~∇~˜v)T], (6.6)and• I◦ is the identity tensor, which is dimensionless.For this combination of models, the conditions for a positive semidefinite subgrid stresstensor (Equation 4.23) can be rewritten [20] asCFI≥√32CFS(6.7)which is comfortably satisfied for the prescribed values of CFSand CFI.39 C subSis often written as C2S, where CS= 1pi(23cK)3/4≈ 0.172 per [85]. The notation adopted here isdeliberately different to allow for the possibility that the coefficient may be negative, which is not possiblefor the square of a real number.766.2. Theory6.2.2 The Germano IdentityAs previously stated in Section 6.1, a static sub-filter model can be used as the basis for anassociated dynamic model in which the modelling coefficients are set locally and instanta-neously. The procedure for generating dynamic models involves selecting an explicit testfilter F̂ (to complement the original grid filter F) and evaluating the inter-filter Leonardstress, defined asL F̂F ≡ τ ŝubF − τ subF̂ (6.8)where• τ ŝubF is the SFS at the scale of the combined test-upon-grid filter F̂, and• τ subF̂ is the result of test applying the test filter F̂ to the sub-grid stress (i.e., to theSFS at the scale of the grid filter F).As observed by Germano et al [33], the Leonard stress is closed by definition, but can also beclosed by applying the static model at both the original grid scale and the explicitly-filteredtest-upon-grid scale. This gives the so-called Germano identity [33], which generalizes tothe variable-density case as [83]Exact LF̂Fucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright1ρρ~v ρ~v̂ −1ρ̂ρ̂~v ρ̂~v =Modelled LF̂Fucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright2[CFSmF̂ − C F̂SmF̂]− 23[CFInF̂ − C F̂InF̂]I◦ (6.9)where mF‡ and nFare intermediates with dimensionality m/(Lt2), defined asmF‡ ≡ ρ(∆F)2 ∣∣∣SF∣∣∣SF‡ nF ≡ ρ(∆F)2 ∣∣∣SF∣∣∣2 (6.10)and mF̂‡ and nF̂are defined in the same way but with the test-upon-grid filter F̂ taking theplace of the grid filter F. If one assumes that the modelling coefficients are scale-independent,i.e.,CFS= C F̂SCFI= C F̂I(6.11)then the Germano identity can, in principle, be used as a condition to tune these coefficientsto match instantaneous, local conditions. The original dynamic closures [33, 82, 83] arebased on the simplifying assumption that the coefficients are sufficiently homogeneous thatthey can be extracted from the test filter operation, to witCFSmF‡̂ ≈ CFSmF‡̂ CFI nF̂ ≈ CFI nF̂. (6.12)776.2. Theory6.2.3 Dynamic Localization with CSE TechniquesAs originally demonstrated by Ghosal et al [80], a rigorous dynamic closure must explicitlyaccount for the fact that modelling coefficients vary in space when evaluating the test-filteredproducts CFSmF‡̂ and CFInF̂ in Equation 6.9. Inspired by CSE ensemble division, this processcan be formalized using a set G of geometric variables g which is collectively expected toaccount for all variation in the Smagorinsky and Yoshizawa coefficients, to witCFS(~x, t) = CFS(G (~x) , t) CFI(~x, t) = CFI(G (~x) , t) . (6.13)In the general case, G = {x, y, z} and the introduction of G changes nothing but notation; inflows with statistically homogeneous directions, a smaller G set may be used. In a turbulentjet, for example, one might assume that all spatial variation can be accounted for by r(distance from jet axis), i.e., G = {r}.A second connection to CSE can be introduced by re-casting the function evaluation inEquation 6.13 as an inner product using the Dirac delta (a technique previously appliedby [86]). The resulting equations areCFS(~x, t) =∫GCFS(G∗, t) · δ (G (~x, t)−G∗) dG∗ (6.14)CFI(~x, t) =∫GCFI(G∗, t) · δ (G (~x, t)−G∗) dG∗ (6.15)where G is the domain of G, i.e., the set of all G values which appear in the spatial domainX :G ≡ {G (~x) : ~x ∈ X} . (6.16)Equation 6.14 recalls the relationship between conditionally- and unconditionally-filteredterms which forms the background of CSE; the Dirac delta replaces CSE's modelled PDF.The practical consequence of this rearrangement is that the functions CF (G∗, t) are inde-pendent of ~x, and can therefore be extracted from the filtering operation in Equation 6.9(assuming a purely spatial LES filter).Paralleling the raw dynamic model, if one assumes that the static sub-filter model appliesat both the grid and test-upon-grid scales with the same coefficient functions, to wit∀G∗ ∈ G :C F̂S(G∗, t) = CFS(G∗, t) C F̂I(G∗, t) = CFI(G∗, t) (6.17)then the values of the coefficient functions can be deduced dynamically. For the Smagorin-sky-Yoshizawa model, the optimal dynamic values are those which satisfy the equations∀~x ∈ X : L F̂‡ (~x, t) = 2∫GCFS(G∗, t) ·M F̂‡ (G∗, ~x, t) dG∗ (6.18)∀~x ∈ X : Tr(L F̂F (~x, t))= −2∫GCFI(G∗, t) ·N F̂ (G∗, ~x, t) dG∗ (6.19)786.2. TheorywhereL F̂‡ (~x, t) =(L F̂F (~x, t))‡= L F̂F (~x, t)−13Tr(L F̂F (~x, t))I◦ (6.20)MF̂‡ (G∗, ~x, t) = mF‡ (~x, t) · δ (G (~x, t)−G∗)−mF̂‡ (~x, t) · δ (G (~x, t)−G∗) (6.21)N F̂ (G∗, ~x, t) = nF (~x, t) · δ (G (~x, t)−G∗)̂− nF̂ (~x, t) · δ (G (~x, t)−G∗) (6.22)and L F̂F , mF‡ , mF̂‡ , nF, and nF̂ are as in Equation 6.9. All terms in Equations 6.18 and 6.19save CFS(G∗, t) and CFI(G∗, t) are closed, so the equations can (in principle) be invertedfor these functions. This closure is reminiscent of existing approaches in which a rigorousaccounting of the test filter's interaction with the coefficient functions produces a FredholmIntegral Equation [80, 86].6.2.4 Stabilization via RegularizationAlthough the assumption that G captures all variation in the Smagorinsky and Yoshizawacoefficients parallels the traditional stabilization procedure of averaging across statisticallyhomogeneous regions, the similarities between Equations 6.18 and 6.19 and the integralequations encountered in CSE suggest an alternate approach: stabilization via regulariza-tion. Inverting Equation 6.18 subject to the regularization constraint that the functionCFS(G∗, t) varies slowly with each geometric argument g∗ ∈ G∗ would increase the amountof input data constraining each point in the solution function, increasing the tendency ofthe solution to approach the positive overall average and reducing the tendency to return ade-stabilizing negative result. Although this could be used in combination with a small G,a sufficiently high regularization might make it possible to apply the above procedure withG = {x, y, z}, i.e., to apply stabilization via regularization only, ignoring the existence ofany statistically homogeneous directions.6.2.5 Conditioning on ScalarsIn Chapter 5, the CSE model was re-formulated to use geometric conditioning variablesin combination with the traditional conditioning scalars. Translating these ideas into thecontext of turbulence modelling suggests a complementary change of conditioning variables,specifically, replacing the geometric variables G of section 6.2.3 with a set KF of fieldsassociated with filter F whose values fall in a corresponding conditional domain K,K ≡{KF (~x, t) : ~x ∈ X , t ∈ T}. (6.23)The static model would then be based on an assumption that KF accounts for all variationin CFSand CFI, to witCFS(~x, t) = CFS(KF (~x) , t)C F̂S(~x, t) = C F̂S(KF̂ (~x) , t)(6.24)CFI(~x, t) = CFI(KF (~x) , t)C F̂I(~x, t) = C F̂I(KF̂ (~x) , t)(6.25)796.2. Theorywhile the dynamic model would be based on an assumption the static sub-filter model appliesat both the grid and test-upon-grid scales with the same coefficient functions, to wit∀K∗ ∈ K :C F̂S(K∗, t) = CFS(K∗, t) C F̂I(K∗, t) = CFI(K∗, t) . (6.26)The optimal dynamic values would then be those satisfying the equations∀~x ∈ X :L F̂‡ (~x, t) = 2∫KCFS(K∗, t) ·M F̂‡ (K∗, ~x, t) dK∗ (6.27)∀~x ∈ X :Tr(L F̂F (~x, t))= −2∫KCFI(K∗, t) ·N F̂ (K∗, ~x, t) dK∗ (6.28)whereMF̂‡ (K∗, ~x, t) = mF (~x, t) · δ(KF (~x, t)−K∗)̂−mF̂ (~x, t) · δ(KF̂ (~x, t)−K∗)(6.29)N F̂ (K∗, ~x, t) = nF (~x, t) · δ(KF (~x, t)−K∗)− nF̂ (~x, t) · δ(KF̂ (~x, t)−K∗)(6.30)and L F̂‡ , mF, mF̂, nF, and nF̂ are as in Equation 6.9. This is nearly identical to the closureproposed in the previous section but for the fact that G and G have been replaced with KFand K, respectively. Apart from this change of variables, the key remaining difference is thatapplying the static model at the test scale requires a different set KF̂ of fields than applyingthe model at the grid scale (which requires KF); for geometric variables, the two sets neednot be distinguished, since the geometric variables are not associated with any filter scale.Potential Benefits of ConditioningDa Silva's analysis of DNS of a plane jet [87] illustrate that applying the dynamic procedurewith grouping based on position relative to the instantaneous shear layer (as opposed to afixed reference) significantly improves a priori predictions of the Smagorinsky coefficient.Although neither the traditional modelling approaches nor the dynamic localization proce-dure described in Section 6.2.3 can effect such a grouping, the conditional procedure can bemade to do so by a judicious choice of conditioning variables. Da Silva's results show thatthe components of the vorticity field effectively undergo step changes at the shear layer,but relatively modest changes within the turbulent and non-turbulent regions; this suggeststhat conditioning on vorticity could effectively separate the instantaneous turbulent andnon-turbulent regions, but could not differentiate between cells close to the shear layer andthose far from it. A more promising alternative would be to condition based on the enstro-phy and the resolved strain rate: the enstrophy could be used to identify whether a givencell fell on the turbulent or non-turbulent side of the instantaneous shear layer, while thestrain rate could be used to determine the distance between the cell and the instantaneous806.2. Theoryshear layer. Regardless of the precise conditioning variables selected, Da Silva's results sug-gest that a conditional dynamic procedure which effectively groups cells based on positionrelative to the instantaneous shear layer has the potential to out-perform dynamic sub-filtermodels which group based on raw position.Choice of Conditioning VariablesTurbulent combustion problems often feature a shear layer which is roughly coincident with amixing layer (in non-premixed combustion) or reaction layer (in premixed combustion). Thissuggests that the filtered mixture fraction Z˜ (in non-premixed combustion) and the filteredprogress variable c˜ (in premixed combustion) might make good conditioning variables whenapplying a dynamic sub-filter model to a turbulent combustion systema further connectionto CSE, which conditions on similar variables.For a non-premixed flame, a conditional dynamic SFS model based on mixture fractionhas two advantages over one based on enstrophy:• mixture fraction changes more smoothly across the shear layer, and• mixture fraction correlates strongly with combustion-generated heat release, whichchanges the density and viscosity and can therefore affect the balance of inertial andviscous forces (i.e., the degree of turbulence).Conditioning on filtered mixture fraction does have one major drawback, however, illustratedin Figure 6.1(a): the field takes on moderate values both in the core of the mixing layer andin the downstream plume. Since the local turbulence intensity differs widely between thesetwo locations, a model which groups them is likely to smear together unrelated conditionsjust as traditional dynamic models do, degrading performance. One alternative option whichmight be expected to improve upon this would be conditioning on the filtered variances ofmixture fraction Z v˜ar = Z 2˜−(Z˜)2, as previously applied by Kaul et al in a dynamic modelfor the un-resolved scalar dissipation of mixture fraction [88]. Unfortunately, this has its owndrawbacks: as illustrated in Figure 6.1(b), the variance assumes low and moderate values onboth sides of the shear layer, and therefore would group the most laminar part of the flowwith the most turbulent part. If both the filtered mixture fraction and its filtered variancewere used as conditioning variables, cells could be discriminated so long as their values ofeither conditioning variable differed; the hybrid choiceKF ={Z˜, Z v˜ar}(6.31)would therefore combine the strengths and suppress the weaknesses of using either conditionalone.For a premixed case, most of the statements from the preceding paragraph apply with cin place of Z 40. A premixed case would therefore be amenable to the conditioning variablesKF ={c˜, c v˜ar}. (6.32)40One notable exception is the fact that the filtered progress variable does not vary across the shear layeras smoothly as the filtered mixture fraction does.816.3. Methods(a) Z˜ (linear scaling) (b) Z v˜ar (logarithmic scaling)Figure 6.1: Visualization of the mixing field in a turbulent non-premixed jet flame.6.3 MethodsGiven that this work is intended to explore synergies between CSE and dynamic sub-filtermodelling, it is natural to choose a test case which features turbulent combustion. SandiaFlame D (described thoroughly in Section E) is selected as it has previously been simu-lated with CSE [64] and has a wealth of experimental measurements of velocity [72] andscalars [71] available for comparison. For this non-premixed case, the conditioning variablesin the conditional dynamic sub-grid model are chosen as in Equation 6.31; the 2-D condi-tioning space is discretized with 10 bins in Z˜ and 5 bins in Z v˜ar, for a total of 50 bins.To minimize computational overhead, Equations 6.27 and 6.28 are inverted only once every10 time steps; this is tantamount to an assumption that the conditional Smagorinsky andYoshizawa functions do not vary rapidly in time. These inversions are performed subject tothe constraint that the solution should vary smoothly with Z˜ and Z v˜ar, which parallels thestabilization technique described in Section 6.2.4.As mentioned earlier, a negative Smagorinsky coefficient can trigger instability. Whilethe conditional dynamic procedure is generally capable of avoiding this condition withoutassistance, a backstop clipping procedure is applied after each inversion, as follows:• largest value of the conditional Smagorinsky coefficient are identified, then• all conditional Smagorinsky coefficients are clipped so that they are no more negativethan the largest is positive.This roughly represents the idea that the coefficients can be locally positive or negative butthe overall average must be positive; in practice, this clipping procedure does not changethe coefficients especially often.As a comparison case, the traditional dynamic closure [33, 82] is applied to the samegrid, with cells grouped based on distance from the jet axis, r (i.e., G = {r}); in the interestsof a fair comprarson this case also uses 50 groups. This comparison case is identical to thatpresented in Chapter 5; all implementation details not discussed above are as described inSection 5.3.826.4. Results05 · 10−20.1〈YCO2 |Z〉z = 7.5djetz = 15djetz = 30djetz = 45djet05 · 10−20.1〈YH2O|Z〉02 · 10−24 · 10−26 · 10−28 · 10−2〈YCO|Z〉02 · 10−34 · 10−3〈YOH|Z〉0 0.2 0.4 0.6 0.8 102 · 10−34 · 10−36 · 10−38 · 10−3Z〈YH2 |Z〉0 0.2 0.4 0.6 0.8 1Z0 0.2 0.4 0.6 0.8 1Z0 0.2 0.4 0.6 0.8 1ZExperiment Traditional ConditionalFigure 6.2: Time-averaged conditional profiles of scalars. Experimental data from [71]. Theshaded area and dashed line mark the premixed flammability limits [76] and stoichiometriccomposition, respectively.6.4 ResultsMean conditional scalar fields are presented in Figure 6.2; conditional profiles of Z, YNO,and T are not presented, since they are not used in the simulation. The profiles of the majorscalars, CO2and H2O, show good agreement with experimental results. The profiles of theminor scalars CO, OH, and H2show some agreement with experiment results, although notto the same degree as the major scalars. As in Section 5.4, the jaggedness of the conditionalOH mass fraction can be explained as an artefact of the TGLDM chemistry. In all cases,the differences between the traditional and conditional closure are negligible.The mean axial velocity and fluctuating kinetic energy are presented in Figure 6.3. Theresults illustrate that the turbulence is predicted well at and near the inlet, but poorlydownstream (profiles at stations further downstream, which are not presented, suggest thatthis trend continues). Once again, however, the results are fairly consistent between the836.5. Discussion0204060〈vz〉 [m/s]z = 7.5djetz = 15djetz = 30djetz = 45djet0 1 2020406080100r/djet12〈~v′ · ~v′〉[m2/s2]0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djetExperiment Traditional ConditionalFigure 6.3: Radial profiles of mean axial velocity and fluctuating kinetic energy. Experi-mental data from [72].traditional and conditional closure; the conditional closure is, perhaps, slightly worse down-stream.The mean and root-mean-square fluctuations of the scalar fields are presented in Fig-ures 6.4 and 6.5 respectively. Given that the turbulence model has not predicted the velocityfield well downstream, it is not surprising that the mixing profile is predicted well near theinlet but less effectively downstream. Profiles of all other scalars are in good agreement withexperiment, and tend to differ only at locations where the mixture fraction is also predictedincorrectly. Modest differences between the predictions of the traditional and conditionaldynamic closures are apparent in the profiles of NO, CO, OH, and H2, although neitherclosure consistently matches the experimental results better than the other.6.5 DiscussionThe key distinction between the conditional dynamic procedure proposed here and tra-ditional dynamic procedures (including that described in Section 6.2.3) is that geometricvariables are a proxy for average degree of turbulence, whereas filtered scalars can be chosenas a proxy for instantaneous degree of turbulence. Since LES resolves some fluctuations,locations with the same average degree of turbulence do not, in general, have the same in-stantaneous degree of turbulence; requiring that turbulence parameters are functions of Gonly effectively smears together information associated with different instantaneous localconditions, reducing the model's ability to respond accurately to local conditions. Allowingturbulence parameters to vary with a set K which better predicts instantaneous, local con-ditions improves the model's ability to respond dynamically while preserving some of thestabilizing effects associated with limiting variation. In the language of optimal estimationtheory [89], conditioning on KF has the potential to produce a better model than condition-ing on G does so long as the irreducible error in modelling the turbulence coefficients asfunctions of KF is lower than that in modelling the turbulence coefficients as functions of G.846.5. Discussion00.20.40.60.81〈Z〉z = 7.5djetz = 15djetz = 30djetz = 45djet05 · 10−20.1〈YCO2〉05 · 10−20.1〈YH2O〉02 · 10−54 · 10−56 · 10−58 · 10−51 · 10−4〈YNO〉02 · 10−24 · 10−26 · 10−2〈YCO〉05 · 10−41 · 10−31.5 · 10−32 · 10−3〈YOH〉01 · 10−32 · 10−33 · 10−3〈YH2〉0 1 205001,0001,5002,000r/djet〈T 〉 [K]0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djetExperiment Traditional ConditionalFigure 6.4: Radial profiles of time-averaged scalars. Experimental data from [71].856.5. Discussion05 · 10−20.10.150.2〈Z′2〉12z = 7.5djetz = 15djetz = 30djetz = 45djet02 · 10−24 · 10−2〈Y ′2CO2〉1202 · 10−24 · 10−2〈Y ′2H2O〉1202 · 10−54 · 10−5〈Y ′2NO〉1201 · 10−22 · 10−23 · 10−2〈Y ′2CO〉1205 · 10−41 · 10−31.5 · 10−3〈Y ′2OH〉1205 · 10−41 · 10−31.5 · 10−32 · 10−3〈Y ′2H2〉120 1 20200400r/djet〈T ′2〉12 [K]0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djetExperiment Traditional ConditionalFigure 6.5: Radial profiles of root-mean-square fluctuations of scalars. Experimental datafrom [71].866.6. Concluding RemarksThis is expected to hold for any intelligently-selected KF.Using mixture fraction and its variance as conditioning variables in the SFS closureincreases the coupling between these fields by allowing for more direct feedback from thetwo former fields to the velocity field. One potential drawback of this is that it makes theSFS closure indirectly dependent on the models used to close the transport of Z˜ and Z v˜ar.Given that SFS closure is a more ubiquitous problem than modelling transport of Z˜ andZ v˜ar, the errors in the latter can be expected to be larger, and could potentially form theweakest link in the chain of models which underlie the current closure. This