Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Mousemi, Arash
Master of Applied Science - MASc
University of British Columbia
The experimental data from the Cambridge-Sandia burner working in nine different configurations are post-processed to explore the effects of coordinate, swirl flow ratio, and stratification factor on the conditionally-averaged reactive scalars. First, the mixture fraction and progress variable are employed to construct a two-dimensional conditional manifold by using all of the data – regardless of the coordinate in the physical domain, swirl ratio, and stratification factor. Moreover, one-point, one-time measurements are utilized to compute the exact joint Probability Density Function (PDF) of the conditioning variables at each point. Having the conditional averages of temperature and mass fractions of reactive scalars in addition to the joint PDF of the conditioning variables, the manifold is applied to calculate the unconditional averages for each of the scalars. The mean values are also obtained by ensemble averaging all the data available for each of the measuring points, and the discrepancies between the values calculated from the two approaches are reported in order to assess the validity of the assumptions underlying the low-dimensional chemistry representations. The results suggest that the two chosen conditioning variables are not sufficient to make the manifold independent of the real domain, so, the normalized total enthalpy is introduced as the third conditioning variable and the process repeated. Results obtained from the three-condition manifold demonstrate that the discrepancies for prediction of the reactive scalars decrease when using the third condition. Moreover, the experimental data are employed to obtain the marginal PDFs for the conditioning variables at various points in the reacting domain. The measurements are then combined from all positions in space to form conditional PDFs of the normalized total enthalpy for various values of the other two variables. The correlation coefficients between the conditioning variables are also investigated. Next, to consider the association between the conditioning variables for modeling, the copula concept is introduced, and the performance of three different copulas are tested. Furthermore, the statistical moments of the conditioning variables are computed from the experimental data at different points and are utilized for modeling the joint PDF of the conditioning variables from two different approaches which are compared.
Evaluation of Low-Order Manifold Representation ofChemistry in Turbulent Reactive FlowsbyArash MousemiB.Sc., Sharif University of Technology, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)August 2021© Arash Mousemi, 2021The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Evaluation of Low-Order Manifold Representation of Chemistry inTurbulent Reactive Flowssubmitted by Arash Mousemi in partial fulfillment of the requirements for thedegree of MASTER OF APPLIED SCIENCE in Mechanical Engineering.Examining Committee:Dr. W. Kendal Bushe, Mechanical EngineeringSupervisorDr. Patrick Kirchen, Mechanical EngineeringSupervisory Committee MemberDr. Gwynn Elfring, Mechanical EngineeringSupervisory Committee MemberiiAbstractThe experimental data from the Cambridge-Sandia burner working in nine differ-ent configurations are post-processed to explore the effects of coordinate, swirlflow ratio, and stratification factor on the conditionally-averaged reactive scalars.First, the mixture fraction and progress variable are employed to construct a two-dimensional conditional manifold by using all of the data – regardless of the co-ordinate in the physical domain, swirl ratio, and stratification factor. Moreover,one-point, one-time measurements are utilized to compute the exact joint Proba-bility Density Function (PDF) of the conditioning variables at each point.Having the conditional averages of temperature and mass fractions of reactivescalars in addition to the joint PDF of the conditioning variables, the manifold isapplied to calculate the unconditional averages for each of the scalars. The meanvalues are also obtained by ensemble averaging all the data available for each ofthe measuring points, and the discrepancies between the values calculated from thetwo approaches are reported in order to assess the validity of the assumptions un-derlying the low-dimensional chemistry representations. The results suggest thatthe two chosen conditioning variables are not sufficient to make the manifold in-dependent of the real domain, so, the normalized total enthalpy is introduced asthe third conditioning variable and the process repeated. Results obtained from thethree-condition manifold demonstrate that the discrepancies for prediction of thereactive scalars decrease when using the third condition.Moreover, the experimental data are employed to obtain the marginal PDFs forthe conditioning variables at various points in the reacting domain. The measure-ments are then combined from all positions in space to form conditional PDFs ofthe normalized total enthalpy for various values of the other two variables. Theiiicorrelation coefficients between the conditioning variables are also investigated.Next, to consider the association between the conditioning variables for modeling,the copula concept is introduced, and the performance of three different copulasare tested. Furthermore, the statistical moments of the conditioning variables arecomputed from the experimental data at different points and are utilized for mod-eling the joint PDF of the conditioning variables from two different approacheswhich are compared.ivLay SummaryThe combustion process is an inseparable component of the energy sector on whichour modern society is built. Nevertheless, the combustion of fossil fuels has manydetrimental impacts on our health and the environment. Having a predictive simu-lation tool can facilitate the optimization of the combustion systems to reduce thefuel consumption and pollutant formation rates also aids with the development ofthe combustion systems working with renewable and clean fuels. This study aimsto improve a computational method to be used for the simulation of the combustionprocess to provide an effective tool for the design of combustion systems.vPrefaceThe contents presented in Chapter 3 were previously published in:• A. Mousemi, W. K. Bushe, and S. Hochgreb. “Evaluation of manifold rep-resentations of chemistry in stratified, swirl-stabilized flames.” Combustionand Flame 229, 111418, 2021.In this paper, I was responsible for conducting the research work and prepar-ing the manuscript. Professor Kendal Bushe supervised the research, revised themanuscript, and provided feedback on the work. Professor Simone Hochgreb pro-vided the experimental data which was utilized in the research. Meanwhile, shereviewed the manuscript and provided feedback on the work. Figures 3.1 and 3.2were also provided by Dr. Hochgreb.Most of the contents presented in Chapter 4 were published in:• A. Mousemi and W. K. Bushe. “The joint probability density function ofmixture fraction, reaction progress variable, and total enthalpy in a stratified,swirl-stabilized turbulent flame.” Physics of Fluids 33.3, 035106, 2021.In this paper, I conducted the whole research work and prepared the manuscript.Professor Kendal Bushe supervised the research, reviewed the manuscript, and pro-vided feedbacks on the manuscript. In Chapter 4, some improvements with respectto this paper are made and presented.Furthermore, some parts of the two aforementioned papers are used in the ab-stract, Chapter 2, and Chapter 5 of this thesis.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Filtration of the Governing Equations . . . . . . . . . . . . . . . 10vii2.3.1 Reynolds Average Navier-Stokes . . . . . . . . . . . . . . 112.3.2 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . 132.4 Combustion Regimes . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Filter-Chemistry Interaction Models . . . . . . . . . . . . . . . . 162.5.1 The Laminar Flamelet Model (LFM) . . . . . . . . . . . 172.5.2 The Probability Density Function (PDF) Transport Model 172.5.3 The Conditional Moment Closure (CMC) Model . . . . . 182.5.4 The Multiple Mapping Conditioning (MMC) Model . . . 192.5.5 The Conditional Source-term Estimation (CSE) Model . . 192.5.6 The Uniform Conditional State Model . . . . . . . . . . . 202.6 Different Presumed PDF Modelling Methods . . . . . . . . . . . 213 The Effect of Heat Transfer on the Low-Order Manifold Represen-tation of Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.1 Effects of the Swirl and Stratification on the Two-DimensionalConditional Manifold . . . . . . . . . . . . . . . . . . . . 363.2.2 Three-Condition Conditional Averages: Employing the DataCollected from Each Specific Flame . . . . . . . . . . . . 413.2.3 Three-Condition Conditional Averages: Employing the DataCollected from All Cases . . . . . . . . . . . . . . . . . . 433.2.4 Effects of Swirl and Stratification on the Three-DimensionalConditional Manifold . . . . . . . . . . . . . . . . . . . . 463.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 The Joint PDF of Mixture Fraction, Progress Variable and Total En-thalpy in Turbulent Flames . . . . . . . . . . . . . . . . . . . . . . . 514.1 Marginal PDF of the Conditioning Variables . . . . . . . . . . . . 524.2 Conditional PDF of Normalized Total Enthalpy . . . . . . . . . . 584.3 Correlations between the Conditioning Variables . . . . . . . . . 644.4 The Copula for Mixture Fraction and Reaction Progress Variable . 68viii4.5 Modeling the Joint Distribution of the Conditioning Variables . . . 774.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 865.1 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 885.4 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90A The Discrepancies from the Different Conditional Averaging Methods 106B Convergence Study for Obtaining the CDF of the Conditioning Vari-ables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137ixList of TablesTable 3.1 Working conditions for Cambridge-Sandia stratified swirl burner.φi and φo denote equivalence ratio of the flow in the inner andouter annuli, respectively. . . . . . . . . . . . . . . . . . . . . 28Table 3.2 Different approaches for computing the conditional averages . 36Table 4.1 The experimental and modeled conditional PDF of normalizedtotal enthalpy calculated from data of all flames. . . . . . . . . 61Table 4.2 The considered PDF modeling approaches. . . . . . . . . . . . 78xList of FiguresFigure 2.1 Schematic of the turbulent energy cascade diagram. . . . . . . 11Figure 2.2 Decomposition of the energy spectrum to the modelled and theresolved portions in the RANS modelling paradigm. . . . . . 13Figure 2.3 Decomposition of the energy spectrum to the modelled and theresolved portions in the LES modelling paradigm. . . . . . . . 13Figure 3.1 Schematic of the Cambridge-Sandia Burner. . . . . . . . . . . 29Figure 3.2 Using reaction progress variable as the only conditioning vari-able for predicting CO mass fraction in a sample point in flamenumber 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 3.3 Two-condition conditionally averaged reactive scalars from allcases; the dashed line presents the stoichiometric mixture frac-tion, and the dash-dot line shows the lower flammability limitof methane-air mixture. . . . . . . . . . . . . . . . . . . . . . 39Figure 3.4 Two-condition conditionally averaged reactive scalars from case11; the dashed line presents the stoichiometric mixture frac-tion, and the dash-dot line shows the lower flammability limitof methane-air mixture. . . . . . . . . . . . . . . . . . . . . . 40xiFigure 3.5 Normalized reactive scalars along the horizontal measurementline located 20 mm above the burner tip in flame number 5. Allof the species mass fractions shown by the lines are the experi-mental ensemble averaged values normalized by the maximumaverage mass fraction; while the triangle symbols represent thenormalized discrepancies for methane mass fraction as definedin Equation 3.6 from the 3Dn method. . . . . . . . . . . . . . 42Figure 3.6 Variation of normalized total enthalpy as a function of radialposition at different axial distances from the burner tip in flame11. The dash-dotted and the dotted lines are representing theradial position of the mixing layer and the flame brush, respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.7 Normalized accumulated discrepancies obtained from differ-ent conditional averaging approaches. . . . . . . . . . . . . . 48Figure 4.1 The experimental and modeled marginal PDF of mixture frac-tion at various radial positions in flame number 11. . . . . . . 54Figure 4.2 The experimental and modeled marginal PDF of reaction progressvariable at various radial positions in flame number 11. . . . . 55Figure 4.3 The experimental and modeled marginal PDF of reaction progressvariable at various radial positions in flame number 11. . . . . 57Figure 4.4 The location of the flame brush in various axial distances fromthe burner tip in the 9 considered flames. . . . . . . . . . . . 58Figure 4.5 The experimental and modeled conditional PDF of normalizedtotal enthalpy calculated from data of all flames. . . . . . . . 59Figure 4.6 PDF of normalized total enthalpy conditioned on Z = 0.575and c = 0.775 from different data collecting strategies for 9flames. Subfigures in the same rows and columns representflames with the same stratification factor and swirl flow ratio,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62xiiFigure 4.7 Significance level for the conditional distribution of normal-ized total enthalpy conditioned on Z = 0.575 and c = 0.775 atsome representative points located 30 mm downstream of theburner tip in flame number 11. . . . . . . . . . . . . . . . . . 63Figure 4.8 The exact and modeled conditional distributions of normalizedtotal enthalpy conditioned on Z = 0.575 and c = 0.775 at twodifferent points in flame number 11 with. (a) α = 1.00 for allfour models (b) α = 0.494 for the “Swirl” approach and= 1.00for the other three models. . . . . . . . . . . . . . . . . . . . 63Figure 4.9 Correlation factor between mixture fraction and reaction progressvariable at different locations in 9 flames. Subfigures in thesame rows and columns represent flames with the same strati-fication factor and swirl flow ratio, respectively. . . . . . . . . 65Figure 4.10 Correlation factor between mixture fraction and normalized to-tal enthalpy at different locations in 9 flames. Subfigures in thesame rows and columns represent flames with the same strati-fication factor and swirl flow ratio, respectively. . . . . . . . . 66Figure 4.11 Correlation factor between reaction progress variable and nor-malized total enthalpy at different locations in 9 flames. Sub-figures in the same rows and columns represent flames with thesame stratification factor and swirl flow ratio, respectively. . . 67Figure 4.12 Schematic of a sample contingency table. . . . . . . . . . . . 70Figure 4.13 Evaluation of KC(t) for the product, Plackett, and Gaussiancopulas at different points of flame number 11 and comparisonwith that of the experimental copula. . . . . . . . . . . . . . . 74Figure 4.14 Evaluation of C(t, t) for the product, Plackett, and Gaussiancopulas at different points of flame number 11 and comparisonwith that of the experimental copula. . . . . . . . . . . . . . . 75Figure 4.15 Evaluation of C(0.5, t) for the product, Plackett, and Gaussiancopulas at different points of flame number 11 and comparisonwith that of the experimental copula. . . . . . . . . . . . . . . 76xiiiFigure 4.16 Normalized discrepancies in the averaged temperature for 9flames; open and filled symbols represent the results obtainedfrom approach 1 and approach 2, respectively. . . . . . . . . . 81Figure 4.17 Normalized discrepancies in the averaged carbon monoxidemass fraction for 9 flames; open and filled symbols representthe results obtained from approach 1 and approach 2, respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 4.18 Normalized accumulated discrepancies of temperature obtainedfrom two PDF models. . . . . . . . . . . . . . . . . . . . . . 83Figure 4.19 Normalized accumulated discrepancies of CO mass fractionobtained from two PDF models. . . . . . . . . . . . . . . . . 83Figure 4.20 Normalized accumulated discrepancies of CO mass fractionobtained from two PDF models when cells with less than 10,15, and 20 data points are ignored for conditional averaging. . 84Figure A.1 Discrepancies in the averaged temperature for 9 flames; openand filled symbols represent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and each specific case (2Dn), respectively. . . . . . . . . . . . 107Figure A.2 Normalized discrepancies in the averaged methane mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the two-condition conditional averages computedfrom all flames (2DAll) and each specific case (2Dn), respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Figure A.3 Normalized discrepancies in the averaged carbon dioxide massfraction for 9 flames; open and filled symbols represent theresults obtained from the two-condition conditional averagescomputed from all flames (2DAll) and each specific case (2Dn),respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109xivFigure A.4 Normalized discrepancies in the averaged carbon monoxidemass fraction for 9 flames; open and filled symbols representthe results obtained from the two-condition conditional aver-ages computed from all flames (2DAll) and each specific case(2Dn), respectively. . . . . . . . . . . . . . . . . . . . . . . . 110Figure A.5 Normalized discrepancies in the averaged hydrogen mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the two-condition conditional averages computedfrom all flames (2DAll) and each specific case (2Dn), respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Figure A.6 Normalized discrepancies in the averaged water mass fractionfor 9 flames; open and filled symbols represent the results ob-tained from the two-condition conditional averages computedfrom all flames (2DAll) and each specific case (2Dn), respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure A.7 Discrepancies in the averaged temperature for 9 flames; openand filled symbols represent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and the three-condition version by using the data of each spe-cific case (3Dn), respectively. . . . . . . . . . . . . . . . . . 113Figure A.8 Normalized discrepancies in the averaged methane mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the two-condition conditional averages computedfrom all flames (2DAll) and the three-condition version by us-ing the data of the specific case (3Dn), respectively. . . . . . . 114Figure A.9 Normalized discrepancies in the averaged carbon dioxide massfraction for 9 flames; open and filled symbols represent theresults obtained from the two-condition conditional averagescomputed from all flames (2DAll) and the three-condition ver-sion by using the data of each specific case (3Dn), respectively. 115xvFigure A.10 Normalized discrepancies in the averaged carbon monoxidemass fraction for 9 flames; open and filled symbols representthe results obtained from the two-condition conditional aver-ages computed from all flames (2DAll) and the three-conditionversion by using the data of each specific case (3Dn), respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure A.11 Normalized discrepancies in the averaged hydrogen mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the two-condition conditional averages computedfrom all flames (2DAll) and the three-condition version by us-ing the data of each specific case (3Dn), respectively. . . . . . 117Figure A.12 Normalized discrepancies in the averaged water mass fractionfor 9 flames; open and filled symbols represent the results ob-tained from the two-condition conditional averages computedfrom all flames (2DAll) and the three-condition version by us-ing the data of each specific case (3Dn), respectively. . . . . . 118Figure A.13 Discrepancies in the averaged temperature for 9 flames; openand filled symbols represent the results obtained from the two-condition (2DAll) and three-condition (3DAll) conditional av-erages computed from all flames, respectively. . . . . . . . . . 119Figure A.14 Normalized discrepancies in the averaged methane mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the two-condition (2DAll) and three-condition(3DAll) conditional averages computed from all flames, re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure A.15 Normalized discrepancies in the averaged carbon dioxide massfraction for 9 flames; open and filled symbols represent the re-sults obtained from the two-condition (2DAll) and three-condition(3DAll) conditional averages computed from all flames, re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121xviFigure A.16 Normalized discrepancies in the averaged carbon monoxidemass fraction for 9 flames; open and filled symbols repre-sent the results obtained from the two-condition (2DAll) andthree-condition (3DAll) conditional averages computed fromall flames, respectively. . . . . . . . . . . . . . . . . . . . . . 122Figure A.17 Normalized discrepancies in the averaged hydrogen mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the two-condition (2DAll) and three-condition(3DAll) conditional averages computed from all flames, re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Figure A.18 Normalized discrepancies in the averaged water mass fractionfor 9 flames; open and filled symbols represent the resultsobtained from the two-condition (2DAll) and three-condition(3DAll) conditional averages computed from all flames, re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Figure A.19 Discrepancies in the averaged temperature for 9 flames; openand filled symbols represent the results obtained from the three-condition conditional averages computed from cases with sim-ilar swirl flow ratio (3DSwirl) and all flames (3DAll), respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Figure A.20 Normalized discrepancies in the averaged methane mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the three-condition conditional averages com-puted from cases with similar swirl flow ratio (3DSwirl) andall flames (3DAll), respectively. . . . . . . . . . . . . . . . . 126Figure A.21 Normalized discrepancies in the averaged carbon dioxide massfraction for 9 flames; open and filled symbols represent theresults obtained from the three-condition conditional averagescomputed from cases with similar swirl flow ratio (3DSwirl)and all flames (3DAll), respectively. . . . . . . . . . . . . . . 