Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
De Graaf, Stefanie
Doctor of Philosophy - PhD
University of British Columbia
A universally applicable user-friendly tool has been developed for numerical simulations of turbulent combustion in gas turbines and is being evaluated in this work. Ultimately the goal is to reliably predict emissions of interest at low computational cost while maintaining a large degree of flexibility with regards to the fuel type and the complexity of the chemical kinetic mechanism as well as the type of combustion that may be present. Premixed, non-premixed as well as partially-premixed combustion could potentially be present in a technical application. Maintaining this versatility gives freedom of design to the user. In this work the criteria necessary for a combustion model suitable for such a tool will be defined and measures of success identified. The Uniform Conditional State (UCS) model allows for the chemistry to be computed a priori and disconnected from physical space using so-called conditioning variables; variables describing specific characteristics, which are deterministic for the combustion process. With that the computational cost is very low and a large degree of flexibility can be achieved compared to other modelling approaches. In theory, the recently developed model of UCS promises to satisfy the required criteria and thus an extensive study was conducted putting this model into practice for the first time. Initially a number of parameters within this modelling approach were defined based on weak assumptions and needed to be evaluated and verified as part of this work as well. With that a valuable report was completed to outline the ability of UCS to serve as a reliable predictive model of turbulence-chemistry interaction in its current state. For instance, while the computational cost indeed proved to be only 20% of that of a comparable simulation in one of the test cases, the accuracy of predictions of emissions such as NOx remains below the aspired target in another test case. The evaluation of this new approach was carried out based on a non-premixed swirl-stabilized model aero engine combustor, a premixed swirl-stabilized flame series and a partially-premixed lifted jet flame, while exploring different modelling parameters and limitations of the modelling approach.
Evaluation of the Uniform Conditional State model forgas turbine combustor designbyStefanie De GraafB.Sc Aerospace Engineering, University of Stuttgart, 2013M.Sc Aerospace Engineering, University of Stuttgart, 2015A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)April 2021c© Stefanie De Graaf, 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 the Uniform Conditional State model forgas turbine combustor designsubmitted by Stefanie De Graaf in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in Mechanical Engineering.Examining Committee:W. Kendal Bushe, Department of Mechanical EngineeringSupervisorCarl Ollivier-Gooch, Department of Mechanical EngineeringSupervisory Committee MemberPatrick Kirchen, Department of Mechanical EngineeringSupervisory Committee MemberGwynn Elfring, Department of Mechanical EngineeringUniversity ExaminerAnthony Wachs, Department of Chemical and Biological EngineeringUniversity ExaminerMatthew Cleary, Faculty of Engineering, University of SydneyExternal ExamineriiAbstractA universally applicable user-friendly tool has been developed for numerical simu-lations of turbulent combustion in gas turbines and is being evaluated in this work.Ultimately the goal is to reliably predict emissions of interest at low computationalcost while maintaining a large degree of flexibility with regards to the fuel type andthe complexity of the chemical kinetic mechanism as well as the type of combus-tion that may be present. Premixed, non-premixed as well as partially-premixedcombustion could potentially be present in a technical application. Maintainingthis versatility gives freedom of design to the user.In this work the criteria necessary for a combustion model suitable for sucha tool will be defined and measures of success identified. The Uniform Condi-tional State (UCS) model allows for the chemistry to be computed a priori anddisconnected from physical space using so-called conditioning variables; variablesdescribing specific characteristics, which are deterministic for the combustion pro-cess. With that the computational cost is very low and a large degree of flexibilitycan be achieved compared to other modelling approaches. In theory, the recentlydeveloped model of UCS promises