issue could bealleviated to some extent by using a similar conditional dynamic model to close the sub-filterfluxes of Z˜ and Z v˜ar and the sub-filter scalar dissipation term in transport of Z v˜ar.Overall, the results demonstrate that the proposed dynamic closure is tractable andstable. While the model's performance is sub-optimal, the fact that the traditional modelproduces near-identical inaccuracies suggests that the root of the problem may be the gridrather than a failing of the model. As previously mentioned in Section 5.5.1, the dynamicprocedure has been shown to perform poorly when cells have pencil-like anisotropy (twodimensions resolved significantly better than the third) [77], as is the case in the downstreamsection of the present grid.6.6 Concluding RemarksThe objective of this chapter was to explore connections between CSE and dynamic sub-filter modelling. In proposing a new conditional dynamic sub-filter model which borrowstechniques from both fields, the study has fulfilled its objectives. Whereas the traditionaldynamic closure stabilizes raw dynamic coefficients by averaging across ensembles of ex-pected statistical homogeneity, the novel variation averages conditionally on some set ofscalars whose local values are expected to correlate with the local degree of turbulence. Theresults suggest that the conditional dynamic model produces results extremely similar tothose of the traditional dynamic model, although the performance of both models is sub-optimal. Future work should compare these models on a more uniform grid, to determinewhether the conditional model can match or exceed the traditional model's performance inscenarios where the traditional model performs well.As in the prior chapter, the conclusions of this work will be re-visited in the context ofthe thesis as a whole in Section 9.1.1.87Chapter 7The Uniform Conditional State(UCS) Model forTurbulence-Chemistry InteractionThis chapter introduces a novel approach for turbulence-chemistry (filter-chemistry) inter-action modelling. The approach is related to conditional moment methods, in which aconditionally-filtered state is calculated, but differs from existing conditional moment meth-ods in that the spatial uniformity of the conditional state is taken as a starting assumption.Although the result of this derivation is equivalent to an existing flamelet manifoldmodel, the project arose out of a desire to extend existing conditional moment methods,and is therefore framed in a conditional moment context. The implied closure is related toexisting conditional moment approaches in that a conditionally-filtered state is calculated,but differs in that the spatial uniformity of the conditional state is taken as a startingassumption.7.1 IntroductionAs discussed in Section 4.3.3, conditional moment approaches to turbulence-chemistry (filter-chemistry) interaction are characterized by the fact that they can be used with any chemistry,apply partial conditioning, and use a modelled PDF. The two main varieties of conditionalmoment approachConditional Moment Closure, or CMC [54, 55, 56], and ConditionalSource-term Estimation, or CSE [57, 58]have each been used with great success, but alsointroduce distinct modelling challenges.7.1.1 Conditional Moment Closure (CMC)In CMC [54, 55, 56], the conditional state is evaluated by solving conditional transportequations. This introduces two major challenges: it effectively adds one independent variableper condition, increasing the dimensionality of the domain, and it requires the modellingof unclosed terms associated with physical gradients of conditional quantities. The formerchallenge can be alleviated by assuming that conditional quantities vary slowly in space;this makes it possible to define conditional quantities on a mesh with relatively coarsespatial resolution, reducing the number of points/cells required for a single-condition CMCsimulation to a tractable level [90]. Some authors have contemplated using more than oneCMC conditioning variable [91, 92, 93, 94, 95], but neither of the above challenges hasbeen addressed sufficiently to make such simulations commonplace; contemporary CMCimplementations usually employ a single conditioning variable.887.2. Additional Background7.1.2 Conditional Source-term Estimation (CSE)In Conditional Source-term Estimation (CSE) [57, 58], the conditional state is evaluated byinverting an integral equation. The inversion procedure frees CSE from the requirement ofsolving conditional transport equations and, by extension, of modelling physical gradients ofconditional quantities. Unlike CMC, CSE has been applied with multiple conditioning vari-ables [61]. CSE models introduce their own challenges; the most notable is that the inversionof integral equations is an imperfect process, and the inevitable associated errors make itextremely difficult to ensure that the deduced conditional composition satisfies conservationof elements. Highly dynamic chemistry modelssuch as full chemical mechanismscan failto generate a solution when applied to non-physical chemical compositions; in practice, thishas limited CSE models to fuels whose chemistry can be adequately described using modelswhich are robust in the face of elemental non-conservation, such as skeletal mechanisms orlow-dimensional manifold methods [64, 96].7.1.3 Research Question and ObjectivesBoth practical CMC and CSE methods require that the conditionally-filtered scalars havesmall gradients in space; this motivated a recent study which demonstrated that experi-mental measurements are consistent with the small-gradient hypothesis, particularly whenmultiple conditions are applied [97]. This revelation inspired the present chapter, whichexamines the theoretical implications of postulating that (with sufficient conditioning) con-ditional fields should be independent of space. In the context of this thesis, the researchquestion underpinning this chapter can be phrased as what are the theoretical implicationsof postulating that the conditional filtering eliminates all spatial dependence?7.2 Additional BackgroundThis section provides background information which is not relevant in motivating this workbut is necessary for contextualizing it. To be more specific, this section• introduces the transport equations and notation which will be referred to in the deriva-tion of Section 7.3, and• provides some background on the Multidimensional Flamelet Manifold (MFM) model,which the model derived in Section 7.3 turns out to be equivalent to despite thedifferent starting assumptions.7.2.1 Species-Like Transport EquationsIt is convenient to define Y+ as the set of of properties with mass fraction-like transportequations,Y+ = Y⋃{Z, c, 1, u, h . . .} . (7.1)where the inclusion of the property 1 represents the fact that the density transport equationcan be viewed as a specialization of the species mass transport equation in which Yα is897.2. Additional Backgroundreplaced with the constant 1 41. The degree to which the actual transport equations for themembers of Y+ match the transport equation for the species mass fraction varies:• for the species mass fractions Y (governed by Equation 2.5) and mixture fraction Z(governed Equation 3.4), the match is exact,• for the field 1, the match is exact (returning Conservation of Mass, Equation 2.4),provided that the diffusive flux and source are defined as~j1 ≡ ~0 ω˙1 ≡ 0 (7.2)• for progress variable c, the match is exact in premixed systems (Equation 3.19), butonly approximate in partially premixed systems (Equation 3.20), although the impactof the additional terms may be small in turbulent combustion simulation [61],• for internal energy u and enthalpy h the match is, in general, approximate (Equa-tions 2.12 and 2.13), but becomes exact if radiative transport is negligible (generally appropriate in non-sooting flames), viscous dissipation is negligible (generally appropriate in all combustion systems), the flow is isochoric (~∇ · ~v = 0), for u, or isobaric (DPDt = 0), for h, and the diffusive flux and source of the energy potentials are defined as∀ψ ∈ {u, h} : ~jψ ≡ ~q ω˙ψ ≡ 0 (7.3)(the sensible internal energy and enthalpy have reaction-associated sources, butthese are outside the scope of this work).The species transport equation (Equation 2.5) can therefore be generalized by enlarging thedomain of applicability from Yα ∈ Y to Y ∈ Y+:∀Y ∈ Y+ : ∂∂t(ρY )´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ · (ρ~vY )´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Advection= −~∇ ·~jY´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Diffusion+ ρω˙Y®Production(7.4)or, in non-conservative form∀Y ∈ Y+ : ρ∂Y∂t+ ρ~v · ~∇Y = −~∇ ·~jY + ρω˙Y (7.5)where• ω˙Y ∈Y+ is the source term due to chemical reactions, with dimensionality Y/t, whichcan be calculated from the reduced thermodynamic state basis as∀Y ∈ Y+ : ω˙Y = ω˙Y(Ψˆrtherm)(7.6)(a specialization of Equation 4.39 to reaction-associated source terms), and41The constant 1 can be viewed as an effective specific mass m/m.907.2. Additional Background• ~jY ∈Y+ is the diffusive flux of Y , with dimensionality Y/(L2t), which in general dependson the gradients of every field in the set D of diffusion-driving properties as∀Y ∈ Y+ : ~jY = −∑ψ∈DρDψY ~∇ψ (7.7)(a generalization of Equation 2.17 to variables beyond species mass fractions) where,for u and h, the coefficients DψY ∈{u,h} have dimensionality L4/(t3ψ) and can be deducedby rearranging the expressions from Section 2.3.3.7.2.2 NotationLike existing conditional moments methods, this work involves conditioning on a set K ofproperties κ. The set of dependent conditioning fields is labelled as K (~x, t), while the set ofindependent dummy conditioning variables corresponding to K is labelled K∗. Functionaldependence on (~x, t) is suppressed except where it is necessary for clarity; K may thereforerepresent the set of properties κ or the set of associated fields κ (~x, t), depending on context.The act of making the substitution K∗ = K (i.e., evaluating each independent dummyconditioning variable κ∗ ∈ K∗ with the corresponding value of the dependent local fieldκ (~x, t) ∈ K (~x, t)) can be represented using an underbracket or by literally replacing K∗with K 42:K∗φK≡Kφ ≡K∗φ∣∣∣∣∣K∗=K(~x,t).7.2.3 The Multidimensional Flamelet (MFM) ModelThe Multidimensional Flamelet Manifold (MFM) model [98] is a generalization of theflamelet manifold concept intended for partially premixed combustion. Within the frame-work laid out in Section 4.2.3, the MFM can be characterized as a flamelet manifold inwhich• the parameterizing variables φ are linear combinations of species mass fractions (theoriginal paper, [98], considers the special case of two parameterizing variables: mixturefraction, Z, and progress variable-defining linear combination, Yc), and• the controlling variables are associated with modelling scalar dissipations (the originalpaper uses nmodel= 3 such variables).The central species transport equation within this model can be written as∀Yα ∈ Y :∂Yα∂t+nφ∑j=1ω˙φj∂Yα∂φj= ω˙Yα +nφ∑i,j=1∂Yα∂φi∂φjχφiφj (7.8)42The traditional notation of a following | and subscript is intentionally avoided, since it requiressignificantly more space in expressions such asK∗a[K∗b]∣∣∣∣∣K∗=K(~x,t), where brackets are necessary to clarify thatthe substitution applies toK∗b but not toK∗a .917.3. Derivationwhere the mass fractions Yα∈S are functions of t and of the parameterizing variables φ butnot of position ~x. As will be shown in Equation 7.48, the derivation of Section 7.3 producesa near-identical transport equation.The MFM chemistry model can be adapted into a flamelet turbulence-chemistry inter-action model (Section 4.3.3) as follows:1. let the set {φ} of parameterizing variables φ serve as the conditioning variables (partialconditioning),2. assume that the joint PDF, P{φ}s˜ub, can be modelled, and3. assume that the conditional scalar dissipations{φ}χφiφj˜ can be modelled.Filtered quantities can then be evaluated by viewing the result of Equation 7.8 as theconditional state and evaluating an inner product with the PDF; as discussed further inSection 7.4.2, this transformation implicitly invokes the flamelet assumption (Section 4.2.3).For clarity, this turbulence-chemistry interaction model will be referred to as Filtered MFM(FMFM), to differentiate it from the pure chemistry model MFM.7.3 DerivationThis derivation is divided into sections as follows:7.3.1 Starting assumptions are described and some of their immediate implications are pre-sented.7.3.2 An initial conditional transport equation (with unclosed diffusion term) is derived.7.3.3 The conditionally-filtered flux term is rearranged, leaving only scalar dissipation-liketerms unclosed.7.3.4 The results of the previous two sections are combined into a conditional transportequation which applies for any diffusive flux law.7.3.5 The special case of Fickian diffusion is explored.7.3.1 AssumptionsThe key assumptions underpinning the following derivation are:Assumption 1: The relevant chemical dynamics can be represented using a reduced chem-istry model in which every element of the reduced thermodynamic state basis (Equa-tion 4.40) has a mass fraction-like transport equationΨˆrtherm ⊆ Y+. (7.9)Assumption 2: There exists a set K of independently variable properties κ for which thefollowing properties hold at all (~x, t):927.3. DerivationProperty 1: The conditionally-filtered thermodynamic state is spatially uniform,i.e., independent of ~x (the uniform conditional state assumption):∀ψ ∈ Ψthermo :K∗ψ˜ = f (K∗, t) .´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶independent of ~x(7.10)It also implies that the spatial derivatives of conditionally-filtered thermodynamicproperties are zero:∀ψ ∈ Ψthermo : ~∇K∗ψ = ~∇K∗ψ˜ = ~0. (7.11)Property 2: Evolution of each un-filtered conditioning variable κ ∈ K is describedby a species mass fraction-like transport equation, i.e.,K ⊆ Y+. (7.12)Property 3: Fluctuations about the conditional state are sufficiently small to permitthe first moment closure approximation [54, 55, 56],∀φ ∈ Ψthermo :K∗φ(Ψˆrtherm)˜≈ φ K∗Ψˆrtherm˜(7.13)Property 4: The joint probability density function (PDF) of all conditioning vari-ables,K∗P˜ (~x, t) ≡ F [ρ (~x, t) · δ (K (~x, t)−K∗)]F [ρ (~x, t)](7.14)can be modelled, for example by assuming a functional form parameterized bytransported moments or by solving a PDF transport equation. This implies theconditional-unconditional relation: unconditionally-filtered quantities can becalculated from conditional ones by evaluating an integral over the conditionaldomain K (inner product),∀φ : φ˜ (~x, t) =∫KK∗P˜ (~x, t) ·K∗φ˜ (~x, t) dK∗. (7.15)An additional property (Property 5) will also be assumed in Section 7.3.4.Assumption 3: EitherAssumption 3(a): the filter kernel has spatial extent only, i.e., a filtered field at timet0 has no dependence on an un-filtered field at time t1 6= t0 (true in most LES),Assumption 3(b): filtered fields are independent of t (true in RANS with a time-average filter), or937.3. DerivationAssumption 3(c): filtered fields are independent of realization (true in RANS withan ensemble-average filter).As illustrated in Section F, any version of this assumption leads to the filter-rearrange-ment relationship∀X,Y 6= f(~x) :K∗f(K∗X)KY =K∗f(K∗X)Y = f(K∗X)K∗Y . (7.16)which is, in a sense, tantamount to a statement that conditional filter is a Reynolds operator.An additional assumption (Assumption 4) will also be introduced in the course of thederivation.Since the PDF has been assumed closed, the combination of Equations 7.13 (specializedas Equations 4.48 and 4.49), 4.46, and 4.47 could be used to close any filtered reactionrate or chemical property in terms of the conditional thermodynamic state. It is thereforedesirable to generate a transport equation describing the evolution of the conditionally-filtered thermodynamic state (conditional transport equation), as such an equation couldprovide closure for the conditional state and (by extension) all filtered terms of interest.7.3.2 Generating Conditional Governing EquationsThe objective of this section is to generate transport equations for the elements of the reducedthermodynamic state basis. Since all elements of the reduced thermodynamic state basishave species-like transport equations (Assumption 1), the required transport equations canall be represented by Equation 7.4 or Equation 7.5. This suggests that conditionally-filteringone of these equations might be a reasonable first step; unfortunately, both equations featurethe velocity field, which is not expected to have convenient properties under conditionalfiltering. It is therefore desirable to eliminate the velocity field before conditionally filtering.Once the velocity field has been eliminated, it is possible to conditionally filter the re-sult. The transport equation that this produces features new terms associated with theconditionally-filtered rate of production and diffusion, which can each be evaluated sepa-rately. Substituting the resulting expressions for the conditionally-filtered source and diffu-sive transport produces an equation with a relatively small number of unclosed terms.Eliminating Velocity from Species TransportThe derivation begins with the non-conservative form of the species mass fraction-styletransport equation (Equation 7.5). Writing each scalar Y as the sum of a conditionally-filtered value and a residual fluctuation gives the expression∀Y ∈ Y+ : ρ ∂∂tKY˜ + ρ~v · ~∇KY˜ + ρ∂∂tYKfl˜uc + ρ~v · ~∇YKfl˜uc = −~∇ ·~jY + ρω˙Y . (7.17)It is convenient to consider the two left-most terms together. Although the conditional-ly-filtered propertiesK∗Y˜ are nominally functions of (~x, t,K∗) (with no actual dependence on947.3. Derivation~x, per Property 1), dependence on K∗ has been replaced with dependence on (~x, t) by thesubstitution K∗ = K (~x, t). The derivatives can therefore be expanded with consideration ofthe chain rule, as∀Y ∈ Y+ : ∂∂tKY˜ =∑κ∈K∂K∗Y˜∂κ∗K∂κ∂t+∂∂tK∗Y˜K(7.18)∀Y ∈ Y+ : ~∇KY˜ =∑κ∈K∂K∗Y˜∂κ∗K~∇κ+~∇K∗Y˜K(7.19)where the cancellation of the last term is justified by Property 1. Combining these results,the two left-most terms in Equation 7.17 can be rewritten as∀Y ∈ Y+ :ρ∂∂tKY˜ + ρ~v · ~∇KY˜ = ρ∂K∗Y˜∂tK+∑κ∈K∂K∗Y˜∂κK[ρ∂κ∂t+ ρ~v · ~∇κ], (7.20)where, per the un-filtered transport equation for κ (which, per Property 2, can also beexpressed in the form of Equation 7.5), the bracketed term can be rewritten as∀κ ∈ K :[ρ∂κ∂t+ ρ~v · ~∇κ]=[ρω˙κ − ~∇ ·~jκ]. (7.21)Substituting this into Equation 7.20 and that result into Equation 7.17 gives∀Y ∈ Y+ :ρ∂K∗Y˜∂tK+∑κ∈K∂K∗Y˜∂κK[ρω˙κ − ~∇ ·~jκ]+ ρ∂∂tYKfl˜uc + ρ~v · ~∇YKfl˜uc = −~∇ ·~jY + ρω˙Y . (7.22)Although the velocity field has not been entirely eliminated, its impact has been confinedto a term involving gradients of fluctuations, which will ultimately be assumed to be smallas a modelling assumption.Conditional Filter ApplicationWith the advection term rearranged, conditionally filtering Equation 7.22 can be expectedto produce tractable terms. Performing this filtering gives∀Y ∈ Y+ :K∗ρ∂K∗Y˜∂tK+∑κ∈KK∗∂K∗Y˜∂κ∗K[ρω˙κ − ~∇ ·~jκ]+ L = −K∗~∇ ·~jY +K∗ρω˙Y (7.23)957.3. DerivationwhereL ≡K∗ρ∂YKfl˜uc∂t+K∗ρ~v · ~∇YKfl˜uc(7.24)is a residual term which will ultimately be neglected. Applying the filter-rearrangementrelationship (Equation 7.16), and basic filter manipulation rules allows Equation 7.23 to berewritten as∀Y ∈ Y+ : K∗ρ∂K∗Y˜∂t+∑κ∈K∂K∗Y˜∂κK∗ρ K∗ω˙κ˜ − K∗~∇ ·~jκ+ L = − K∗~∇ ·~jY + K∗ρ K∗ω˙Y˜ . (7.25)The conditionally-filtered reaction source terms are already closed via the first momentclosure approximation (Equation 4.48); closure for the conditionally-filtered divergence ofthe diffusive flux vectors would close this transport equation.7.3.3 Conditionally-Filtered Divergence of Diffusive FluxThe conditionally-filtered divergence of the diffusive flux can be evaluated by applying theconstituent operations in order: evaluate the flux, then evaluate its divergence, then applythe conditional filter.The FluxPer Equation 7.6, the original un-filtered diffusive flux depends on DY ∈Y+ψ∈D and ψ, which canbe expressed as∀Y ∈ Y+.