127xviiFigure A.22 Normalized discrepancies in the averaged carbon monoxidemass fraction for 9 flames; open and filled symbols repre-sent the results obtained from the three-condition conditionalaverages computed from cases with similar swirl flow ratio(3DSwirl) and all flames (3DAll), respectively. . . . . . . . . 128Figure A.23 Normalized discrepancies in the averaged hydrogen mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the three-condition conditional averages com-puted from cases with similar swirl flow ratio (3DSwirl) andall flames (3DAll), respectively. . . . . . . . . . . . . . . . . 129Figure A.24 Normalized discrepancies in the averaged water mass fractionfor 9 flames; open and filled symbols represent the resultsobtained from the three-condition conditional averages com-puted from cases with similar swirl flow ratio (3DSwirl) andall flames (3DAll), respectively. . . . . . . . . . . . . . . . . 130Figure A.25 Discrepancies in the averaged temperature for 9 flames; openand filled symbols represent the results obtained from the three-condition conditional averages computed from cases with sim-ilar stratification factor (3DStratification) and all flames (3DAll),respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Figure A.26 Normalized discrepancies in the averaged methane mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the three-condition conditional averages com-puted from cases with similar stratification factor (3DStratifi-cation) and all flames (3DAll), respectively. . . . . . . . . . . 132Figure A.27 Normalized discrepancies in the averaged carbon dioxide massfraction for 9 flames; open and filled symbols represent theresults obtained from the three-condition conditional averagescomputed from cases with similar stratification factor (3DStrat-ification) and all flames (3DAll), respectively. . . . . . . . . . 133xviiiFigure A.28 Normalized discrepancies in the averaged carbon monoxidemass fraction for 9 flames; open and filled symbols representthe results obtained from the three-condition conditional av-erages computed from cases with similar stratification factor(3DStratification) and all flames (3DAll), respectively. . . . . 134Figure A.29 Normalized discrepancies in the averaged hydrogen mass frac-tion for 9 flames; open and filled symbols represent the resultsobtained from the three-condition conditional averages com-puted from cases with similar stratification factor (3DStratifi-cation) and all flames (3DAll), respectively. . . . . . . . . . . 135Figure A.30 Normalized discrepancies in the averaged water mass fractionfor 9 flames; open and filled symbols represent the results ob-tained from the three-condition conditional averages computedfrom cases with similar stratification factor (3DStratification)and all flames (3DAll), respectively. . . . . . . . . . . . . . . 136Figure B.1 Normalized accumulated discrepancies of temperature obtainedby using different number of subdivisions in each bin for inte-gration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138xixList of SymbolsLatin AlphabetSymbol Description SI UnitAr Pre-exponential coefficient in the Arrhenius relation –c Reaction progress variable –C Copula –Cexp Experimental copula –ci ith random conditioning variable –cˆγiγ j Number of concordant pairs for γi and γ j –Di j Binary diffusivity between ith and jth species m2/sDmiMixture-averaged mass diffusioncoefficient of ith speciesm2/sDT,i Thermal diffusion coefficient of ith species kg/(m.s)Dˆγiγ j Number of discordant pairs for γi and γ j –e Specific internal energy J/kgEa,r Molar activation energy of rth reaction J/molEk Normalized accumulated error –F/A Fuel to air mass ratio –fi Body forces on ith species per unit mass N/kgh Normalized Total enthalpy –htot Specific total enthalpy J/kgI 3×3 Identity matrix –Jαβconditional diffusive flux associatedwith scalar α and conditioning variable β kg/(m2.s)k Specific turbulent kinetic energy m2/s2kb,rArrhenius reaction coefficient ofrth reaction in the backward direction–xxSymbol Description SI UnitKC,r Equilibrium coefficient of rth reaction –k f ,rArrhenius reaction coefficient ofrth reaction in the forward direction–Kγiγ j joint cumulative distribution function of γi and γ j –l Characteristic length scale ml0 Largest eddy’s length scale mlk Kolmogorov eddy length scale mM Mean vector –p Pressure PaP Joint probability density function –qHeat flux by radiation, conduction, and concentrationgradient- and thermophoresis driven-transportkg/s3qrad Heat flux due to radiation kg/s3R Rotation rate tensor 1/sRXY Correlation factor between X and Y –Re Reynolds number –Rek Kolmogorov Reynolds number –Ret Turbulent Reynolds number –Ru Ideal gas constant J/(mol.K)S Strain rate tensor 1/st Time sT Temperature KTe Temperature of combustion products in equilibrium KTu Temperature of unburnt reactant Kt0 Largest eddy’s time scale stk Kolmogorov eddy time scale su Velocity m/sU0 Largest eddy’s velocity scale m/suk Kolmogorov eddy velocity scale m/sVi Diffusive velocity of ith species m/sWi Molar mass of ith species kg/molXi Mole fraction of ith species –Yi Mass fraction of ith species –xxiSymbol Description SI UnitYi,Direct Mass fraction of ith species from ensemble averaging –Yi,Est Mass fraction of ith species estimated from the UCS model –Yi,maxMaximum mass fraction of ith species measured in theflow field–Z Mixture fraction –Greek SymbolsSymbol Description SI Unitα Significance level –αr Pre-exponential coefficient in the Arrhenius relation –γi Non-random field for ith conditioning variable –δ Non-random field for normalized total enthalpy –ε Specific turbulent energy dissipation rate W/kgζ Non-random field for mixture fraction –η Non-random field for progress variable –θXY Cross product ratio for two random variables X and Y –λ Thermal conductivity W/(m.K)µ Dynamic viscosity Pa.sν Kinematic viscosity m2/sν ′i,rStoichiometric coefficient of ith species in rth reaction onthe reactant side–ν ′′i,rStoichiometric coefficient of ith species in rth reactionon the product side–νT Turbulent kinematic viscosity m2/sξ Vorticity 1/sρ Density kg/m3Σ Covariance matrix –τ Viscous stress tensor Paφ equivalence ratio –ω˙i Chemical source-term of ith specie kg/(m3.s)xxiiOperatorsOperator Description~Ξ A three-dimensional vector of ΞΞ Averaged or filtered value of ΞΞ˜ Density-weighted average or filtered value of ΞΞ′ Fluctuations of Ξ around the mean valueΞ′′ Fluctuations of Ξ around the density-weighted mean value〈Ξ1|Ξ2〉 Averaged value of Ξ1 conditioned on Ξ2xxiiiList of AcronymsCDF – Cumulative Distribution FunctionCFD – Computational Fluid DynamicsCMC – Conditional Moment ClosureCSE – Conditional Source-term EstimationDNS – Direct Numerical SimulationFGM – Flamelet Generated ManifoldILDM – Intrinsic Low-Dimensional ManifoldISAT – In-Situ Adaptive TabulationLEM – Linear Eddy ModelLES – Large Eddy SimulationLFM – Laminar Flamelet ModelLIF – Laser Induced FluorescenceMFM – Multidimensional Flamelet ManifoldMMC – Multiple Mapping ConditioningPCA – Principal Component AnalysisPCM-FPI – Presumed Conditional Moments-Flame Prolongation of IntrinsicPDF – Probability Density FunctionPLIF – Planar Laser Induced FluorescenceRANS – Reynolds Averaged Navier-StokesSGS – SubGrid ScaleSMLD – Statistically Most Likely DistributionTGLDM – Trajectory Generated Low-Dimensional ManifoldUCS – Uniform Conditional StatexxivAcknowledgmentsFirstly, I want to thank my parents who sacrificed their own joy for me. I alsoappreciate all the help of my beloved brother throughout my whole life. Withoutmy family, I couldn’t achieve whatever I have right now. Meanwhile, I am grate-ful for the supports of my best friends Sohrab Entezami, Amir Chizfahm, HosseinMoghimianavval, Mohammad Kazem Meidani, and Mahdi Izadi.I also want to present my special thanks to my kind and patient supervisorDr. W. Kendal Bushe who helped a lot in preparing this thesis. To Dr. SimoneHochgreb and Dr. Robert Barlow, I am grateful for providing the experimentaldata from the Cambridge-Sandia burner and reviewing our paper entitled “Evalua-tion of manifold representations of chemistry in stratified, swirl-stabilized flames”.Furthermore, I appreciate the financial support of the Natural Sciences and Engi-neering Research Council (NSERC) of Canada.xxvChapter 1IntroductionThe motivations behind also the main objectives of the present study are discussedin the first two sections and the overall skeleton of the document is presented in thelast section of this chapter.1.1 MotivationsThe combustion process has a deep root in the civilization of humankind and stillplays a key role in the energy conversion processes that are building the modernworld as we recognize. However, the life of many species on earth is rapidly get-ting close to a critical point which is mainly related to the combustion of fossilfuels [73]. As of 2019, more than 80% of the worldwide energy usage was pro-vided by oil, coal, and gas which have considerable contributions to the emission ofpollutants and greenhouse gases [57]. In addition to the problems associated withclimate change, the limited resources of fossil fuels encourage the transition to theuse of renewable and clean energy sources. Nevertheless, such transitions usuallyhappen rather slowly, therefore, finding solutions to optimize the already-existingcombustion systems for reducing the fuel consumption and pollutant formationrates is inevitable.The renewable resources of energy have the potential to completely replace thefossil fuels for electricity generation [30, 56, 64]. Nonetheless, the more challeng-ing mission is to find a replacement for the role of fossil fuels in the international1energy trades also in the heavy-duty propulsion applications. The combustion ofbiofuels [28], hydrogen [24], and even metal powders [6] have been proposed assome of the alternate scenarios. Both optimization of the existing combustionchambers and the development of the combustion systems working with such alter-native fuels necessitate a deep understanding of combustion physics. The existingexperimental approaches for studying the combustion process and designing newcombustion systems are really expensive and time-consuming. Meanwhile, exper-imentation on many complicated systems with harsh operating conditions such asgas turbine combustion chambers and rocket engines does not provide sufficient in-formation about the flow field of the reacting flow due to the limited optical accessand the three-dimensional nature of the flow field. Thus, the computation methodscan play a crucial role in these developments by providing a predictive tool to beutilized in the design process.The coexistence of fluid dynamics, chemical reactions, and heat transfer –which are amongst the most complicated problems in classical physics – in a com-bustion process makes the simulation of such phenomena cumbersome. Simulatingchemically reactive flow fields only based on the fundamental governing equationsis prohibitively computationally expensive for most industrial cases, and it makesthe application of simplifying models unavoidable during the simulations. Similarto many other problems, compromising between the accuracy and the simplifica-tion level is a challenge in the developed models. The primary objective of thisproject is to improve the prediction power of a combustion model while keepingthe computational cost tractable.1.2 ObjectivesThe interaction between the chemical reactions and turbulence is one of the mainsources of complexity in the simulation of turbulent flames. In the current incar-nation of a novel combustion model named Uniform Conditional State (UCS), onegenerates and stores a chemistry table to be utilized during the CFD calculations ina turbulent reactive flow field. In this model, a few mapping parameters – known asthe conditioning variables by the research community – are employed to effectivelylower the order of the chemistry dynamics. As the chemistry can be represented in2a low-dimensional table, this approach considerably mitigates the CFD computa-tional cost and provides a practical predicting tool for the design process. In thisproject, the experimental data from the Cambridge-Sandia burner are employed toevaluate the UCS combustion model and investigate the effect of adding to the di-mensions of the chemistry table on the accuracy of the predictions. Furthermore,the statistical properties of the conditioning variables are studied based on the mea-surements from this burner. The work in this thesis indirectly facilitates the processof designing and optimization of the combustion systems by improving the UCScombustion model as a powerful predictive tool.1.3 OutlineIn the next chapter, the fundamentals of turbulence and combustion science arereviewed. Meanwhile, the main challenges associated with the simulation of tur-bulent reactive flows and the main modelling streams in this field are introduced.More details regarding the UCS combustion model are also presented in Chapter 2.Next, in Chapter 3, the Cambridge-Sandia burner is introduced as the test case andthe experimental measurements from this burner are employed for evaluation of theUCS model. Moreover, the effect of the history of heat transfer on the chemistrytabulation is examined in this chapter. In Chapter 4, the statistical properties of theconditioning variables are investigated via post-processing the experimental datafrom the Cambridge-Sandia burner. Finally, the conclusions from the presentedresearch are drawn in Chapter 5, and some possible future works are proposed inthat chapter.3Chapter 2BackgroundThe fundamental knowledge in the fluid mechanics and combustion science as theprerequisite for understanding the research work explained in Chapters 3 and 4 arepresented here. The governing equations which describe the state of a chemicallyreactive fluid flow, an introduction to turbulence, and the need for filtering the setof equations are presented in Sections 2.1, 2.2, and 2.3, respectively. Next, thedifferent combustion regimes and various chemistry reduction techniques are in-troduced in Section 2.4. Then, the filter-chemistry interaction problem is discussedin Section 2.5 and the mainstream modelling paradigms in this field of research areintroduced. Finally, the different presumed Probability Density Function (PDF)models are reviewed in Section 2.6.2.1 Governing EquationsThe continuity equation accounting for the conservation of mass, the Navier-Stokesequations describing the conservation of momentum, the internal energy transportequation representing the conservation of energy, and the species continuity equa-tion for the chemically reactive fluid flows are∂ρ∂ t+∇.(ρ~u) = 0, (2.1)4ρ∂~u∂ t+ρ~u.∇~u =−∇p+∇.τ +ρN∑k=1Yk~fk, (2.2)ρ∂e∂ t+ρ~u.∇e =−∇.~q− p∇.~u+τ : ∇~u+ρN∑k=1Yk~fk.~Vk, (2.3)ρ∂Yk∂ t+ρ~u.∇Yk =−∇.(ρ~VkYk)+ ω˙k, (2.4)where the bold symbols are used for representing tensors throughout this docu-ment; ρ is the density; ~u denotes the velocity vector; p is the pressure field; τis the viscous stress tensor; N denotes the number of the species; ~fk is the bodyforces exerted on kth species per unit mass; e denotes the specific internal energyof the mixture; ~Vk represents the diffusive velocity of kth species; ~q is the heatflux by radiation, conduction, and transport due to the concentration gradient andthe thermophoresis effect. Yk and ω˙k represent the mass fraction and the chemicalsource-term of kth species, respectively.The terms which need constitutive relations are as follows:• The viscous stress tensor: This term accounts for the bulk impact of therandom molecular motions on the macroscopic kinematic properties of theflow field. Considering a Newtonian fluid flow, this tensor can be expressedasτ = µ[(∇~u)+(∇~u)T]− 23µ(∇.~u)I , (2.5)where µ is the dynamic viscosity and I is the 3×3 identity matrix.• The diffusive velocity vector: This parameter is associated with the trans-port of the species by mechanisms other than the bulk velocity of the flow(~u). The species concentration gradient and the Soret (thermophoresis) effecthave the main contributions to this velocity. Considering only the impact ofthe concentration gradients and applying the Fick’s first law, the diffusive5velocity can be obtained from:Xk~Vk =−Dmk ∇Xk, (2.6)where Dmk is the mixture-averaged diffusion coefficient for kth species.• The heat flux vector: The contributions of heat conduction, heat diffusiondue to the diffusion of species, thermal diffusion, and radiation to the energycontent of the fluid are digested in the vector ~q. This term can be expressedas~q =−λ∇T +ρN∑i=1hiYi~Vi+RuTN∑i=1N∑j=1(X jDT,iWiDi j)(~Vi−~Vj)+~qrad , (2.7)where λ is the thermal conductivity; hi is the enthalpy of ith species; Ru de-notes the ideal gas constant; DT,i and Di j are the thermal diffusion coefficientof ith species and the binary diffusivity between ith and jth species, respec-tively; X j is the mole fraction of jth species, and Wi represents the molarmass of ith species;~qrad is the heat flux due to radiation.• The chemical source-term: The random collisions between different chem-ical molecules result to the progression of the chemical reactions. There-fore, the reaction rates are increasing functions of the concentration of thecontributing chemical species also the mixture temperature. The chemicalsource-term for kth species involved in R elementary reactions can be obtainviaω˙k =Wk×R∑r=1{(ν ′′k,r−ν ′k,r)[k f ,rN∏j=1(X j pRuT)ν ′k,r− kb,rN∏j=1(X j pRuT)ν ′′k,r]}, (2.8)where for kth species in the rth reaction, ν ′′k,r and ν′k,r represent the stoichio-metric coefficients on the product and reactant sides, respectively; k f ,r and6kb,r can be calculated via the Arrhenius equation ask f ,r(T ) = ArTαr exp(−Ea,rRuT), (2.9)kb,r(T ) =k f ,rKC,r. (2.10)Here Ar and αr are the pre-exponential coefficients; Ea,r denotes the activa-tion energy in the forward direction and KC,r is the equilibrium coefficient ofrth elementary reaction.2.2 TurbulenceThe set of governing equations to be solved for the simulation of chemically-reactive flows are highly non-linear and the solution shows a chaotic behaviour inmost of the applicable cases. This behaviour is associated with turbulence which isregarded as one of the major unsolved problems in classical physics. The Reynoldsnumber, which represents the ratio of the convective to the viscous forces actingon the fluid, can be employed as a criterion for identifying the transition from alaminar to a turbulent regime. This parameter is defined asRe≡ ρUlµ, (2.11)where U denotes the magnitude of the characteristic velocity and l is a character-istic length scale.The presence of vorticity is another feature of turbulent flows. By consid-ering the transport equation for the vorticity vector (define as ~ξ ≡ ∇×~u) in anincompressible flow field, two important mechanisms impacting the dynamics ofturbulence can be described. That is expressed in Equation 2.12.∂~ξ∂ t+~u.∇~ξ = ~ξ .S+µρ∇2~ξ (2.12)7Here S ≡ 12[(∇~u)+(∇~u)T]is the strain rate tensor. By defining the tensor A ≡S11 0 00 S22 00 0 S33, the first term on the right hand side of Equation 2.12 can bere-expressed as~ξ .S = ~ξ .A+~ξ .(S−A), (2.13)where the first term on the right hand side represents the stretch and the secondterm accounts for the tilting of the vortexes due to the strain of the fluid flow.The vortex stretching mechanism accelerates the vortical motions and based on theconservation of angular momentum, smaller structures are formed from the largerscales [152]. On average, the turbulent energy is transferred from the larger to thesmaller scales and is mainly dissipated by the small eddies via viscous effects. Onthe other hand, the vortex tilting mechanism distributes the turbulent energy in dif-ferent directions. These two mechanisms generate a three-dimensional flow fieldcontaining a wide range of length, time, and velocity scales.Identifying the range of the scales present in a turbulent flow is of great impor-tance for conducting numerical simulations. Pope employed a two-point correla-tion for defining the length scale of the largest eddies (l0) [127]. Meanwhile, theorder of the velocity scales associated with the largest vortices (U0) can be relatedto the specific turbulent kinetic energy (k) through:U0 =O(√23k). (2.14)Next, the time scale of the largest eddies can be obtained from the associated ve-locity and length scales as:t0 =l0U0, (2.15)and the turbulent Reynolds number can be defined asRet ≡ ρU0l0µ . (2.16)8As previously described, the small-scale eddies are formed from the larger vor-tices via vortex stretching. Eventually, the accelerating forces exerted by the strainof the flow on the vortices balance with the viscous shear forces and no smallereddy can be formed. The dynamics of the smallest scales are fast enough that thedirectional information from the mean flow gets lost, thus, the small vortices be-come isotropic with universal statistical properties. The process through which theturbulent energy is transferred from the large scales to the small structures and isdissipated by the viscous forces is known as “Energy Cascade”. Based on the Kol-mogorov’s hypothesis [82, 83], the specific dissipation rate of the turbulent energy(ε) and the kinematic viscosity (ν ≡ µ/ρ) are the only parameters that affect thecharacteristics of the sufficiently small vortices. Therefore, the length, velocity,and time scale of the smallest eddy – known as the Kolmogorov eddy – can bedescribed in terms of these two variables:lk =(ν3ε)1/4, (2.17)uk = (εν)1/4, (2.18)tk =lkuk. (2.19)Since this Kolmogorov length scale is typically orders of magnitude larger than themolecular scales, the turbulent structures are usually within the continuum range.The Reynolds number associated with the Kolmogorov scale can be obtained asRek =uklkν=(εν)1/4ν(ν3ε)1/4= 1, (2.20)where the unity value suggests the balance of the convective and the viscous forcesat this scale.In the Direct Numerical Simulation (DNS) of turbulent flows, the fundamentalgoverning equations are solved directly, and all of the spatial and temporal scalesneed to be resolved. The ratio of the size of the largest eddies to that of the smallest9flow structures presents a reasonable approximation for the number of the compu-tational cells to be utilized during the calculations. By approximating the turbulentenergy dissipation rate as ε ∼U30 /l0, this ratio can be estimated asl0lk∼ l0ε1/4ν3/4∼ l3/40 U3/40ν3/4= Re3/4t . (2.21)This calculation presents a rough approximation of the number of the cells inonly one direction of the field. As described previously, turbulent flows are three-dimensional, hence, the number of the computational cells should grow by Re9/4t .On the other hand, the number of time steps during the calculations rises by theratio of the slowest time scales to the fastest one. This ratio can be estimated ast0tk∼ l0ε1/2U0ν1/2∼ l1/20 U1/20ν1/2= Re1/2t . (2.22)Therefore, the total computational effort grows by Re11/4t . These approximationssuggest that DNS becomes prohibitively expensive in turbulent flows with highReynolds number, and the application of turbulent models is generally unavoidable.2.3 Filtration of the Governing EquationsAs explained in the previous section, a wide range of scales are present in a turbu-lent flow field and the turbulent energy is transferred from the larger to the smallerscales to be dissipated by the viscous effects. This energy can be plotted in a log-log scale against different wave numbers (defined as 2pil ) in a turbulent flow fieldas shown in Figure 2.1. Three distinct regions can be observed on this diagram. Inthe energy-containing region, large eddies that inherit the mean flow characteristicsare present. The vortices of the scales near to the Kolmogorov eddy lie within theviscous dissipation range where the viscous forces are dominant. The inertial sub-range is considered as the region between the energy containing and the viscousdissipation regions. The turbulent energy is transferred to the small-scale eddiesthrough the eddies in this range. Based on the Kolmogorov’s hypothesis, the iner-tial subrange and the viscous dissipation region are independent of the mean flowtopology [82, 83].10Figure 2.1. Schematic of the turbulent energy cascade diagram.The presence of such energy cascading structures suggests that the turbulentflows despite of being chaotic, represents some level of predictability which ispromising for modeling the turbulence phenomenon. The application of either atemporal or a spatial filter on the set of governing equations is a common approachfor reducing the computational cost of the simulations. In this section, the ReynoldsAverage Navier-Stokes (RANS) paradigm in which the equations are temporallyfiltered and the Large Eddy Simulation (LES) method in which spatial filters areemployed are described.2.3.1 Reynolds Average Navier-StokesIn a turbulent flow field, each of the variables can be decomposed into a time aver-aged and a fluctuating term as is expressed for a representative field (Ξ):Ξ= Ξ+Ξ′. (2.23)Here Ξ denotes the time-averaged value and Ξ′ is the deviation of the representativefield itself from the mean value. Another approach is to apply a density-weightedaverage known as the Favre-averaging method to decompose the variables asΞ= Ξ˜+Ξ′′, (2.24)11where Ξ˜ ≡ ρΞρ and Ξ′′ denotes the deviation of the representative field from thedensity-weighted average of this variable. Favre-averaging removes the closureproblem for the density-scalar and the density-velocity correlations. By averagingthe governing equations and substituting Equation 2.24 for different variables wehave:∂ρ∂ t+∇.