to satisfy the required criteria and thus an ex-tensive study was conducted putting this model into practice for the first time. Ini-tially a number of parameters within this modelling approach were defined basedon weak assumptions and needed to be evaluated and verified as part of this workas well. With that a valuable report was completed to outline the ability of UCSto serve as a reliable predictive model of turbulence-chemistry interaction in itscurrent state. For instance, while the computational cost indeed proved to be only20% of that of a comparable simulation in one of the test cases, the accuracy ofpredictions of emissions such as NOX remains below the aspired target in anotheriiitest case.The evaluation of this new approach was carried out based on a non-premixedswirl-stabilized model aero engine combustor, a premixed swirl-stabilized flameseries and a partially-premixed lifted jet flame, while exploring different modellingparameters and limitations of the modelling approach.ivLay SummaryThe development of new combustion technology is essential to ensuring a sustain-able future with a carbon-neutral transport sector by 2050 – in particular for theaerospace sector. To enable the industry to maintain a high degree of freedom inearly stage developments, a user-friendly tool for modelling combustion with awide range of applicability is needed. This includes flexibility regarding the fueltype and therefore the complexity of the chemistry involved as well as the geometryof the combustor and whether the fuel and oxidizer are mixed in advance. A newapproach called the Uniform Conditional State (UCS) model was recently devel-oped and promises to fulfill these and other important criteria in theory. The goalof this dissertation was to develop a framework in which UCS could be used in nu-merical simulations of combustion and to evaluate its applicability and versatilitywith regards to the parameters important for gas turbines design.vPrefaceThe contents of chapter 3 are published in:• S. De Graaf, L. de Guillebon, M. Konle, and W.K. Bushe. Evaluation ofthe Uniform Conditional State method for turbulence-chemistry interactionmodelling of swirl-stabilized flames. ASME Turbo Expo, GT2020-14354,2020.In this publication I was responsible for conducting all parts of the research andpreparing the manuscript as well as the conference presentation. Professor KendalBushe supervised the research, reviewed the manuscript and presentation and pro-vided feedback. Dr. Marco Konle and Ludovic de Guillebon assisted with adviceon the research and reviewed the manuscript and presentation and provided feed-back.The contents of chapter 4 are published in:• S. De Graaf, L. de Guillebon, S.R. Taylor, M. Konle, and W.K. Bushe. Im-provements to the Uniform Conditional State model to predict swirl-stabilizedflames. AIAA SciTech Forum, 2021.and has also been submitted as:• S. De Graaf, L. de Guillebon, S.R. Taylor, M. Konle, and W.K. Bushe. Studyto improve the Uniform Conditional State model for prediction of premixedcombustion.In this publication I was responsible for conducting all parts of the research andpreparing the manuscript as well as the conference presentation. Professor KendalviBushe supervised the research, reviewed the manuscript and presentation and pro-vided feedback. Dr. Marco Konle assisted with advice on the research and re-viewed the manuscript and presentation and provided feedback. Ludovic de Guille-bon initiated the idea for the equivalence ratio study and reviewed the manuscriptand provided feedback. Sonia Taylor assisted with processing of simulation resultsand reviewed the manuscript and provided feedback as well.The contents of chapter 5 have been submitted for publication under:• S. De Graaf, M. Konle, and W.K. Bushe. Prediction of premixed, non-premixed and partially-premixed flames with Uniform Conditional Stateturbulence-chemistry interaction model.In this publication I was responsible for conducting all parts of the research andpreparing the manuscript. Professor Kendal Bushe supervised the research, re-viewed the manuscript and presentation and provided feedback. Dr. Marco Konleassisted with advice on the research and reviewed the manuscript and providedfeedback.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Fluid State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Kinematic State . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Thermodynamic State . . . . . . . . . . . . . . . . . . . 7viii2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 RANS vs. LES . . . . . . . . . . . . . . . . . . . . . . . 