∀ψ ∈ D :ρDYψ =KρDYψ +[ρDYψ] Kflucψ =Kψ˜ + ψKfl˜uc(7.26)whereKρDYψ andKψ˜ can be evaluated as functions of the conditional state according to Equa-tion 7.13. This implies that∀Y ∈ Y+ : ~jY = −∑ψ∈DKρDYψ ~∇Kψ˜ −∑ψ∈D[ρDYψ] Kfluc ~∇Kψ˜−∑ψ∈DKρDYψ ~∇ψKfl˜uc −∑ψ∈D[ρDYψ] Kfluc ~∇ψKfl˜uc. (7.27)967.3. DerivationDivergence of the FluxThe first component of the divergence of the flux can be evaluated using vector calculusidentities as∀Y ∈ Y+ :−∑ψ∈DKρDYψ ~∇Kψ˜ = −∑ψ∈D[~∇KρDYψ]·[~∇Kψ˜]−∑ψ∈D[KρDYψ][~∇ · ~∇Kψ˜](7.28)Although the conditionally-filtered coefficientsK∗ρDY ∈Y+ψ∈D and driving forcesK∗ψ˜ are nominallyfunctions of (~x, t,K∗) (with no actual dependence on ~x, per Property 1), dependence onK∗ has been replaced with dependence on (~x, t) by the substitution K∗ = K (~x, t). Thederivatives can therefore be expanded with consideration of the chain rule, as∀Y ∈ Y+.∀ψ ∈ D :~∇KρDYψ =∑κ∈K∂K∗ρDYψ∂κ∗K~∇κ+~∇K∗ρDYψK(7.29)∀ψ ∈ D : ~∇Kψ˜ =∑κ∈K∂K∗ψ˜∂κ∗K~∇κ+ ~∇K∗ψ˜K(7.30)∀ψ ∈ D : ~∇ · ~∇Kψ˜ =∑κ∈K∑κ′∈K∂2K∗ψ˜∂κ∗∂κ′∗K~∇κ · ~∇κ′ +∑κ∈K∂K∗ψ˜∂κ∗K∇2κ (7.31)where the cancelled terms are zero by Property 1. By substituting these results back intoEquation 7.28 and rearranging, one can recover the expression∀Y ∈ Y+ :−∑ψ∈DKρDYψ ~∇Kψ˜ =−∑ψ∈D∑κ,κ′∈K∂∂κ∗ K∗ρDYψ ∂K∗ψ˜∂κ′∗K~∇κ · ~∇κ′ −∑ψ∈D∑κ∈KK∗ρDYψ∂K∗ψ˜∂κ∗K∇2κ.(7.32)977.3. DerivationConditionally-Filtered Divergence of the FluxSubstituting Equation 7.32 into Equation 7.27, conditionally filtering, applying the filter-re-arrangement relationship (Equation 7.16), and rearranging the sums produces the equation∀Y ∈ Y+.∀K∗ ∈ K :K∗~∇ ·~jY = −∑κ,κ′∈K∂∂κ∗∑ψ∈DK∗ρDYψ∂K∗ψ˜∂κ′∗ K∗~∇κ · ~∇κ′ −∑κ∈K∑ψ∈DK∗ρDYψ∂K∗ψ˜∂κ∗ K∗∇2κ+ Y(7.33)where∀Y ∈ Y∗ :Y ≡ −K∗~∇ ·∑ψ∈D[ρDYψ] Kfl˜uc~∇Kψ −K∗~∇ ·∑ψ∈DKρDYψ ~∇ψKfl˜uc −K∗~∇ ·∑ψ∈D[ρDYψ] Kfl˜uc~∇ψKfl˜uc(7.34)is a residual term which will ultimately be neglected. Equation 7.33 can be written morecompactly if one defines the shorthand∀Y ∈ Y+.∀κ ∈ K. ∀K∗ ∈ K :K∗JYκ ≡ −∑ψ∈DK∗ρDYψ∂K∗ψ˜∂κ∗. (7.35)whereK∗Jα∈SY ∈Y+ has dimensionality m/(Ltκ) and can be labelled as the conditional diffusiveflux associated with scalar Y and dummy condition κ∗. This term follows the same form asthe original diffusive flux did (Equation 7.6), except with the spatial gradient∂∂xireplacedby a conditional gradient∂∂κ∗43. Substituting this back into Equation 7.33 gives a fairlystraightforward relationship for the conditional diffusive flux,∀Y ∈ Y+.∀K∗ ∈ K :K∗~∇ ·~jY =∑κ,κ′∈K∂K∗JYκ′∂κ∗K∗~∇κ · ~∇κ′ +∑κ∈KK∗JYκK∗∇2κ+ Y . (7.36)When Y is neglected, this expression has only two unclosed terms:K∗~∇κ · ~∇κ′ (with |K|2components, of which at most12(|K|2 + |K|)are independent) andK∗∇2κ (with |K| compo-nents).43For a given Yα ∈ Y+, the set of |K| distinctK∗Jακ∈K can be viewed as the components of a conditionaldiffusive flux vector which exists in |K|-dimensional conditional space K.987.3. Derivation7.3.4 General Conditional Transport EquationSubstituting the final, expanded expression for the conditional diffusion (Equation 7.36) intothe intermediate conditional transport equation (Equation 7.25) gives the general conditionaltransport equation,∀Y ∈ Y+.∀K∗ ∈ K :K∗ρ∂K∗Y˜∂t+∑κ∈KK∗ρK∗ω˙κ˜∂K∗Y˜∂κ∗+  =K∗ρK∗ω˙Y˜ −Gradient Product Diffusion, GPDucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∑κ,κ′∈K∂ K∗JYκ′∂κ∗−∑κ′′∈K∂K∗Jκ′′κ′∂κ∗∂K∗Y˜∂κ′′∗ K∗~∇κ · ~∇κ′−∑κ∈K K∗JYκ − ∑κ′′∈KK∗Jκ′′κ∂K∗Y˜∂κ′′∗ K∗∇2κ´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Laplacian Diffusion, LD, (7.37)where = L +∑κ∈K∂K∗Y˜∂κ∗κ − Y (7.38)is a residual term (in which κ is defined by Equation 7.34 just as Y is). The fluctuationterms L, κ, and Y must each average to zero when scaled by the PDF and integrated overall of conditioning space; this motivates an additional assumption, modelled after that ofConditional Moment Closure [55].Assumption 4: Transport terms associated with conditionally-filtered derivatives of con-ditional fluctuations have a negligible impact on conditional transport.This assumption implies that  can be neglected.The domain of Equation 7.37 is nominally all t ∈ T , ~x ∈ X , and K∗ ∈ K, but all termssaveK∗~∇κ · ~∇κ′ andK∗∇2κ have been assumed to be independent of ~x. It is implausible thatthe conditional state could remain uniform if its transport equation featured non-uniformtransport effects; this motivates an additional assumed property of the set of conditioningvariables:Property 5: The termsK∗~∇κ · ~∇κ′ andK∗∇2κ are spatially uniform, i.e., independent of ~x(the Uniform Conditional Scalar Dissipation assumption).Under this assumption, all terms in Equation 7.37 are independent of ~x, so it is effectively notan independent variable. Given appropriate initial and boundary conditions, this equationprovides the desired closure for every element of the conditional thermodynamic state basis.997.3. Derivation7.3.5 Special CasesThe general conditional transport equation (Equation 7.37) can be specialized by prescribinga specific diffusive flux relationship. The two cases explored here are both Fickian; in thefirst, each species has its own diffusivity relative to the mixture average (as is common inmodels of hydrocarbon combustion at relatively low pressure), while in the second, there isa single global diffusivity.Fickian Diffusion, Species-Specific DiffusivitiesThe Fickian model with species-specific diffusivities gives the diffusive flux law∀Y ∈ Y+ : ~jY = −ρDY ~∇Y (7.39)where DY ∈Y+ is the diffusivity of scalar Y , with dimensionality L2/t. This model can befit into the general diffusive flux law (Equation 7.6) with the assignments D = Y+ andDYψ = DY δψY (where δψY is the Kronecker delta). Substituting these equations into thedefinition of the conditional diffusive flux component (Equation 7.36) gives the simplification∀Y ∈ Y+.∀κ ∈ K. ∀K∗ ∈ K :K∗JYκ ≡ −∑ψ∈DK∗ρDYψ∂K∗ψ˜∂κ∗= −∑ψ∈DK∗ρDY δψY ∂K∗ψ˜∂κ∗= −K∗ρDY ∂K∗Y˜∂κ∗. (7.40)For conditioning variables, the conditional diffusive flux components can be simplifiedeven further:∀κ, κ′ ∈ K.∀K∗ ∈ K :K∗Jκ′κ = −K∗ρDκ′ ∂K∗κ˜′∂κ∗= −K∗ρDκ′δκ′κ, (7.41)where∂K∗κ˜′∂κ∗ = δκ′κ becauseK∗κ˜′ = κ′∗ and the elements of K are independently variable. Asdemonstrated more explicitly in Sections F.4 and F.5, substituting both of these back forthe GPD (Gradient Product Diffusion) and LD (Laplacian Diffusion) terms in the generalconditional transport equation (Equation 7.37) produces the relationsGPD = −∑κ,κ′∈K K∗ρDY ∂2K∗Y˜∂κ∗∂κ′∗+∂∂κ∗(K∗ρDY −K∗ρDκ′)∂K∗Y˜∂κ′∗ K∗~∇κ · ~∇κ′ (7.42)LD = −∑κ∈K[K∗ρDY −K∗ρDκ] ∂K∗Y˜∂κ∗ K∗∇2κ. (7.43)Substituting these back into the general conditional transport law (Equation 7.37) givesthe conditional transport equation for the case of Fickian diffusion with species-specific1007.3. Derivationdiffusivities:∀Y ∈ Y+.∀K∗ ∈ K :K∗ρ∂K∗Y˜∂t+∑κ∈KK∗ρK∗ω˙κ˜∂K∗Y˜∂κ∗=K∗ρK∗ω˙Y˜ +∑κ,κ′∈K K∗ρDY ∂2K∗Y˜∂κ∗∂κ′∗ K∗~∇κ · ~∇κ′+∑κ,κ′∈K∂∂κ∗(K∗ρDY −K∗ρDκ′)∂K∗Yα˜∂κ′∗ K∗~∇κ · ~∇κ′+∑κ∈K[K∗ρDY −K∗ρDκ] ∂K∗Y˜∂κ∗ K∗∇2κ. (7.44)Fickian Diffusion, Single Global DiffusivityIf one assumes that there is a single, state-independent diffusivity, i.e.∀κ ∈ K. ∀α ∈ S : Dκ = DY ≡ D, (7.45)then the last two terms in the conditional transport equation for Fickian diffusion withspecies-specific diffusivities (Equation 7.44) cancel out, leaving the simplified equation,∀Y ∈ Y+.∀K∗ ∈ K :K∗ρ∂K∗Y˜∂t+∑κ∈KK∗ρK∗ω˙κ˜∂K∗Y˜∂κ∗=K∗ρK∗ω˙Y˜ +∑κ,κ′∈K K∗ρD ∂2K∗Y˜∂κ∗∂κ′∗ K∗~∇κ · ~∇κ′ (7.46)Because the diffusivity is a global constant, it can be freely commuted with the filteringoperation; taking advantage of this property, and dividing away the conditional densitygives the equation∀Y ∈ Y+.∀K∗ ∈ K :∂K∗Y˜∂t+∑κ∈KK∗ω˙κ˜∂K∗Y˜∂κ∗=K∗ω˙Y˜ +∑κ,κ′∈K∂2K∗Y˜∂κ∗∂κ′∗K∗D~∇κ · ~∇κ′. (7.47)The combination of variables which is conditionally-filtered in the final term is then equiv-alent to the scalar dissipation rate, χκκ′ , with dimensionality 1/t, which was previouslydefined in Equation 3.27. Substituting this definition gives the final conditional transportequation for the case of Fickian diffusion with a single global diffusivity,∀Y ∈ Y+.∀K∗ ∈ K :∂K∗Y˜∂t+∑κ∈KK∗ω˙κ˜∂K∗Y˜∂κ∗=K∗ω˙Y˜ +∑κ,κ′∈K∂2K∗Y˜∂κ∗∂κ′∗K∗χκκ′ . (7.48)1017.3. DerivationEquation 7.48 follows the same form as the Multidimensional Flamelet Manifold (MFM)equation of [98], but differs in that it applies to conditionally-filtered quantities (which aremeaningful in all flows), as opposed to coordinate-transformed quantities (which are onlywell-defined when there is a perfect mapping between the chosen coordinates and the un-filtered statei.e., in laminar flows or turbulent flamelets)44. It should be noted that theMFM derivation of [98] begins with an assumption of Fickian diffusion and later enforces aunity Lewis number assumption, effectively giving a Fickian diffusion with a single globaldiffusivity. There is no theoretical reason why the MFM derivation could not be generalizedto an arbitrary diffusive flux; doing so would, presumably, recover Equation 7.37.7.3.6 SummaryThis derivation is based on four main assumptions.Assumption 1: The relevant chemical dynamics can be represented using a reducedchemistry model in which every element of the reduced thermodynamic state Ψˆrthermhas a species-like transport equation.Assumption 2: There exists a set K of conditioning variables κ satisfying the followingproperties:Property 1: the conditionally-filtered thermodynamic state is uniform (the uniformconditional state assumption for which the model is named),Property 2: evolution of each un-filtered conditioning variable κ is described by atransport equation of the same form as that of a species mass fraction,Property 3: fluctuations about the conditional state are sufficiently small to permitthe first moment closure approximation of CMC [54, 55, 56], andProperty 4: the joint probability density function (PDF) of all conditioning vari-ables can be modelled,Property 5: the conditionally-filtered termsK∗~∇κ · ~∇κ′ andK∗∇2κ are spatially uniform,i.e., independent of ~x (the Uniform Conditional Scalar Dissipation assumption).Assumption 3: One of the following holds:Assumption 3(a): the filter kernel has spatial extent only, i.e., a filtered field attime t0 has no dependence on an un-filtered field at time t1 6= t0 (true in mostLES),Assumption 3(b): filtered fields are independent of time t (true in RANS with atime-average filter), orAssumption 3(c): filtered fields are independent of realization (true in RANS withan ensemble-average filter).44This, and other distinctions between UCS and MFM, are discussed more thoroughly in Sections 7.4.2and 7.4.2.1027.3. DerivationAssumption 4: Transport terms associated with the conditionally-filtered derivatives ofconditional fluctuations have a negligible impact on the conditional transport, as inCMC [55].Given these assumptions, the filtered source term can be closed in terms of the modelledPDF and the conditional state, as∀ψ ∈ Ψˆrchem :ω˙ψ˜ (~x, t) =∫KPK∗s˜ub (~x, t) · ω˙ψ K∗Ψˆrtherm˜(t) dK∗ (7.49)where the conditional state can be evaluated by solving conditional transport equations. Inthe general case where the diffusive flux can be written as∀Y ∈ S : ~jY = −∑ψ∈DρDψY ~∇ψ, (7.6 repeated)(where each D is the set of thermodynamic properties whose gradient drives diffusion), theseequations take the form∀Y ∈ Y+.∀K∗ ∈ K :K∗ρ∂K∗Y˜∂t+∑κ∈KK∗ρK∗ω˙κ˜∂K∗Y˜∂κ∗=K∗ρK∗ω˙Y˜ −Gradient Product Diffusion, GPDucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∑κ,κ′∈K∂ K∗JYκ′∂κ∗−∑κ′′∈K∂K∗Jκ′′κ′∂κ∗∂K∗Y˜∂κ′′∗ K∗~∇κ · ~∇κ′−∑κ∈K K∗JYκ − ∑κ′′∈KK∗Jκ′′κ∂K∗Y˜∂κ′′∗ K∗∇2κ´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Laplacian Diffusion, LD, (7.50)but, when the diffusion is Fickian and there is a single global diffusivity (~jY = −ρD~∇Y ),the equations simplify further to∀Y ∈ Y+.∀K∗ ∈ K :∂K∗Y˜∂t+∑κ∈KK∗ω˙κ˜∂K∗Y˜∂κ∗=K∗ω˙Y˜ +∑κ,κ′∈K∂K∗Y˜∂κ∗∂κ′∗K∗χκκ′ . (7.48 repeated)In both cases, the domain of the transport equation is nominally all t ∈ T , ~x ∈ X , andK∗ ∈ K, but all terms are independent of ~x so it is effectively not an independent variable,and•K∗ω˙Y ∈Y+˜ (which includesK∗ω˙κ∈K˜ since K ⊆ Y+) are trivially closed by evaluating thereaction rate functions using the conditionally-filtered fields as arguments,1037.4. Discussion•K∗JY ∈Y+κ∈K (which includesK∗Jκ′∈Kκ∈K since K ⊆ Y+), the conditional diffusive flux associatedwith scalar Y and conditioning variable κ, is defined byK∗JYκ ≡ −∑ψ∈DK∗ρK∗DYψ˜∂K∗ψ˜∂κ∗(7.51)where DY ∈Y+ψ∈D and ψ can be evaluated by substituting the conditional state into ther-modynamic functions, and• K∗χκκ′ is the conditional scalar dissipation, defined asK∗χκκ′ ≡K∗D~∇κ · ~∇κ′. (7.52)If the initial conditional state is known and thermodynamic models and a reaction mecha-nism have been specified, the only unclosed terms in this system are conditionally-filteredfunctions of spatial gradients of conditioning variables,K∗~∇κ · ~∇κ′ andK∗∇2κ (or K∗χκκ′ , for thesimple Fickian case). This closure is termed the Uniform Conditional State (UCS) model,in light of the fact that the key starting assumption is uniformity of the conditional state.7.4 DiscussionThe derivation above raises two key questions:• Can the system be adapted into a workable combustion model?• How does the result relate to existing combustion models?These questions are both addressed in their own section below.7.4.1 Viability of the UCS SystemThe UCS system can be adapted into a combustion model so long as• there exists a set of conditioning variables which satisfies the rather stringent require-ments of Assumption 2, and• the conditionally-filtered products of gradients can be effectively closedThese issues are each discussed in their own section below.1047.4. DiscussionChoice of Conditioning VariablesThe key assumption underpinning the UCS model is that conditional thermodynamic prop-erties are spatially uniform (Property 1). Although this assumption appears restrictive, it isdemonstrably true that a set of conditioning variables with this property exists: the set ofall thermodynamic properties (including species mass fractions) produces trivially-definedconditional quantities corresponding to delta distributions in conditional space and uniformdistributions in physical space. In practice, the minimal set satisfying this property is likelymuch smaller. The combination of mixture fraction, progress variable, and total enthalpy,for example, includes one degree of freedom to account for each of three leading order effectsinfluencing the flame (mixing, reactions, and heat transfer), and could therefore plausiblyaccount for the phenomena which most strongly effect spatial variation in properties. Com-plicated fuels could potentially require multiple progress variables to fully capture reactiondynamics.One aspect of combustion which is unlikely to be captured by a small set of conditioningvariables is the slow chemistry, involving species such as NO; these aspects of the chemistryare generally decoupled from the dominant reaction pathway, and are therefore not typicallycaptured well by the standard conditioning variables. Several modifications are available toaccount for this.• The Age-Extended Progress Variable proposed by Grout [99] could be adopted as aconditioning variable (instead of or as well as the original progress variable); this couldimprove predictions of all slow species by accounting for the dependence of their massfractions on residence time in the reaction zone.• Following the work of Wang et al [64], slow species could be evolved by solving trans-port equations, with the reaction source term provided by the UCS closure, ratherthan by directly prescribing mass fractions using the UCS closure.Closure for Conditional Scalar DissipationsThere are three potential approaches to closing the UCS system:1. Assume that the conditional scalar dissipation doesn't have much impact on the con-ditional state, and simply model it using something plausible. This leaves the con-ditional field independent of the spatial field and makes it possible to simulate as apre-processing step, giving the lowest computational cost. The most computationallyefficient way to implement this approach would be to pre-compute the inner productof the manifold with joint PDFs of the conditioning variables modelled via presumedforms, producing a table that would depend only on the inputs used to model the jointPDF (e.g. the filtered mean and variance of each conditioning variable). While thistable could end up being very large in terms of the number of entries, it would onlyneed to contain the source terms of the conditioning variables and any other scalarsthat might be evolved in the physical space.2. Assume that the conditional scalar dissipation has a significant impact on the con-ditional state, and attempt to choose something which matches the instantaneous1057.4. Discussionconditions. In this approach, the conditional and spatial fields must be evolved si-multaneously (giving the highest computational cost), and the conditional scalar dis-sipation must be deduced using the spatial field. One possible approach would be toperform an integral inversion as in CSE:(a) estimate the un-resolved component of the [unconditional] scalar dissipation usingan existing model,(b) add the resolved component, generating an estimated total unconditional scalardissipation,(c) invert the integral equations∀κ, κ′ ∈ K : χκκ′ (~x, t) =∫KPK∗sub (~x, t)K∗χκκ′(t) dK∗ (7.53)for each required scalar dissipation χκκ′ using the modelled probability densityfunction, which is closed per Property 4. Integral equations of this form areroutinely inverted in CSE [57, 58]the two-dimensional inversion required forthe conditioning variables chosen in this study is identical to that solved in thework of [61].The conditional scalar dissipation could also be deduced via alternative techniques,such as a PDF-weighted approach [100].3. Assume that the conditional scalar dissipation has a moderate impact on the condi-tional state and adopt a hybrid approach:(a) apply a plausible model for the conditional scalar dissipation (as in method 1),(b) evolve the conditional system to steady state (without evolving the spatial sys-tem),(c) use the conditional solutions to evolve the RANS/LES field to steady45state(without evolving the conditional system),(d) use the RANS/time-averaged LES fields to invert for a refined conditional scalardissipation (as in method 2), and(e) repeat from step 2 until the predictions are sufficiently converged.This approach has an intermediate computational cost.The results of Bushe [97], which examined the doubly-conditional profiles of scalars in aseries of jet flames with variable Reynolds numbers, are compatible with the hypothesis thatthe conditionally-filtered state has weak dependence on Reynolds number and, by extension,scalar dissipation. This suggests that closure 1 may be viable, in the non-premixed jet flamesconsidered; this hypothesis will be tested in Chapter 8.45statistically steady, in the case of LES1067.4. Discussion7.4.2 Connections to Existing Chemistry and Combustion ModelsThe UCS model can be viewed as a manifold chemistry reduction method in that it pro-duces a low-dimensional system representing chemistry. Within the field of filter-chemistry(turbulence-chemistry) interaction models, it is intimately related to conditional momentmodels, but can also be connected to flamelet and PDF models.REDIM, Reaction Manifold, and Flamelet Manifold ChemistryThe UCS model can be cast as a REDIM model by viewing the instantaneous conditionalstate as a manifold, suspended in chemical state space and parameterized by the conditioningvariables. The character of this manifold varies somewhat depending on how the conditionalscalar dissipation is closed.1. When the conditional scalar dissipation is assumed to be unimportant, and modelledusing a single time-invariant function, the UCS manifold can be viewed as an |K|-dimensional REDIM manifold in which the effect of diffusion on chemistry is capturedby the conditional scalar dissipation model, much like standard flamelet manifoldssuch as MFM [98]. In this limit, the UCS manifold is equivalent to a slice extractedfrom the (|K| + nmodel)-dimensional MFM manifold by fixing the nmodelmodellingparameters which parameterize the conditional scalar dissipation model. Ultimately,if the scalar dissipation has a small impact on the conditional solution, then MFMcould be carried out with nmodel= 0, and the two manifolds would become identical.In the limit where reactions dominate over all effects, the conditional scalar dissipationmodel is truly irrelevant. The UCS manifold is then a pure reaction manifold, andshould coincide exactly with other such manifolds (ILDM [35, 101], TGLDM [73, 102],etc.).2. When the conditional scalar dissipation is assumed to be very important, and is evolveddynamically, the UCS model can be viewed as a REDIM manifold which considers theinstantaneous diffusion in the real, turbulent flamei.e., a turbulent