(ρ~˜u) = 0, (2.25)ρ∂~˜u∂ t+ρ~˜u.∇~˜u =−∇p+∇.τ˜ +ρN∑k=1Y˜k~fk−∇.(ρ ~˜u′′~u′′), (2.26)ρ∂ e˜∂ t+ρ~˜u.∇e˜ =−∇.~q− p∇.~u+τ : ∇~u+ρN∑k=1˜Yk~fk.~Vk−∇.(ρ ~˜u′′e′′), (2.27)ρ∂Y˜k∂ t+ρ~˜u.∇Y˜k =−∇.(ρ~˜VkYk)+ ω˙k−∇.(ρ ~˜u′′Y ′′k ). (2.28)In the averaged set of equations, there are some terms for which no transport equa-tion is directly solved and are unclosed. Among them, the Reynolds stress tensor(ρ ~˜u′′~u′′) and the averaged chemical source-term (ω˙k) have attracted much attentionin the research community. Discussion on modelling the later term is reserved forSection 2.5.One of the mainstreams in modelling the Reynolds stress tensor is to assumethat the last term on the right hand side of Equation 2.26 acts like a diffusiveterm with a turbulent viscosity coefficient (νT ) to be modelled. The mixing length[61, 130], Spalart-Allmaras [144], k−ω [160], k−ε [133], and the Reynolds stress[51] models are some of the approaches proposed in the context of RANS. All ofthese models make use of several unknown parameters which need to be tuned viaasymptotic analyses or comparisons to experimental or DNS data. This lack ofuniversality is the main downside of the models proposed in the context of RANS.The decomposition of the energy spectrum in the RANS modelling method is il-lustrated in Figure 2.2. This schematic suggests that all of the scales are modelled12in RANS, so, we may expect relatively low accuracy levels from this paradigm.Figure 2.2. Decomposition of the energy spectrum to the modelled and theresolved portions in the RANS modelling paradigm.2.3.2 Large Eddy SimulationLES is based on the spatial filtration of the governing equations to resolve theenergy containing scales in the computational domain and model the impact ofthe small flow structures. The decomposition of the energy spectrum in the LESmodelling context is illustrated in Figure 2.3. Each of the filtered fields can beobtained from the convolution of the corresponding field (Ξ) and a low-pass filteras:Ξ(~x, t)≡∫ΩG(~x∗,~x, t).Ξ(~x∗, t)d~x∗, (2.29)where Ω is the three-dimensional computational domain and G is the filter kerneldefined on Ω.Figure 2.3. Decomposition of the energy spectrum to the modelled and theresolved portions in the LES modelling paradigm.Similar to RANS, the Favre-filtered fields (defined as Ξ˜= ρΞρ for a representa-tive variable Ξ) can be employed for expressing the filtered set of governing equa-13tions:∂ρ∂ t+∇.(ρ~˜u) = 0, (2.30)ρ∂~˜u∂ t+ρ~˜u.∇~˜u =−∇p+∇.τ˜ +ρN∑k=1Y˜k~fk +∇.[ρ(~˜u~˜u−~˜u~u)], (2.31)ρ∂ e˜∂ t+ρ~˜u.∇e˜ =−∇.~q− p∇.~u+τ : ∇~u+ρN∑k=1˜Yk~fk.~Vk+∇.[ρ(~˜ue˜−~˜ue)], (2.32)ρ∂Y˜k∂ t+ρ~˜u.∇Y˜k =−∇.(ρ~˜VkYk)+ ω˙k−∇.[ρ(~˜uY˜k−~˜uYk)]. (2.33)Again, some unclosed terms appear in the set of filtered equations which containthe subgrid scale (SGS) information of the flow field. Several attempts for mod-elling the SGS stress tensor ρ(~˜u~˜u−~˜u~u), as one of the most important terms inthe set of filtered equations, have been made. The Smagorinsky [141], dynamicSmagorinsky [45], and conditional dynamic Smagorisnky [54] models are some ofapproaches that minimize the number of tunable parameters. Even though LES ismore computationally expensive, since only the impacts of the small-scale eddiesneed to be modelled, the models proposed in the context of LES are more universaland this approach has a superior prediction power compared to RANS.2.4 Combustion RegimesA series of self-sustaining chemical reactions releasing energy is known as com-bustion. A global reaction can be used for representing this process as is shown forthe methane-air combustion in Equation 2.34.CH4 + 2 (O2 + 3.76 N2) CO2 + 2 H2O + 7.52 N2. (2.34)14Although the global reactions only summarize the set of chemical reactions in anidealized format, it is crucial for defining a stoichiometric mixture. A mixture isconsidered as stoichiometric if the fuel to air mass ratio is that calculated from theglobal reaction. In case of having excess air or fuel, the mixture is known as fuel-lean or fuel-rich, respectively. In this regard, equivalence ratio is a useful parameterdefined as:φ =(F/A)(F/A)stoich, (2.35)where (F/A) is the fuel to air mass ratio in the corresponding mixture and the sub-script “stoich” denotes that ratio in the stoichiometric condition. Each fuel-oxidizermixture has a flammability limit, and combustion often can not be sustained in mix-tures with an equivalence ratio too far from 1.The level of mixedness of the reactants significantly affects the combustionprocess. If the fuel and oxidizer are fully mixed together prior to taking part inthe combustion reactions, the generated flame is called premixed. A variable rep-resenting the progress of the chemical reactions plays a key role in describing thepremixed combustion. The different approaches for defining such parameter arediscussed in the next chapter. In a premixed combustion, the flame front movestoward the reactants with a propagation speed which is remarkably affected by thepresence of turbulence. The turbulent eddies by stretching the flame front increasethe interface area and enhance the scalar gradients which results to higher propa-gation speeds.If the fuel and oxidizer are injected separately to the combustion zone, non-premixed (diffusion) combustion occurs. As the mixing process is slow comparedto the chemical reactions, diffusion and mixing effects are dominant in this com-bustion regime. Therefore, a parameter accounting for the level of mixedness offuel and air at each point of the domain is a key parameter for characterizing dif-fusion flames. With the purpose of having such parameter, Bilger [9] proposed avariable defined asZ =2(YC−YC,o)/WC +(YH −YH,o)/(2WH)− (YO−YO,o)/WO2(YC, f −YC,o)/WC +(YH, f −YH,o)/(2WH)− (YO, f −YO,o) , (2.36)15where Z denotes mixture fraction as the variable representing the local mixinglevel; the subscripts C, H and O correspond to the carbon, hydrogen and oxygenelements, respectively; the subscript “f” corresponds to the elemental mass frac-tions in the fuel stream and the subscript “o” is for the oxidizer stream. In most ofthe real cases, both premixed and non-premixed combustion modes coexist. Thismixed mode is known as partially premixed combustion.As previously described, the global reaction represents an idealization of thereal chemical reactions in the combustion process. However, in practice, thereare numerous chemical species and reactions that are controlling the combustiondynamics. Application of the full chemical mechanism for the simulation of tur-bulent flames is computationally expensive due to the wide range of time scales inchemistry and high dimensionality of the problem. The Intrinsic Low-DimensionalManifold (ILDM) [94], In-Situ Adaptive Tabulation (ISAT) [126], Trajectory Gen-erated Low-Dimensional Manifold (TGLDM) [58, 128, 129], and the FlameletGenerated Manifold (FGM) [109] are some of the proposed approaches for chem-istry reduction. Nevertheless, such models are not necessarily compatible withfiltration of the equations which arises the filter-chemistry closure problem for thesimulation of turbulent reactive flows.2.5 Filter-Chemistry Interaction ModelsAs discussed in the previous sections, There can be a vast range of scales and nu-merous species and chemical reactions in a turbulent reacting flow field. As such,either RANS or LES filtering is needed to reduce the required resolution for tem-poral and spatial discretization cost. At the same time, the chemical kinetic mech-anism needs to be reduced in order to lower the dimensionality of the problem.These necessities lead to a closure problem in determining the chemical source-term appearing in the RANS/LES species transport equation (ω˙k). This problemhas motivated many researchers in this field to seek functional filter-chemistrymodels [12].Several approaches for modeling filter-chemistry interaction can be classified[35] as 1) separation of scales and 2) separation of model elements which give de-scriptions for the moments of reactive parameters with regard to scalar description16in state-space and model the distribution function. The laminar flamelet approachis one of the most popular paradigms which employs a separation of scales as-sumption for modeling [15, 112, 113, 156]. On the other hand, there are PDFmethods [33, 34, 52, 125] and Conditional Moment Closure (CMC) models [11,78, 79, 102] which can be categorized in the second class.2.5.1 The Laminar Flamelet Model (LFM)In laminar flamelet-based approaches, it is assumed that the time and length scalesof turbulence are large compared to chemistry scales such that the flame reactionzone is thin compared to Kolmogorov scale [12, 115]. This model implies thatthe turbulent flame can be considered to be a collection of laminar flame pocketsinteracting with the turbulent flow field, and the complex chemical structure of theflame can be separated from the fluid flow dynamics [35]. While this is viable formany flames, in flow fields with high levels of turbulence intensity, the Kolmogorovscale can become comparable or even finer than the fuel consumption layer; it caneven penetrate the inner layer of the flame [115]. Thus, the range of validity oflaminar flamelet models is limited – some argue that it is limited to sufficientlylarge turbulence scales with low intensities [12] while others [114] argue that therange of validity is wider than this, however, it is undisputed that there is a limitto the range of validity. The model still appears to provide stable solutions evenoutside of its range of validity, which presents the danger that uninformed usersmay use it outside of its range of validity, with potentially misleading results.2.5.2 The Probability Density Function (PDF) Transport ModelThe PDF transport approach was firstly introduced by Lundgren [92] who derived,modelled, and solved a transport equation for the velocity joint PDF in a chemi-cally inert turbulent flow field. Due to the simplicity of the proposed models forthe unknown terms, Lundgren was able to solve the set of equations via analyticalmethods. Dopazo and O’Brien [32–34] and Pope [119] developed the compositionPDF transport equation which was subsequently employed in a variety of turbu-lent combustion simulations [47, 69, 70, 105, 120]. As in the general case, toomany reactive scalars may need to be tracked, the standard approaches such as fi-17nite difference, finite element, or finite volume fail to efficiently be employed forthe solution of the PDF transport equation. Nevertheless, Monte-Carlo methods[27, 50, 124] are best suited for simulations with high dimensionality. In Monte-Carlo methods, the computational cost rises only linearly with the number of di-mensions involved in the problem. Therefore, in the shadow of the Monte-Carloapproach, the velocity-composition joint PDF was solved for a thermal wake [1],and premixed [2, 86] and diffusion flames [67, 107].In the PDF transport filter-chemistry model, a transport equation is solved forjoint PDF of the hydrodynamic and thermochemical state of the reacting flow. Thecalculated PDF gives one-point, one-time statistics for reactive scalars with per-fect closure the chemical source-term in the filtered transport equation. Althoughthis model addresses very general statistical properties of the flow field for pre-mixed, non-premixed, and partially premixed flames, its high dimensionality andunclosed mixing terms [39, 127, 147] are major obstacles for employing the modelin industrial cases. Monte-Carlo methods can be helpful for reducing the compu-tational cost. Nevertheless, this model is still computationally expensive even aftersimplifications.2.5.3 The Conditional Moment Closure (CMC) ModelThe key idea of the CMC approach is that fluctuations in reactive scalars arecorrelated with fluctuations in the value of a few specific scalars which are thenused as conditioning variables (e.g. mixture fraction) [80]. Fluctuations aroundthe averaged scalars conditioned on these conditioning variables are small, andconditionally-filtered source-terms can be closed. By solving the conditionally-averaged species and energy transport equations, and modeling the PDF for con-ditioning variables, the filtered chemical source-terms can be calculated. Addingnew conditioning variables to the equations increases the dimensionality of theproblem. Furthermore, it is necessary to model diffusion; while the common ap-proaches to modeling diffusion (such as the modified Curl mixing model [68])are suitable for simple Fickian diffusion, they are unable to cope with full, multi-component diffusion, pressure diffusion and thermophoresis phenomena that canbecome significant at the high pressures and temperatures seen in many real com-18bustion chambers. In addition, a first-order approximation for the filtered condi-tional source-terms using only one conditioning variable is not suitable in regionswith extinction/re-ignition of the flame [134]. In order to address the latter prob-lem, second-order closure [80] and multi-variable CMC [8, 10, 17, 19, 84] methodshave been tried, however, none of these approaches has been found to fully resolvethe problem.2.5.4 The Multiple Mapping Conditioning (MMC) ModelThe MMC approach [22, 81] was introduced to combine the advantages from theCMC and PDF transport paradigms, for which both stochastic and deterministicversions were introduced. MMC closes the joint PDF of the conditioning variablesand the conditional scalar dissipation terms, and has been successfully applied tosimulate swirl-stabilized [60] and stratified [42] turbulent flames. By reducing thenumber of the reference variables, this model lowers the dimensions of the jointPDF for which a transport equation is required to be solved. Even though thisfeature makes MMC a relatively cheaper model compared to the PDF transport ap-proach, it is still considered among the relatively high-cost filter-chemistry closuresolutions, particularly for flames in the flamelet regime.2.5.5 The Conditional Source-term Estimation (CSE) ModelIn order to avoid solving conditionally-averaged transport equations, the CSE modelwas introduced [18, 145] in which the conditional averages are calculated from theinversion of integral equations. In a recent work comparing CSE with a CMCmodel for a particular problem, it was shown that CSE provides a more accuratesolution while also being less time-consuming [87]. Yet, in CSE, errors inevitablyresult from inversion of the integral equations, and it is not straightforward to guar-antee conservation of elements. This problem has limited CSE applications tothe chemistry reduction strategies that intrinsically guarantee element conserva-tion [53].192.5.6 The Uniform Conditional State ModelHendra and Bushe recently introduced the Uniform Conditional State (UCS) modelas another off-shoot of CMC [53]. The idea in this approach comes from the as-sumption that with a sufficient number of conditioning variables, spatial gradientsof the conditional averages are sufficiently small as to be negligible. This waspreviously verified in the context of CSE by post-processing experimental datafrom the Sandia/TUD and Sydney swirl burners [16]. By neglecting the spatialgradients, transport equations in the conditional manifold are decoupled from thephysical domain, and a first-order closure for the filtered conditional source-termis sufficient. The filtered chemical rate can be closed through:˜˙ωi(~x, t) = ∫ςω˙i(〈T |~γ〉,〈~Y |~γ〉,〈ρ|~γ〉)P˜(~γ;~x, t)d~γ, (2.37)where ω˙ denotes chemical source term, ˜˙ωi(~x, t) is the Favre-averaged chemicalsource-term in a specific location and time for ith reactive scalar, 〈α|β 〉 is the con-ditional Favre-average of α with conditioning variable β ; ~Y is the species massfraction vector; ~γ denotes the vector of non-random conditioning variables, and P˜is the filtered one-point, one-time joint probability density function for the condi-tioning variables. The governing PDE for conditional averages in the conditionalspace is represented by Equation 2.38 which is surprisingly identical to the equa-tion obtained in the context of the Multidimensional Flamelet Manifold (MFM)[104] where the same terms appear in the transport equation for the reactive scalarsin the reduced-order composition space. Furthermore, in both approaches, it is pos-sible to use as many variables as desirable to define the low-dimensional chemistrymanifold. However, the two models differ in that UCS deals with conditionally-filtered scalars rather than coordinate-transformed variables. More detailed discus-sions on the similarities between these two models are presented in [53]. The UCS20equation is:ρ|~γ ∂ 〈Yi|~γ〉∂ t+nc∑j=1ρ|~γ 〈ω˙c j |~γ〉∂ 〈Yi|~γ〉∂γ j= ρ|~γ〈ω˙i|~γ〉−nc∑l,m=1[∂Jim∂γl−nc∑j=1∂Jc jm∂γl∂ 〈Yi|~γ〉∂γ j]·~∇cl ·~∇cm|~γ−nc∑m=1[Jim−nc∑j=1Jc jm∂ 〈Yi|~γ〉∂γ j].∇2cm|~γ,(2.38)where α|β represents the average value of α conditioned on β , c j denotes jthrandom conditioning variable, Jαβ is the conditional diffusive flux associated withreactive scalar of α and conditioning variable β ; ~∇cl.