132.3 Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Combustion Theory . . . . . . . . . . . . . . . . . . . . 152.3.2 Combustion Modelling . . . . . . . . . . . . . . . . . . . 222.3.3 Uniform Conditional State Model . . . . . . . . . . . . . 312.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Non-Premixed Combustion – Swirl-Stabilized Flame . . . . . . . . . 433.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Ethylene Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 523.4.1 Flow Fields . . . . . . . . . . . . . . . . . . . . . . . . . 523.4.2 Flame Structure and Temperature . . . . . . . . . . . . . 533.4.3 Species Analysis . . . . . . . . . . . . . . . . . . . . . . 583.5 Concluding Thoughts . . . . . . . . . . . . . . . . . . . . . . . . 594 Premixed Combustion – Swirl-Stabilized Flame . . . . . . . . . . . 634.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 714.4.1 PDF Study . . . . . . . . . . . . . . . . . . . . . . . . . 734.4.2 Equivalence Ratio Variation . . . . . . . . . . . . . . . . 774.4.3 Flame Shape . . . . . . . . . . . . . . . . . . . . . . . . 814.5 Concluding Thoughts . . . . . . . . . . . . . . . . . . . . . . . . 865 Partially-Premixed Combustion – Jet Flame . . . . . . . . . . . . . 895.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 93ix5.4.1 Lift-off Height . . . . . . . . . . . . . . . . . . . . . . . 935.4.2 Temperature and Emission Predictions . . . . . . . . . . . 975.5 Concluding Thoughts . . . . . . . . . . . . . . . . . . . . . . . . 1036 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 1076.1 Evaluation based on Industry Criteria . . . . . . . . . . . . . . . 1076.1.1 Computational Cost . . . . . . . . . . . . . . . . . . . . 1076.1.2 Ease of Integration . . . . . . . . . . . . . . . . . . . . . 1096.1.3 Versatility with regards to Combustion Types . . . . . . . 1096.1.4 Flexibility towards fuel types . . . . . . . . . . . . . . . . 1106.1.5 Accuracy of Emission Predictions . . . . . . . . . . . . . 1106.2 Current Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . 112Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A Interpolation in 4D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127B User Guide for UCS software . . . . . . . . . . . . . . . . . . . . . . 129B.1 Step-by-step: How to simulate the chemistry with UCS? . . . . . 129B.2 Step-by-step: How to use UCS in a CFD simulation? . . . . . . . 137xList of TablesTable 2.1 List of mechanisms used in the course of this work . . . . . . . 20Table 2.2 State-of-the-Art combustion model evaluation . . . . . . . . . 30Table 3.1 Inflow conditions for FIRST burner . . . . . . . . . . . . . . . 45Table 3.2 Comparison of simulations published for FIRST burner . . . . 46Table 3.3 Boundary conditions of investigated chemistries . . . . . . . . 47Table 3.4 UCS parameters for this investigation . . . . . . . . . . . . . . 48Table 4.1 Equivalence ratio series for validation study . . . . . . . . . . 66Table 4.2 UCS parameters of BRS burner simulations compared to FIRSTburner simulation . . . . . . . . . . . . . . . . . . . . . . . . 68Table 5.1 Cabra flame boundary conditions from experiment . . . . . . . 91Table 5.2 UCS parameters of Cabra flame simulations compared to FIRSTand BRS burner simulation . . . . . . . . . . . . . . . . . . . 93Table 6.1 Quantitative analysis of UCS as per current status and ultimategoals with regards to the five Criteria for industry application . 108Table 6.2 UCS parameters tested in this study . . . . . . . . . . . . . . . 110xiList of FiguresFigure 2.1 Illustration of the energy cascade . . . . . . . . . . . . . . . . 14Figure 2.2 Borghi Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.3 Turbulent time scales in comparison with chemical time scales(adopted from Warnatz et al. [122]) . . . . . . . . . . . . . . 20Figure 2.4 Diagram of conditional H2O mass fraction . . . . . . . . . . 28Figure 2.5 Flow chart schematic for using UCS as a predictive method ina CFD simulation . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.6 Examples of β -distribution for various values of mean and seg-regation values for γi = Z . . . . . . . . . . . . . . . . . . . . 41Figure 2.7 Four cases of Modified laminar flamelet PDF; Image: Salehi etal. [100] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.1 Schematic of geometry of FIRST burner [38] . . . . . . . . . 44Figure 3.2 Computational domain and mesh for numerical simulation ofFIRST burner . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.3 Chemical source-term of progress variable in conditional space 49Figure 3.4 UCS solution for cases A-D (as per Tab. 3.3) at their respectivestoichiometric mixture fraction . . . . . . . . . . . . . . . . . 50Figure 3.5 Contour plot of the magnitude of velocity . . . . . . . . . . . 53Figure 3.6 Axial velocity profiles of UCS simulation and experiment atdifferent locations downstream . . . . . . . . . . . . . . . . . 54Figure 3.7 Temperature along the centerline: Comparison with differentsimulations from the literature and experimental data . . . . . 55xiiFigure 3.8 Temperature comparison of UCS simulation with the meanexperimental data Tmean and the mode temperature Tmp withit’s uncertainty as well as the simulation results available fromother publications at different locations z downstream . . . . . 56Figure 3.9 Contour plot of temperature on x− z−plane . . . . . . . . . . 57Figure 3.10 Comparison of UCS simulation against results published byEberle et al. [29] for temperature . . . . . . . . . . . . . . . . 58Figure 3.11 Comparison of UCS simulation against results published byFranzelli et al. [36] . . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.12 Comparison of UCS simulation against results published byEberle et al. [29] . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 3.13 Qualitative comparison of YOH of UCS simulation with resultspublished by Eberle et al. [29] . . . . . . . . . . . . . . . . . 61Figure 4.1 Schematic of the experimental test rig of the BRS burner [66]including geometric dimensions in [mm] . . . . . . . . . . . 64Figure 4.2 Simulation domain for CFD simulation using UCS as a predic-tive method . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.3 Complete combustion according to Burke-Schuhmann [125]and equilibrium computed with Cantera for YCO2 and T plot-ted over Z . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 4.4 UCS solution for temperature with clipping functions based on(a) GRI-3.0 Mechanism [103] and (b) UBC-3.0 mechanism [51] 70Figure 4.5 UCS chemistry solution for various equivalence ratios φ as la-beled in the figures for both chemical mechanisms . . . . . . 72Figure 4.6 Contour plots of the axial velocity field prediction of UCS forthe case of φ = 0.77 with comparison of β -PDF and the MLF-PDF for progress variable and experimental image . . . . . . 74Figure 4.7 Line plot of the axial velocity field prediction of UCS using thetwo different JPDF approaches compared to the experimentalmeasurements and other simulations . . . . . . . . . . . . . . 76xiiiFigure 4.8 Contour plots of the axial velocity field prediction of UCS forthe case of φ = 0.67 with comparison of β -PDF and the MLF-PDF for progress variable and experimental image . . . . . . 77Figure 4.9 Average temperature on exit plane for all simulated cases andvariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 4.10 Overall thermal power produced in the combustor for all sim-ulated cases using the UCS approach as per Table 4.1 . . . . . 79Figure 4.11 Progress variable c at the exit plane plotted over mixture frac-tion Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 4.12 Average NO in ppm by mass on the exit plane for all BRSseries simulations . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.13 Line plot of the axial velocity field prediction of UCS usingthe UBC mechanism and the GRI mechanism compared to theexperimental measurements and other simulations for φ = 0.77at different downstream locations . . . . . . . . . . . . . . . 83Figure 4.14 Contour plots of the enthalpy source term predicted by UCSwith the GRI mechanism using the β -PDF approach and theMLF-PDF approach for the PDF of progress variable and UCSwith the UBC mechanism compared to the LES simulation re-sults of Tay-Wo-Chong et al. [110] . . . . . . . . . . . . . . . 84Figure 4.15 Contour plots of the temperature for φ = 0.77 predicted byUCS with the GRI mechanism using the β -β PDF approachand the β -MLF PDF approach and also the prediction of UCSwith the UBC mechanism using the β -MLF PDF approachcompared to the LES simulation results of Tay-Wo-Chong etal. [109] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.1 (a) Burner schematic [19] and (b) a photograph of the experi-ment for the lifted methane-air flame in vitiated coflow . . . . 90Figure 5.2 Numerical domain of Cabra burner . . . . . . . . . . . . . . . 91Figure 5.3 Contour plots of Simulation #1 (baseline) with (a) instanta-neous temperature and (b) mean temperature (left) and meanOH mass fraction (right) . . . . . . . . . . . . . . . . . . . . 