REDIM. Whilethis suggests that the UCS manifold might be well-suited to account for the impact ofdiffusion on chemistry (notably outside of the flamelet regime, where flamelet-basedmanifolds might struggle), it comes at a cost: in a fully-coupled simulation, the UCSmanifold evolves dynamically, and therefore cannot be pre-tabulated as in traditionalREDIM approaches. The manifold can thus be viewed as (|K|+1)-dimensional, wherethe last dimension (time) must be probed as the simulation progresses.3. In the hybrid approach, the UCS manifold remains dynamic, and thus (|K| + 1)-dimensional, but responds less promptly to changes in the instantaneous flow condi-tions. Indeed, in this mode, one could vary the frequency of updates to the conditionalfield and even test when the manifold can be approximated as being static, switchingto the computationally-efficient mode of approach 1 as and when deemed appropriate.Conditional Moment Turbulence-Chemistry Interaction ModelsThe UCS model can be viewed as both a generalization and specialization of ConditionalMoment Closure [56]: a generalization in that the number of conditioning variables is arbi-1077.4. Discussiontrary, but a specialization in that the conditional state is required to be uniform. Despiteits similarity to CMC, the UCS model is (for real systems46) distinct in several ways:1. it does not introduce any unclosed terms associated with the physical gradients ofconditional quantities,2. it requires more conditioning variables to satisfy its preconditions than CMC does,and3. it implies that there are distinct physical and conditional domains which are onlyloosely coupled through the modelled PDF:• conditional fields exist in the conditional domain K, and influence the physicalfields through the reaction rate only, while• unconditional fields exist in the spatial domain X , and influence the conditionalfields through the scalar dissipation only.The final distinction is quite convenient: the instantaneous conditional domain is not(|K| + 3)-dimensional as in CMC, but rather |K|-dimensional. As a consequence of this,the computational complexity of the UCS model scales much more gently with the numberof conditioning variables than CMC does.As compared to CSE (which shares distinction 1 in the previous paragraph), the chiefadvantage of the UCS model is that it naturally enforces conservation of elements, and istherefore not restricted to fuels whose chemistry can be captured by skeletal mechanisms ormanifold chemistry.Flamelet Turbulence-Chemistry Interaction Models (FTCIMs)Given the similarities between the UCS and MFM transport equations, the flamelet turbu-lent-chemistry interaction model most directly comparable to UCS would be FMFM (Sec-tion 7.2.3), with UCS's conditioning variables K selected as the controlling variables andan additional nmodelparameterizing variables as inputs to the scalar dissipation model. Ona practical level, the implementations of these models would be similar or identicaleithercould potentially be closed via any of the closures listed in Section 7.4.1. On a theoreticallevel, however, the interpretations of the model are very distinct.• MFM is based on a coordinate transformation, which is only appropriate if the localspecies mass fractions and scalar dissipations predict the local thermodynamic stateperfectly. This assumption is valid within a laminar flame, but can only be invoked ina turbulent flame under the flamelet assumption.• UCS is based on conditional filtering; while it requires that the conditioning variablespredict the thermodynamic state well, it allows for the possibility that conditionalfiltering fails to resolve some residual fluctuations. To be more specific, UCS requiresthat these conditional fluctuations are small enough to permit the first moment closure46In the corner case of a zero-dimensional spatial domain, no conditional quantity could ever exhibit spatialvariation and the UCS assumption is trivially satisfied; UCS and MFM then become identical.1087.5. Concluding Remarksapproximation (Property 3 of Section 7.3) and that conditional transport associatedwith gradients of these fluctuations is negligible (Assumption 4 of Section 7.3). Theseassumptions are valid within laminar flames and under flamelet assumptions, but canalso hold for turbulent flames which are not in the flamelet regime, making UCSapplicable to a wider range of flames than MFM47.A related distinction can be made between the un-closed scalar dissipation terms:• in MFM, the scalar dissipation to be closed is the un-filtered scalar dissipation associ-ated with the coordinate transformation, which could be equated to the conditionally-filtered scalar dissipation in a turbulent flame by invoking the flamelet assumption,but would more traditionally be directly modelled; whereas• in UCS, the scalar dissipation to be closed is natively the conditionally-filtered scalardissipation.Considering the fact that the UCS and FMFM models invoke different modelling as-sumptions, the fact that their implementations converge in some scenarios is somewhatstartling. The convergence suggests that flamelet and conditional moment methods may,in some situations, be more compatible than is generally acknowledged. It is also possi-ble (although far from convincingly demonstrated) that there may be relatively universalmodelling relationships which transcend any one set of starting assumptions.PDF Turbulence-Chemistry Interaction ModelsIn the limit where each scalar is a conditioning variable, all conditional quantities aretrivially-defined and the true challenge is instead modelling the joint PDF; this providesan indirect way of motivating PDF methods. The method could also be adapted to borrowthe PDF transport techniques, potentially allowing for improved modelling of the joint PDF(which can become an extremely complicated task as the number of conditioning variablesincreases).7.5 Concluding RemarksThe objective of this chapter was to examine the theoretical implications of postulating that,with sufficient conditioning, conditional fields should be independent of space. In deriving anew turbulence-chemistry interaction model based on a uniform conditional state, the studyhas fulfilled its objectives. The new UCS approach is related to conditional moment methods,in which a conditionally-filtered state is calculated, and its implementation can, in somescenarios, converge with that of a turbulence-chemistry model based on the MultidimensionalFlamelet Manifold approach.As in previous research chapters, the conclusions of this work will be re-visited in thecontext of the thesis as a whole in Section 9.1.1.47The UCS derivation also explicitly calculates the error term which is neglected under Assumption 4,allowing for future validation and generalization. The error term neglected by assuming Property 3 couldalso be estimated via perturbation theory.109Chapter 8Application of the UniformConditional State (UCS) Model toNon- and Slightly-PremixedCombustionThis chapter tests the Uniform Conditional State model derived in Chapter 7.8.1 IntroductionA novel conditional moment method for turbulent combustion simulation, termed the Uni-form Conditional State (UCS) model, was derived in Chapter 7. While there are theoreticalreasons to expect that this model could successfully simulate turbulent combustion, it isnot, as yet, validated.8.1.1 Objectives and Research QuestionThe primary objective of this chapter is to test the UCS model. Within the context of thisthesis, the associated research question can be phrased as what are the practical implicationsof postulating that the conditional filtering eliminates all spatial dependence?8.2 Study ScopeThe scope of the study can be defined by identifying• the specialization of the UCS model which will be implemented,• the test flame that it will be applied to, and• the hypotheses which will be tested.These components are each discussed in their own sub-section below.8.2.1 UCS SpecializationFormally, the Uniform Conditional State model described in Section 7.3 constitutes a tem-plate from which an entire family of specializations (each corresponding to a different setK of conditioning variables κ and a different model for the diffusive flux ~jY ∈Ψˆrtherm) can beinstantiated. Since there are infinite such specializations, an exhaustive test is impossible;1108.2. Study Scopethis work examines the specialization corresponding to simple Fickian diffusion (with equalspecies diffisuvities) and conditioning upon mixture fraction and progress variable, i.e.K = {Z, c} K∗ = {Z∗, c∗} (8.1)where Z∗ and c∗ are independent dummy variables corresponding to the mixture fractionand progress variable respectively, and both have domain [0, 1], to witK = {[0, 1], [0, 1]} . (8.2)The progress variable is defined based on the carbon dioxide mass fraction:c ≡ YCO2Y ccCO2(Z)(8.3)where Y ccCO2(Z) is the carbon dioxide mass fraction which would be observed at completecombustion (cc) at the local mixture fraction Z. Normalization by complete combustionis selected over normalization by chemical equilibrium since it allows for the possibilitythat mixing in the mixture fraction dimension of conditional space may drive the localcomposition beyond chemical equilibrium. Since Y ccCO2(Z) varies with Z, the transportequation for c (Equation 3.20) contains scalar dissipation-based terms, and therefore doesnot match the form of the species transport equation as is assumed in UCS. As discussedin [61], these terms have a small but non-zero impact on turbulent jet flames; in this work,as in [61], they are neglected in the interests of simplicity.8.2.2 Test FlameAs previously discussed in Sections 7.2.3 and 7.4.2, the UCS transport Equation (Equa-tion 7.48) is effectively identical to the MFM transport Equation [98]. Given this, theresults of Nguyen et al. [98] (which considered a near-fully premixed turbulent flame) andWu et al. [103] (which considered a variety of laminar flames spanning the full continuum ofpremixedness), can also be said to demonstrate the UCS model in their respective contexts.To complement the previous MFM simulations, the present work considers a turbulent flamewhich falls opposite to that considered in the work of Nguyen et al. along the continuumof premixedness (Figure 3.3): the Sandia flame series of turbulent, nominally non-premixedflames (described thoroughly in Section E), which exhibit slightly premixed behaviour inthe highest-Reynolds number cases. The flame series is attractive because:• it includes multiple related flames spanning a wide range of Reynolds numbers• it has extensive experimental data [71] available for comparison, and• most of the key properties required for chosen specialization of the UCS model tobe applicable have been demonstrated by previous authors. In particular, a recentstudy [97] of experimental measurements [71] found that assuming a single, spatially-invariant conditional state could represent the experimental data to good accuracyfor all reactive scalars other than NO, suggesting that the conditional state is indeeduniform as assumed in Property 1 of Section 7.3.1.1118.3. MethodsThere is therefore reason to believe that this test case can be successfully simulated withthe chosen specialization of the UCS model. Although the full series contains seven flames,the first three are decidedly laminar and have not been extensively studied experimentally;this study therefore focuses on Sandia flames C through F.8.2.3 HypothesesThe first hypothesis of this work is that the chosen specialization of the UCS model canpredict the properties of the non-premixed/slightly-premixed Sandia flame series with goodaccuracy.A second hypothesis was inspired by the results of Bushe [97], which demonstrate that asingle doubly-conditional state can describe the measurements of Sandia flames C, D, E, andF with good accuracy48a rather surprising result, given that the Reynolds numbers of theseflames vary by a factor of over 3. The variation in Reynolds number implies variation inconditional scalar dissipation; the lack of variation in conditional state implies some degreeof insensitivity to conditional scalar dissipation. This observation is at odds with Nguyen etal.'s conclusion that the MFM manifold is highly sensitive to scalar dissipation. There aretwo potential explanations for this discrepancy:1. re-examination of Nguyen et al.'s figures suggests that, for most fields (with the notableexception of the conditional source of progress variable), the variation with scalardissipation appears to be modest, and2. Nguyen et al.'s observed variation of conditional source of progress variable with scalardissipation is less likely to impact the conditional chemical state in slightly premixedcombustion than in near-fully premixed combustion, since the source of progress vari-able becomes a less determinative parameter as one progresses along the continuumfrom premixed to non-premixed.Due to feedback within the conditional system, it is possible that even the conditionalreaction rate will show only modest variation with conditional scalar dissipation in thepartially-premixed system. The second hypothesis is, therefore: the chosen specializationof the UCS model is substantially insensitive to variations in conditional scalar dissipation,and thus it is not important that the conditional scalar dissipation be modelled accurately.8.3 MethodsAs discussed in Section 7.4.1, there are three potential closures for the conditional scalardissipation terms which appear in the UCS transport equation:1. the simple closure, in which scalar dissipation is assumed to have a small impact onthe conditional state and the spatial and conditional systems are decoupled;2. the full closure, in which the scalar dissipation is assumed to have a significant impacton the conditional state and the spatial and conditional systems are fully coupled; and48except for NO1128.3. Methods3. the hybrid closure, in which the scalar dissipation is assumed to have a modestimpact on the conditional state and the spatial and conditional systems are looselycoupled.Given the hypothesis that the conditional scalar dissipation is unimportant, the simpleclosure is applied. This implies that the chemistry can be pre-calculated by solving steady-state versions of the UCS transport equations with a static scalar dissipation, and theresulting conditional profiles can be used on-the-fly in a RANS/LES simulation by takingan inner product with the modelled PDF. This closure is particularly convenient for theSandia flame series as all flames have identical chemistry; it implies that only one conditionalsolution is required, and that the differences between the flames in the series can be accountedfor by the PDF alone. In order to focus more finely on the accuracy/inaccuracy of the UCSmodel, no attempt is made to model the joint PDF used in this comparison; it is generatedfrom experimental measurements. Previous authors [104, 105] have demonstrated that thisprobability density function can be modelled, but using the best-case experimental PDFeliminates PDF modelling error as a confounding variable and allows any shortcomings inthe predictions to be attributed to the UCS model alone. The use of an experimentalPDF implies that a full RANS/LES simulation is not necessary; the test need only solvethe the UCS transport equations in conditional space and perform post-processing usingthe experimental PDF. This makes it feasible to generate and evaluate several distinctconditional solutions to the UCS transport equations.8.3.1 Conditional Case RegimenA variety of grids, chemistry models, and scalar dissipation models are each analyzed byevolving the UCS transport equation (Equation 7.48) until steady state is reached.GridsIn the Sandia flame series, pure fuel is in fact only 25 % methane by volume (the remainderbeing air); this composition is therefore selected as the reference defining Z = 1. The domain,basis, and associated boundary conditions are illustrated in Figure 8.1. Following the MFMwork of [98], a reference grid is defined by dividing the conditional domain into 300 bins inmixture fraction (concentrated near the stoichiometric composition) and 100 bins in progressvariable (uniform spacing). To explore the impact of grid resolution, a coarser 150× 50 andfiner 600× 200 grid are also defined.ChemistryPrevious tests of the MFM model have used relatively simple reduced and skeletal mech-anisms [98, 103]. To expand on this, the present study considers a more detailed skeletalmechanism and two full mechanisms. All five mechanisms are described more fully inTable 8.1.1138.3. MethodsZ∗c∗0 101(a) Domain and basisUn-reacted MixtureComplete Combustion ProductsPureAirPure Fuel(25 % CH4,75 % Airby vol.)(b) Boundary ConditionsFigure 8.1: Conditional domain, basis, and boundary conditions. All subsequent colourplots use the same axis limits. The premixed flammability limits [76] and stoichiometriccomposition are marked with dashed and dotted lines, respectively.Table 8.1: Relevant chemical mechanismsMechanism Species Reactions Reference Used in. . .ARM2 19 15 [106] [98]DRM-19 19 91 [107] [103]Lu and Law 30 184 [36] this workGRI Mech 3.0 53 325 [38] this workUBC Mech 3.0 71 379 [42] this workScalar Dissipation ModelsFollowing the MFM work of [98], the cross scalar dissipation{Z∗,c∗}χZc˜ is approximated aszero. This can be justified by assuming that gradients in Z and c are independent; resultsof [108] have shown that these gradients are weakly correlated in the unburnt region, andnon-correlation is a necessary (but not sufficient) condition for independence. Also followingthe work of [98], the scalar dissipation of mixture fraction,{Z∗,c∗}χZZ˜ , is approximated as∀ {Z∗, c∗} ∈ {[0, 1], [0, 1]} :{Z∗,c∗}χZZ˜ = kZ exp[−2 (erfc−1(2Z∗))2] (8.4)where kZ values of 12.5 s−1, 25 s−1, and 37.5 s−1 are considered. The scalar dissipationof progress variable,{Z∗,c∗}χcc˜ , is modelled in two distinct ways. The first model uses thatobserved in an un-strained laminar premixed flame at the same mixture fraction, with twomodifications:• outside of the premixed flammability limits (where no un-strained premixed flame canexist), the scalar dissipation is estimated by interpolating:1148.3. Methods at lean compositions, the scalar dissipation is interpolated along a line of constantc∗ connecting the lean premixed flammability limit and the Z = 0 boundary(which has the fixed scalar dissipation 0 s−1), and at rich compositions, the scalar dissipation is interpolated along a line of constantc∗ connecting the rich premixed flammability limit and the Z = 1 boundary(which has the fixed scalar dissipation 0 s−1),and• the scalar dissipation is clipped to a minimum value of 15 s−1, to ensure stability (lowervalues were found to not produce stable solutions with the given solver).The resulting profile has an approximate range of 15 s−1 to 50 s−1; the alternative modelsselected are therefore the constant models{Z∗,c∗}χcc˜ = 15 s−1and{Z∗,c∗}χcc˜ = 50 s−1chosen fortheir simplicity, not for their accuracy. The resulting scalar dissipation profiles are illustratedin Figure 8.2.0 s−1 50 s−1(a) Constant at 15 s−1 (b) Clipped Flamelet (c) Constant at 50 s−1Figure 8.2: Model profiles of conditional scalar dissipation of progress variable,{Z∗,c∗}χcc˜ . Thedomain of and vertical lines in each sub-plot are as in Figure 8.1a.Final Case RegimenConsidering all combinations of grid, mechanism, and scalar dissipation model would givea total of 81 test cases. For simplicity, a single combination of parameters (described inTable 8.2) is taken as a reference case, and different variations about this case are consideredas independent sub-studies. A total of three such sub-studies are carried out:1. grid convergence is assessed by coarsening and refining the mesh,2. dependence on chemical mechanism is assessed by considering the Lu and Law [36]and UBC Mech 3.0 [42] chemical mechanisms, and1158.3. Methods3. dependence on scalar dissipation is assessed by varying the two scalar dissipationmodels independently, producing the 5 cases identified in Table 8.3.Table 8.2: Reference case conditionsParameter ValueGrid 300× 100Chemical Mechanism GRI Mech 3.0kZ 25 s−1{Z∗,c∗}χcc Clipped flamelet profileTable 8.3: Names and series styles for the combinations of scalar dissipation models consid-ered. All 5 associated cases use a 300× 100 grid and GRI Mech 3.0 chemistry. The centralReference case is intentionally identical to that described in Table 8.2.