~∇cm|~γ and ∇2cm|~γ are re-lated to the conditional scalar dissipation rate – unclosed terms which should bemodeled. In addition to solving Equation 2.38, the joint probability distribution ofthe conditioning variables is required to be modeled at every spatial point and timestep in order to determine the filtered chemical source-term through Equation 2.37.In actual implementations of the UCS model, the current practice is to assumethat the chemistry reaches a pseudo-steady-state with respect to the scalar dissi-pation rates, such that one can solve Equation 2.38 in advance, storing the resultas a manifold representation of the chemistry, exactly as one does in MFM ap-proaches. As such, the approaches are equivalent; and in either one, the effect is toassume that the chemistry can be described by a low-dimensional manifold that isa function only of a small number of conditioning variables.2.6 Different Presumed PDF Modelling MethodsThe presumed PDF method has been vastly employed in the context of flameletand CMC paradigms (including CSE and UCS as branches of CMC) to address thechemical source-term closure problem. As previously described, in such methods,the attempt is to describe the reactive scalars in terms of some passive parame-ters – named as conditioning variables in the CMC-based methods – and model aone-point, one-time joint PDF for the passive scalars according to their statisticalmoments. In this section, various approaches for modeling the joint PDF of the21conditioning variables are surveyed, and works where these methods have beenutilized are reviewed.Mixture fraction and progress variable are two conditioning variables that havefrequently been used in turbulent combustion simulations. The general approachfor modelling the corresponding joint PDF is to assume that these variables arestatistically independent. Therefore, by modeling the marginal PDFs for each con-ditioning variable, the joint PDF can be obtained by:P˜(ζ ,η) = P˜(ζ )P˜(η), (2.39)where ζ and η are representing two components of γi as the non-random field formixture fraction and reaction progress variable, respectively.In the context of CSE, Bushe [16] showed that adding total enthalpy as a thirdconditioning variable significantly improves the predictions of reactive scalars forthe Sydney swirl burner wherein there is significant heat transfer to and from thebluff body behind which the flame is stabilized. This finding is valid in both theUCS and MFM contexts [100]. The statistical properties of this parameter areimportant for modeling its distribution. Total enthalpy is usually assumed to beindependent of mixture fraction and reaction progress variable, and its PDF hasbeen taken to be in the form of a Dirac-delta centered on its mean value [96, 136],thus, only the first statistical moment of this variable is utilized for constructing themarginal PDF.A bimodal behaviour for the distribution of scalar mixing was previously re-ported [43, 143]. For modeling the PDF of this variable, a Gaussian distribution[91] and the Dirac-delta function [77] are suggested which have been proven to notwork well except in the final stages and very early phases of the mixing process,respectively. Instead, a well-established marginal PDF form for mixture fraction isthe β -function which necessitates information about the first two moments of thisvariable [46, 66, 88, 132, 158]:P(ζ ) =ζα−1(1−ζ )β−1Γ(α+β )Γ(α)Γ(β ), (2.40)22where Γ is the gamma function and α and β are β -PDF shape parameters thatare dependent on the first two statistical moments of mixture fraction. Z˜ and Z˜′′2respectively representing the mean and variance of the mixture fraction, the param-eters of the β -PDF can be obtained throughα = Z˜[Z˜(1− Z˜)Z˜′′2−1];β = (1− Z˜)[Z˜(1− Z˜)Z˜′′2−1]. (2.41)The Favre-averaged transport equations for the mean and variance of mixture frac-tion are required to model the corresponding PDF [118, 146, 149]:∂ρZ˜∂ t+∂ (ρ u˜iZ˜)∂xi=∂∂xi(ρD∂Z∂xi−ρu′′i Z′′), (2.42)∂ρZ˜′′2∂ t+∂ (ρ u˜iZ˜′′2)∂xi=∂∂xi(ρD∂Z′′2∂xi−ρu′′i Z′′2)−2ρu′′i Z′′∂ Z˜∂xi−2ρD(∂Z′′∂xi)2, (2.43)where a simple Fickian diffusion with a constant molecular diffusivity (D) is as-sumed for the whole field.The β -function has also been employed in the context of Presumed ConditionalMoments-Flame Prolongation of Intrinsic low-dimensional manifold (PCM-FPI)for modeling the marginal PDF of reaction progress variable (c) [38, 55]. Thispresumed function has been utilized for the simulation of premixed and partiallypremixed flames by other researchers [13, 31, 106, 111, 157]. The transport equa-tions for the mean and variance of progress variable are [118, 149]:∂ρ c˜∂ t+∂ (ρ u˜ic˜)∂xi=∂∂xi(ρD∂c∂xi−ρu′′i c′′)+ ω˙c, (2.44)23∂ρ c˜′′2∂ t+∂ (ρ u˜ic˜′′2)∂xi=∂∂xi(ρD∂c′′2∂xi−ρu′′i c′′2)−2ρu′′i c′′∂ c˜∂xi−2ρD(∂c′′∂xi)2+2c′′ω˙c. (2.45)Jin et al. [71] found that using the β -function as a model for the PDF of reactionprogress variable leads to significant errors in predictions of mean reaction ratesin CSE when applied to premixed turbulent combustion. This motivated develop-ment of alternatives. One of these is the laminar flame PDF, originally proposedby Bray et al. [14] and then modified by Jin et al. [71] to expand its region of ap-plicability to cover all possible values of mean and variance of progress variable.This was found [87, 138] to be a significant improvement on the β -function inCSE calculations of turbulent premixed flames, but there was a concern expressedthat, because the function is based on a laminar flame, it may only be valid in theflamelet regime. This motivated development of the LEM-PDF, wherein a seriesof Linear Eddy Model (LEM) [97] calculations are used as the base function forthe PDF and these are spatially filtered to match mean and variance, with the fil-tered solution tabulated [154]. Tsui et al. [155] found that this LEM-PDF matchedthe experimentally measured one, even when swirl was added to the turbulent flow(LEM, being a one-dimensional model, cannot account for swirl).Barlow et al. [5] studied the joint statistical properties of mixture fraction andreaction progress variable – as the most commonly used conditioning variables –in a pilot-stabilized recessed burner. Their results revealed that there exist somecorrelations between the conditioning variables especially in regions with near-homogenous mixtures. However, compositional inhomogeneity and local flameextinctions deteriorate this correlation [26]. Tian and Lindstedt [153] applied ajoint composition-enthalpy transported PDF/Finite Volume method and came tothe same conclusions about the correlation characteristics between these two con-ditioning variables. Furthermore, Kamal et al. [74] made use of the measurementdata from the Cambridge-Sandia swirl burner to investigate the statistical proper-ties for local equivalence ratio – which is representative of mixture fraction – andprogress of reaction, and observed a correlation between these two variables. Groutet al. [49] used direct numerical simulation of a stratified premixed flame to study24the characteristics of a flame in the wrinkled flamelet regime. They reported aweak correlation between the two conditioning variables which can be reasonablyignored where only the endpoints of the reaction progress variable PDF are used.Nonetheless, if the interior part of the PDF is considered for modeling, the effectsof correlation may be significant. In their simulations, the assumption of statisticalindependence resulted in an overestimated probability of being within the flameinterior for the richest regions while it was underestimated for the regions farthestfrom stoichiometric conditions.One can conclude that the statistical dependence of the two conditioning vari-ables plays a key role in the simulations of mixed mode combustion. Libby andWilliams [89] introduced a two-delta function for the joint PDF of mixture frac-tion and progress variable. This idea was further developed by Ribert et al. [131]who assumed a positive covariance between mixture fraction and progress variable.Their method implied that the covariance between the conditioning variables is pro-portional to the square root of the product of their variances. Thus, in their method,the transport equation for the covariance did not need to be solved. Nevertheless,the aforementioned assumption stems from the presumed double-delta PDF formwhich is ad hoc and inconsistent with the physics of combustion. Hence, the trans-port equation for the covariance (Z˜′′c′′) likely needs to be solved [118, 146, 149]:∂ρZ˜′′c′′∂ t+∂ (ρ u˜iZ˜′′c′′)∂xi=∂∂xi(ρD∂Z′′c′′∂xi−ρui′′Z′′c′′)−ρui′′c′′∂ Z˜∂xi−ρui′′Z′′ ∂ c˜∂xi −2ρD(∂c′′∂xi)(∂Z′′∂xi)+2Z′′ω˙c. (2.46)Pope [122, 123] proposed the Statistically Most Likely Distribution (SMLD)method for modeling the PDF of conditioning variables. This method gives a struc-tured platform for using the correlation factor between the conditioning variablesand any number of their statistical moments to build the PDF, and does not em-ploy any presumed shape for the distribution. In this approach, one maximizes theuncertainty of the outcome to find the most probable shape of the PDF. Ihme andPitsch [62] employed this approach for modeling the conditional PDF of reactionprogress variable conditioned on mixture fraction. They showed that this approach25does not work better than a presumed β -PDF unless at least four statistical mo-ments are used for modeling the most likely PDF. This is because extra momentsare inherently imposed by the presumed functions. The need for higher statisticalmoments of the conditioning variables and the complexity of the SMLD approachincrease the computational cost of the simulations.Darbyshire and Swaminathan [29] made use of the Plackett family of copulas[116] to construct a joint PDF of mixture fraction and reaction progress variablethrough taking their covariance into account. Coexistence of premixed and non-premixed combustion modes in the leading edge of lifted flames gives rise to thepresence of triple flames which challenge the validity of the statistical indepen-dence of Z and c. Therefore, having a PDF model based on the correlation betweenthese two conditioning variables is crucial for predicting the flame lift-off height.Ruan et al. [135] and Chen et al. [21] employed the copula concept for model-ing the PDF of mixture fraction and reaction progress variable to simulate liftedturbulent jet flames in the context of the laminar flamelet model, and reported im-provements in the accuracy of the modeled PDF compared to the results obtainedby ignoring the correlations.26Chapter 3The Effect of Heat Transfer onthe Low-Order ManifoldRepresentation of ChemistryIn this chapter, the validity of the assumption that a low-dimensional manifold caneffectively represent the chemistry in a turbulent flame is investigated by looking tothe measurement data from the Cambridge-Sandia burner which uses methane asthe fuel and air as the oxidizer working at various swirl numbers and stratificationlevels. First, we start with introducing the properties and working conditions of theburner and the methodology employed for predicting the mean value of reactivescalars using the manifold. Then, the results for different approaches applied tocalculate the manifold are compared, and finally, this chapter is summarized inSection 3.4.3.1 MethodologyThe crucial simplification necessary to assume a conditional manifold for the chem-istry of a turbulent flame is the assumption of negligible spatial gradients for theconditional averages. In the present chapter, this assumption is examined viaprocessing measurements from the Cambridge-Sandia burner [36, 150, 151, 161]working in nine different conditions. These conditions differ in the imposed swirl27and level of stratification, as tabulated in Table 3.1. The other purpose of this chap-ter is to investigate the impact of swirl and stratification on the conditional averagesas is addressed in the following sections. The high-quality measurements made fordifferent thermo-hydrodynamical properties of the flow and a wide range of swirland stratification indices under which the experimental data are collected made thisburner a proper candidate for studying the behaviour of the conditionally-averagedreactive scalars. We use the case numbers assigned in the papers originally describ-ing the data [36, 150, 151, 161].As illustrated in Figure 3.1 which represents a schematic of the selected burner,there are a large co-flow of pure filtered air which was intended to prevent ambientair from entering the reaction zone, and two concentric outer and inner annuli forsupplying fuel/air mixtures. The stratification factor is defined as the ratio of equiv-alence ratio in the inner annulus to that for the flow passing through the outer one.The inner annulus is fed with the mixture of reactants which has passed throughplenum A where structure of the flow made in supply lines is broken down beforeentering the burner. Besides, plenum B and C are arranged to feed the outer annu-lus with the fuel/air mixture. The flow which gets into plenum C passes through theswirler before entering the outer annulus; while there is no swirling force inducedto the flow which enters plenum B. The swirl flow ratio is the ratio of mass flowpassing through plenum C to that passing through plenum B [150].Table 3.1. Working conditions for Cambridge-Sandia stratified swirl burner.φi and φo denote equivalence ratio of the flow in the inner and outerannuli, respectively.Case* Swirl Factor Ratio Stratification Factor φi φo1 0.00 1 0.75 0.752 0.25 1 0.75 0.753 0.33 1 0.75 0.755 0.00 2 1.00 0.506 0.25 2 1.00 0.507 0.33 2 1.00 0.509 0.00 3 1.125 0.37510 0.25 3 1.125 0.37511 0.33 3 1.125 0.37528In all cases, the measurements were made in different axial and radial positionswith at least 300 samples per location via Rayleigh scattering employed for captur-ing temperature, Raman scattering and CO-LIF for measuring the mole fractions –and hence mass fractions – of CO2, CO, CH4, H2, H2O, O2, and simultaneous OH-PLIF for obtaining the flame normal solid angle and spatial gradients of the reactivescalars. Extra shots were taken in the vicinity of the oscillating flame brush wherethe measured quantities showed strong fluctuations around their mean values. Theavailable measurements allow us to calculate the conditionally-averaged reactivescalars and joint PDF of the conditioning variables in different locations in orderto evaluate the unconditional averages of reactive scalars using the manifold ap-proach.Figure 3.1. Schematic of the Cambridge-Sandia Burner.In actual simulations using the UCS or MFM filter-chemistry models, the con-ditional chemical source-terms are required to be extracted from the conditionalmanifolds – by solving Equation 2.38 in the conditional space – and be combinedin Equation 2.37 for calculating the unconditionally averaged chemical source-terms. Next, the obtained averaged source-terms are used to integrate the filteredspecies and energy transport equations in time to compute the averaged reactivescalars at each point. In this study, as the information for the chemical source-29terms cannot readily by calculated from the measurements, the averaged values forthe reactive scalars are obtained via Equation 3.1. The results from this approachshow the dependence of the conditional averages on the spatial coordinate, swirlratio, and stratification level which address the main goal of this chapter.Y k,Est(~x, t) =∫ς〈Yk|~γ〉P(~γ;~x, t)d~γ (3.1)The simplest approach would consider only one conditioning variable for con-structing the conditional manifolds. Figure 3.2 shows how predicting the carbonmonoxide mass fraction in the radial position of 18.77 mm and the axial distance of30 mm from the burner tip in flame number 7 would work when reaction progressvariable is utilized as the only conditioning variable. Here the conditional averagesare calculated from the measurements from all nine flames, and 30 isometric binsare used for discretizing the one-dimensional conditional space. The predicted COmass fraction from this approach is 0.0149, while directly averaging the CO massfraction from all of the shots taken from this point suggest this averaged mass frac-tion to be 0.2512. Since stratification is present in the flow field, one could expectvery poor predictions from this one-condition approach. This is the motivation forwanting to consider a higher-dimensional manifold: merely including mixture frac-tion as a second conditioning variable improves the prediction considerably, as willbe shown shortly. In the subsequent sections, two-condition and three-conditionversions of the low-dimensional manifold are applied for calculating the averagesof reactive scalars. In the former, mixture fraction and progress variable are se-lected as the conditioning variables to construct two-dimensional conditional man-ifolds. In such cases, the random conditioning field takes the form of {ci}= {Z,c}.Unconditionally-averaged reactive scalars are calculated by Equation 3.2:Y k,Est(~x, t) =∫ 10∫ 10〈Yk|ζ = Z,η = c〉P(Z,c;~x, t)dZ dc, (3.2)30Figure 3.2. Using reaction progress variable as the only conditioningvariable for predicting CO mass fraction in a sample point in flamenumber 7.In the three-condition version, normalized total enthalpy is added as a third condi-tioning variable, and the conditional space becomes three-dimensional. Then, theunconditionally-averaged scalars are obtained throughY k,Est(~x, t) =∫ 10∫ 10∫ 10〈Yk|ζ = Z,η = c,δ = h〉P(Z,c,θ ;~x, t)dZ dcdh, (3.3)where δ is the sample-space for the dimensionless total enthalpy (h).In order to assemble the conditional domain, the instantaneous single-pointmeasurement data are utilized to calculate one-point, one-time values of mixturefraction, progress variable and normalized total enthalpy. Bilger’s definition forcalculating the mixture fraction from mass fractions of the probed species is used.The mixture fractions of zero and unity are assigned to pure air and the richestentry flow through all of the cases with equivalence ratio of 1.125, respectively.For the progress variable, it is plausible to normalize the measured CO2 massfractions with the maximum attainable value in each point [16], however, higherreliability of the temperature measurements compared to those for the species massfractions encouraged Sweeney et al. [150] to employ a thermal progress variable.31Even though employing a temperature-based definition for calculating reactionprogress variable in numerical simulations compels solving a transport equationfor temperature, which is more difficult than solving for species mass fractions,here it is more beneficial to utilize a thermal progress variable. The most importantmotivation is associated with much clearer measurements for the temperature field[150]. On the other hand, using only one species mass fraction may fail to fullyrepresent the progress of the chemical reactions for more complex fuels. There-fore, more scalars are required for tracking the reaction progress variable in theflow field [20, 139, 159], which results in uncertainty propagation for calculatingprogress variable from the experimental data in the present study. Furthermore, asone of the conditioning variables becomes a function of more scalars, the diffusionterm appearing in the transport equation of the conditioning variable becomes morecomplicated to be modeled in numerical simulations. On the other hand, as shownby Kamal et al. [75], the oxygen- and (CO+CO2+H2)-based reaction progress vari-ables are both homeomorphic to the temperature-based definition of this variablefor combustion of methane. So, one can conclude that there exists a bijective func-tion which maps manifolds generated from either of these definitions to the otherone. Moreover, Kronenburg [84] employed sensible enthalpy as a temperature-based conditioning variable for double-condition CMC simulation of nonpremixedturbulent flames which demonstrates the feasibility of employing a temperature-based progress variable for numerical simulations. In the present study, the localflame temperature is normalized by the temperature at the equilibrium condition,and the progress variable is calculated by Equation 3.4:c =T (~x)−TuTe(φ)−Tu , (3.4)Here T (~x) denotes local measured temperature, Tu is temperature of the unburntgas (the entry gas flow); Te denotes temperature of products at the equilibrium con-dition which is a function of local equivalence ratio, and varies from point to pointfor stratified flames. The values of this conditioning variable were provided by theexperimental researchers.Total enthalpy accounts for the history of heat transfer that the fluid experiencesprior to reaching the measuring point. The hot products of combustion release heat32to the surroundings through radiation and convection, and the available time forthat heat transfer varies considerably (i.e. fluid caught in the recirculation zonebehind the bluff-body experiences more heat transfer than other fluid will, for ex-ample). Meanwhile, unburnt reactants absorb some of this heat when passing nextto the hot components of the burner before taking part in the reactions. These phe-nomena make a distribution for the total enthalpy of the fluid. This conditioningvariable extracts the effects of the chemical reactions from the heat transfer mecha-nisms on the temperature-based progress variable. Again, as previously discussed,since the species mass fraction-based progress variables are homeomorphic to thethermal progress variable for the combustion of methane, we expect that employ-ing normalized total enthalpy as a new conditioning variable would have similarimpact when other definitions are utilized for calculating reaction progress vari-able. Non-adiabatic manifolds were previously employed in the context of laminarflamelet [25, 93] and CMC [137] models. In the present work, knowing the temper-ature and mass fraction of the major species at each measuring point, Cantera [48]is employed to calculate the spontaneous one-point specific total enthalpy for eachmeasurement. Next, this parameter is normalized throughh =htot −htot,minhtot,max−htot,min , (3.5)where htot is the local specific total enthalpy; htot,min and htot,max are the minimumand maximum specific total enthalpies calculated in the flow field, respectively.In three-condition cases, the conditional manifold takes the form of a cube withunity edge length and three axes of mixture fraction, thermal progress variable, andnormalized total enthalpy; 50 discrete intervals (or ”bins”) in the direction of mix-ture fraction, 30 bins for reaction progress variable, and 50 bins for the dimension-less total enthalpy are used to discretize the conditional manifold, so, 75000 “cells”are made in the conditional domain. The conditional averages for temperature andspecies mass fractions are obtained by assigning every data point to one of theformed cells and ensemble averaging the value of reactive scalars in each of these.Conditional averaging over cells with insufficient number of data points imposesunrealistically small fluctuations around the mean value which results in underpre-diction of the model discrepancies, therefore, cells with less than 10 data points33are ignored in the computations. The joint PDF for each point of the flow fieldis calculated through counting the number of measurements which fall in each ofthe cells in the conditional domain, and then, normalizing the obtained numbers bytotal count of measurements found at the corresponding point. Thus, the obtainedjoint PDF is in a discrete format as described in [125]. Conditional averages andjoint PDFs are used to calculate the unconditionally-averaged reactive scalars viaEquation 3.3. The unconditional averages are also obtained by directly ensembleaveraging the value of reactive scalars in each location from as many measure-ments as available for that point. Finally, the results from the manifold approachare compared to those from direct averaging. In this work, the discrepancies in thetemperature field are reported in kelvin, while, in order to be able to compare thefunctionality of the model for predicting the mass fraction of various species, thediscrepancies in the mass fractions are normalized by the largest detected value foreach of the averaged values throughout the domain:Y ′k =Y k,Est −Y k,DirectY k,max, (3.6)where Y ′k is the normalized discrepancy in Yk; Y k,Est and Y k,Direct are the estimatedunconditionally-averaged mass fractions calculated from the manifold and directlyaveraging, respectively, and Y k,max is the largest averaged value for kth scalar mea-sured in the corresponding flame.3.2 ResultsTo demonstrate the spatial invariance of the conditional averages, in each of thecases they are calculated from all of the points in the physical domain regard-less of their coordinate. In the calculations, no simplification is considered forthe joint PDF of the conditioning variables, so, the discrepancies in the value ofunconditionally-averaged scalars represent the errors attributable to either the cal-culation of conditional averages or uncertainties in the measurement techniques.Obviously, if the conditional averages were computed only from the data at thespecific location in the space, measurement errors would be the only source for thedeviations.34Various approaches for obtaining the conditionally-averaged reactive scalarsare summarized in Table 3.2. In the first practice which is denoted as 2DAll,measurements from all 9 flames are piled up to form a general two-dimensionalconditional domain for each of the scalars. In this case, it is assumed that the con-ditional averages are independent of not only the spatial coordinate but also theswirl flow ratio and stratification factor; while in the 2Dn method, the conditionalaverages are calculated from the measurements made for the specific case withcertain swirl and stratification indexes for each of the flames. Next, in the 3Dncase, the measurement data for each of the flames are used to build up 9 indepen-dent three-dimensional conditional manifolds per reactive scalar. It is expected thatthis strategy will give