94xivFigure 5.4 Contour plots of Simulation #2 with (a) instantaneous temper-ature and (b) mean temperature (left) and mean OH mass frac-tion (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 5.5 Contour plots of Simulation #3 with (a) instantaneous temper-ature and (b) mean temperature (left) and mean OH mass frac-tion (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 5.6 Contour plots of Simulation #4 with (a) instantaneous temper-ature and (b) mean temperature (left) and mean OH mass frac-tion (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 5.7 UCS predictions of different variables along the centerline fromfour different simulations . . . . . . . . . . . . . . . . . . . . 99Figure 5.8 Temperature prediction at different downstream locations fromfour different simulations . . . . . . . . . . . . . . . . . . . . 101Figure 5.9 CO mass fraction prediction at different downstream locationsfrom four different simulations . . . . . . . . . . . . . . . . . 102Figure 5.10 OH mass fraction prediction at different downstream locationsfrom four different simulations . . . . . . . . . . . . . . . . . 104Figure 5.11 NO mass fraction prediction at different downstream locationsfrom four different simulations . . . . . . . . . . . . . . . . . 105Figure A.1 Schematic for UCS table lookup interpolation . . . . . . . . . 128xvAcronymsCFD Computational Fluid DynamicsCMC Conditional Moment ClosureCSE Conditional Source-term EstimationDCSE Doubly-Conditional Source-term EstimationDNS Direct Numerical SimulationEDC Eddy-Dissipation ConceptEVM Eddy Viscosity modelsFGM Flamelet-Generated ManifoldFIRST Fuel Injection Research for Sustainable TransportFPV Flamelet/progress variableFSD Flame Surface DensityILDM Intrinsic Low-Dimensional ManifoldISAT In-Situ Adaptive TabulationJPDF joint PDFUCS Uniform Conditional StateURANS Unsteady Reynolds-Averaged Navier-StokesxviLES Large-Eddy SimulationPDF Probability Density FunctionRANS Reynolds-Averaged Navier-StokesSGS Subgrid ScaleTGLDM Trajectory-Generated Low-Dimensional ManifoldxviiList of SymbolsLatin SymbolsSymbol Description SI Unita Speed of sound [m/s]Cmix Tunable parameter of PaSR model [-]c progress variable [-]Da Damko¨hler number [-]g gravitational force [m/s2]h Enthalpy [J/kg]i Spacial indexjα,i molecular diffusive flux of species α in dimension i [kg/(m2 s)]Kn Knudsen number [-]k Turbulent kinetic energy [m2/s2]kc Conditional scalar dissipation constant of c [1/s]kZ Conditional scalar dissipation constant of Z [1/s]l Characteristic length scale [m]ls Smallest characteristic length scale [m]L mean free path [m]m˙ Mass flow [kg/s]Ma Mach number [-]P Probability density function [-]Pth Thermal power [W]p Pressure [Pa]xviiiSymbol Description SI Unitri Inner radius [m]ro Outer radius [m]Re Reynolds number [-]Ret turbulent Reynolds number [-]SL Laminar flame speed [m/s]Seg Segregation [-]T Temperature [K]t Time [s]U Internal energy [J]u Velocity [m/s]ui Velocity in i-th dimension of the vector #»x [m/s]uL Velocity of turbulent structure of integral length scale [m/s]x Position in physical dimension x [m]xi Position in i-th dimension of the vector #»x [m]#»x Spacial position in (x,y,z) coordinates [m]Y Mass fraction [-]Z Mixture fraction [-]xixGreek SymbolsSymbol Description SI Unitα Index for speciesΓ Gamma functionγ First independent non-random conditioning variable [-]δ Kronecker delta functionε Rate of dissipation of turbulent kinetic energy k [m2/s3]ζ Second independent non-random conditioning variable [-]η Kolmogorov length [m]κ Reacting volume fraction [-]λ Air-fuel equivalence ratio [-]µ Dynamic viscosity [kg/(m s)]ν Kinematic viscosity [m2/s]ρ Density [kg/m3]τ Time scale [s]τη Kolmogorov time scale [s]τL Time scale associated with eddy of integral length [s]τC Chemical time scale [s]τi j Viscous stress tensor [kg/(m s2)]φ Fuel-air equivalence ratio [-]χ Scalar dissipation [1/s]ω Specific rate of dissipation [s]ω˙ Chemical source term [1/s]Other SymbolsSymbol Description SI UnitD Diffusivity or molecular diffusion constant [m2/s]L Integral length scale [m]O Order of magnitudeP Joint PDFS Set of all species present in the compositionxxOperatorsSymbol Description#»u Vector of u with 3 independent dimensions∂ui Partial derivative of uiui Mean value of uiu′i Fluctuating component of uiu˜i Favre averaged mean value of uiu′′i Favre fluctuating component of ui (u′′i = ui− u˜i)dui Derivative of uixxiAcknowledgmentsFirst of all, I would like to thank my parents for all that they have done for me andfor allowing me to pursue my dreams. In the darkest of hours of the past five years, Iwas lucky enough to receive continuous encouragement from various women – rolemodels