{Z∗,c∗}χcckZ12.5 s−1 25 s−1 37.5 s−1Constant High χccat 50 s−1Clipped Low χZZ Reference High χZZFlameletConstant Low χccat 15 s−18.3.2 SimulationsSimulations are carried out using a customized finite-volume solver based on the open-sourcepackage OpenFOAM [74] (version 3.0.1). Chemistry and conditional transport are coupledin a segregated way; for each time step:• A discrete integrator evolves the chemical state according to the chemical mechanism,subdividing the time step (if necessary) to respect the chemical time scale. The pres-sure and enthalpy fields are held constant during this integration.• For each species except the last, an effective average reaction rate over the time stepis calculated, then used to step the conditional transport equation forward by therequired time step.• The last species' mass fraction is set in such a way as to ensure that the species massfractions sum to one, and thus respect conservation of mass.The size of the time steps is fixed at 1× 10−7 s to satisfy Courant-Friedrichs-Lewy (CFL)-style conditions associated with conditional advection and diffusion.1168.3. MethodsAll simulations use second-order approximations for gradients (OpenFOAM's faceLim-ited Gauss linear 1 scheme), divergences (Gauss linear), Laplacians (Gauss linear cor-rected), and time advance (backward and rodas23 [109] for transport and reaction rateintegration, respectively).8.3.3 Post-ProcessingAs mentioned earlier, no attempt is made to model the joint PDF used in this comparison;simultaneous experimental measurements of Z and c are instead used to effectively con-struct an experimental PDF49. In practice, an experimental PDF is not calculated explicitlycalculated and then used as an input to an inner product operation; the two operationsare instead combined into one. Predicted singly-conditional and radial profiles are thereforegenerated from predicted doubly-conditional averages and the experimental measurementsof [71] as∀Z∗ ∈ [0, 1] : 〈ψ〉 (z, Z∗) = 1nmeas(z, Z∗)nmeas(z,Z∗)∑i=1{Z∗,c∗}ψ˜{Zi,ci}(8.5)〈ψ〉 (z, r) = 1nmeas(z, r)nmeas(z,r)∑i=1{Z∗,c∗}ψ˜{Zi,ci}(8.6)where i indexes the single-shot experimental measurements of Z and c at the relevant down-stream distance z and dummy mixture fraction Z∗ or downstream and distance z and radiusr, respectively (only data points for which 0 ≤ Z ≤ 1 and 0 ≤ c ≤ 1 are considered). Theseoperations effectively evaluate the inner product of the doubly-conditional average with aninfinitely finely resolved PDF generated from the experimental measurements. Compari-son profiles are generated from the raw experimental measurements of [71] by binning theexperimental data according to Z and c and averaging,∀Z∗ ∈ [0, 1] : 〈ψ〉 (z, Z∗) = 1nmeas(z, Z∗)nmeas(z,Z∗)∑i=1ψi (8.7)〈ψ〉 (z, r) = 1nmeas(z, r)nmeas(z,r)∑i=1ψi, (8.8)where i indexes the single-shot experimental measurements ψi, Zi, and ci as in Equations 8.5and 8.650. Both of these averaging operations effect an ensemble- or equivalently time-averaging of experimental data; this simulation can therefore be viewed as emulating aRANS calculation using experimental data, and the conditional state as a conditionallytime-averaged field. This is consistent with the fact that only steady-state solutions to theUCS transport equations are considered.49Although there are minor differences in the definitions of Z and c between simulations and experiments(the simulation assumes equal species diffusivities, but in reality there is some preferential transport), theseare expected to be negligible.50Excluding measurements with invalid Z or c produces experimental profiles which differ slightly fromthose published with the raw measurements [71]1178.4. ResultsTo complement the detailed singly-conditional and radial profiles, it is convenient to beable to define a single metric which quantifies solution accuracy. For some set of doubly-conditional predictions{Z∗,c∗}ψ˜ of a given scalar ψ, a normalized error Eψ (quantifying devi-ation from the experimental measurements of [71]) can be defined asEψ ≡ 1ψ99√√√√√ 1nmeasnmeas∑i=1ψi − {Z∗,c∗}ψ˜ (Zi, ci)2(8.9)where• ψ99 is the 99th percentile of the raw measurements of ψ, and• i indexes the single-shot experimental measurements ψi, Zi, and ci at all downstreamdistances and radii (only data points for which 0 ≤ Z ≤ 1 and 0 ≤ c ≤ 1 areconsidered).Because the two-condition conditional average cannot represent every experimental mea-surement perfectly, even the ideal conditional average generated directly by binning theexperimental measurements has a non-zero normalized error. In the language of optimal es-timator theory [89], the normalized error in the experimental measurements represents theirreducible error associated with the assumption that a single doubly-conditional averagecan represent all measurements; the ratio of the normalized error from a given simulationto that inherent in the measurements can, in turn, be viewed as an indicator of the qualityof the simulation. This metric is, of course, an extremely coarse-grained measure of perfor-mance, but offers an attractive way of comparing simulations because it is both objectiveand quantitative.8.4 ResultsThe results of the three sub-studies (grid convergence, chemical mechanism, and scalardissipation are each presented in their own sub-section.8.4.1 Grid Convergence Sub-StudyFigures 8.3 and 8.4 illustrate doubly-conditional profiles of some scalars of interest, andFigure 8.5 illustrates normalized errors associated with each grid. Although the doubly-conditional profiles show that refining from a 150 × 50 grid to a 300 × 100 grid leads tonoticeable smoothing of most fields, the normalized errors suggest that the discretization-associated discontinuities in the 150 × 50 grid have a negligible impact on overall abilityto match experimental data. Taken together, these results suggest that even the coarse150× 50 mesh is grid-converged. Despite this, the 300× 100 mesh is used in the remainingsub-studies for consistency with prior work.1188.4. Results150× 50 300× 100 600× 200YH2O00.129YCO00.085YOH00.0055YH200.005Figure 8.3: Variation of doubly-conditional profiles with grid resolution (YH2O, YCO, YOH,and YH2). All parameters other than grid resolution are as in the reference case (Table 8.2);domains and vertical lines are as in Figure 8.1a.1198.4. Results150× 50 300× 100 600× 200YNO04 · 10−5YC2H200.015T300 K2350 Kω˙c−1700 s−10 s−110 000 s−1Figure 8.4: Variation of doubly-conditional profiles with grid resolution (YNO, YC2H2 , T ,and ω˙c). All parameters other than grid resolution are as in the reference case (Table 8.2);domains and vertical lines are as in Figure 8.1a.1208.4.ResultsYH2O YCO YOH YH2 T YNO0%5%10%15%20%25%ScalarNormalizedError,EExperimental150× 50300× 100600× 200Figure 8.5: Normalized error (Equation 8.9) associated with various scalars and grid resolutions. All parameters other than gridresolution are as in the reference case (Table 8.2).1218.4. Results8.4.2 Chemical Mechanism Sub-StudyFigures 8.6 and 8.7 illustrate doubly-conditional profiles of some scalars of interest, and Fig-ure 8.8 illustrates normalized errors associated with each mechanism. The results suggestthat, despite the wide difference in complexity (Table 8.1), the doubly-conditional scalarprofiles associated with the three mechanisms all exhibit strikingly similar patterns (Fig-ures 8.6 and 8.7). The UBC mechanism predicts a more intense reaction zone (with higherpeak values of YCO, YOH, YH2 , and ω˙c) at slightly lower c∗than the other mechanisms.On the other hand, the peak value of YNO is lower for the UBC mechanism than for theGRI mechanism (the Lu and Law mechanism offers no prediction). Considering ability tomatch experimental data (as represented by the normalized error, Figure 8.8), the Lu andLaw mechanism and GRI mechanism are essentially indistinguishable (with the exception ofNO, where the Lu and Law mechanism offers no prediction), while the UBC mechanism isinvariably less successful. The similarity of the results suggest that all three models capturethe broad trends in methane-air combustion reasonably well; for simplicity, only the GRIMech 3.0 mechanism (which offers intermediate computational cost and appears to give thebest overall accuracy) is considered in other sub-studies.1228.4. ResultsLu and Law GRI Mech 3.0 UBC Mech 3.0YH2O00.129YCO00.085YOH00.0055YH200.005Figure 8.6: Variation of doubly-conditional profiles with chemical mechanism (YH2O, YCO,YOH, and YH2). All parameters other than chemical mechanism are as in the reference case(Table 8.2); domains and vertical lines are as in Figure 8.1a.1238.4. ResultsNot presentin mechanismLu and Law GRI Mech 3.0 UBC Mech 3.0YNO04 · 10−5YC2H200.015T300 K2350 Kω˙c−1700 s−10 s−110 000 s−1Figure 8.7: Variation of doubly-conditional profiles with chemical mechanism (YNO, YC2H2 ,T , and ω˙c). All parameters other than chemical mechanism are as in the reference case(Table 8.2); domains and vertical lines are as in Figure 8.1a.1248.4.ResultsYH2O YCO YOH YH2 T YNO0%5%10%15%20%25%30%35%ScalarNormalizedError,EExperimentalLu and LawGRI Mech 3.0UBC Mech 3.0Figure 8.8: Normalized error (Equation 8.9) associated with various scalars and chemical mechanisms. The Lu and Law mechanismdoes not include NO; the EYNOplotted for this mechanism is calculated based on the implied prediction YNO= 0. All parametersother than chemical mechanism are as in the reference case (Table 8.2).1258.4. Results8.4.3 Scalar Dissipation Sub-StudyFigures 8.9 through 8.16 illustrate doubly-conditional profiles of YH2O, YCO, YOH, YH2 , YNO,YC2H2 , T , and ω˙c, respectively. The results generally illustrate that varying scalar dissipationchanges details of the predicted fields, but does not alter the broad trends. For the scalesof variations considered,• YH2O, YCO, YOH, YH2 , T , and ω˙c appear to be more sensitive to changes in χZZ thanchanges in χcc, while• NO and C2H2 show noticeable sensitivity to both scalar dissipations.Figures 8.17 through 8.24 illustrate singly-conditional profiles of YH2O, YCO, YOH, YH2 ,YNO, YC2H2 , T , and ω˙c, respectively; Figures 8.25 through 8.32 illustrate radial profiles ofthe same scalars. For the scalar dissipation models considered considered,• overall ability to match experimental data is good, with predictions of H2O and Tstanding out as particularly good and predictions of OH and NO standing out ascomparatively poor, and• sensitivity of predicted profiles to scalar dissipation is variable: for H2O, OH, NO, and T , it is quite modest (although often less so for thehigher-Reynolds number flames), for CO, it is modest, for H2, it is noticeable, and for C2H2 and ω˙c, it is significant.The normalized errors, which attempt to represent solution accuracy with a single valueper scalar field, are presented in Figure 8.33. There is noticeable variation in the normalizederror associated with the various solutions. Overall, the high χcc case could be labelled asbest since it gives the lowest or second-lowest normalized error for all cases, but its marginof victory is slim. As expected, the normalized error in NO is highest (illustrating that itis predicted comparatively poorly), butsurprisinglythe normalized error in OH is onlymarginally lower. Overall, the results are consistent with Bushe's previous observation thata doubly-conditional average can represent the experimental measurements of some (but notall) species to good accuracy [97].1268.4. Results0 0.129Figure 8.9: Variation of doubly-conditional profiles of YH2O with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1278.4. Results0 0.085Figure 8.10: Variation of doubly-conditional profiles of YCO with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1288.4. Results0 0.0055Figure 8.11: Variation of doubly-conditional profiles of YOH with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1298.4. Results0 0.005Figure 8.12: Variation of doubly-conditional profiles of YH2 with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1308.4. Results0 4 · 10−5Figure 8.13: Variation of doubly-conditional profiles of YNO with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1318.4. Results0 0.015Figure 8.14: Variation of doubly-conditional profiles of YC2H2 with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1328.4. Results300 K 2350 KFigure 8.15: Variation of doubly-conditional profiles of T with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1338.4. Results−1700 s−1 10 000 s−1Figure 8.16: Variation of doubly-conditional profiles of ω˙c with scalar dissipation model.Cases are arranged as in Table 8.3 and domains and vertical lines are as in Figure 8.1a.1348.4.Results05 · 10−20.1FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet05 · 10−20.1FlameD05 · 10−20.1FlameE0 0.2 0.4 0.6 0.8 105 · 10−20.1Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Experimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.17: Singly-conditioned profiles of YH2O (generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1358.4.Results02 · 10−24 · 10−26 · 10−28 · 10−2FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet02 · 10−24 · 10−26 · 10−28 · 10−2FlameD02 · 10−24 · 10−26 · 10−28 · 10−2FlameE0 0.2 0.4 0.6 0.8 102 · 10−24 · 10−26 · 10−28 · 10−2Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Experimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.18: Singly-conditioned profiles of YCO (generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1368.4.Results02 · 10−34 · 10−36 · 10−3FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet02 · 10−34 · 10−36 · 10−3FlameD02 · 10−34 · 10−36 · 10−3FlameE0 0.2 0.4 0.6 0.8 102 · 10−34 · 10−36 · 10−3Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Experimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.19: Singly-conditioned profiles of YOH (generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1378.4.Results02 · 10−34 · 10−3FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet02 · 10−34 · 10−3FlameD02 · 10−34 · 10−3FlameE0 0.2 0.4 0.6 0.8 102 · 10−34 · 10−3Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Experimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.20: Singly-conditioned profiles of YH2 (generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1388.4.Results02 · 10−54 · 10−56 · 10−5FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet02 · 10−54 · 10−56 · 10−5FlameD02 · 10−54 · 10−56 · 10−5FlameE0 0.2 0.4 0.6 0.8 102 · 10−54 · 10−56 · 10−5Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Experimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.21: Singly-conditioned profiles of YNO (generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1398.4.Results02 · 10−34 · 10−36 · 10−38 · 10−3FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet02 · 10−34 · 10−36 · 10−38 · 10−3FlameD02 · 10−34 · 10−36 · 10−38 · 10−3FlameE0 0.2 0.4 0.6 0.8 102 · 10−34 · 10−36 · 10−38 · 10−3Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Reference Low χcc High χcc Low χZZ High χZZFigure 8.22: Singly-conditioned profiles of YC2H2 (generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1408.4.Results01,0002,000FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet01,0002,000FlameD01,0002,000FlameE0 0.2 0.4 0.6 0.8 101,0002,000Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Experimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.23: Singly-conditioned profiles of T , in K (generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1418.4.Results−2,00002,0004,000FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet−2,00002,0004,000FlameD−2,00002,0004,000FlameE0 0.2 0.4 0.6 0.8 1−2,00002,0004,000Z∗FlameF0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗0 0.2 0.4 0.6 0.8 1Z∗Reference Low χcc High χcc Low χZZ High χZZFigure 8.24: Singly-conditioned profiles of ω˙c , in s−1(generated via Equations 8.5 and 8.7). Case labels correspond with those inTable 8.3. The premixed flammability limits [76] and stoichiometric composition are marked by a shaded background and verticaldashed line, respectively.1428.4.Results05 · 10−20.1FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet05 · 10−20.1FlameD05 · 10−20.1FlameE0 1 205 · 10−20.1r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetExperimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.25: Radial profiles of YH2O(generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1438.4.Results02 · 10−24 · 10−2FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet02 · 10−24 · 10−2FlameD02 · 10−24 · 10−2FlameE0 1 202 · 10−24 · 10−2r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetExperimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.26: Radial profiles of YCO(generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1448.4.Results01 · 10−32 · 10−33 · 10−34 · 10−3FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet01 · 10−32 · 10−33 · 10−34 · 10−3FlameD01 · 10−32 · 10−33 · 10−34 · 10−3FlameE0 1 201 · 10−32 · 10−33 · 10−34 · 10−3r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetExperimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.27: Radial profiles of YOH(generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1458.4.Results01 · 10−32 · 10−33 · 10−3FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet01 · 10−32 · 10−33 · 10−3FlameD01 · 10−32 · 10−33 · 10−3FlameE0 1 201 · 10−32 · 10−33 · 10−3r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetExperimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.28: Radial profiles of YH2(generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1468.4.Results01 · 10−52 · 10−53 · 10−54 · 10−5FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet01 · 10−52 · 10−53 · 10−54 · 10−5FlameD01 · 10−52 · 10−53 · 10−54 · 10−5FlameE0 1 201 · 10−52 · 10−53 · 10−54 · 10−5r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetExperimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.29: Radial profiles of YNO(generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1478.4.Results01 · 10−32 · 10−33 · 10−34 · 10−3FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet01 · 10−32 · 10−33 · 10−34 · 10−3FlameD01 · 10−32 · 10−33 · 10−34 · 10−3FlameE0 1 201 · 10−32 · 10−33 · 10−34 · 10−3r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetReference Low χcc High χcc Low χZZ High χZZFigure 8.30: Radial profiles of YC2H2(generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1488.4.Results05001,0001,5002,000FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet05001,0001,5002,000FlameD05001,0001,5002,000FlameE0 1 205001,0001,5002,000r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetExperimental Reference Low χcc High χcc Low χZZ High χZZFigure 8.31: Radial profiles of T , in K (generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1498.4.Results01,0002,0003,000FlameCz = 7.5djetz = 15djetz = 30djetz = 45djetz = 60djetz = 75djet01,0002,0003,000FlameD01,0002,0003,000FlameE0 1 201,0002,0003,000r/djetFlameF0 1 2 3r/djet0 2 4 6r/djet0 2 4 6 8r/djet0 5 10r/djet0 5 10r/djetReference Low χcc High χcc Low χZZ High χZZFigure 8.32: Radial profiles of ω˙c, in s−1(generated via Equations 8.6 and 8.8). Case labels correspond with those in Table 8.3.1508.4.ResultsYH2O YCO YOH YH2 T YNO0%5%10%15%20%25%30%ScalarNormalizedError,EExperimentalReferenceLow χccHigh χccLow χZZHigh χZZFigure 8.33: Normalized error (Equation 8.9) associated with various scalars and scalar dissipation models. Series labels correspondwith cases in Table 8.3.1518.5. Discussion8.5 DiscussionIn addition to assessing the two hypotheses, this section considers two questions raised bythe results above, specifically:• What are the implications of the grid convergence and chemical mechanism studies?• Why are the predictions of OH almost as poor as those of NO?