the most accurate results for each of the flames because, inthat, heat transfer is also taken into account for assembling the conditional domain,moreover, no assumption is made for swirl and stratification invariance of the con-ditional averages. Subsequently, in the 3DAll approach, data from all cases aregathered to employ a three-condition version of the manifold for calculating theaveraged reactive scalars. A comparison of the obtained discrepancies with thosecomputed from the two-condition version represents the significance of adding nor-malized total enthalpy as the third conditioning variable. In Subsection 3.2.4, theeffect of the stratification on the conditional field is explored. As appeared in Ta-ble 3.1, for each single swirl ratio, there are three flames with different stratificationfactors. By using the measurements made for various stratification factors but witha single particular swirl ratio for building the three-dimensional conditional do-main – the case named as 3DSwirl – the discrepancies show the dependence ofconditional averages on the stratification factor. Furthermore, in order to observethe impact of swirl on the conditional averages, for each specific stratification fac-tor, flames with different swirl flow ratios are used for computing the conditionally-averaged scalars (the 3DStratification case).35Table 3.2. Different approaches for computing the conditional averagesMethod ConditioningVariables Comment2DAll Z, cAll 9 cases in one global two-dimensionalconditional manifold per scalar2Dn Z, c9 independent two-dimensionalconditional manifolds per scalar3Dn Z, c, h9 independent three-dimensionalconditional manifolds per scalar3DAll Z, c, hAll 9 cases in one global three-dimensionalconditional manifold per scalar3DSwirl Z, c, h3 independent three-dimensionalconditional manifolds per scalar for caseswith common swirl flow ratio3DStratification Z, c, h3 independent three-dimensionalconditional manifolds per scalar for caseswith common stratification factor3.2.1 Effects of the Swirl and Stratification on the Two-DimensionalConditional ManifoldFirstly, in a two-dimensional manifold, the conditional averages obtained for tem-perature and the mass fractions of CH4, CO2, CO, H2, and H2O are computedthrough the data collected from all 9 cases. This is test 2DAll in Table 3.2. Thecontours of conditionally-averaged scalars in the conditional space are illustratedin Figure 3.3. It can be seen that despite gathering the data from various radialand axial positions in the flames with different swirl ratios and stratification fac-tors, there are still two regions in the conditional domain in which no measurementhas been found. The presence of the top-left region suggests that it is improbableto have a relatively complete reaction in the mixtures with local equivalence ra-tios well below the lower flammability limit of methane. On the other hand, thebottom-right empty region shows that if the mixture fraction lies within the lim-its of flammability, the unburnt reactants become unstable and start to burn, so,methane is consumed and progress variable rises. Therefore, CH4 mass fractionpeaks near the lower flammability limit of the methane-air mixture. The contours36for conditionally-averaged temperature, water mass fraction, and carbon dioxidemass fraction – which represent the progress of the chemical reactions – generallyhave a similar appearance. Their maximum values are obtained in the vicinity ofthe stoichiometric mixture fraction and progress variable of 1. Carbon monoxideand hydrogen which are intermediate species in the reaction chain also have similaroverall behavior in the conditional domain.In order to specify the error that arises by gathering the data from differentswirl ratios and stratification factors, for each of the flames, the two-condition con-ditional averages are also computed from the measurements collected from eachspecific case. This is test 2Dn in Table 3.2. As seen in Figure 2.2 which shows theresults from case 11 as a representative sample, the conditional manifolds sharerelatively similar basic characteristics for the reactive scalars in each flame, how-ever, insufficiency of the data for obtaining the conditional averages results in moreempty cells in the conditional domain. It is worth noting that for the fully premixedflames, the conditional domain effectively collapses to one dimension, since thereis vanishingly little variation in Z in such cases.Having the conditional averages calculated from both approaches, the discrep-ancies in the unconditionally-averaged reactive scalars are calculated in every sin-gle location as discussed in Section 3.1. The results for the reactive scalars arepresented in Figures A.1 to A.6 for all of the 9 flames. Differences between thediscrepancies computed from both methods demonstrate that neglecting the effectsof swirl and stratification on the conditional manifold in the two-dimensional ver-sion imposes relatively large errors in predicting the averaged reactive scalars. Atthis stage, as the differences between the results of the two methods are not uniformfor all of the cases, it is not straightforward to evaluate the impact of consideringswirl and stratification on the general functionality of the model. Thus, a normal-ized accumulated discrepancy is introduced in Section 3.3 as a metric for each ofthe cases so one can see the overall effect of each assumption that is made for con-structing the conditional manifold.The crucial criterion for having a practical UCS model is that it should pro-duce small discrepancies with no functional dependence on the location of thepoints. In most of the cases, however, the obtained discrepancies represent func-tional changes along different radial positions, which is most clear in the results of37CO mass fraction. The amplitude of the functional dependence is generally largerfor cases calculated from the 2DAll approach compared to those from 2Dn, and itcan be inferred that the swirl flow ratio and stratification factor are affecting thetwo-dimensional conditional averages. Nonetheless, for the worst predictions, theamplitude of the normalized discrepancies is about 10%, so, even the first versionof the UCS model is working reasonably well.The results for the temperature field show that, even though the definition ofprogress variable is based on the temperature field itself, the discrepancies arestrongly affected by the method used for computing the conditional averages. Thisis due to the relatively coarse grid used for discretizing the reaction progress di-mension. When using higher resolution in this direction, the discrepancies fortemperature decline significantly and become rather independent of the UCS ver-sion utilized for conditional averaging. This occurs because the changes in temper-ature field are linearly correlated with those of reaction progress variable, which istracked directly as a conditioning variable. One can expect that if carbon dioxideor water mass fraction were used for calculating the chemical progress variable,the same phenomenon would occur for those reactive scalars, as again, their fluc-tuations would become perfectly aligned with those of the progress variable. Nev-ertheless, in such cases, the discrepancies for other scalars would rise due to thelarger uncertainties in the value of these scalars which are employed for calculatingthe reaction progress variable.38Figure 3.3. Two-condition conditionally averaged reactive scalars from allcases; the dashed line presents the stoichiometric mixture fraction, andthe dash-dot line shows the lower flammability limit of methane-airmixture.39Figure 3.4. Two-condition conditionally averaged reactive scalars from case11; the dashed line presents the stoichiometric mixture fraction, and thedash-dot line shows the lower flammability limit of methane-airmixture.403.2.2 Three-Condition Conditional Averages: Employing the DataCollected from Each Specific FlameMeasurements from each one of the flames are used to construct 9 three-dimensionalconditional manifolds for the selected reactive scalars (temperature and mass frac-tions of CO2, CO, CH4, H2, and H2O), and each conditional manifold is specifiedfor the corresponding case with particular swirl and stratification. These corre-spond to test 3Dn. In this approach, the conditional averages are computed fromdata which all have the same swirl flow ratio and stratification factor. Meanwhile,the normalized total enthalpy is utilized as the third conditioning variable alongwith mixture fraction and thermal progress variable to track the history of the fluidparticles’ heat transfer. The discrepancies that result from this approach are shownnext to the errors computed from the two-condition version with data collectedfrom the 2DAll method in Figures A.7 to A.12. The results show that the discrep-ancies for the three-condition manifold are generally smaller than the deviationsobtained from the two-condition one.In the results for the methane mass fraction predictions, it is worth noting thatthe discrepancies – even for the 3Dn as the most accurate conditional averagingapproach – seem to blow up at the flame edges. To further investigate the reasonbehind such large deviations, the mean mixture fraction and reaction progress vari-able values for the points at x = 20mm and located on the left hand side (from theviewpoint of the measurement window in the experiments) of the burner center-line in flame number 5 are illustrated in Figure 3.5. Here r and x are the radialand axial coordinates of the measuring points, respectively; while d is the diam-eter of the bluff-body. This flame and this particular region are representative ofthe cases in which the predictions of methane mass fraction show relatively highdiscrepancies: the same conclusions can be derived for other similar cases. Theresults show that the mean mixture fraction tends to remain slightly greater thanzero for the edge of the flow field while the mean progress variable is effectivelyzero in that region. Even though we are only looking at the first statistical mo-ment of the conditioning variables, this observation suggests that the UCS modelshould predict some methane mass fraction for a nearly zero progress variable anda non-zero mixture fraction in this part of the flow field. To seek an explanation forsuch behaviour of the conditioning variables in this particular region, the normal-41ized ensemble averaged mass fractions of CO2, CO, CH4, H2, and H2O along withthe normalized discrepancies for methane mass fraction from the 3Dn approach –presented by the triangle symbols – are shown in this figure. Here the mean speciesmass fractions are directly obtained from the experimental data and are normalizedby the maximum measured average in the corresponding flame for the particularspecies. These results show that most probably the detection of water here leadsto non-zero mixture fraction values at the edges. This behaviour can be regardedeither as a result of experimental errors or preferential diffusion of the species. Asdiscussed by Barlow et al. [4], hydrogen and water molecules tend to diffuse fasterthan other species to the reactant side in swirl or bluff-body stabilized flames, andthis effect gets more important at the base of the corresponding flames. This non-zero mixture fraction accompanied by very small progress variable values, resultsin the over-prediction of methane mass fractions in such regions.-2 -1.5 -1 -0.5 0r/d-0.4-0.200.20.40.60.811.2Normalized Reactive Scalarx = 20mm; Flame Number 5 cmean Zmean YCO2 YCO YCH4 YH2 YH2O 20 Y'CH4Figure 3.5. Normalized reactive scalars along the horizontal measurementline located 20 mm above the burner tip in flame number 5. All of thespecies mass fractions shown by the lines are the experimentalensemble averaged values normalized by the maximum average massfraction; while the triangle symbols represent the normalizeddiscrepancies for methane mass fraction as defined in Equation 3.6from the 3Dn method.It can be seen in the figures that predictions for the mass fractions of carbondioxide and carbon monoxide generally show sharper sensitivity to adding the nor-42malized total enthalpy as the third conditioning variable. Since the formation andconsumption reactions for these species are related to slow-rate chemistry, andthey depend on the temperature field, taking the history of heat transfer into ac-count for conditional averaging enhances the predictions for their mass fractions,substantially. On the other hand, the consumption rate of CH4 is more independentof temperature, so, discrepancies in the mass fraction of methane do not show asignificant dependence on the method which is employed for computing the con-ditional averages.As mentioned previously, an accurate approach for computing the conditionalaverages should not result in any bias in the values of errors. Even the discrepan-cies calculated by using the three-condition conditional averages computed for thespecific swirl ratios and stratification factors show a functional dependence on thecoordinate. Nevertheless, the amplitude of the functional dependence is generallyreduced for the new method. In spite of very small discrepancies for the approach,reliance of the calculations for the conditional averages on swirl and stratificationdeprives us of constructing a conditional manifold that is independent of the geom-etry of the combustion system. By using a filter-chemistry model that disconnectsthe conditional domain from real space, a multidimensional lookup table can beformed to represent the filtered chemical source-term as a function of statisticalmoments of the conditioning variables. This vastly decreases the computationalcost of the model. Therefore, in the upcoming parts, the impact of swirl and strati-fication on the three-condition conditional averages is investigated.3.2.3 Three-Condition Conditional Averages: Employing the DataCollected from All CasesVariations of the first two statistical moments of normalized total enthalpy at vari-ous points in flame number 11 are presented in Figure 3.6 as a representative case.Here the solid and the dashed lines show the mean and one standard deviationabove and below from the mean value of normalized enthalpy, respectively. Mean-while, the radial position of the mixing layer and the flame brush are illustrated bythe dash-dotted and dotted lines, respectively. One can observe that at small axialdistances from the bluff-body behind which a recirculation zone is formed, there isa relatively sharp change for this variable as a function of the radial coordinate. The43fluid which takes part in the chemical reactions has a higher temperature and losesenergy via different heat transfer mechanisms. On the other hand, some share ofthis energy is transferred to the unreacted fluid. Near the flame brush where thereis approximately the same probability for the fluid to be taken part in the reactionsand not being combusted, the standard deviation is larger than in other zones in theflow field. The functional dependence of normalized total enthalpy on the radialcoordinate gets smaller for further downstream positions due to diffusion. Never-theless, this variation is still considerable, and one can expect that the heat transferhistory can still have a substantial impact on the conditional manifold.Data from all flames with different swirl and stratification levels are used forconstructing a universal three-dimensional conditional manifold for each one of thereactive scalars to test the functionality of the 3DAll approach. The assumption isthat by using mixture fraction, progress variable, and normalized total enthalpy asthe conditioning variables, the effects of coordinate, swirl ratio, and stratificationfactor on the conditional averages become negligible, and one conditional manifoldper scalar can be used for all cases. The discrepancies for the reactive scalars ob-tained from 3DAll, when compared to the errors from the 2DAll case (as is shownin Figures A.13 to A.18), suggest that the errors are reasonably lowered by addingnormalized total enthalpy as a third conditioning variable.440 0.5 1 1.5 2 2.5 3 3.5 4 4.50.80.850.9x = 60 mm0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.80.850.9x = 50 mm0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.80.850.9x = 40 mm0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.80.850.9x = 30 mm0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.80.850.9x = 20 mm0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.80.850.9x = 10 mmr/dNormalized Total EnthalpyFlame Number 11Figure 3.6. Variation of normalized total enthalpy as a function of radialposition at different axial distances from the burner tip in flame 11. Thedash-dotted and the dotted lines are representing the radial position ofthe mixing layer and the flame brush, respectively.For most of the cases, the discrepancies in the temperature, carbon dioxideand water mass fraction fields which are representative for the completeness ofthe combustion reactions suggest that the 2DAll method underestimates the valuesof these reactive scalars in the inner part of the reactive flow field, in which thefluid is recirculated. Conversely, this approach overestimates these values at theedge of the flame. The flow field structure can be found in [151]. We concludefrom these comparisons that heat transfer clearly has an important impact on theconditional averages: the fluid that is trapped in the recirculation zone loses a sig-nificant fraction of energy to the burner bluff body, and this phenomenon gives riseto under-prediction of the reaction progress variable for the corresponding fluid.On the other hand, the fluid at the edge of the reacting flow field absorbs someshare of the energy which is lost by the combustion products as it passes next tothe heated solid components of the burner. This results in overestimation of the re-action progress variable in these regions, and over-prediction of the product species45mass fractions as a direct consequence. By the same logic, in most of those cases,the carbon monoxide mass fractions are predicted as to be too high in the recir-culating zones, and too low at the edge of the flames. The results show that byconsidering the normalized total enthalpy as a third conditioning variable, the dis-crepancies are effectively lowered for all of these reactive scalars.Nevertheless, there are cases in which these observations for the inner parts andthe edge of the flame can not be made, and even after considering the heat transfereffects, there are some errors in the conditional averaging approach which can berelated to some other phenomena that are not still considered (such as strain rate)and the assumption of vanishingly small gradients in the conditional averages. Thisassumption is more susceptible to inaccuracy near the flame brush where extremechanges in the reaction progress variable in such regions impose high gradients forthe progress of the reaction and hence, substantial dissipation rates for this scalar –as one can see the effect of the scalar dissipation rate on the conditional averagesin Equation 2.38.3.2.4 Effects of Swirl and Stratification on the Three-DimensionalConditional ManifoldThe Cambridge-Sandia burner is designed in a way that can produce differentranges of stratification and swirl in the reacting flow field. The diversity of thevalues for these two parameters in the stabilized flames brings the opportunity tospecifically investigate their impact on the conditional averages. At first, data fromthe cases with the same swirl flow ratio are gathered to construct three differentconditional manifolds for each reactive scalar to test 3DSwirl. The utilized datafor conditional averaging are collected from cases with different stratification fac-tors, so, the discrepancies obtained from this approach specify the effect of strat-ification on the variation of the conditional averages. The discrepancies from the3DSwirl method, when compared to the 3DAll method, (as shown in Figures A.19to A.24) share similar characteristics, and it can be inferred that unlike with thetwo-dimensional manifold, stratification does not impact the three-condition con-ditional averages considerably.Next, data from the cases with the same stratification factor but dissimilar swirlflow ratios are employed to build three different conditional manifolds per reactive46scalar for testing 3DStratification. This approach for computing the conditionalaverages identifies the dependence of the three-condition conditional averages onthe swirl flow ratio. By comparing the results obtained from the 3DStratificationand 3DAll approaches (as shown in Figures A.25 to A.30), one can deduce thatthe three-condition conditional averages are rather independent of the stratificationfactor. Hence, in the 3DAll approach, it is reasonably accurate to envisage theequations in the conditional manifold to be decoupled from the real domain.3.3 DiscussionAs mentioned earlier, since the effects of changing the method of conditional aver-aging are not uniform for various positions in the domain, it is difficult to evaluatethe accuracy of the models and draw a comparison between them. In order tocompare their general performances, a single metric is required to represent theaccumulated errors obtained for each reactive scalar. In this study, the normalizedRMS discrepancies from all of the points are added up to assign a general errorscale associated with each reactive scalar for each of the models. Equation 3.7 isadopted for defining the metric.Ek =√1npnp∑i=1Y ′k,i2 (3.7)Here np is the total number of measurement points in the corresponding flame, andY ′k,i is the normalized discrepancy for kth scalar which is described in Equation 3.6and is calculated in position i.The normalized accumulated discrepancies for temperature and mass fractionof methane, carbon dioxide, carbon monoxide, and water are illustrated in Fig-ure 3.7 for all nine flames and six conditional averaging methods. The results forhydrogen mass fraction are not shown due to high experimental uncertainties forthis scalar. As discussed previously, discrepancies in methane mass fraction donot show considerable differences by changing the model. On the other hand, car-bon dioxide and carbon monoxide mass fractions, compared to other scalars, showrelatively stronger sensitivities to adding the third conditioning variable. For in-stance, in flame number 7, the 3DAll method results in accumulated errors smaller47than one-third and one-half of the 2DAll discrepancies for carbon dioxide and car-bon monoxide, respectively. Although the 3Dn model gives the smallest errors foralmost all of the cases, the influence of separating the data on the basis of swirlflow ratio and stratification factor is not as significant as that of considering thehistory of heat transfer as the third