and friends – whom I wish to acknowledge here: Jutta Wiemers, SabineHeller, Janine Holzkamp, Dr. Susanne Klaus and Olga O¨zkaya. Their support isinvaluable in so many regards.A special thank you to my supervisor Professor W. Kendal Bushe for his con-sistent support and patience. I want to thank him for never failing to provide adviceand for believing in the success of the project – regardless of what the challengeswere.I want to thank Professors Patrick Kirchen and Carl Ollivier-Gooch for provid-ing valuable feedback and supporting me at different stages of this work. To SoniaR. Taylor, I am grateful for her continuous efforts and motivation throughout herMaster’s research. It was always a joy to work with her. In particular, I want tothank Dr. Graham H. Hendra whose preceding research was of significant impor-tance and whose expertise I greatly appreciated in the early years of my work. Notonly was he responsible for the original development of the methodology I adoptedand used, but he was the most patient, helpful and kindest colleague and teacher Ihave had the pleasure of working with.I would like to acknowledge the financial support of Natural Sciences and En-gineering Research Council of Canada (NSERC). Essential was also the accessto supercomputing facilities such as the Mammouth Paralle`le II supercomputer ofUniversite` of Sherbrooke and the WestGrid computing center, which is where mostof the simulations were performed.xxiiLast but not least, I want to thank MTU Aero Engines AG and MTU Main-tenance Canada for their support in the past 5 years. Not only am I grateful forthe financial support, but our cooperation was an essential driving force for thisresearch. Dr. Marco Konle and Ludovic de Guillebon were invaluable in continu-ously motivating me but also providing the expertise and advice needed at varioustimes throughout the project. In particular, I want to acknowledge the opportunitiesto work together hand in hand in Munich and pushing our development to its limitsby challenging its application to real combustor applications.xxiiiExperience is what you getwhen you don’t get what you want.xxivChapter 1Introduction1.1 MotivationIt is indisputable that climate change will soon affect our daily lives significantly –even in developed countries. Heat-waves and drought during the past several yearshave been connected to a 20% spike in the price of wheat in European and U.S.markets during a three-week period in 2018 [24]. According to the EmergencyEvent Database (EM-DAT), the frequency of natural disasters recorded has in-creased almost three-fold comparing the period of 1975 to 1984 with over 1,300incidents to more recent records of over 3,900 incidents from 2005 to 2014 [54].For the past decade the transport sector – and the aviation industry in particular –have been in the spotlight and pressure is being applied to reduce the emissionof pollutants and the fuel consumption of this sector. However, even with someradical technological advances, efficiency gains have not offset the expected traf-fic growth or achieve the decarbonisation targets set by the aviation industry forthe year 2020 [4]. While many areas of energy consumption are being addressedby more sustainable forms of energy conversion into and from electric energy, theaerospace industry is not able to apply the same technology in the foreseeable fu-ture. To stand by the commitments made for 2050 by the aviation industry, drop-infuels such as biodiesel and synthetic fuels are expected to play an important role[53], but even hydrogen combustion in combination with electric power genera-tion or other hybrid constellations are being discussed as long term solutions. The1development of new combustion technology – be it a new fuel or a new genera-tion of propulsion engines – is a two-fold problem and thus there are two sourcesof motivation for carrying out the work proposed in this dissertation: science andengineering. So far, numerical simulation of combustion has been a tool used tolearn about the science behind already existing technology – the physics and chem-istry – but hardly gets used by someone developing a new combustor – let alone anew technology. Why is that?To imagine the incredible complexity of the issue, an analogy1 might help.One of the local mountains here in Vancouver, BC – Seymour Mountain – iscurrently 1,449m high. The ratio between the height of Seymour mountain andwoman of average height hiking on its soil is about the same as that between thelargest and the smallest scales of turbulence in a combustor flow field. In orderto correctly