• In light of the results, do all of the UCS closures proposed in Section 7.4.1 remainviable?8.5.1 Assessment of HypothesesHypothesis 1: The chosen specialization of the Uniform Conditional Statemodel can predict the properties of the Sandia flame series with good accuracyThe results generally support of this hypothesis. Predictions of the major scalars and radicalsfollow the correct trends and are generally a good match for experimental measurements,although there are exceptions. The predicted profiles of OH and NO are least accurate; inthe case of NO, this can be explained as a consequence of the fact that the species does notsatisfy the uniform conditional state property (Property 1 of Section 7.3.1).Hypothesis 2: The chosen specialization of the Uniform Conditional Statemodel is substantially insensitive to variations in conditional scalar dissipationThe results are consistent with previous MFM results [98] in that they generally refutethis hypothesis. Although profiles of several scalars show very weak or weak sensitivity tovariations in scalar dissipation, the source of progress variable and the profiles of the sootprecursor C2H2show significant sensitivity to scalar dissipation. This work also neglectsan effect which would likely introduce additional scalar dissipation dependence to all terms:coupling between the progress variable and its conditional source. For typical presumed PDFmodels, the full coupling is somewhat complicated: the conditional source term influencesthe filtered progress variable and its variance (through their respective spatial transportequations), which influences the PDF (through the PDF model), which influence the uncon-ditionally filtered predictions (through the conditional-unconditional relation). Althoughthis feedback has the potential to reinforce or damp the dependence on scalar dissipationwhich is observed in the source of progress variable, the extreme nonlinearity of the reactionrate source term suggests that an amplification effect is more likely. Ultimately, only a fullRANS/LES simulation which accounts for this coupling can fully assess the dependence ofpredictions on scalar dissipation.8.5.2 Grid Convergence and Chemical Mechanism StudiesThe grid convergence study suggests that a 150×50mesh is grid converged, and (since coarsergrids were not considered) leaves open the possibility that an even coarser mesh mightalso show convergence. The chemical mechanism study suggests that the UBC chemicalmechanism is not worth the associated increase in computational complexity (and, indeed,1528.6. Concluding Remarksgives worse predictions), and also that the Lu and Law mechanism is an excellent substitutefor the full GRI mechanism provided that one is not interested in NO chemistry. Takentogether, these results suggest that future simulations of the Sandia flame series should usea conditional mesh with a resolution no finer than 150× 50 and either the GRI mechanism(if NO is of interest) or the Lu and Law mechanism (if NO is not of interest).8.5.3 Predictions of OHThe normalized errors in Figure 8.33 suggest that, overall, predictions of OH are almostas poor as those of NO. It was expected that predictions of the latter would be poorNOis produced and consumed by relatively slow chemical mechanisms, so its mass fraction issensitive to residence time in the reaction zone, and the assumption that filtering condi-tionally on K = {Z, c} eliminates all spatial variation is questionablebut, since OH is partof the main reaction pathway, it was not expected that its predictions would be similarlypoor. The simplest explanation for this is that the normalized error provides a very coarse-grained measure of performance; the more detailed conditional and radial comparisons (inFigures 8.19 vs. 8.21 and 8.27 vs. 8.29, respectively) provide a more detailed comparison,and illustrate that, station-by-station, predictions of OH are obviously superior to those ofNO. This suggests that it may not be meaningful to compare normalized error values be-tween scalars, and overall underscores the fact that no single scalar can provide a completerepresentation of a multi-faceted data set.8.5.4 Closure for the UCS ModelThe partial refutation of Hypothesis 2 suggests thatat least for the case consideredthesimple closure of Section 7.4.1 is unlikely to be successful in a full RANS or LES calculation,and the UCS model must instead be implemented using the full or hybrid closure. Thissuggests a model which not only has theoretical differences from FMFM (as previouslydiscussed in Section 7.4.2), but also features a significantly different implementation fromprior FMFM solvers due to its reliance on an integral inversion.8.6 Concluding RemarksThe objective of this chapter was to test the UCS model. The extensive results of Sec-tion 8.4 fulfill this objective. The results generally confirm the hypothesis that the UCSmodel can predict the properties of the Sandia flame series, but refute the hypothesis thatthe model is substantially insensitive to variations in the conditional scalar dissipation. Fu-ture work should include full RANS/LES simulations to account for feedback between thePDF of progress variable and its rate of production, and to thereby more fully assess bothhypotheses.As in all other research chapters, the conclusions of this work will be re-visited in thecontext of the thesis as a whole in Section 9.1.1.153Part IIIConclusions and References154Chapter 9ConclusionsAs laid out in Section 1.2, this work has several objectives. At the highest level, it aims toimprove the quality (applicability, computational efficiency, and/or accuracy) of combustionsimulation tools and thereby facilitate the development of improved combustion technology.At the intermediate level, the various chapters each aim to answer research questions relatedto the general theme of conditional filtering.Chapter 5: what are the implications of treating position as a conditioning variable inCSE?Chapter 6: what are the implications of conditioning on scalars rather than position inthe dynamic Smagorinsky model?Chapter 7 what are the theoretical implications of assuming that conditioned fields areindependent of position?Chapter 8 what are the practical implications of assuming that conditioned fields areindependent of position?These objectives are each mirrored by conclusions below; the intermediate-level conclusionsare presented in Section 9.1, while the high-level conclusions are reserved for Section 9.2.9.1 Chapter-Level9.1.1 Chapter 5The theoretical analysis of Section 5.2 provides thorough documentation of both thetraditional CSE algorithm and the new CSE with Geometric Conditioning Variables (CSE-GCV) algorithm, which is based on the idea of using position (or, more generally, functionsof position) as a conditioning variable. The derivation illustrates that treating position as aconditioning variable produces an ensemble-free variant of CSE, and thereby eliminates thetheoretical and practical issues associated with traditional CSE. The results of Section 5.4illustrate that CSE-GCV can successfully simulate a turbulent non-premixed flame justas traditional CSE can, and the analysis of Section 5.5 demonstrates that CSE-GCV isgenerally no more expensive (and, indeed, less expensive, typically by a factor of about 3)than traditional CSE. Overall, then, the answer to the chapter-level research question isthat treating position as a conditioning variable in CSE produces a successful CSE variantwhich is more self-consistent, more straightforward to implement, and less computationallydemanding than traditional CSE.1559.1. Chapter-LevelThe main limitation of this work is that CSE-GCV was validated using only one test case(the non-premixed Sandia D flame) and that results were not directly compared to thoseproduced by traditional CSE. Future work could explore other test cases and could performa direct comparison of the results produced by the two algorithms; given that traditionalCSE and CSE-GCV both produce conditional fields which vary slowly in space (althoughthe details of the variation differ), it is natural to hypothesize that the two algorithms shouldproduce essentially identical results in essentially all flames. Given the theoretical benefitsof CSE-GCV and the relatively low likelihood that its predictions could ever be appreciablyworse than those of traditional CSE, it is recommended that future CSE simulations preferthe CSE-GCV algorithm over the traditional CSE algorithm.9.1.2 Chapter 6The theoretical analysis presented in Section 6.2.5 suggests that conditioning on determina-tive scalars rather than position can more effectively group related cells, thereby capturingthe spatial dependence of turbulence modelling parameters more effectively. The resultsof Section 6.4 illustrate that, for the case considered, this theoretical advantage does nottranslate into a practical improvement in prediction quality. Despite this, the conditionaldynamic model is a success in that it is both tractable and stable. Overall, then, theanswer to the chapter-level research question is that conditioning on scalars rather thanposition in the dynamic Smagorinsky model has theoretical benefits which do not translateinto practical benefits in the case considered.The concept of a conditional dynamic turbulence model has the potential to evolve intoits own sub-field of LES sub-filter modelling; there are many sets of conditioning variableswhich could give a plausible turbulence model, and therefore many potential specializa-tions of the conditional dynamic turbulence model concept. This work conditioned on bothmixture fraction and its variance in a non-premixed flame; future work could condition onprogress variable and its variance in a premixed flame, on enstrophy and resolved strain ratein an arbitrary turbulent flow, or on some other set of determinative variables.The main limitation of this work is that both the traditional and conditional dynamicprocedures performed sub-optimally in the given test case; comparing the procedures' pre-dictions would be more meaningful in a situation where the traditional dynamic procedureperforms well. The sub-optimal performance is thought to be a consequence of the pencil-like cells which appear in the downstream region of the given grid; future work couldpotentially address this limitation by considering a more uniform or pancake-like grid.9.1.3 Chapter 7The derivation of Section 7.3 illustrates that assuming uniformity of conditional state canproduce a turbulence-chemistry interaction model. In light of the discussion of Section 7.4.2,the answer to the chapter-level research question is that assuming conditional fields areindependent of position has several theoretical consequences:• a conditional moment approach need not model terms associated with physical gradi-ents of conditional quantities,1569.2. Overall• there are distinct conditional spatial and conditional domains which are only looselycoupled through the modelled PDF, and• in some situations, the implementation of the resulting model UCS can coincide withthat of turbulence-chemistry models based on the Multidimensional Flamelet Manifoldapproach.The final consequence is particularly interesting because it suggests that, in some situations,there may be relatively universal modelling relationships which transcend any one set ofstarting assumptions.Like the conditional dynamic turbulence model, the UCS model can be specialized inmany ways; there are many plausible sets of conditioning variables and diffusive flux models.Given this, and the theoretical attractiveness of a turbulence-chemistry interaction modelwhich is compatible with any chemistry model and not restricted to any particular pre-mixedness level or combustion regime, the UCS model has the potential to evolve into itsown sub-field of turbulence-chemistry interaction modelling.The main limitation of this work is that it assumes that all thermodynamic variablesof interest have species-like transport equations, which is not strictly true for the progressvariable (unless the combustion is full premixed), internal energy, or enthalpy. Future workcould address this limitation by introducing extra terms to account for those present intransport of progress variable, internal energy, or enthalpy; the MFM temperature equa-tion [98] would provide a useful reference in this endeavour. A second limitation is theassumption that transport terms associated with gradients of conditionally-filtered fluctua-tions of are negligible. This assumption is relatively uncontroversial, as it is already invokedin the well-established CMC model, but it could be explicitly tested (using DNS data) forthe enlarged set of conditioning variables which is required for UCS.9.1.4 Chapter 8The results of Section 8.4 illustrate that the UCS model can predict many of the properties ofthe Sandia flame series to good accuracy. The answer to the chapter-level research questionis therefore that, for the non-premixed case and the conditioning variables considered, thereis one major practical consequence to assuming that conditional fields are independent ofposition: reasonably accurate calculations predictions can be generated by solving the UCStransport equations and using an experimental PDF.The main limitation of this work is the reliance on an experimental PDF, which eliminatespotential coupling between the PDF of progress variable and its source term. Future workcould address this limitation by performing full RANS/LES simulations.9.2 OverallThe highest-level objective of this work was to improve the quality (applicability, computa-tional efficiency, and/or accuracy) of combustion simulation tools and thereby facilitate thedevelopment of improved combustion technology. Each of the chapters has contributed tofulfilling this objective:1579.2. Overall• the CSE-GCV algorithm of Chapter 5 improves the applicability and computationalefficiency of the CSE approach,• the conditional dynamic sub-filter modelling approach of Chapter 6 improves the ap-plicability and (potentially) the accuracy of LES,• the UCS model of Chapters 7 and 8 improves the applicability and, potentially, theefficiency and accuracy of turbulence-chemistry interaction modelling.These techniques could, in future, be applied in combustion simulation tools for industrialdesign, and would thereby aid in the development of improved combustion technology. Thehighest-level objective is therefore fulfilled in full.9.2.1 Future WorkWhile the chapters of this thesis each individually address facets of turbulent combustionmodel, together they suggest a single turbulent combustion modelling approach which hasthe potential to supersede several more specialized techniques. This approach would be basedon an LES solver, using the full or hybrid UCS closure for turbulence-chemistry interactionmodelling and a conditional dynamic model to close all sub-filter terms. Combining the UCSclosure with a conditional dynamic model for the sub-filter scalar dissipation eliminates manyopportunities for tuning, giving a more objective and thus theoretically palatable model. Thecombination potentially allows for significant feedback between the scalar dissipation andall other fields; testing is required to assess whether this coupling tends to bring the systemto a stable solution (potentially representing some kind of universal equilibrium) or resultsin oscillatory or divergent behaviour.One of the models that LES-UCS-conditional dynamic closure proposed above couldsupersede is CSE, including CSE-GCV. Although the CSE-GCV algorithm represents asignificant improvement over traditional CSE, it continues to rely on manifold chemistry ap-proaches as traditional CSE does; where manifold chemistry is appropriate, CSE-GCV wouldoffer a potentially simpler closure than the UCS approach, but in a general scenariowherethe chemistry cannot be reduced to a low-dimensional manifoldonly the UCS approach isapplicable. The UCS approach can nonetheless be viewed as a descendant or evolution ofCSE methods, since the full and hybrid UCS closures rely on the integral inversion techniqueswhich form the backbone of CSE.Given a functional LES-UCS-conditional dynamic solver, it would become possible toexplore a number of interesting questions related the model's performance, such as• is the full UCS closure required, or is the hybrid closure sufficient?• if the hybrid closure is sufficient, how many update cycles are required before theconditional scalar dissipation stabilizes?• how does varying the number of conditioning variables impact predictions?• how does varying the identities of conditioning variablesin particular, the definitionof progress variableimpact predictions?1589.2. OverallAnother option which could be explored is whether it is necessary to use a progress variableat all; conditioning on the progress variable-defining field Yc instead of the progress vari-able (which is defined as a normalized version of Yc) could potentially eliminate the scalardissipation-like terms from the original spatial transport equation. Eliminating these termswould make the UCS closure more exact, at the cost of introducing cross scalar dissipa-tion terms to the conditional transport equation. It would also be enlightening to considerhow the UCS system performs when species that do not vary monotonically across normalflames, such as OH, are selected as conditioning variables; this could be assessed using ex-perimental or DNS data. 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Piloted CH4/Air Flames C, D, E, and F  Release 2.1, 2007.167Appendices168Appendix ANotationA.1 Mathematical ObjectsThe notation for mathematical objects described in Table A.1. The superscript ◦ on theidentity tensor denotes the fact that the identity tensor is isotropic; this notation is intro-duced in Section A.2.2.Table A.1: Mathematical Object NotationClass Vector Notation Einstein NotationScalar x x[Physical] Vector ~x xi[Rank-Two] Tensor x xij[Rank-Two] Identity TensorI◦ δij(Kronecker Delta)A.2 Mathematical OperatorsA.2.1 Basic Vector and Tensor OperationsThe notation for basic vector and tensor operations is described in Tables A.2 through A.4.Table A.2: Magnitude NotationOperation Vector Notation Einstein NotationMagnitude of Vector |~a| = √~a · ~a |ai| = √akakMagnitude of Tensor |a| =√a : a |aij | = √aklakl169A.2. Mathematical OperatorsTable A.3: Vector and Tensor Product NotationOperation Vector Notation Einstein NotationScalar Multiplication c = ab or a · b c = ab or a · b~c = a~b or a ·~b ci = abi or a · bic = ab or a · b cij = abij or a · bijInner Product c = ~a ·~b c = aibi~c = a ·~b ci = aijbj~c = ~a · b cj = aibijDouble Inner Product c = a : b c = aijbjiOuter Product ~c = a~b ci = abic = ~a~b cij = aibjTable A.4: Tensor Operation NotationOperation Vector Notation Einstein NotationTrace b = Tr(a)= a : I◦ b = aii = aijδijTranspose b = aT bij = ajiA.2.2 Tensor DecompositionThe decomposition of a tensor a into isotropic, symmetric-and-traceless, and anti-symmetriccomponents is represented asa =IsotropicComponentucurly(a)◦+Symmetric,TracelessComponentucurly(a)‡´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶(a)F®SymmetricComponent+Anti-SymmetricComponentucurly(a)†These operations are described in more detail in Table A.5.Table A.5: Tensor Decomposition Operator NotationComponent Extracted Vector Notation Einstein NotationSymmetric(a)F= 12(a+ aT)(aij)F = 12 (aij + aji)Antisymmetric(a)†= 12(a− aT) (aij)† = 12 (aij − aji)Isotropic(a)◦= 13 Tr(a)I◦ (aij)◦ = 13akkδijSymmetric, Traceless(a)‡=(a)F − (a)◦ (aij)‡ = (aij)F − (aij)◦170A.2. Mathematical OperatorsThe rationale for this choice of symbols is:• ◦ is the same in all directions, i.e., isotropic• † is not vertically symmetrical and does not include ◦; an anti-symmetric tensor isnot symmetrical about the main diagonal and has no isotropic component• ‡ is vertically symmetrical and does not include ◦; a symmetric tensor is symmetricalabout the main diagonal and has no isotropic component• F can be viewed as a stylized combination of ‡ and ◦; a symmetric tensor is the sumof a symmetric, traceless tensor and an isotropic tensorWhen these same symbols appear as subscripts rather than superscripts, they indicatethat the named matrix has the relevant property:• a◦ is an isotropic matrix• a‡ is a symmetric, traceless matrix• aF is a symmetric matrix• a† is an antisymmetric matrix171A.2. Mathematical OperatorsA.2.3 Vector CalculusThe notation for vector calculus operations is described in Table A.6.Table A.6: Vector Calculus NotationOperation Vector Notation Einstein NotationGradient ~y = ~∇x yi = ∂∂xixy = ~∇~x yij = ∂∂xixjDivergence y = ~∇ · ~x y = ∂∂xixi~y = ~∇ · x yj = ∂∂xixijLaplacian y = ∇2x y = ∂∂xi∂xix~y = ∇2~x yi = ∂∂xj∂xj xiMaterial Derivative y = DDtx y =DDtx~y = DDt~x yi =DDtxi172Appendix BFilter-Related DefinitionsB.1 Filtering OperationsB.1.1 The Basic (Volume-Weighted) FilterEach field φ (~x, t) can be assigned a corresponding filtered field φ (~x, t):∀~x ∈ X .