conditioning variable. Furthermore, the 3Dnapproach binds the conditional manifold to the geometrical characteristics of thephysical space. Nevertheless, in 3DAll, conditional averages are considered to beindependent of stratification, swirl, and any geometrical property of the real do-main, thus, the chemical source-terms for various species can be obtained from apre-calculated lookup table.Normalized Accumulated Error (%)Flame Number 101234Flame Number 201234Flame Number 301234 2DAll2Dn3DAll3Dn3DSwirl3DStratificationFlame Number 501234Flame Number 601234Flame Number 701234Flame Number 9Temperature YCH4YCO2YCO YH2O01234Flame Number 10Temperature YCH4YCO2YCO YH2O01234Flame Number 11Temperature YCH4YCO2YCO YH2O01234Figure 3.7. Normalized accumulated discrepancies obtained from differentconditional averaging approaches.One can imagine using other sets of conditioning variables to separate the con-ditional manifold from the real domain. Principal Component Analysis (PCA)[110, 148] may be employed to find the set of conditioning variables which havethe highest correlations with the reactive scalars. It is expected that this approachwould result in smaller discrepancies for the unconditional averages; nonetheless,as discussed in Section 3.1, when more reactive scalars combine to define a con-48ditioning variable, the diffusion term for this variable becomes more complex andthe source-term for more scalars should combine to define the source-term associ-ated with the conditioning variable. Meanwhile, in most of the cases, the principalcomponents suggested by the PCA approach do not have a direct physical inter-pretation, and modeling their PDFs is not straightforward. Local equivalence ratio,the degree of progression of the chemical reactions, total enthalpy, and strain rateare four of the most important physical properties that affect the chemical rates in areactive flow field. For this burner, mixture fraction, reaction progress variable, andnormalized total enthalpy were utilized to track the effects of stratification, finite-rate chemistry, and the history of heat transfer on the reactive scalars, respectively.It is plausible to employ the scalar dissipation rates as additional conditioning vari-ables [19] to consider the effect of the flame straining on the conditional averages.However, the three conditioning variables used in this work can not be replacedby scalar dissipation rate since this variable accounts for a physical phenomenonwhich is different from those covered by mixture fraction, reaction progress vari-able, and total enthalpy. Meanwhile, as shown in [148], scalar dissipation fails topredict extinction/re-ignition where the history of this variable plays a key role.Here three simple scalars were utilized as the conditioning variables, and they leadto errors that are comparable to measurement accuracy, so it is not useful to addan extra variable which cannot be measured very accurately. All in all, we suggestthe scalar dissipation rate as a fourth conditioning variable which will improve theaccuracy of the model at the cost of adding another dimension to the conditionalmanifold and the joint PDF which is required to be modeled, although an assess-ment of this fourth conditioning variable is left for future works.3.4 SummarySingle-point, one-time measurements from the Cambridge-Sandia burner have beenused to evaluate the two- and three-condition Uniform Conditional State filter-chemistry model. Six different manifolds were tested to investigate the effectsof swirl and stratification on the conditional averages. Firstly, mixture fraction andreaction progress variable were used as the conditioning variables to predict theaveraged reactive scalars. The results showed that the two-condition conditional49averages are strongly dependent on the swirl and stratification levels. So, as thenext step, normalized total enthalpy was added as a third conditioning variable toaccount for the history of heat transfer in constructing the conditional manifolds.The results demonstrate that the discrepancies are reduced significantly for massfractions of carbon dioxide and carbon monoxide which are two of the most impor-tant combustion pollutants while the predictions for other reactive scalars are not asconsiderably improved. As an example, the normalized accumulated discrepancyfor carbon monoxide mass fraction in flame number 7 is reduced from 2.95% in the2DAll method to 1.42% for the 3DAll approach which translates to approximately52% improvement in the predictions. However, the prediction improvement formethane mass fraction is only about 13% for this flame.50Chapter 4The Joint PDF of MixtureFraction, Progress Variable andTotal Enthalpy in TurbulentFlamesIn this chapter, the statistical properties of mixture fraction, reaction progress vari-able, and normalized total enthalpy are investigated as conditioning variables bypost-processing the measurement data from the Cambridge-Sandia burner. Themethodology of obtaining the marginal PDFs of each conditioning variable and adiscussion of the results are presented in the next section. Next, the conditionalPDF of the normalized total enthalpy conditioned on specific values of the mix-ture fraction and progress variable are presented. Then, in order to examine theassumption of statistical independence of the conditioning variables, their correla-tion factors are calculated at various locations and in flames with different swirl andstratification indices. Next, the copula which relates the marginal PDFs of mixturefraction and reaction progress variable are extracted from the experimental data,and the performance of three well-known copulas are investigated with the aim ofmodeling the joint PDF. Subsequently, a joint PDF model based on the copula con-cept for taking the correlation between the conditioning variables into account and51a bimodal Gaussian function for the marginal PDF of the normalized total enthalpyis tested on the three-condition version of UCS. The obtained results are comparedto those from a β -β -δ PDF model in which the correlations between the condi-tioning variables are ignored, a β -function is used for modeling both the marginalPDFs of mixture fraction and reaction progress variable, and a Dirac-delta is em-ployed for modeling the marginal PDF of the normalized total enthalpy. The workis summarized in the last section.4.1 Marginal PDF of the Conditioning VariablesWhen either a copula is used or the statistical dependence of the conditioning vari-ables is neglected, the marginal distribution of each of the conditioning variablesplays a crucial role in modeling the joint PDF. As mentioned earlier, the β -functionis widely employed to model the marginal PDF of the mixture fraction and reactionprogress variable. Moreover, the variance of the normalized total enthalpy is usu-ally assumed to be negligible so a Dirac-delta function is normally used for mod-eling the marginal PDF of this variable. In this section, the high-resolution mea-surements of the Cambridge-Sandia burner are employed to obtain the marginalPDF of mixture fraction, reaction progress variable, and normalized total enthalpyat various locations in flames with different swirl flow ratios and stratification fac-tors. The database consists of low- and high-resolution sets of measurements forvarious locations in the space. The low-resolution measurements cover more loca-tions in space whereas the high-resolution data set has measurements from somelimited coordinates with higher spatial resolution and many repetitions. As thelow-resolution measurements are collected from more coordinates in space, theyshow smaller biasing so they are used here for computing the conditional averagesof the reactive scalars and conditional PDF of the normalized total enthalpy. How-ever, the high-resolution data set is employed for obtaining the local properties ofthe reactive flow, such as the marginal PDF of the conditioning variables.At each point, the mean and variance of the mixture fraction and reactionprogress variable are computed from the measurement data, and the required pa-rameters for constructing the β -PDF are calculated via Equation 2.41 for each ofthe two conditioning variables. The experimental and modeled marginal PDFs of52mixture fraction and reaction progress variable for several representative coordi-nates in flame number 11 are shown in Figures 4.1 and 4.2, respectively1.The β -function effectively captures bimodality of the PDFs of these two vari-ables, however, the shape of the experimental PDFs of mixture fraction suggeststhat, in general, two statistical moments are not sufficient to precisely model itsmarginal PDF. This inconsistency between the β -PDF and the experimental distri-bution of mixture fraction may be due to preferential diffusion [3, 4, 90, 101, 142]and bulk flow effects which violate the conservation of this conditioning variable.Meanwhile, the nearly symmetrical shape of the β -function fails to accurately cap-ture the marginal PDF of progress variable which suggests that either the modifiedlaminar flame-based approach or LEM calculations would be preferred for model-ing this PDF. The results for the progress variable show that there is a low probabil-ity of having complete reactions (c = 1); as such, the effects of heat loss [16, 36],preferential diffusion [4], and the limited reaction time [138] results in consider-ably lower gas temperatures compared to those of full chemical equilibrium.In combusting flows, depending on the nature of the fuel and the geometry ofthe domain, both radiative and convective heat transfer modes can be present. Thenormalized total enthalpy is representative of the history of this heat transfer for thefluid prior to reaching the measuring point. The hot products of combustion canlose energy to the surroundings and to the solid parts of the burner so their totalenthalpy is reduced. On the other hand, as the incoming flows of reactants pass byhot components of the burner, they can gain some heat such that their total enthalpywould be increased. Meanwhile, the gases that do not take part in the chemical re-actions are colder than the combustion products so they can absorb some energyand their total enthalpy can rise. These phenomena suggest that the distribution oftotal enthalpy is likely not entirely independent of the reaction progress variableand mixture fraction.1Here r is the radial position and x is the axial coordinate of the measuring points; while d is thediameter of the bluff body.530.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00246810121416P(Z)r/d = 1.355; x = 30mmExperimentModeled -PDF0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.001234567r/d = 1.453; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.000.511.522.5P(Z)r/d = 1.527; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.000.511.522.5r/d = 1.576; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Z00.511.522.533.5P(Z)r/d = 1.649; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Z00.511.522.533.544.55r/d = 1.723; x = 30mmFigure 4.1. The experimental and modeled marginal PDF of mixture fractionat various radial positions in flame number 11.The experimental marginal PDFs of the normalized total enthalpy are presentedfor various radial positions in flame number 11 with solid lines in Figure 4.3. ThePDFs look like the bimodal Gaussian distribution in which two modes correspond-ing to the higher and relatively lower total enthalpies exist. At the inner points ofthe flame, where the fluid with a higher probability of participating in the combus-tion reactions is present, the flow is more likely to lose energy so the mode relatedto the larger total enthalpies has a much smaller peak. In the vicinity of the oscil-lating turbulent flame brush, each measuring shot has a nearly equal probability of540.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0051015P(c)r/d = 1.355; x = 30mmExperimentModeled -PDF0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0024681012r/d = 1.453; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0012345678P(c)r/d = 1.527; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.002468101214r/d = 1.576; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0c0510152025P(c)r/d = 1.649; x = 30mm0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0c05101520253035r/d = 1.723; x = 30mmFigure 4.2. The experimental and modeled marginal PDF of reactionprogress variable at various radial positions in flame number 11.being taken from the reacted fluid and non-reacted fluid, thus, the peak amplitudesfor two modes become similar. The position of the flame brush in various axialdistances from the burner tip in all 9 flames are shown in Figure 4.4. Furthermore,at the points outside the combustion zone, most of the shots are taken from thenon-reacted flow, so the mode corresponding to the lower total enthalpies has asmaller peak which vanishes sufficiently far from the reaction zone.In order to model the distribution of the normalized total enthalpy as a condi-tioning variable with a bimodal Gaussian PDF, five parameters depending on its55first five statistical moments are required:P(h) =κσ1√2pie−12(h−µ1σ1)2+1−κσ2√2pie−12(h−µ2σ2)2, (4.1)where κ , σ1, µ1, σ2, and µ2 are related to the statistical properties of the variablethrough:h = κµ1+(1−κ)µ2, (4.2)h′′2 = κ[σ21 +(µ1−h)2]+(1−κ)[σ22 +(µ2−h)2], (4.3)h′′3 = κ[3(µ1−h)σ21 +(µ1−h)3]+(1−κ)[3(µ1−h)σ22 +(µ2−h)3], (4.4)h′′4 = κ[3σ41 +6(µ1−h)2σ21 + (µ1−h)4]+(1−κ)[3σ42 +6(µ2−h)2σ22 + (µ2−h)4] , (4.5)h′′5 = κ[15σ41(µ1−h)+10(µ1−h)3σ21 + (µ1−h)5]+(1−κ)[15σ42(µ2−h)+10(µ2−h)3σ22 + (µ2−h)5] , (4.6)where h =∫ 10 hP(h)dh and h′′i =∫ 10(h−h)i P(h)dh. This set of equations needsa numerical method to obtain the PDF parameters in terms of the statistical mo-ments of the normalized total enthalpy. Firstly, Newton’s method is employed,however, in some of the cases, the solution vector shows high sensitivity to theinitial guess for σ1 and σ2. In particular, the results for the conditions in whichσ1 = σ2 = 12√h˜′′2 and σ1 = σ2 = 13√h˜′′2 are taken as the initial guesses for thenumerical algorithm are shown in Figure 4.3 and denoted as the modeled bimodalGaussian 1 and 2, respectively. These results identify one of the challenges that onemay encounter when modeling a bimodal Gaussian PDF by the statistical moments560.70 0.80 0.90 1.00051015202530354045P(h)r/d = 1.355; x = 30mmExperimentModeled Bimodal Gaussian 1Modeled Bimodal Gaussian 2Optimized Bimodal Gaussian0.70 0.80 0.90 1.000510152025303540r/d = 1.453; x = 30mm0.70 0.80 0.90 1.000510152025303540P(h)r/d = 1.527; x = 30mm0.70 0.80 0.90 1.000102030405060r/d = 1.576; x = 30mm0.70 0.80 0.90 1.00h01020304050607080P(h)r/d = 1.649; x = 30mm0.70 0.80 0.90 1.00h0102030405060708090r/d = 1.723; x = 30mmFigure 4.3. The experimental and modeled marginal PDF of reactionprogress variable at various radial positions in flame number 11.of the normalized total enthalpy. In an alternative approach, the whole shape of themarginal PDF is taken into consideration and parameters in Equation 4.1 are op-timized through a genetic algorithm to have the minimum accumulated differencebetween the experimental PDF and the modeled one. The results of this methodare also illustrated in Figure 4.3; they show an excellent agreement with the mea-surements, however, this approach is impractical in simulations where only a fewmoments are available for the conditioning variables.570.5 1 1.5 2 2.5r/d10203040506070x (mm)Flame Brush PositionFlame 1Flame 2Flame 3Flame 5Flame 6Flame 7Flame 9Flame 10Flame 11Figure 4.4. The location of the flame brush in various axial distances fromthe burner tip in the 9 considered flames.4.2 Conditional PDF of Normalized Total EnthalpyAs was mentioned in the previous section, the marginal distribution of normalizedtotal enthalpy strictly depends on the position of the corresponding point relative tothe reaction zone. Therefore, it is expected that the PDF of this variable conditionedon specific values of mixture fraction and reaction progress variable would showless complexity compared to the marginal distribution. In order to have a sufficientnumber of data points in all of the cells, each of the mixture fraction and progressvariable dimensions in the conditional manifold are discretized by 20 isometricbins. Thus, i and j which represent indices of a measuring point in the directions ofmixture fraction and reaction progress variable, respectively, and can be calculatedthus:ipoint =b20Zpointc+1; i f Zpoint 6= 120; i f Zpoint = 1 (4.7a)jpoint =b20cpointc+1; i f cpoint 6= 120; i f cpoint = 1 (4.7b)Firstly, the low-resolution data set for all nine flames with different stratificationand swirl levels are sorted to extract data points with the same i and j indices. Foreach specific pair of (i, j), a set of points with various values of normalized total en-thalpy can be found. Since the number of the available points significantly changes58for different values of mixture fraction and reaction progress variable, a standardapproach is required for determining the number of bins in the direction of normal-ized total enthalpy based on the corresponding data. Freedman and Diaconis [41]suggested that the bin width which minimizes the discrepancy between the actualand the modeled PDF is bN−1/3; where b is a constant and N denotes the numberof available measurements. In this study, for each of the cases, the coefficient b isoptimized in order to have the smallest difference between the experimental con-ditional PDF and the modeled Gaussian PDF based on the mean and variance ofthe data set for normalized total enthalpy. The PDFs of this variable conditionedon various values of mixture fraction and reaction progress variable are shown inFigure 4.5. The results show that a Gaussian PDF with a small variance is a rea-sonably accurate model for this distribution.0.80 0.85 0.90 0.95020406080100120P(h|Z=,c=) = 0.175; = 0.375ExperimentModeled Gaussian PDF0.80 0.85 0.90 0.950102030405060 = 0.375; = 0.3750.80 0.85 0.90 0.95h05101520253035P(h|Z=,c=) = 0.775; = 0.5750.80 0.85 0.90 0.95h0510152025303540 = 0.575; = 0.775Figure 4.5. The experimental and modeled conditional PDF of normalizedtotal enthalpy calculated from data of all flames.In order to study the effects of swirl and stratification on the conditional PDFof normalized total enthalpy, different strategies for stacking up the data and col-lecting the measurements with particular values of mixture fraction and reaction59progress variable are employed here. These approaches are summarized in Ta-ble 4.1. In the method “All”, which was previously employed for obtaining thegraphs in Figure 4.5, the effect of swirl flow ratio and stratification factor on theconditional PDF of normalized total enthalpy is neglected, so all of the availablelow-resolution data are utilized to find the conditional PDF. As prescribed in Ta-ble 3.1, for each of the cases, 2 other flames with the same values of stratificationfactor and different swirl flow ratios can be found. In the method “Stratification”,by collecting the measurement data from all flames with identical stratification fac-tor, 3 different conditional PDFs are obtained for each pair of (i, j). On the otherhand, in order to observe the impact of the swirl level on the conditional PDF, foreach specific swirl flow ratio, flames with different stratification factors are used forcomputing the conditional PDF (the “3D Stratification” approach). Meanwhile, inthe method “Case n”, only the measurements from a specific flame are gathered toobtain the conditional PDF, therefore, 9 independent PDFs are computed per (i, j)pair. The results of these four methods are compared in Figure 4.6 for a mixturefraction of 0.575 and reaction progress variable of 0.775 (results at other valuesof mixture fraction and progress variable are similar to these). As the effects ofswirl and stratification indices on the conditional PDFs are not totally ignored inthe “Case n” method, one can expect that this approach would result in the mostrealistic conditional distributions for the points in one particular flame. Neverthe-less, interestingly, the results shown in Figure 4.6 suggest that the conditional PDFsfrom these four methods are pretty close to each other, so, the swirl flow ratio andstratification factor have only minor effects on the conditional PDFs of normalizedtotal enthalpy.To explore the dependence of the conditional PDF on the spatial coordinate, thehigh-resolution data are used to examine the goodness of fit for the PDFs obtainedfrom the four approaches presented in Table 4.1. Even though the high-resolutionmeasurements are made for only a limited set of points in the field, they are pre-ferred due to their having more measurements at each point, providing sufficientdata to extract the conditional PDFs. Here, conditional mean and variance of nor-malized total enthalpy are calculated from each of the four approaches to constructa Gaussian distribution for each of these methods. Then, all four of the constructedPDFs are compared to the actual conditional distributions of h obtained from the60Table 4.1. The experimental and modeled conditional PDF of normalizedtotal enthalpy calculated from data of all flames.Method CommentAllAll 9 cases used for one globalconditional PDF for each pair of (i, j)StratificationCases with common stratification factor employedfor 3 independent conditional PDF for each pair of (i, j)SwirlCases with common swirl flow ratio employed for 3independent conditional PDF for each pair of (i, j)Case n9 independent conditionalPDF for each pair of (i, j)measurements at some representative points. The χ2 test [117] is used as a met-ric to compare the expected number of points falling in each of the bins with theobserved number. As in each of the tests, the measurements from only one pointare gathered, most of the values for mixture fraction and progress variable arerare, thus, the corresponding conditional PDFs can be examined for only a fewpairs of (i, j). The significance levels (α) – a parameter lying between 0 to 1 forwhich higher values represent better fitness of the data to the considered distribu-tion [23] – for the conditional PDF of h conditioned on Z = 0.575 and c= 0.775 atsome representative points in flame number 11 are illustrated in Figure 4.7. Theseresults suggest that for most of the considered points, all four approaches have sig-nificance levels higher than 0.9 which shows very good agreeableness between themodeled distributions and the experimental data. On the other hand, there are somepoints in which the “Swirl” model results in lower significance levels. The exactand modeled conditional PDFs for two of the points – one with the significancelevel of 1.00 for all four models and one with the significance level of 0.494 for the“Swirl” approach as one of the worst found cases – are presented in Figure 4.8. Ascan be seen, even in the worst cases which are rare in the whole data set, the mod-eled distribution is not too different from the exact PDF. Other points in each of the9 flames show much more affability between the four models and the experimentaldata.61Figure 4.6. PDF of normalized total enthalpy conditioned on Z = 0.575 and c = 0.775 from different data collectingstrategies for 9 flames. Subfigures in the same rows and columns represent flames with the same stratificationfactor and swirl flow ratio, respectively.621.4 1.5 1.6 1.7 1.8r/d0.50.60.70.80.91Flame 11; x = 30 mm; = 0.575; = 0.775AllStratificationSwirlCase nFigure 4.7. Significance level for the conditional distribution of normalizedtotal enthalpy conditioned on Z = 0.575 and c = 0.775 at somerepresentative points located 30 mm downstream of the burner tip inflame number 11.0.70 0.75 0.80 0.85 0.90 0.95h0510152025303540P(h|Z = 0.575,c = 0.775)Flame 11; r/d = 1.4939; x = 30 mm(a)ExactAllStratificationSwirlCase n0.70 0.75 0.80 0.85 0.90 0.95h0510152025303540P(h|Z = 0.575,c = 0.775)Flame 11; r/d = 1.4123; x = 30 mm(b)ExactAllStratificationSwirlCase nFigure 4.8. The exact and modeled conditional distributions of normalizedtotal enthalpy conditioned on Z = 0.575 and c = 0.775 at two differentpoints in flame number 11 with. (a) α = 1.00 for all four models (b)α = 0.494 for the “Swirl” approach and = 1.00 for the other threemodels.The conditional statistical properties of normalized total enthalpy show negligi-ble dependence – from the modeling view point – on spatial coordinate, swirl flowratio, and stratification factor. Therefore, if available, one can store the conditionalmeans and variances of this variable against various values of mixture fraction andprogress variable, and employ Equation 4.8 to calculate the joint PDF of mixture63fraction, progress variable, and normalized total enthalpy for simulations. Possibleapproaches for obtaining the conditional statistical properties of this variable areleft for future studies.P(Z,c,h) = P(h|Z,c)P(Z,c). (4.8)4.3 Correlations between the Conditioning VariablesFor the conditioning variables to be assumed statistically independent, one nec-essary condition is having zero covariance between each pair of variables whichrequires that the correlation factors – defined as Rγiγ j =Cov(γi,γ j)√Var(γi)Var(γ j)– must bezero, as well. The low-resolution data are used to compute the variances of theconditioning variables also the covariances and correlation factors between themat various points in all 9 cases. The correlation factors are shown in Figures 4.9to 4.11. These graphs demonstrate that the correlations between the conditioningvariables are not negligible – especially in the vicinity of the flame brush wherethe chemical reactions are in progress. Even in the regions distant from the os-cillating flame brush, covariances are slight; nevertheless, the small values of thevariances in these regions result in considerable non-zero correlation factors be-tween the conditioning variables. Therefore, as reported earlier [26, 49, 74, 153],correlations between the conditioning variables are important and should be takeninto account when modeling the joint PDF.Considering the results in Figures 4.9 to 4.11, one can observe that the correla-tion factors for the flames with identical swirl flow ratio show more similarity as afunction of location. From this, it can be inferred that the swirl flow ratio influencesthe correlation factors between the conditioning variables. A thorough explanationfor the behavior of the correlation factor as a function of coordinate is out of thescope of this research and will be left for future work.64Figure 4.9. Correlation factor between mixture fraction and reaction progress variable at different locations in 9flames. Subfigures in the same rows and columns represent flames with the same stratification factor and swirlflow ratio, respectively.65Figure 4.10. Correlation factor between mixture fraction and normalized total enthalpy at different locations in 9flames. Subfigures in the same rows and columns represent flames with the same stratification factor and swirlflow ratio, respectively.66Figure 4.11. Correlation factor between reaction progress variable and normalized total enthalpy at different locationsin 9 flames. Subfigures in the same rows and columns represent flames with the same stratification factor andswirl flow ratio, respectively.674.4 The Copula for Mixture Fraction and ReactionProgress VariableAn n-dimensional copula (C) is a function from In = [0,1]n to I = [0,1] with thefollowing properties [103]:1. For every~u = (u1,u2, ...,un) in In,C(~u) = 0 if at least one component of~u is 0, (4.9a)if all components of~u except uk are 1, then C(~u) = uk; (4.9b)2. For every ~a = (a1,a2, ...,an) and~b = (b1,b2, ...,bn) where ak ≤ bk holds forall values of k ∈ {1,2, ...,n}, we have:VC([~a,~b])≥ 0; (4.10)here[~a,~b]= [a1,b1]× [a2,b2]× ...× [an,bn] and VC([~a,~b])is its C-volumedefined as:VC([~a,~b])=∑sgn(~c)C (~c) , (4.11)where each component of~c is either ak or bk, and sgn(~c) is given bysgn(~c) =1; if ck = ak holds for an even number of k′s−1; if ck = ak holds for an odd number of k′s . (4.12)The significance of the copula concept is highlighted through Sklar’s theorem [140]which states that for any n-dimensional distribution function K, with marginal Cu-mulative Distribution Functions (CDFs) F1,F2, ...,Fn, there exists an n-dimensionalcopula C such that for any~x = (x1,x2, ...,xn) with real components,K(x1,x2, ...,xn) =C(F1(x1),F2(x2), ...,Fn(xn)). (4.13)Therefore, by knowing the marginal distribution of the conditioning variables andusing a proper copula, one can construct the joint distribution – and hence the68joint PDF – of the corresponding variables. The product copula ∏(u1,u2, ...,un) =u1u2...un is employed for obtaining the joint distribution of statistically indepen-dent variables. Furthermore, the Plackett family form another class of copulas.They have been used for the simulation of turbulent reacting flows [21, 29, 135]due to their simplicity. The bivariate Plackett copulas are based on the assumptionof a constant cross product ratio (θXY ) for the corresponding two variables X andY with the marginal CDFs of F and G. For a specific pair of (p,q), this ratio isdefined asθXY =Pr[X ≤ p;Y ≤ q].Pr[X > p;Y > q]Pr[X ≤ p;Y > q].Pr[X > p;Y ≤ q] , (4.14)where the numerator describes the product of the probabilities when X and Y areboth smaller or greater than p and q, respectively; while the denominator is theproduct of the probabilities when only one of the variables is greater than eitherp or q. This equality can be expressed according to the copula and the marginalCDFs asθXY =C (F(x),G(y)) [1−F(x)−G(y)+C (F(x),G(y))][F(x)−C (F(x),G(y))] [G(y)−C (F(x),G(y))] ; (4.15)solving this equation for C gives two roots; Mardia [95] showed that one of thosetwo roots is never a copula, so, the joint CDF can be obtained with the other root,ie.:K(x,y) =C (F(x),G(y)) ={[1+(θXY −1)(F(x)+G(y))]−√[1+(θXY −1)(F(x)+G(y))]2−4F(x)G(y)θXY (θXY −1)}/[2(θXY −1)] .(4.16)The number of observed points in each of the four quadrants obtained by linesparallel to the axes in real two-dimensional space and passing through a point (p,q)can be used to construct a contingency table for a sample set and estimate the crossproduct ratio. Mardia [95] suggested that an optimum choice for (p,q) is the sam-ple median vector (i.e. F(p) = G(q) = 0.5). The schematic of a sample contin-69𝑁3𝑁1 𝑁2𝑁4𝑋𝑌𝑝𝑞11Figure 4.12. Schematic of a sample contingency table.gency table is shown in Figure 4.12, where Ni specifies the number of data points inthe ith quadrant. Having the contingency table, one can compute the cross productratio as θXY = N1N4N2N3 . Darbyshire and Swaminathan [29] introduced an algorithmfor estimating the cross product ratio from the correlation factor; however, in thisstudy, the single-shot data for mixture fraction and reaction progress variable aredirectly used for calculating this ratio at each point. The concept of cross productratio and the Plackett family of copulas can be generalized for higher dimensions[72, 98]; nevertheless, as the dimensionality increases, no analytic solution can befound for the value of the copula – like the one in Equation 4.16 – for each value ofthe conditioning variables; as such, a numerical method is required for obtainingthe copula which adds to the difficulty of using the Plackett family for modelingthe joint distribution of more than two variables.Metaelliptical copulas form another class of multivariate functions which showhigh flexibility for modeling multivariate distributions. An n-dimensional randomvector ~X = (X1,X2, ...,Xn) is considered to have an elliptical distribution with pa-rameters ~M , Σ, and g if and only if it can be expressed in this form [37]:~XT = ~M T +RA~UT , (4.17)70where ~M T is the transpose of the mean vector, Σ is the covariance matrix, A isan n× n matrix such that Σ = AAT , ~U is the coordinate of a random point on thesurface of the n-dimensional unit sphere, and R is a random variable with the PDFoffg(r) =2pin/2Γ(n/2)rn−1g(r2); (4.18)g : [0,∞)→ [0,∞) representing the generator function of the distribution, the jointPDF of ~X is obtained as Pg (~x) = |Σ|−1/2g[(~x− ~M )Σ−1(~x− ~M )T], and an n-dimensional metaelliptical copula with such a generator is defined as the joint dis-tribution of the vector (Ψ1,Ψ2, ...,Ψn), where, for any k ∈ {1,2, ...,n}:Ψk = Pr(Xk√Σkk≤ x)=12+pi(n−1)/2Γ(n−12 )∫ x0∫ ∞u2(y−u2)(n−3)/2g(y)dydu. (4.19)In this work, the Gaussian copula is chosen as a representative of the metael-liptical class of copulas for modeling the joint distribution of mixture fraction andreaction progress variable. In its n-dimensional version, the corresponding gener-ator takes the form of g(t) = (2pi)−n/2exp(−t/2), and the copula can be writtenasCℜ(u1,u2, ...,un) =Φℜ(Φ(−1)(u1),Φ(−1)(u2), ...,Φ(−1)(un)), (4.20)whereℜ ∈ [−1,1]n×n is a correlation matrix, Φℜ is the multivariate CDF of a nor-mal distribution with the covariance matrix of ℜ, and Φ(−1) is the inverse CDFof a standard normal. ℜ is a symmetric matrix with the diagonal components ofunity, therefore, only n(n−1)/2 superdiagonal elements are required to specify thecorrelation matrix and the copula. Genest et al. [44] introduced maximization ofthe log-pseudo likelihood as an approach for determining the superdiagonal corre-lations, however, this method is far from application in the simulation of turbulentreacting flows.Kendall’s τ and Spearman’s ρ [76] are two scale-invariant measures of asso-ciation which can be used to estimate the correlation matrix. These two metricsmeasure the level of “concordance” between two random variables, besides, it can71be shown that they are independent of the marginal distributions and are only re-lated to the copula which forms their joint CDF [103]. Meanwhile, applying strictlyincreasing transformations on the variables does not change these two measures.Kendall’s τ and Spearman’s ρ for each pair of random variables in a multivariatenormal distribution (Φℜ) can be approximated by the elements of the correlationmatrix through [59, 85]:τ kl =2pisin−1(ℜkl); ρ kl =6pisin−1(ℜkl/2), (4.21)where τkl and ρkl respectively denote Kendall’s τ and the Spearman’s ρ for the kthand lth random variables in a multivariate normal distribution with the covariancematrix ofℜ.Since Φ(−1)(F(γi)) is almost always a strictly increasing function of the con-ditioning variable γi with the marginal CDF of F , the sample versions of eitherKendall’s τ or Spearman’s ρ for the conditioning variables can be utilized to esti-mate the components of the correlation matrix through Equation 4.21. In this study,Kendall’s τ for mixture fraction and progress variable are computed fromτ Zc =CˆZc− DˆZcCˆZc+ DˆZc, (4.22)where CˆZc and DˆZc specify the number of concordant and discordant pairs, respec-tively. (ζi,ηi) and (ζ j,η j) denote two observations from the (Z,c) space; they aresaid to be concordant if and only if (ζi−ζ j)(ηi−η j)> 0, and they are considereddiscordant whenever (ζi−ζ j)(ηi−η j)< 0.In this study, the high-resolution measurements are used to obtain the copulawhich forms the joint distribution of mixture fraction and reaction progress vari-able. The marginal CDF of these two variables being strictly increasing – andhence invertible – the copula at each point can be obtained throughCexp(u1,u2) = KZc(F(−1)Z (u1),F(−1)c (u2)), (4.23)where Cexp is the experimental copula obtained from the measurements at eachpoint; FZ and Fc are respectively the marginal distributions of mixture fraction and72reaction progress variable, and KZc denotes their joint CDF. In order to decide ona proper copula to be used for the simulations of turbulent reacting flows, somemetrics are required to draw comparisons between the experimental copula andthe modeled ones; KC(t) = Pr[C(u1,u2) ≤ t] is used as one of the standards hereto evaluate the performance of the product, Plackett, and Gaussian copulas. Theresults for several points in flame 11 are shown in Figure 4.13. Moreover, C(t, t)and C(0.5, t) for these copulas are presented in Figures 4.14 and 4.15, respectively.The graphs suggest that the product copula shows no flexibility for following theexperimental copula and its results significantly mismatch those from the experi-ment at points not close to the centerline of the burner. In general, the results forthe Gaussian copula acceptably fit to the experiments – especially at points notclose to the flame brush. Meanwhile, the Plackett copula has poor performancein regions near the centerline, however, it follows the experimental results betterthan the Gaussian copula in the vicinity of the flame brush. Therefore, a possiblescenario for combustion modeling might be to use the Gaussian copula whereverthe variance of the reaction progress variable – representative of closeness to theflame brush – is smaller than a certain threshold, and switch to the Plackett familyin regions where this variance is higher. The results for other flames are consistentwith these conclusions.730.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.331; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.396; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.461; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.527; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.592; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.657; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.723; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81KC(t)r/d = 1.788; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct CopulaFigure 4.13. Evaluation of KC(t) for the product, Plackett, and Gaussian copulas at different points of flame number11 and comparison with that of the experimental copula.740.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.331; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.396; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.461; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.527; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.592; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.657; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.723; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.20.40.60.81C(t,t)r/d = 1.788; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct CopulaFigure 4.14. Evaluation of C(t, t) for the product, Plackett, and Gaussian copulas at different points of flame number11 and comparison with that of the experimental copula.750.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.50.6C(0.5,t)r/d = 1.331; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.50.6C(0.5,t)r/d = 1.396; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.5C(0.5,t)r/d = 1.461; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.50.6C(0.5,t)r/d = 1.527; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.50.6C(0.5,t)r/d = 1.592; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.50.6C(0.5,t)r/d = 1.657; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.50.6C(0.5,t)r/d = 1.723; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct Copula0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t00.10.20.30.40.50.6C(0.5,t)r/d = 1.788; x = 30mmExperimentGaussian CopulaPlackett CopulaProduct CopulaFigure 4.15. Evaluation of C(0.5, t) for the product, Plackett, and Gaussian copulas at different points of flamenumber 11 and comparison with that of the experimental copula.764.5 Modeling the Joint Distribution of the ConditioningVariablesIn the three-condition version of the UCS model, using mixture fraction, reac-tion progress variable, and normalized total enthalpy as conditioning variables, theconditional manifold takes the form of a cube with unity edge length which is in-dependent of the spatial coordinate, swirl flow ratio, and stratification factor. Inthis study, 50 isometric bins are used for discretizing each of mixture fraction andnormalized total enthalpy axes, and the direction associated with progress variableis discretized by 30 identical bins to form 75000 three-dimensional “cells” in themanifold. The low-resolution measurement data from all points in the 9 flames arecollected and every measurement point is assigned to one of the cells with match-ing values of the conditioning variables. The conditional averages are computedby ensemble averaging the reactive scalars in each of the cells. Since conditionalaveraging in cells with an inadequate number of measurements will contribute toan underestimation of the model discrepancies, cells with less than 10 data pointsare ignored in the computations – The sensitivity of our results to this criterionis discussed in this section. Next, in order to map the conditional manifold intoreal space, two approaches for modeling the joint distribution of the conditioningvariables are employed here:• Distributions of mixture fraction and reaction progress variable are assumedto follow the β -function, while the normalized total enthalpy is assumed tobe a Dirac-delta PDF centered on its mean value. The statistical dependenceof the conditioning variables is ignored, so, the product copula introduced inthe previous section is used for constructing the joint distribution.• The β -function is again used for modeling the marginal PDFs of mixturefraction and reaction progress variable, while the bimodal Gaussian PDFis employed for normalized total enthalpy. A three-dimensional Gaussiancopula is utilized for constructing the joint distribution of the conditioningvariables. This copula is selected due to its simplicity and compatibility withthree conditioning variables.77The characteristics of these two PDF modeling approaches are summarized in Ta-ble 4.2.Table 4.2. The considered PDF modeling approaches.Method P(Z) P(c) P(h) Copula1 β -PDF β -PDF δ -PDF Product copula2 β -PDF β -PDF BimodalGaussian PDFThree-dimensionalGaussian copulaThe first two statistical moments of mixture fraction and reaction progress vari-able in addition to the mean value of normalized total enthalpy are extracted fromall of the available measurements in each of the points and for each flame to cal-culate the parameters required for modeling the joint PDF in approach 1. Then, byintegrating the joint PDF over mixture fraction, progress variable, and normalizedtotal enthalpy, the joint CDF is obtained for different values of the conditioningvariables. Subsequently, by evaluating the CDF in 8 vertices of each conditionalcell, the probability for the conditioning variables to fall into the correspondingcell is given by the C-volume of the cell as described in Equations 4.11 and 4.12,where C is the product copula here. In approach 2, in addition to the first twomoments of mixture fraction and progress variable, in order to avoid the problemregarding the high sensitivity to the initial guess for obtaining the parameters of abimodal Gaussian PDF from five statistical moments of normalized total enthalpy –as pointed out in Section 4.1 – the whole marginal PDF of this variable is extractedat each point to optimize the corresponding parameters and model the PDF. Fur-thermore, Kendall’s τ is calculated directly from the measurements for each pairof the conditioning variables. These two steps make the second approach imprac-tical for the simulation of turbulent combustion where only the statistical centraland mixed moments of the conditioning variables are available. The probability ofthe conditioning variables is again calculated from the value of the joint CDF atthe vertices and C-volume of the corresponding cell. However, in this method, thethree-dimensional Gaussian copula is used as C. Regardless of the method usedfor modeling the distribution of the conditioning variables, the unconditionally-78averaged reactive scalars are obtained throughY l,UCS(~x) =50∑k=130∑j=150∑i=1〈Yl|~γ〉VC(celli, j,k), (4.24)where Y l,UCS is the unconditional average of the lth reactive scalar computed fromthe UCS model, and VC(celli, j,k) is the C-volume of the conditional cell formed bythe ith, jth, and kth intervals in the Z, c, and h directions, respectively.To evaluate the performance of these two joint PDF models, the predicted un-conditional averages from each model are compared to those from the directlyaveraging approach, and the discrepancies at each point are calculated. In thisstudy, only the results for the temperature field and CO mass fraction are presentedas two of the most important reactive scalars. The discrepancies for temperatureare reported in Kelvin; while those for CO mass fraction are normalized by themaximum value of this scalar throughout the domain:Y ′CO =YCO,UCS−YCO,DirectYCO,max. (4.25)The discrepancies as functions of spatial coordinate for the two joint distribu-tion modeling approaches are shown in Figures 4.16 and 4.17. The Simpson in-tegration method is employed for obtaining the CDF of the conditioning variablesfrom their PDFs. The number of subdivisions utilized for this numerical integrationapproach has a significant impact on the performance of the models. The resultsfor method 1 presented in [99] were obtained by employing one subdivision ineach of the cells. A convergence analysis for the integration scheme is presented inAppendix B. In this section, only the converged results are presented. The resultssuggest that although the first approach is simpler, it has a better performance inthe regions far from the center line. As the differences between the discrepanciesfrom the two approaches are not uniform for various positions in the field, a met-ric is needed to specify the accumulated errors obtained for the temperature andCO mass fraction predictions. In this work, the normalized RMS discrepanciesfrom all of the points are summed to evaluate the overall performance of each PDF79modeling approach:Ek =√1npointnpoint∑i=1Y ′k,i2, (4.26)where npoint is the total number of measurement points in