∀t ∈ T :φ (~x, t) ≡ F [φ (~x, t)] ≡∫T∫XGF (~x, ~x∗, t, t∗) · φ (~x∗, t∗) d~x∗dt∗ (B.1)where• X is the entire space domain,• T is the entire time domain,• F is the filter operator, which can only act on and return a field51,• GF is the filter kernel, which is (in this work) always non-negative∀~x, ~x∗ ∈ X .∀t, t∗ ∈ T : GF (~x, ~x∗, t, t∗) ≥ 0, (B.2) normalized,∀~x ∈ X . ∀t ∈ T :∫T∫XGF (~x, ~x∗, t, t∗) d~x∗dt∗ = 1, (B.3)(implying [110] that the filter operation has no impact on uniform fields), and conservative,∀~x∗ ∈ X .∀t∗ ∈ T :∫T∫XGF (~x, ~x∗, t, t∗) d~xdt = 1,(implying that the integral of a filtered function is the same as that of the corre-sponding un-filtered function [110]).51as opposed to (for example) the square operator, which can act on and return a field or a single value173B.1. Filtering OperationsFor notational compactness, one can also define the filter integral operator,∀A : G(~x,~x∗,t,t∗)[A] ≡∫T∫XGF (~x, ~x∗, t, t∗) ·Ad~x∗ dt∗. (B.4)which can be used to write filtering more compactly as∀φ : φ = G(~x,~x∗,t,t∗)[φ (~x, t)] . (B.5)The filter is a linear operator,∀φ, ψ : F [φ (~x, t) + ψ (~x, t)] = F [φ (~x, t)] + F [ψ (~x, t)] (B.6)∀a, φ : F [aφ (~x, t) ] = aF [φ (~x, t) ] , (B.7)and (although it is not always rigorously true) is typically assumed to commute with timeand space derivatives:∀φ : ~∇φ ≈ ~∇φ ∂∂tφ ≈ ∂∂tφ. (B.8)B.1.2 The Favre (Mass-Weighted) FilterEach field φ (~x, t) can be assigned a corresponding Favre-filtered field φ˜ (~x, t) by the operation∀~x ∈ X .∀t ∈ T :φ˜ (~x, t) ≡ F˜ [φ (~x, t)] ≡ F [ρ (~x, t) · φ (~x, t)]F [ρ (~x, t)]. (B.9)where F˜ is the Favre filter operator, which is linear, and can be viewed as a modification ofthe basic filter in which volume-weighting is replaced with mass-weighting. The associatedkernel is thenGF˜ (~x, ~x∗, t, t∗) =ρ (~x∗, t∗) ·GF (~x, ~x∗, t, t∗)ρ (~x, t). (B.10)For notational compactness, one can define the Favre filter integral operator,∀A : G˜(~x,~x∗,t,t∗)[A] ≡∫T∫XGF˜ (~x, ~x∗, t, t∗) ·Ad~x∗ dt∗. (B.11)which can be used to write Favre filtering more compactly as which can be used to writefiltering more compactly as∀φ : φ˜ = G˜(~x,~x∗,t,t∗)[φ (~x, t)] . (B.12)The Favre filter is a linear operator,∀φ, ψ : F˜ [φ (~x, t) + ψ (~x, t)] = F˜ [φ (~x, t)] + F˜ [ψ (~x, t)] (B.13)∀a, φ : F˜ [aφ (~x, t) ] = aF˜ [φ (~x, t) ] , (B.14)174B.1. Filtering Operationsbut commutation of the basic filter with time and space derivatives does not imply commu-tation of the Favre filter with the same:~∇φ ≈ ~∇φ 6=⇒ ~∇φ˜ ≈ ~∇φ˜ (B.15)∂∂tφ ≈ ∂∂tφ 6=⇒ ∂∂t φ˜ ≈∂∂tφ˜. (B.16)B.1.3 Conditional FiltersGiven a set K of conditioning variables κ, one can define• K, the set of all allowable values of K or conditional domain,• K (~x, t), the set of corresponding dependent fields κ (~x, t), satisfying∀~x ∈ X . ∀t ∈ T : K (~x, t) ∈ K (B.17)• K∗, the set of corresponding independent variables κ∗, which may take on any valuein K 52.In a similar vein, for some conditioning variable κ (κ ∈ K), κ (~x, t) and κ∗ can be defined asthe corresponding dependent field independent variable, respectively. Conditionally-filteredand conditionally-Favre-filtered fields are then written asK∗φ (~x, t) ≡K∗F [φ (~x, t)] ≡ F[φ (~x, t) ·∏κ∈K δ (κ (~x, t)− κ∗)]F[∏κ∈K δ (κ (~x, t)− κ∗)](B.18)K∗φ˜ (~x, t) ≡K∗F˜ [φ (~x, t)] ≡ F[ρ (~x, t) · φ (~x, t) ·∏κ∈K δ (κ (~x, t)− κ∗)]F[ρ (~x, t) ·∏κ∈K δ (κ (~x, t)− κ∗)] (B.19)Conditionally-filtered fields are, formally, functions of K∗ (in addition to (~x, t)); for nota-tional compactness, and in recognition of the fact that the K∗ dependence is inherentlyconnected to the filtering operation, functional dependence on K∗ is represented by placingthe group above the filter annotation, rather than within trailing parentheses. For a givenK∗, the conditional filters can be viewed as modifications of the basic and Favre filter inwhich only points at which K (~x, t) = K∗ are considered. The associated kernels are thenGK∗F (~x, ~x∗, t, t∗) =GF (~x, ~x∗, t, t∗) ·∏κ∈K δ (κ (~x, t)− κ∗)F[∏κ∈K δ (κ (~x, t)− κ∗)](B.20)GK∗F˜ (~x, ~x∗, t, t∗) =GF (~x, ~x∗, t, t∗) · ρ (~x, t) ·∏κ∈K δ (κ (~x, t)− κ∗)F[ρ (~x, t) ·∏κ∈K δ (κ (~x, t)− κ∗)] (B.21)=GF˜ (~x, ~x∗, t, t∗) ·∏κ∈K δ (κ (~x, t)− κ∗)F˜[∏κ∈K δ (κ (~x, t)− κ∗)] . (B.22)52Formulae featuring K∗ are implicitly defined ∀K∗ ∈ K, just as formulae featuring ~x or t are implicitlydefined ∀~x ∈ X or ∀t ∈ T respectively.175B.1. Filtering OperationsThe conditional filters are linear operators,∀φ, ψ :K∗F [φ (~x, t) + ψ (~x, t)] =K∗F [φ (~x, t)] +K∗F [ψ (~x, t)] (B.23)∀a, φ :K∗F [aφ (~x, t) ] = aK∗F [φ (~x, t) ] (B.24)∀φ, ψ :K∗F˜ [φ (~x, t) + ψ (~x, t)] =K∗F˜ [φ (~x, t)] +K∗F˜ [ψ (~x, t)] (B.25)∀a, φ :K∗F˜ [aφ (~x, t) ] = aK∗F˜ [φ (~x, t) ] , (B.26)butas with the Favre filtercommutation of the basic filter with time and space derivativesdoes not imply commutation of either conditional filter with the same:~∇φ ≈ ~∇φ 6=⇒ K∗~∇φ ≈ ~∇K∗φ or K∗~∇φ˜ ≈ ~∇K∗φ˜(B.27)∂∂tφ ≈ ∂∂tφ 6=⇒ K∗∂∂tφ ≈ ∂∂tK∗φ orK∗∂∂t φ˜ ≈∂∂tK∗φ˜ . (B.28)As illustrated using the time-average filter in Figure B.1, conditional filtering can accountfor more variation in the original data; the conditional filter has a smaller impact than theun-conditional filter.• Figure B.1(a) illustrates the unconditional time average: each point represents a mea-surement of YH2O and the time average is located at the centre of mass of the pointcloud (marked by a single large dot). The root mean square deviation about thismean is marked by horizontal bars. The fact that these bars are far from the meanillustrates that, at many instants, the difference between the un-filtered and filteredvalue is quite largei.e., the filter has a very large impact.• Figure B.1(b) illustrates the conditional time average: each point represents a simul-taneous measurement of YH2O and Z, and the conditional time average is the line ofbest fit through the point cloud (marked by a solid line). The conditional root meansquare deviation about this mean is marked by dashed lines. The fact that these linesare close to the conditional mean (as compared to bars in the unconditional plot) il-lustrates that the difference between the un-filtered and conditionally-filtered value isrelatively smalli.e., the conditional filter has a comparatively small impact.176B.2. Special Filtered Fields00.050.10.15(a) UnconditionalYH2O0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.050.10.15(b) Conditional (upon mixture fraction, Z)ZYH2OFigure B.1: Unconditional and conditional ensemble averaging of experimental measure-ments of Sandia Flame D (a turbulent non-premixed flame) at z = 30djetand r = 12 mm.Data from [71].B.2 Special Filtered FieldsB.2.1 Sub-Filter FluxesFor some field φ, the sub-filter flux is defined asj subφ ≡ ρ · G˜(~x,~x∗,t,t∗)[(~v (~x∗, t∗)− ~˜v (~x, t))(φ (~x∗, t∗)− φ˜ (~x, t))](B.29)= ρ(~vφ˜− ~˜vφ˜). (B.30)This term represents advection of φ which is not resolved by the filter F.B.2.2 Filtered Moments of a Single FieldFor n ≥ 1, the nth raw moment of some field A is simply An˜. The nth central moment of Ais defined asAMn˜ (~x, t) ≡ G˜(~x,~x∗,t,t∗)[(A (~x∗, t∗)− A˜ (~x, t))n](B.31)177B.2. Special Filtered FieldsThis can be related to raw moments by using the binomial theorem:AMn˜ (~x, t) = G˜(~x,~x∗,t,t∗)[n∑k=0(nk)(−1)kAn−k (~x∗, t∗) A˜k (~x, t)](B.32)=n∑k=0(nk)(−1)k G˜(~x,~x∗,t,t∗)[An−k (~x∗, t∗)]A˜k (~x, t) (B.33)=n∑k=0(nk)(−1)kAn−k˜ (~x, t) A˜k (~x, t) (B.34)Extracting the k = 0 term from the sum provides a direct relationship between the raw andcentral nth moments:AMn˜ = An˜ +n∑k=1(nk)(−1)kAn−k˜A˜k (B.35)An˜ = AMn˜ −n∑k=1(nk)(−1)kAn−k˜A˜k. (B.36)The sub-filter component of An (i.e., An˜ − A˜n) can then be evaluated by subtracting A˜n:An˜ − A˜n = AMn˜ −n∑k=1(nk)(−1)kAn−k˜A˜k − A˜n (B.37)These definitions are consistent when n = 1, but this case is not usually considered since itis quite uninteresting:• the first central moment is 0,• the first raw moment is the filtered field, and• the un-resolved component of A1 is 0.B.2.3 Filtered Mixed Moments of a Set of FieldsFor some set of nψ fields ψ, the number of moments of order n ≥ 1 is(nψ+n−1n), althoughn of these involve only one of the fields53. Using one multi-index m (with |m| = n) torepresent the combination of powers which defines each such moment, the raw moments aresimply ψm˜. The central moments are defined asψMm˜ (~x, t) ≡ G˜(~x,~x∗,t,t∗)[(ψ (~x∗, t∗)− ψ˜ (~x, t))m]. (B.38)53In particular, no moment involves more than one field when n = 1178B.2. Special Filtered FieldsThese can be related to raw moments by using the multi-binomial theorem:ψMm˜ (~x, t) = G˜(~x,~x∗,t,t∗) ∑0≤k≤m(mk)(−1)kψm−k (~x∗, t∗) ψ˜k (~x, t)(B.39)=∑0≤k≤m(mk)(−1)k G˜(~x,~x∗,t,t∗)[ψm−k (~x∗, t∗)]ψ˜m (~x, t) (B.40)=∑0≤k≤m(mk)(−1)kψm−k˜ (~x, t) ψ˜k (~x, t) (B.41)Extracting k = 0 term from the multi-index sum provides a direction relationship betweeneach raw raw moment and the corresponding central moment:ψMm˜ = ψm˜ +∑1≤k≤m(mk)(−1)kψm−k˜ψ˜k (B.42)ψm˜ = ψMm˜ −∑1≤k≤m(mk)(−1)kψm−k˜ψ˜k, (B.43)The sub-filter component of ψm (i.e., ψm˜ − ψ˜m) can then be evaluated by subtracting ψ˜m:ψm˜ − ψ˜m = ψMm˜ −∑1≤k≤m(mk)(−1)kψm−k˜ψ˜k − ψ˜m. (B.44)B.2.4 Filtered Probability Density Functions (PDFs)Given the definition of the conditioning variables and associated sets in Section B.1.3, onecan define the volume- and mass-weighted sub-filter probability density functionsbothcommonly referred to as the Probability Density Function (PDF)asPK∗sub (~x, t) ≡ F[∏κ∈Kδ (κ (~x, t)− κ∗)](B.45)PK∗s˜ub (~x, t) ≡ F[ρ (~x, t) ·∏κ∈K δ (κ (~x, t)− κ∗)]F [ρ (~x, t)]. (B.46)Filtered probability density functions are, formally, functions of K∗ (in addition to (~x, t));following the notational convention adopted in Section B.1.3 for conditional filtering, func-tional dependence on K∗ is represented by placing the group above the filter annotation,rather than within trailing parentheses.Given these definitions, and those of Section B.1.3, it is straightforward to demonstratethat conditionally- and unconditionally-filtered fields are related through the PDFs:φ (~x, t) =∫KK∗φ (~x, t) · PK∗sub (~x, t) dK∗ (B.47)φ˜ (~x, t) =∫KK∗φ˜ (~x, t) · PK∗s˜ub (~x, t) dK∗. (B.48)179B.3. Reynolds FiltersThe PDF can also be related to mixed moments of the conditioning variables; for somemulti-index m, the corresponding raw and central mixed moments can be evaluated ascm˜ =∫K(c∗)m · PK∗s˜ub (~x, t) dK∗ (B.49)cm(m)˜ =∫K(c∗ − c˜ (~x, t))m · PK∗s˜ub (~x, t) dK∗. (B.50)It therefore follows that knowledge of the PDF provides complete information about thefiltered conditioning variables and their conditioned moments.B.3 Reynolds FiltersIn general, a Reynolds operator f is defined by the fact that it satisfies the relationship∀A,B : f (f (A)B) = f (A) f (B) . (B.51)It follows that a Reynolds filtera filter which is also a Reynolds operatorsatisfies∀A,B : A ◦B = A ◦B (B.52)where ◦ is some product operator (e.g. if A and B are vectors ◦ could be either the innerour outer product). This relationship can also be generalized to combinations of Favre andnon-Favre filtering, asA ◦ B˜ = ρA ◦Bρ=ρBρ◦A = A ◦ B˜ (B.53)A˜ ◦B = ρA ◦Bρ=ρA ◦Bρ= A˜ ◦B (B.54)A˜ ◦B =ρρA◦Bρρ=ρAρ ◦ ρBρ= A˜ ◦ B˜. (B.55)Reynolds filters exhibit several properties which do not hold for arbitrary filters, as detailedbelow.• Idempotence: repeated applications after the first have no effect:A = A · 1 = A · 1 = A (B.56)A˜ = A · 1˜ = A · 1˜ = A (B.57)A˜ = A˜ · 1 = A˜ · 1 = A˜ (B.58)˜˜A = A˜ · 1 = A˜ · 1˜ = A˜. (B.59)• Vanishing Filtered Fluctuations: filtered fluctuations are identically zero:A fluc = A−A = A−A = 0 (B.60)A fl˜uc˜ = A− A˜˜ = A˜− A˜ = 0. (B.61)180B.3. Reynolds Filters• filtered products can be expanded:A ◦B = A ◦B +Afluc ◦B fluc (B.62)A ◦ B˜ = A˜ ◦ B˜ +A fl˜uc ◦B fl˜uc, (B.63)and• sub-filter terms can be rearranged:A ◦B −A ◦B = AB fluc = BA fluc = AflucB fluc (B.64)A ◦ B˜ − A˜ ◦ B˜ = AB fl˜uc˜ = BA fl˜uc˜ = A fl˜ucB fl˜uc. (B.65)181Appendix CFilter-Related IdentitiesC.1 Filter-Function CommutationDefine the shorthandG(~x,~x∗,t,t∗)[a(b, c . . . )] ≡∫T∫XG (~x, ~x∗, t, t∗) · a(b, c . . . ) d~x∗ dt∗ (C.1)where the operation has no impact unless one of the arguments of a (i.e., b, c . . . ) has somefunctional dependence on ~x∗ or t∗ 54. By extension, defineG˜(~x,~x∗,t,t∗)[a(b, c . . . )] ≡ 1ρ (~x, t)G(~x,~x∗,t,t∗)[ρ (~x∗, t∗) a(b, c . . . )] . (C.2)Both of these filter integral operators are linear.Function Arguments at any PointSuppose that the state is represented using ψ1, ψ2, . . . ψn. In a continuum, all of these statevariables are fields, so they can be written as∀ψi ∈ Ψˆthermo :ψi (~x∗, t∗) = ψi(~x, t) + ∆ψi (~x, ~x∗, t, t∗) (C.3)where ∆ψi can be defined as a difference (and, for fields ψi (~x, t) which are analytic at (~x, t),can be evaluated using a multi-dimensional Taylor series).Function at any PointAt some point (~x∗, t∗) the local value of property φ can be evaluated asφ (~x∗, t∗) = φ(Ψˆthermo (~x∗, t∗))(C.4)but, using Equation C.3, this can be rewritten asφ (~x∗, t∗) = φ(Ψˆthermo (~x, t) + ∆Ψˆthermo (~x, ~x∗, t, t∗))(C.5)54If a is independent of the integration variables, it can be pulled out of the integral; the integral thenevaluates to 1 because the filter kernel is normalized.182C.1. Filter-Function Commutationwhichassuming the state function φ is analytic at Ψˆthermo (~x, t)can be expanded as aTaylor series:φ (~x∗, t∗) = φ(Ψˆthermo (~x, t))+∑α∈Nn0|α|≥1(∆ψ (~x, ~x∗, t, t∗))αα!(∂αφ∂ψα)∣∣∣∣Ψˆthermo(~x,t)(C.6)where α is a multi-index composed of n integral values, and terms involving α follow multi-index notation standards:|α| =n∑i=1αi (C.7)xα =n∏i=1xαii (C.8)α! =n∏i=1αi! (C.9)∂α∂ψα=∂|α|∂ψα11 ∂ψα22 . . . ∂ψαnn(C.10)Function of Filtered ArgumentsThe Favre filtered field arguments ψi can be evaluated as:ψ˜i (~x, t) =∫T∫XG (~x, ~x∗, t, t∗) · ψi (~x∗, t∗) d~x∗ dt∗ (C.11)= G˜(~x,~x∗,t,t∗)[ψi (~x, t) + ∆ψi (~x, ~x∗, t, t∗)] (C.12)= ψi (~x, t) + G˜(~x,~x∗,t,t∗)[∆ψi (~x, ~x∗, t, t∗)]´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶(=−ψ fluc(~x,t))(C.13)So the state function φ evaluated using the local Favre filtered properties can be rewrittenasφ(Ψˆthermo˜ (~x, t))= φ(Ψˆthermo (~x, t) + G˜(~x,~x∗,t,t∗)[∆Ψˆthermo (~x, ~x∗, t, t∗)]). (C.14)Assuming the state function φ is analytic at Ψˆthermo (~x, t), this can be expanded as a Taylorseries as:φ(Ψˆthermo˜ (~x, t))= φ(Ψˆthermo (~x, t))+∑α∈Nn0|α|≥1(G˜(~x,~x∗,t,t∗)[∆ψ (~x, ~x∗, t, t∗)])αα!(∂αφ∂ψα)∣∣∣∣Ψˆthermo(~x,t)(C.15)183C.1. Filter-Function CommutationFare Filtered Function at any PointThe Favre filtered property φ can be evaluated asφ˜ (~x, t) = φ(Ψˆthermo (~x, t))˜=∫T∫XG (~x, ~x∗, t, t∗) · φ(Ψˆthermo (~x∗, t∗))d~x∗ dt∗but, substituting Equation C.6,φ˜ (~x, t) = G˜(~x,~x∗,t,t∗)[φ(Ψˆthermo (~x, t))]+ G˜(~x,~x∗,t,t∗)∑α∈Nn0|α|≥1(∆ψ (~x, ~x∗, t, t∗))αα!(∂αφ∂ψα)∣∣∣∣Ψˆthermo(~x,t)and, recognizing that terms which aren't functions of ~x∗ or t∗ can be pulled out of the filterintegral operation (and, for the first term, that the filter is normalized),φ˜ (~x, t) = φ(Ψˆthermo (~x, t))+∑α∈Nn0|α|≥1G˜(~x,~x∗,t,t∗)[(∆ψ (~x, ~x∗, t, t∗))α]α!(∂αφ∂ψα)∣∣∣∣Ψˆthermo(~x,t)CommutatorThe difference between the filtered property and the state function evaluated using filteredarguments isφ(Ψthermo (~x, t))˜− φ(Ψthermo˜ (~x, t))=∑α∈Nn0|α|≥21α!(∂αφ∂ψα)∣∣∣∣Ψˆthermo(~x,t)·[G˜(~x,~x∗,t,t∗)[(∆ψ (~x, ~x∗, t, t∗))α]−(G˜(~x,~x∗,t,t∗)[∆ψ (~x, ~x∗, t, t∗)])α ](C.16)where the last term is the sub-filter component of a raw filtered moment, and the case of|α| = 1 has be removed from the sum because∀α ∈ Nn0 , |α| = 1 :G˜(~x,~x∗,t,t∗)[(∆ψ (~x, ~x∗, t, t∗))α] =(G˜(~x,~x∗,t,t∗)[∆ψ (~x, ~x∗, t, t∗)])α. (C.17)184C.2. Limiting Values of Sub-Filter TermsSince the Taylor series converges for analytic function and φ(Ψˆthermo)has been assumed tobe analytic, the terms in the infinite series must approach zero; it is therefore acceptable totruncate the series at some |α|maxwithout introducing significant error (although the valueof |α|maxdepends on everything: the state function, the argument fields, the filter, and thelevel of precision required).Examples of sufficient (but not necessary) conditions for a negligible difference include:• the Favre filter operation has no impact on the function arguments ψi (i.e., ψ˜i = ψi),for example because: the arguments ψi are completely uniform in time and space∀ψi ∈ Ψˆthermo : ∆ψi = 0 (C.18)(a trivial solution), or the filter is local in time and space (the value of the kernel tends to zero as dis-tance from (~x, t) tends to infinity), and the function arguments are completelyuniform across the region where the kernel is non-negligible (the local filter vol-ume at (~x, t))G˜(~x,~x∗,t,t∗)[(∆ψ (~x, ~x∗, t, t∗))α] = G˜(~x,~x∗,t,t∗)[∆ψ (~x, ~x∗, t, t∗)] = 0, (C.19)or• the function φ(Ψˆthermo) is locally planar near Ψˆthermo (~x, t)∀α ∈ Nn0 , 2 ≤ |α| ≤ |α|max: (∂αφ∂ψα)∣∣∣∣Ψˆthermo(~x,t)= 0, (C.20)where the fact that derivatives with |α| > |α|maxneed not respect the equality impliesthat there is some volume in state space [centred at Ψˆthermo (~x, t)] beyond which thestate function φ need not be planar.C.2 Limiting Values of Sub-Filter TermsMost sub-filter terms can be written in terms of a differenceA ◦ B˜ − A˜ ◦ B˜ (C.21)where ◦ is a generic commutative operator, i.e.∀A,B : A ◦B = B ◦A. (C.22)When reasoning about the possible values of this difference, it is convenient to rewrite it asthe result of applying the filter integral operator to a single term.185C.2. Limiting Values of Sub-Filter TermsC.2.1 Generic Sub-Filter TermGeneralizing a rearrangement presented in [20], begin by subtracting and adding A˜ ◦ B˜:A ◦ B˜ − A˜ ◦ B˜ = A ◦ B˜ − A˜ ◦ B˜−A˜ ◦ B˜ + A˜ ◦ B˜´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶+0(C.23)Selectively re-writing filtered quantities in terms of the filter integral operation,A ◦ B˜ (~x, t)− A˜ (~x, t) ◦ B˜ (~x, t) (C.24)= G˜(~x,~x∗,t,t∗)[A (~x∗, t∗) ◦B (~x∗, t∗)]− G˜(~x,~x∗,t,t∗)[A (~x∗, t∗)] ◦ B˜ (~x, t) (C.25)− A˜ (~x, t) ◦ G˜(~x,~x∗,t,t∗)[B (~x∗, t∗)] + A˜ (~x, t) ◦ B˜ (~x, t) G˜(~x,~x∗,t,t∗)[1] . (C.26)The filter is normalized, so the filter integral has no impact on terms which are functions of(~x, t) only and they can be brought into and out of the integral as convenient:A ◦ B˜ (~x, t)− A˜ (~x, t) ◦ B˜ (~x, t) (C.27)= G˜(~x,~x∗,t,t∗)[A (~x∗, t∗) ◦B (~x∗, t∗)]− G˜(~x,~x∗,t,t∗)[A (~x∗, t∗) ◦ B˜ (~x, t)](C.28)− G˜(~x,~x∗,t,t∗)[A˜ (~x, t) ◦B (~x∗, t∗)]+ G˜(~x,~x∗,t,t∗)[A˜ (~x, t) ◦ B˜ (~x, t)]. (C.29)Since the filter integral is a linear operator, the sum of the integrals is the integral of thesum; that sum can be factored, givingA ◦ B˜ (~x, t)− A˜ (~x, t) ◦ B˜ (~x, t)= G˜(~x,~x∗,t,t∗)[(A (~x∗, t∗)− A˜ (~x, t))◦(B (~x∗, t∗)− B˜ (~x, t))]. (C.30)This difference will only be negligible if either A or B stays near its Favre filtered valuethroughout the filter volume (the volume over which the filter kernel is non-negligible).If one of the fields (taken, without loss of generality, as A) is bounded according toAmin≤ A ≤ Amax, then it is possible to derive limits on the value of the sub-filter term:Amin≤ A ≤ Amax(C.31)Amin◦B ≤ A ◦B ≤ Amax◦B (C.32)Amin◦ B˜ ≤ A ◦ B˜ ≤ Amax◦ B˜ (C.33)Amin◦ B˜ − A˜ ◦ B˜ ≤ A ◦ B˜ − A˜ ◦ B˜ ≤ Amax◦ B˜ − A˜ ◦ B˜. (C.34)If ◦ is distributive, to witx ◦ (y + z) = x ◦ y + x ◦ z, (C.35)then this can be rewritten asB˜ ◦(Amin− A˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶≤0≤ A ◦ B˜ − A˜ ◦ B˜ ≤ B˜ ◦(Amax− A˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶≥0(C.36)186C.2. Limiting Values of Sub-Filter TermsC.2.2 Sub-Filter Mass FluxApplying Equation C.36 with A as a species mass fraction, mixture fraction, or progressvariable (all of which are constrained to [0, 1]), B as the velocity, and ◦ as the scalar product,∀φ ∈ (Y ∪ {Z, c}) : −ρ~˜vφ˜ ≤ ~j subφ ≤ ρ~˜v(1− φ˜). (C.37)C.2.3 Sub-Filter StressApplying Equation C.30 with A and B as the velocity and ◦ as the outer product,τ subF (~x, t) = ρ (~x, t) G˜(~x,~x∗,t,t∗)[(~v (~x∗, t∗)− ~˜v (~x, t))2](C.38)This term differs from that in [20] in that the filter kernel has been scaled by ρ (~x∗, t∗) aspart of the Favre filter integral operator, but the density must must be positive55, so theconclusion is unchanged: τ subF must always be positive semi-definite.C.2.4 VariancesSimple MinimumApplying Equation C.30 with A and B as some scalar field φ and ◦ as the scalar product,φ fl˜uc (~x, t) = G˜(~x,~x∗,t,t∗)[(φ (~x∗, t∗)− φ˜ (~x, t))2]. (C.39)The square of some real quantity is non-negative and the filter kernel is non-negative, sothe integrand of the filter integral operator is never negative; it follows that variance mustalways be non-negative.Full BoundsApplying Equation C.36 with A and B as some scalar field φ and ◦ as the scalar product,φ˜ ◦(φmin− φ˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶≤0≤ φ fl˜uc ≤ φ˜ ◦(φmax− φ˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶≥0(C.40)Combining Equation C.39,max(0, φ˜ ◦(φmin− φ˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶≤0)≤ φ fl˜uc ≤ φ˜ ◦(φmax− φ˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶≥0. (C.41)55In the limit as ρ approaches 0, the continuum assumption becomes unreasonable and many other as-sumptions break down; requiring ρ > 0 here is no more restrictive than the continuum assumption.187Appendix DTransport Equations for Filter-BasedTurbulence ModelsD.1 Transport of Filtered QuantitiesD.1.1 Mass BalanceThe filtered mass balance can be written as:∂∂tρ®LocalChange+ ~∇ ·(ρ~˜v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ResolvedAdvection= 0 (D.1)where all fields are filtered versions of fields which appear in the un-filtered balance (Equa-tion 2.4).D.1.2 Momentum BalanceThe filtered momentum balance can be written as:ρD˜Dtv˜ucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρ~˜v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ ·(ρ~˜v~˜v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ResolvedAdvection= −~∇ · τ subF´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶UnresolvedAdvection+ ~∇ · σF´udcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces+ ρ~g®BodyForce. (D.2)where τ subF is the sub-filter stress,τ subF ≡ ρ · G˜(~x,~x∗,t,t∗)[(~v (~x∗, t∗)− ~˜v (~x, t))(~v (~x∗, t∗)− φ˜ (~x, t))], (D.3)= ρ(~˜v~v − ~˜v~˜v)(4.15 repeated)and all other fields are filtered versions of fields which appear in the un-filtered balance(Equation 2.9).188D.1. Transport of Filtered QuantitiesD.1.3 Energy BalanceThe filtered energy balance can be written in terms of the specific total energy, e = u+ ke,LocalChangeucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρe˜) +ResolvedAdvectionucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright~∇ ·(ρ~˜ve˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ρD˜Dte˜=UnresolvedAdvectionucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright−~∇ · ~q sube +HeatFluxucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright(−~∇ · ~q)+RadiationucurlyQ+ ~∇ ·(σF · ~v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces+ ρ~g · ~˜v®BodyForce, (D.4)the specific internal energy, u,ρD˜Dtu˜ucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρu˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ ·(ρ~˜vu˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ResolvedAdvection= −~∇ · ~q subu´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶UnresolvedAdvection+(−~∇ · ~q)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶HeatFlux+ Q®Radiation+σF : ~∇~v´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces, (D.5)or the specific enthalpy, h,LocalChangeucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρh˜)+ResolvedAdvectionucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright~∇ ·(ρ~˜vh˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ρD˜Dth˜=UnresolvedAdvectionucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright−~∇ · ~q subh +HeatFluxucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright(−~∇ · ~q)+RadiationucurlyQ+ σF : ~∇~v + ∂P∂t+ ~∇ · (P~v)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶SurfaceForces. (D.6)where the ~q subφ terms are defined by Equation 4.18, but can also be written as~q subφ ≡ ρ · G˜(~x,~x∗,t,t∗)[(~v (~x∗, t∗)− ~˜v (~x, t))(φ (~x∗, t∗)− φ˜ (~x, t))](D.7)= ~vφ˜− ~˜vφ˜ (4.18 repeated)and all other fields are filtered versions of fields which appear in the un-filtered balances(Equations 2.11 to 2.13).189D.1. Transport of Filtered QuantitiesD.1.4 Mass Fraction, Mixture Fraction, and Progress VariableThe filtered balance for the species mass fractions, the mixture fraction, and the progressvariable can all be written as∀φ ∈ (Y ∪ {Z, c}) :ρD˜Dtφucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρφ˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ ·(ρ~˜vφ˜)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ResolvedAdvection= −~∇ ·~j subφ´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶UnresolvedAdvection−~∇ ·~jφ´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Diffusion+ ρω˙φ˜®Production+ ρQφ˜®Interactionwith Z field(D.8)where• ~j subφ is the sub-filter flux,~j subφ ≡ ρ · G˜(~x,~x∗,t,t∗)[(~v (~x∗, t∗)− ~˜v (~x, t))(φ (~x∗, t∗)− φ˜ (~x, t))](D.9)= ~vφ˜− ~˜vφ˜, (4.13 repeated)• the interaction term is zero when φ is the mixture fraction or a species mass fraction:∀φ ∈ (Y ∪ {Z}) : Qφ˜ = 0 (D.10)but more complicated when φ is the progress variable:Qc˜ =(∂Yc∂c)−1Z(∂2Yc∂Z2)c;cχZZ + 2(∂Yc∂c)−1Z(∂2Yc∂Z∂c)c;ZχZc˜ (D.11)where the partial derivatives are defined in Section 3.3.3 and the term vanishes in thecase of fully premixed combustion,• all other terms are filtered versions of fields which appear in the un-filtered balances(Equations 2.5, 3.4, and 3.20), and• the reaction source term drops out in the case of mixture fraction, since ω˙Z = 0.D.1.5 Other QuantitiesFiltered transport equations for other quantities, such as temperature, pressure, and entropy,can also be derived [18], but they are not of interest in this work.190D.2. Transport of Filtered VariancesD.2 Transport of Filtered VariancesIn some contexts, it is useful to track the variance associated with some scalar field and filter,defined as the filtered second central moment (Section B.2.2) but more succinctly writtenas the commutator of the square of operation and the filter operation [111],∀φ : φ v˜ar ≡ G˜(~x,~x∗,t,t∗)[(φ (~x∗, t∗)− φ (~x, t))2]= φ˜2 − φ˜2, (D.12)which has dimensionality equal to the square of the dimensionality of φ 56. When the filterkernel is always non-negative (as is assumed in this work), then the values of the varianceare limited by0 ≤ φ v˜ar ≤ φ˜(φmax− φ˜)(D.13)where φmaxis the maximum physically possible value of φ. This maximum is, in general,infinity (making the upper bound on φ v˜ar also infinity), but may take on special values forφ which have a maximum possible value by definition, such as species mass fractions.In this work, the variance of species mass fractions, the mixture fraction, and the progressvariable are of interest. All of these variables are constrained to be on the interval [0, 1], andtherefore their variances are limited by∀φ ∈ (Y ∪ {Z, c}) :0 ≤ φ v˜ar ≤ φ˜(1− φ˜). (D.14)The transport equation for any one of these variables can be written as∀φ ∈ (Y ∪ {Z, c}) :ρD˜Dtφ v˜arucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright∂∂t(ρφ v˜ar)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶LocalChange+ ~∇ ·(ρ~˜vφ v˜ar)´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶ResolvedAdvection=UnresolvedAdvectionucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright−~∇ ·~j subφ v˜arDiffusionucurlyleftudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyright−~∇ ·~jφ v˜ar +Interactionwith filterand Z fielducurlyleftudcurlymodudcurlymodudcurlymodudcurlymoducurlymidudcurlymodudcurlymodudcurlymodudcurlymodudcurlymoducurlyrightρQ subφ−2~∇φ˜ ·~j subφ − 2ρχ subφφ´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Dissipation+ ρω˙φ v˜ar´udcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¸udcurlymodudcurlymodudcurlymodudcurlymodudcurlymodudcurlymod¶Reaction-FilterInteraction(D.15)where56Note that, if the square of a vector is interpreted as an outer product, then τ subF = ρ~vv˜ar; if it isinterpreted as an inner product, then k sub = 12~v v˜ar.191D.2. Transport of Filtered Variances• ~j subφ v˜aris the sub-filter flux of φ v˜ar, with dimensionality m/(L2t),∀φ ∈ (Y ∪ {Z, c}) :~j subφ v˜ar≡ ρ · G˜(~x,~x∗,t,t∗)[(~v (~x∗, t∗)− ~˜v (~x, t))·[(φ (~x∗, t∗)− φ˜ (~x, t))2 − φ v˜ar (~x, t)]] (D.16)= ρ[(~vφ2˜ − ~˜vφ˜2)− 2φ˜(~vφ˜− ~˜vφ˜)], (D.17)which formally represents transport due to advection that the filter F has not resolved,but plays the same role in the transport as a diffusive flux, and can therefore can beviewed as an effective turbulent diffusive flux;• ~jφ v˜ar is the transport of φ v˜ar due to diffusion of φ, which can be viewed as diffusion ofφ v˜ar (dimensionality m/(L2t)),∀φ ∈ (Y ∪ {Z, c}) :~jφ v˜ar ≡ 2ρ G˜(~x,~x∗,t,t∗)[(φ (~x∗, t∗)− φ˜ (~x, t))·(∆~vφ (~x∗, t∗)−∆~vφ˜ (~x, t))](D.18)= 2ρ(φ∆~vφ˜ − φ˜∆~vφ)(D.19)= 2(φ~jφ − φ˜~jφ)(D.20)where∀φ : ∆~vφ ≡~jφρ= ~vφ − ~v (3.25 repeated)which is separated from the sub-filter diffusive flux because, in the case where diffu-sion of all species is Fickian with a single global diffusivity, it takes the same form asa Fickian diffusive flux;• Q subφ (with dimensionality m/(L3t)) represents the generation (or destruction) of vari-ance due to interactions between the φ field, the mixture fraction field, and the filter,which is zero when φ is the mixture fraction or a species mass fraction:∀φ ∈ (Y ∪ {Z}) : Q subφ = 0 (D.21)but takes a complicated form when φ ≡ c:Q subc = c(∂Yc∂c)−1Z(∂2Yc∂Z2)c;cχZZ˜ + 2c(∂Yc∂c)−1Z(∂2Yc∂Z∂c)c;ZχZc˜− c˜(∂Yc∂c )−1Z (∂2Yc∂Z2 )c;cχZZ − 2c˜(∂Yc∂c )−1Z ( ∂2Yc∂Z∂c)c;ZχZc (D.22)192D.2. Transport of Filtered Varianceswhere the partial derivatives were defined in Section 3.3.3, and the term is equal tozero in fully premixed combustion;• ~j subφ is given by Equation D.9;• χ subφφ is the sub-filter scalar dissipation of φ with itself (dimensionality 1/t), defined by∀φ ∈ (Y ∪ {Z, c}) :χ subφφ ≡ −(∆~vφ · ~∇φ˜−∆~vφ˜ · ~∇φ˜)(D.23)= −1ρ(~jφ · ~∇φ−~jφ · ~∇φ˜)(D.24)where∀φ : ∆~vφ ≡~jφρ= ~vφ − ~v (3.25 repeated)which is generally expected to be positive57; and• ω˙φ v˜ar is the source of φ v˜ar due to production of φ by chemical reactions (dimensionality1/t),∀φ ∈ (Y ∪ {Z, c}) :ω˙φ v˜ar ≡ 2 · G˜(~x,~x∗,t,t∗)[(φ (~x∗, t∗)− φ˜ (~x, t))·(ω˙φ (~x∗, t∗)− ω˙φ˜ (~x, t))](D.25)= 2(φω˙φ˜ − φ˜ω˙φ˜)(D.26)which is zero when φ ≡ Z.D.2.1 SimplificationsIn the case of RANS, the filter kernel has time extent only, so within a filter integral G˜(~x,~x∗,t,t∗)one can make the substitutionφ (~x∗, t∗)− φ˜ (~x, t) = φ (~x, t∗)− φ˜ (~x) (D.27)= φ fl˜uc (~x, t∗) (D.28)57Note that the definition of χ subφφ is not entirely consistent with that used in other sub-filter terms: thedrift velocity ∆~vZ replaces the bulk velocity ~v, and the last term is ~∇φ˜ rather than ~∇φ˜. In general, ~∇φ˜ and~∇φ˜ are distinct, but it is common to assume that substituting one for the other is acceptable as part of aclosure for the resolved diffusive flux term in the transport equation for φ˜. This assumption also allows χ subφφto be written as∀φ ∈ (Y ∪ {Z, c}) :χ subφφ ≡ G˜(~x,~x∗,t,t∗)[(∆~vφ (~x∗, t∗)−∆~vφ˜ (~x, t))·(~∇φ (~x∗, t∗)− ~∇φ˜ (~x, t))].193D.2. Transport of Filtered Varianceswhich is useful becauseG˜(~x,~x∗,t,t∗)[φ fl˜uc (~x, t∗)ψ fl˜uc (~x, t∗) . . .]= φ fl˜uc (~x, t)ψ fl˜uc (~x, t) . . . (D.29)i.e. all terms can be represented without invoking the filter integral operator. This permitsseveral simplifications:∀φ ∈ Z : φ v˜ar = φ fl˜uc2 (D.30)∀φ ∈ Z : ~j subφ v˜ar= ρ[~v fl˜uc(φ fl˜uc)2]. (D.31)∀φ ∈ Z : ~jφ v˜ar = 2ρφ fl˜uc∆~v fl˜ucφ (D.32)If one further assumes that~∇φ˜ ≈ ~∇φ˜ for all φ (as is typically necessary to close filtereddiffusion terms), then∀φ ∈ Z : χ subφφ = −(∆~v fl˜ucφ · ~∇Z fl˜uc)(D.33)If Fickian diffusion with a single global diffusivity D is also assumed, then the diffusion andsubgrid scalar dissipation terms simplify further:~jZ v˜ar = ρD~∇(Z fl˜uc)2˜(D.34)χ subZZ = D~∇Z fl˜uc · ~∇Z fl˜uc. (D.35)194Appendix EThe Sandia Non-Premixed FlameSeriesE.1 DescriptionThe Sandia non-premixed flame series [112] consists of several related non-premixed pilotedflames assigned letters A through F. The geometry, chemistry, and velocity parameters aredescribed in the following subsections, while a visualization of the mixing field for one flamein the series is presented as Figure E.1E.1.1 GeometryAll flames have radial symmetry and consist of a central fuel jet surrounded by an annularpilot and air coflow, with dimensions as described in Table E.1.Table E.1: Key geometric parameters defining the Sandia flame series [113].DiameterComponent Internal ExternalFuel Jet 0 mm 7.2 mmPilot 7.7 mm 18.2 mmAir Coflow 18.9 mm 150 mm**The air coflow has a square cross section with side length 30 cm.E.1.2 ChemistryLike the geometry, the chemistry is the same for all flames in the series:• the central fuel stream consists of 25 % methane and 75 % air (by volume) at 294 Kand 0.993 atm,• the pilot stream consists of the products of premixed combustion, and can be approx-imated as the output of a burner with φF/A= 0.77 at 1880 K, and• the coflow consists of air at 291 K and 0.993 atm.E.1.3 VelocityThe velocities of the various inlet streams vary from flame to flame as described in Table E.2.195E.2. Experimental Measurementsz = 15djetz = 30djetz = 45djetFigure E.1: Visualization of the mixing field of Flame D, including measurement locations.Red corresponds with pure fuel (25 % methane by volume) and blue with pure air.Table E.2: Key velocities defining the Sandia flame series [112].Bulk Velocity (m/s)Flame Main Jet Pilot CoflowA 2.44 none 0.9B 18.2 6.8 0.9C 29.7 6.8 0.9D 49.6 11.4 0.9E 74.4 17.1 0.9F 99.4 22.8 0.9E.2 Experimental MeasurementsExperimental measurements of many scalars [71] and velocities [72] are available for flamesC through F. Measurements are performed on a relatively small number of downstreamplanes (some of which are illustrated in Figure E.1); on each plane, the radial resolution isrelatively fine.196Appendix FUniform Conditional State (UCS)IdentitiesF.1 From Assumption 3(a) to Equation 7.16To keep track of functional dependencies, it is convenient to define the shorthandQ(K∗, t) ≡ f(K∗X). (F.1)The left-most term in the filter-rearrangement relationship (Equation 7.16) can then berepresented asK∗f(K∗X)KY =F [Q (K, t) · Y · δ (K−K∗)]F [δ (K−K∗)] . (F.2)As a consequence of the delta function, the only terms which will contribute to the filteredresult are those at points where K = K∗; it is therefore possible to replace Q (K, t) withQ (K∗, t) without changing the value of the expression:K∗f(K∗X)KY =F [Q (K∗, t) · Y · δ (K−K∗)]F [δ (K−K∗)] =K∗f(K∗X)Y (F.3)where the right-most equality illustrates that the term which has been generated aboveis now the middle term in the filter-rearrangement relationship (Equation 7.16), i.e., thefirst equality of interest has been demonstrated. The filter operation in the numerator ofEquation F.3 can be written out explicitly asF [Q (K∗, t) · Y · δ (K−K∗)]=∫X G (~x, ~x∗) ·Q (K∗, t) · Y (~x∗, t) · δ (K (~x∗, t)−K∗) d~x∗∫X G (~x, ~x∗) d~x∗(F.4)where there is no integral over time because the filter kernel, G, has been assumed to havespatial extent only. Because Q (K∗, t) is independent of ~x∗, it can be extracted from the197F.2. From Assumption 3(b) to Equation 7.16integral:F [Q (K∗, t) · Y · δ (K−K∗)]=Q (K∗, t)∫X G (~x, ~x∗) · Y (~x∗, t) · δ (K (~x∗, t)−K∗) d~x∗∫X G (~x, ~x∗) d~x∗(F.5)=Q (K∗, t)F [Y · δ (K−K∗)] (F.6)Substituting this back into Equation F.3 gives the equationK∗f(K∗X)Y = Q (K∗, t)F [Y · δ (K−K∗)]F [δ (K−K∗)] = f(K∗X)K∗Y (F.7)where the right-most equality illustrates that the term which has been generated above isnow the right-most term in the filter-rearrangement relationship (Equation 7.16), i.e., thesecond equality of interest has been demonstrated. Demonstrating these equalities separatelyproves the filter-rearrangement relationship (Equation 7.16).F.2 From Assumption 3(b) to Equation 7.16To keep track of functional dependencies, it is convenient to define the shorthandQ(K∗) ≡ f(K∗X). (F.8)where Q is not a function of t because the filtered fields have been assumed to be independentof t. The left-most term in the filter-rearrangement relationship (Equation 7.16) can thenbe represented asK∗f(K∗X)KY =F [Q (K) · Y · δ (K−K∗)]F [δ (K−K∗)] . (F.9)As a consequence of the delta function, the only terms which will contribute to the filteredresult are those at points where K = K∗; it is therefore possible to replace Q (K) with Q (K∗)without changing the value of the expression:K∗f(K∗X)KY =F [Q (K∗) · Y · δ (K−K∗)]F [δ (K−K∗)] =K∗f(K∗X)Y (F.10)where the right-most equality illustrates that the term which has been generated aboveis now the middle term in the filter-rearrangement relationship (Equation 7.16), i.e., the198F.3. From Assumption 3(c) to Equation 7.16first equality of interest has been demonstrated. The filtered quantity in the numerator ofEquation F.10 can be written out explicitly asF [Q (K∗) · Y · δ (K−K∗)]=∫τ∫X G (~x, ~x∗, t, t∗) ·Q (K∗) · Y (~x∗, t∗) · δ (K (~x∗, t∗)−K∗) d~x∗ dt∗∫τ∫X G (~x, ~x∗, t, t∗) d~x∗ dt∗.(F.11)Because Q (K∗) is independent of ~x∗ and t∗, it can be extracted from the integral:F [Q (K∗) · Y · δ (K−K∗)]=Q (K∗)∫τ∫X G (~x, ~x∗, t, t∗) · Y (~x∗, t∗) · δ (K (~x∗, t∗)−K∗) d~x∗ dt∗∫τ∫X G (~x, ~x∗, t, t∗) d~x∗ dt∗(F.12)=Q (K∗) · F [Y · δ (K−K∗)] . (F.13)Substituting this back into Equation F.10 gives the equationK∗f(K∗X)Y = Q (K∗)F [Y · δ (K−K∗)]F [δ (K−K∗)] = f(K∗X)K∗Y (F.14)where the right-most equality illustrates that the term which has been generated above isnow the right-most term in the filter-rearrangement relationship (Equation 7.16), i.e., thesecond equality of interest has been demonstrated. Demonstrating these equalities separatelyproves the filter-rearrangement relationship (Equation 7.16).F.3 From Assumption 3(c) to Equation 7.16To keep track of functional dependencies, it is convenient to define the shorthandQ(K∗, t) ≡ f(K∗X). (F.15)The left-most term in the filter-rearrangement relationship (Equation 7.16) can then berepresented asK∗f(K∗X)KY =F [Q (K, t) · Y · δ (K−K∗)]F [δ (K−K∗)] . (F.16)As a consequence of the delta function, the only terms which will contribute to the filteredresult are those at points where K = K∗; it is therefore possible to replace Q (K) with Q (K∗)without changing the value of the expression:K∗f(K∗X)KY =F [Q (K∗, t) · Y · δ (K−K∗)]F [δ (K−K∗)] =K∗f(K∗X)Y (F.17)199F.4. Equations 7.40 and 7.41 to Equation 7.42where the right-most equality illustrates that the term which has been generated aboveis now the middle term in the filter-rearrangement relationship (Equation 7.16), i.e., thefirst equality of interest has been demonstrated. The filtered quantity in the numerator ofEquation F.17 can be written out explicitly asF [Q (K∗, t) · Y · δ (K−K∗)] =∑iQi (K∗, t) · Yi (~x∗, t∗) · δ (Ki (~x∗, t∗)−K∗)∑i δ (Ki (~x∗, t∗)−K∗)(F.18)where Qi (K∗, t) is independent of realization i,Qi (K∗, t) = Q (K∗, t) (F.19)because Q (K∗, t) is a deterministic function ofK∗X andK∗X is independent of realization. Itfollows that Q (K∗, t) can be extracted from the sum:F [Q (K∗, t) · Y · δ (K−K∗)] = Q (K∗, t)∑i Yi (~x∗, t∗) · δ (K (~x∗, t∗)−K∗)∑i δ (K (~x∗, t∗)−K∗)(F.20)= Q (K∗, t) · F [Y · δ (K−K∗)] . (F.21)Substituting this back into Equation F.17 gives the equationK∗f(K∗X)Y = Q (K∗, t) · F [Y · δ (K−K∗)] = f(K∗X)K∗Y (F.22)where the right-most equality illustrates that the term which has been generated above isnow the right-most term in the filter-rearrangement relationship (Equation 7.16), i.e., thesecond equality of interest has been demonstrated. Demonstrating these equalities separatelyproves the filter-rearrangement relationship (Equation 7.16).F.4 Equations 7.40 and 7.41 to Equation 7.42In Equation 7.37, the Gradient Product Diffusion (GPD) is defined as.GPD =∑κ,κ′∈K∂ K∗JYκ′∂κ∗−∑κ′′∈K∂K∗Jκ′′κ′∂κ∗∂K∗Y˜∂κ′′∗ K∗~∇κ · ~∇κ′. (F.23)Substituting Equations 7.40 and 7.41 givesGPD =∑κ,κ′∈K ∂∂κ∗− K∗ρDY ∂K∗Y˜∂κ′∗− ∑κ′′∈K∂∂κ∗(−K∗ρDκ′′δκ′κ′′)∂K∗Y˜∂κ′′∗ ·K∗~∇κ · ~∇κ′. (F.24)200F.5. Equations 7.40 and 7.41 to Equation 7.43The Dirac delta δκ′κ′′ is zero except when κ′ ≡ κ′′, so the second sum can be simplified:GPD =∑κ,κ′∈K ∂∂κ∗− K∗ρDY ∂K∗Y˜∂κ′∗− ∂∂κ∗(−K∗ρDκ′)∂K∗Y˜∂κ′∗ K∗~∇κ · ~∇κ′. (F.25)Applying the product rule to the first term and rearranging then gives the desired result:GPD = −∑κ,κ′∈K K∗ρDY ∂2K∗Y˜∂κ∗∂κ′∗+∂∂κ∗(K∗ρDY)∂K∗Y˜∂κ′∗− ∂∂κ∗(K∗ρDκ′)∂K∗Y˜∂κ′∗ ·K∗~∇κ · ~∇κ′ (F.26)= −∑κ,κ′∈K K∗ρDY ∂2K∗Y˜∂κ∗∂κ′∗+∂∂κ∗(K∗ρDY −K∗ρDκ′)∂K∗Y˜∂κ′∗ K∗~∇κ · ~∇κ′ (7.42 repeated)F.5 Equations 7.40 and 7.41 to Equation 7.43In Equation 7.37, the Laplacian Diffusion (LD) is defined as,LD =∑κ∈K K∗JYκ −∑κ′∈KK∗Jκ′κ∂K∗Y˜∂κ′∗ K∗∇2κ. (F.27)Substituting Equations 7.40 and 7.41, givesLD =∑κ∈K− K∗ρDY ∂K∗Y˜∂κ∗−∑κ′∈K(−K∗ρDκ′δκκ′)∂K∗Y˜∂κ′∗ K∗∇2κ. (F.28)The Dirac delta δκκ′ is zero except when κ ≡ κ′, so the second sum can be simplified:LD = −∑κ∈K K∗ρDY ∂K∗Y˜∂κ∗−( K∗ρDκ) ∂K∗Y˜∂κ∗ K∗∇2κ. (F.29)Algebra then gives the desired result:LD = −∑κ∈K[K∗ρDY −K∗ρDκ] ∂K∗Y˜∂κ∗ K∗∇2κ. (7.43 repeated)201

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