the corresponding flame,and Y ′k,i is the normalized discrepancy which is calculated in position i.The normalized accumulated discrepancies for temperature and carbon monox-ide mass fraction are presented in Figures 4.18 and 4.19, respectively. In general,the results suggest that the first approach – which is also less complicated andcomputationally cheaper – works better in almost all of the cases. Meanwhile, aspreviously mentioned, the second approach is not practical for the simulations dueto its need for extra information about the statistical properties of the conditioningvariables which are not easily accessible in the simulations.80-3 -2 -1 0 1 2 3r/d-300-200-1000100200300400500Discrepancy in Temperature (K)Flame Number 1 , , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm x = 70 mm-3 -2 -1 0 1 2 3 4r/d-500-400-300-200-1000100200300400500Discrepancy in Temperature (K)Flame Number 2 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3 4r/d-300-200-1000100200300400500Discrepancy in Temperature (K)Flame Number 3 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3r/d-400-300-200-1000100200300400500600Discrepancy in Temperature (K)Flame Number 5 , , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm x = 70 mm-3 -2 -1 0 1 2 3r/d-600-400-2000200400600Discrepancy in Temperature (K)Flame Number 6 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3 4r/d-600-400-2000200400600Discrepancy in Temperature (K)Flame Number 7 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3r/d-500-400-300-200-1000100200300400500Discrepancy in Temperature (K)Flame Number 9 , , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm x = 70 mm-3 -2 -1 0 1 2 3r/d-600-400-2000200400600Discrepancy in Temperature (K)Flame Number 10 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3 4 5r/d-500-400-300-200-1000100200300400500Discrepancy in Temperature (K)Flame Number 11 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mmFigure 4.16. Normalized discrepancies in the averaged temperature for 9 flames; open and filled symbols represent theresults obtained from approach 1 and approach 2, respectively.81-3 -2 -1 0 1 2 3r/d-0.15-0.1-0.0500.050.10.15Normalized Discrepancy in CO Mass FractionFlame Number 1 , , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm x = 70 mm-3 -2 -1 0 1 2 3 4r/d-0.1-0.0500.050.10.150.2Normalized Discrepancy in CO Mass FractionFlame Number 2 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3 4r/d-0.0500.050.10.150.2Normalized Discrepancy in CO Mass FractionFlame Number 3 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3r/d-0.2-0.15-0.1-0.0500.050.1Normalized Discrepancy in CO Mass FractionFlame Number 5 , , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm x = 70 mm-3 -2 -1 0 1 2 3r/d-0.25-0.2-0.15-0.1-0.0500.050.10.15Normalized Discrepancy in CO Mass FractionFlame Number 6 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3 4r/d-0.2-0.15-0.1-0.0500.050.10.15Normalized Discrepancy in CO Mass FractionFlame Number 7 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3r/d-0.3-0.25-0.2-0.15-0.1-0.0500.050.1Normalized Discrepancy in CO Mass FractionFlame Number 9 , , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm x = 70 mm-3 -2 -1 0 1 2 3r/d-0.35-0.3-0.25-0.2-0.15-0.1-0.0500.050.1Normalized Discrepancy in CO Mass FractionFlame Number 10 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mm-3 -2 -1 0 1 2 3 4 5r/d-0.25-0.2-0.15-0.1-0.0500.050.1Normalized Discrepancy in CO Mass FractionFlame Number 11 , , , , , , x = 10 mm x = 20 mm x = 30 mm x = 40 mm x = 50 mm x = 60 mmFigure 4.17. Normalized discrepancies in the averaged carbon monoxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from approach 1 and approach 2, respectively.82Normalized Accumulated Error for TemperatureFlame 1 Flame 2 Flame 3 Flame 5 Flame 6 Flame 7 Flame 9 Flame 10 Flame 110123456789Normalized Accumulated Error (%)Method 1Method 2Figure 4.18. Normalized accumulated discrepancies of temperature obtainedfrom two PDF models.Normalized Accumulated Error for CO Mass FractionFlame 1 Flame 2 Flame 3 Flame 5 Flame 6 Flame 7 Flame 9 Flame 10 Flame 1100.511.522.533.544.55Normalized Accumulated Error (%)Method 1Method 2Figure 4.19. Normalized accumulated discrepancies of CO mass fractionobtained from two PDF models.In order to investigate the effect of the minimum number of data points in theconditional cells being considered in conditional averaging, in some of the tests,this minimum number is increased from 10 to 15 and 20 for comparison. Theresults for the normalized accumulated discrepancies of CO mass fraction in flame83number 1 are illustrated in Figure 4.20 as a representative case. These resultssuggest that the normalized discrepancies show much smaller sensitivity to thiscriterion than the employed joint PDF model. Moreover, as long as this minimumnumber is kept similar for both of the PDF models, they are comparable. The sameconclusions can be drawn for any considered reactive scalar in any of the flames.Altogether, these results indicate that, even given extra information about thestatistics of the conditioning variables – information that would be very difficult toobtain in a live simulation of a turbulent flame – one struggles to get a sufficientlyaccurate model for the joint PDF of the conditioning variables in a stratified, swirl-stabilized turbulent flame. Just how significant the impact of the errors in predictingthe joint PDF might be in a live simulation will have to be evaluated by performingsuch a simulation, which will be the subject of future work. Another alternativethat may need to be explored would be to solve the joint PDF transport equationfor the conditioning variables, using a Monte Carlo-like method [108, 121]. Thiswill also be a topic for future research.Normalized Accumulated Error for CO Mass Fraction in Flame 1Method 1 Method 2012345Normalized Accumulated Error (%)Min = 10Min = 15Min = 20Figure 4.20. Normalized accumulated discrepancies of CO mass fractionobtained from two PDF models when cells with less than 10, 15, and20 data points are ignored for conditional averaging.844.6 SummaryThe measurements from the Cambridge-Sandia burner working in different lev-els of swirl and stratification were employed to obtain the statistical properties ofmixture fraction, reaction progress variable, and normalized total enthalpy as con-ditioning variables. The marginal PDFs of these variables were extracted and com-pared to some previously used models. The conditional PDF of normalized totalenthalpy was extracted and found to be nearly Gaussian. In addition, the depen-dence of the conditional mean and variance of this variable on spatial coordinate,stratification factor, and swirl flow ratio was found to be small. Subsequently, thecorrelation factors for the conditioning variables were calculated in different loca-tions. Meanwhile, the functionality of a few copulas were compared for modelingthe joint PDF of mixture fraction and progress variable. Finally, two different ap-proaches for modeling the joint PDF of the conditioning variables were introducedand applied for obtaining the averaged scalar fields.85Chapter 5Conclusions and Future WorkIn this chapter, the conclusions from the research works presented in Chapters 3and 4 are summarized. Moreover, the general conclusions from the project andsome suggestions for future works are presented.5.1 Chapter 3In the Cambridge-Sandia burner, collecting the measurement data from flames withdifferent swirl and stratification levels showed that the three-condition conditionalaverages were largely independent of these two indices. It would appear that, byconsidering the history of heat transfer with three-condition tabulation, the chem-istry table becomes robust enough that it does not change significantly with vari-ation of either the swirl level or the degree of stratification. The results suggestthat the conditionally-averaged reactive scalars and chemical source-terms can bestored in a pre-processed lookup table and be used during the solution of RAN-S/LES filtered transport equations. This approach significantly reduces the compu-tational cost of UCS – suggesting that the MFM approach would be valid for theseflames – and should lead to more accurate results than the two-condition versionof these models.The study in Chapter 3 can be contemplated as a follow up for the previ-ous work on the spatial gradient of the conditional averages [16] where it wasdemonstrated that by using a sufficient number of conditioning variables, the con-86ditional averages in the Sandia/TUD and Sydney swirl burners show negligibledependence on the spatial coordinate and Reynolds number. Here the Cambridge-Sandia burner, which is more representative of an industrial burner, was utilizedto study the impact of swirl and stratification levels on the conditionally-averagedreactive scalars. Independence of the conditional averages from spatial coordi-nate, Reynolds number (found in [16]), and swirl flow ratio (shown in Chapter 3)suggest that when three conditioning variables are employed, it is sufficiently ac-curate to assume that the flow topology and geometrical boundary conditions havenegligible impact on the conditional averages. Furthermore, from Chapter 3, rel-ative independence of the three-condition conditionally-averaged scalars from thestratification factor shows that the level of premixing also has small effects on theconditional manifold. Moreover, it was demonstrated in Chapter 3 that the his-tory of heat transfer is not only important near the solid components of the burnerwhere the flow field is highly non-adiabatic, but also it remains influential at down-stream positions. Finally, it is important to note that all of the aforementioned im-provements with three-condition chemistry tabulation approach rely on having anaccurate three-dimensional joint PDF model for the conditioning variables whichclearly indicates the need for more research in this particular aspect of the mod-elling.5.2 Chapter 4By post-processing the measurement data from the Cambridge-Sandia burner, itwas found that there are significant departures from the β -function in the marginalPDFs of both mixture fraction and reaction progress variable. The experimentalresults suggested that the marginal PDF of normalized total enthalpy has the ap-pearance of a bimodal Gaussian distribution, and its shape is highly dependent onthe measuring location with respect to the flame brush. Furthermore, the shape ofthe conditional PDF of total enthalpy was found to be Gaussian with small depen-dence on the swirl and stratification indices.The results for the correlation factor between the conditioning variables in dif-ferent locations revealed that the conditioning variables are statistically dependent,and their correlations should be taken into account in modeling. The functionality87of the product, Plackett, and Gaussian copulas were examined for modeling theassociation between mixture fraction and progress variable. It was found that thePlackett family of copulas works better in the vicinity of the flame brush, while theGaussian copula has a better performance in regions not very close to the flamebrush.Finally, two different models for the joint PDF of the conditioning variableswere introduced and applied for obtaining the averaged scalar fields via the three-condition version of the UCS model. In the first approach, the β -function was em-ployed for modeling the marginal PDFs of mixture fraction and reaction progressvariable, and the marginal PDF of normalized total enthalpy was modeled by aDirac-delta function. Meanwhile, the correlations between the conditioning vari-ables were ignored, so, a product copula was utilized for constructing the jointPDF. In the second approach, again the β -function was employed for modeling themarginal PDFs of mixture fraction and progress variable, while the marginal PDFof normalized total enthalpy was modeled by a bimodal Gaussian PDF. A three-dimensional Gaussian copula was used to construct the joint distribution. The re-sults showed that the first approach provides overall better predictions. Moreover,the second approach can not be easily used in the simulation of turbulent react-ing flow due to complexity and too much of information it requires. Altogether, itwould appear that a more practical model is required to improve the predictions ofthe joint PDF. This research provided a basis for investigating other PDF modelsin future works.5.3 General ConclusionsThis thesis was destined to evaluate some possible improvements in the predictionpower of the UCS filter-chemistry model to be applied for the simulation of turbu-lent reactive flows. The experimental measurements from the Cambridge-Sandiaburner were utilized to study the effect of adding normalized total enthalpy as anew conditioning variable on the predictions. Moreover, the experimental datawere employed to study the statistical properties of the conditioning variables withthe aim of introducing more accurate joint PDF models for these variables in fu-ture works. It is expected that the findings of this work will play a key role in the88development of more efficient combustion chambers to put brakes on the climatechange issue.5.4 Future WorksThe impact of adding normalized total enthalpy as a new conditioning variableshould be evaluated in the results of CFD calculations. However, it is expectedthat this will add two major problems to the simulation process: complicationsin modelling the three-dimensional joint PDF of the conditioning variable and theneed for a considerable size in RAM for loading the chemistry table. As demon-strated in this study, many of the widely used assumptions for modeling the jointPDF of the conditioning variables are not realistic. Future works should considermore accurate PDF models in the context of UCS. It can also be contemplated tosolve a transport equation for the joint PDF of the conditioning variables instead ofemploying a presumed shape for this PDF. 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ISSN00102180. → pages 27, 28105Appendix AThe Discrepancies from theDifferent Conditional AveragingMethodsHere the normalized discrepancies in the prediction of reactive scalars from var-ious conditional averaging approaches for all of the nine configurations of theCambridge-Sandia burner are presented.106Figure A.1. Discrepancies in the averaged temperature for 9 flames; open and filled symbols represent the resultsobtained from the two-condition conditional averages computed from all flames (2DAll) and each specific case(2Dn), respectively.107Figure A.2. Normalized discrepancies in the averaged methane mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and each specific case (2Dn), respectively.108Figure A.3. Normalized discrepancies in the averaged carbon dioxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the two-condition conditional averages computed from all flames(2DAll) and each specific case (2Dn), respectively.109Figure A.4. Normalized discrepancies in the averaged carbon monoxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the two-condition conditional averages computed from all flames(2DAll) and each specific case (2Dn), respectively.110Figure A.5. Normalized discrepancies in the averaged hydrogen mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and each specific case (2Dn), respectively.111Figure A.6. Normalized discrepancies in the averaged water mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and each specific case (2Dn), respectively.112Figure A.7. Discrepancies in the averaged temperature for 9 flames; open and filled symbols represent the resultsobtained from the two-condition conditional averages computed from all flames (2DAll) and the three-conditionversion by using the data of each specific case (3Dn), respectively.113Figure A.8. Normalized discrepancies in the averaged methane mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and the three-condition version by using the data of the specific case (3Dn), respectively.114Figure A.9. Normalized discrepancies in the averaged carbon dioxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the two-condition conditional averages computed from all flames(2DAll) and the three-condition version by using the data of each specific case (3Dn), respectively.115Figure A.10. Normalized discrepancies in the averaged carbon monoxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the two-condition conditional averages computed from all flames(2DAll) and the three-condition version by using the data of each specific case (3Dn), respectively.116Figure A.11. Normalized discrepancies in the averaged hydrogen mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and the three-condition version by using the data of each specific case (3Dn), respectively.117Figure A.12. Normalized discrepancies in the averaged water mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition conditional averages computed from all flames (2DAll)and the three-condition version by using the data of each specific case (3Dn), respectively.118Figure A.13. Discrepancies in the averaged temperature for 9 flames; open and filled symbols represent the resultsobtained from the two-condition (2DAll) and three-condition (3DAll) conditional averages computed from allflames, respectively.119Figure A.14. Normalized discrepancies in the averaged methane mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition (2DAll) and three-condition (3DAll) conditional averagescomputed from all flames, respectively.120Figure A.15. Normalized discrepancies in the averaged carbon dioxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the two-condition (2DAll) and three-condition (3DAll) conditionalaverages computed from all flames, respectively.121Figure A.16. Normalized discrepancies in the averaged carbon monoxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the two-condition (2DAll) and three-condition (3DAll) conditionalaverages computed from all flames, respectively.122Figure A.17. Normalized discrepancies in the averaged hydrogen mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition (2DAll) and three-condition (3DAll) conditional averagescomputed from all flames, respectively.123Figure A.18. Normalized discrepancies in the averaged water mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the two-condition (2DAll) and three-condition (3DAll) conditional averagescomputed from all flames, respectively.124Figure A.19. Discrepancies in the averaged temperature for 9 flames; open and filled symbols represent the resultsobtained from the three-condition conditional averages computed from cases with similar swirl flow ratio(3DSwirl) and all flames (3DAll), respectively.125Figure A.20. Normalized discrepancies in the averaged methane mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the three-condition conditional averages computed from cases with similarswirl flow ratio (3DSwirl) and all flames (3DAll), respectively.126Figure A.21. Normalized discrepancies in the averaged carbon dioxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the three-condition conditional averages computed from cases withsimilar swirl flow ratio (3DSwirl) and all flames (3DAll), respectively.127Figure A.22. Normalized discrepancies in the averaged carbon monoxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the three-condition conditional averages computed from cases withsimilar swirl flow ratio (3DSwirl) and all flames (3DAll), respectively.128Figure A.23. Normalized discrepancies in the averaged hydrogen mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the three-condition conditional averages computed from cases with similarswirl flow ratio (3DSwirl) and all flames (3DAll), respectively.129Figure A.24. Normalized discrepancies in the averaged water mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the three-condition conditional averages computed from cases with similarswirl flow ratio (3DSwirl) and all flames (3DAll), respectively.130Figure A.25. Discrepancies in the averaged temperature for 9 flames; open and filled symbols represent the resultsobtained from the three-condition conditional averages computed from cases with similar stratification factor(3DStratification) and all flames (3DAll), respectively.131Figure A.26. Normalized discrepancies in the averaged methane mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the three-condition conditional averages computed from cases with similarstratification factor (3DStratification) and all flames (3DAll), respectively.132Figure A.27. Normalized discrepancies in the averaged carbon dioxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the three-condition conditional averages computed from cases withsimilar stratification factor (3DStratification) and all flames (3DAll), respectively.133Figure A.28. Normalized discrepancies in the averaged carbon monoxide mass fraction for 9 flames; open and filledsymbols represent the results obtained from the three-condition conditional averages computed from cases withsimilar stratification factor (3DStratification) and all flames (3DAll), respectively.134Figure A.29. Normalized discrepancies in the averaged hydrogen mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the three-condition conditional averages computed from cases with similarstratification factor (3DStratification) and all flames (3DAll), respectively.135Figure A.30. Normalized discrepancies in the averaged water mass fraction for 9 flames; open and filled symbolsrepresent the results obtained from the three-condition conditional averages computed from cases with similarstratification factor (3DStratification) and all flames (3DAll), respectively.136Appendix BConvergence Study for Obtainingthe CDF of the ConditioningVariablesAs described in Section 4.5, the Simpson integration approach is utilized for cal-culating the CDF of the conditioning variables from their PDFs. As the β -PDFshows sharp changes in some regions of its domain, the number of subdivisions forintegrating this function plays a key role in the accuracy of the model. Here from1 to 16 isometric subdivisions were tested for obtaining the integrals. The resultsin Figure B.1 show that with 16 subdivisions in each bin, the accumulated errorsfor the temperature field in flame number 1 converge. The results for other casesis similar. Thus, this integration scheme is utilized for comparing the two PDFmodeling approaches.137Normalized Accumulated Error for Temperature1 2 4 8 16Number of Subdivisions in each Bin051015Normalized Accumulated Error (%)Figure B.1. Normalized accumulated discrepancies of temperature obtainedby using different number of subdivisions in each bin for integration.138
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Bushe, W. K.
Evaluation of low-order manifold representation of chemistry in turbulent reactive flows
Text