predict this flow field both the largest and the smallest of those lengthscales need to be resolved or modelled. Now imagine watching this person hikingup the mountain, while simultaneously watching the trees grow. The ratio betweenthese two speeds is about the same ratio as that of the fastest and the slowest timescale of chemical reactions occurring during combustion. Amongst the slowestprocesses can be the production of nitrogen oxides NOX , a harmful pollutant andindirect greenhouse gas, under certain conditions. To be able to predict the emis-sions of a device such as a gas turbine correctly all the scales of the physics andchemistry of the reacting flow field need to be captured. Not only that, but thereare many parameters and variables that the production rate of emissions dependson and those parameters are closely intertwined with the characteristics of the tur-bulence in that particular location. The accurate prediction of NOX emissions isespecially difficult because it requires the resolution of very small time scales andat the same time its production rate is highly dependent on thermophysical proper-ties such as temperature. It is as though one wanted to predict the exact amount ofgrowth of one particular tree in a given day. The growth rate is not a fixed value,but rather it depends on the conditions of the soil at that time, the amount of rainand sun as well as the surrounding ecosystem.1The analogy with nature is based on average values found in literature for female height andaverage pine tree growth rate as well as average walking speed of humans. Average combustorturbulence scales can be found in Lubbock and Oldfield [74].2Numerical simulations are carried out using a virtual representation of reality –the computational domain – and solving transport equations of the fluid on a grid,which maps this domain. The resolution of this domain determines the resolutionof the length scales that can be calculated directly. The smaller the turbulent lengthscales, the smaller the grid cell size, but with that the smaller the time steps ofevery iteration of the calculation. In other words, the more resolution, the morecomputational cost is associated with the simulation. Many decades of scientificresearch have provided astonishingly accurate insights into the physics and chem-istry of turbulent reacting flows and with that models have become available thatcan provide the details of the smaller scales to some degree.In the following section a brief overview of the state of the art of combustionmodelling found in academia and industry is given and the criteria derived to benecessary to consider a model for use in industrial research and development arelisted.1.2 State of the artA variety of flavors of combustion modelling techniques are available today. Theycan be categorized into four groups and while withholding the details of thesemethods until Sec. 2.3.2, an overview of their most significant advantages and dis-advantages can be summarized as follows:1. Algebraic models+ low computational cost+ easily accessible and often pre-installed in commercial software− no or very little knowledge of emissions2. Flamelet models+ low computational cost+ some are accessible to the general research communityo individual models are designed for different types of combustion (pre-mixed / non-premixed / partially-premixed)3− only applicable to flamelet regime (more on that in Sec. 2.3.1)3. Conditional Moment Closure (CMC) models+ sufficient level of resolution for chemistry and physics achievableo significant adjustments necessary to be able to simulate different typesof combustion (premixed / non-premixed / partially-premixed)− inaccessible and highly non-trivial in implementation− high computational cost4. PDF transport models+ highly accurate details on emission production and flow structures− lots of experience and know-how required for implementation and ap-plication− very high computational costThe models associated with high computational cost generally are being usedonly in academic research environments and their application is focused on labo-ratory scale flames and simple fuels. When a company wants to analyze an alreadyexisting combustor and evaluate different options for improvements to reduce fuelconsumption or emissions, they are usually limited to choose from one of the firsttwo options as the computational domain is much larger than most academic testcases and their computational resources are limited. However these applicationsare usually outside the flamelet regime – flamelet models are not applicable – andalgebraic models provide inaccurate information about emissions.These observations point to five important criteria for a suitable combustionmodel:1. Affordable overall computational cost for simulating real engine combus-tors;2. Accessibility to industry and compatibility with the existing software envi-ronment;43. Applicability to premixed, non-premixed and partially premixed combus-tion;4. Versatility with regards to fuel types and chemical mechanisms;5. Accuracy of qualitative and quantitative prediction of flame characteristicsand emissions;The Uniform Conditional State (UCS) [49, 50] model has recently been devel-oped and shows promise to fulfill all of these criteria – at least in theory. Hendraand Bushe [50] were able to verify the theoretical hypothesis through analysis ofdata from laboratory scale flames. The goal of this present work is to develop aframework in which UCS could be used as a predictive method for CFD simula-tions and to evaluate the UCS method with respect to the five criteria above.1.3 OutlineFirstly, the basic knowledge fundamental to the research presented here is ex-plained in Chapter 2. In the spirit of Criteria 3 previously described, the researchwas structured by combustion type and so is this thesis. With that the evaluation ofUCS started with the simulation of a non-premixed swirl-stabilized model aero en-gine combustor in Chapter 3. The same methodology was then applied to a seriesof premixed swirl-stabilized methane flames and improvements were made leadingto the results presented in Chapter 4. Given the findings of these two investigationsa partially-premixed lifted jet flame with vitiated coflow was chosen as a third testcase and gave an opportunity to further examine a number of parameters of themodelling approach. This research is described in Chapter 5. Altogether the inves-tigation of those three test cases gives enough information to form a preliminarypicture of the new modelling approach UCS with regards to the five criteria listedabove. Overall conclusions as well as a summary of the quantitive and qualitativeevaluation of the UCS model on the combustion spectrum is presented in Chapter 6.5Chapter 2BackgroundIn this chapter the basic physical principals which build the foundation for numer-ical simulation of fluid dynamics in combustion systems and which are relevantfor the research conducted in this thesis are outlined. The goal of numerical sim-ulations of combustion applications is to gain information that is not accessiblethrough experiments because a) the area of interest is physically hidden from viewand the conditions of temperature and pressure don’t allow optical access or theinsertion of measurement devices or b) it is too expensive to build prototypes toinvestigate a series of parameters and identify the best one. For industrial applica-tions the locations of critical importance could be – for instance: the exit plane ofthe combustor, the area closest to the combustor liner, the injector nozzle, etc. InSec. 2.1 the characteristics of the fluid state are outlined and in Sec. 2.2 the physicsof determining the states in the numerical domain is described. The fundamentalsof combustion theory and combustion modelling build on this knowledge furtherand are explained in Sec. 2.3.2.1 Fluid StateThe immediate outcome of conducting these numerical simulations is the predic-tion of the state of a fluid. This state can be defined from two different viewpoints:the kinematic state and the thermodynamic state. Both of these states may containan infinite number of properties, however a subset of these properties already de-6fines the state to an extent where all other properties can be evaluated utilizing the“state functions”. The total state of an observed medium is defined as the set ofall properties.2.1.1 Kinematic StateThe kinematic state is the set of all properties associated with motion and is de-fined by 6 independent variables – typically composed by the 3 independent com-ponents of position, #»x , and velocity, #»u .2.1.2 Thermodynamic StateThe thermodynamic state is the set of all thermodynamic properties and as suchis independent of the reference frame. It can be divided into the chemical statebasis – defined by the composition – and the physical state basis with 2 indepen-dent components. Thereby the thermodynamic state basis is made up by the massfraction of (S−1) components – to define the chemical state – along with pressure,p, or density, ρ , and temperature, T , internal energy, U , or enthalpy, h, as prop-erties of the physical state. S is the collective domain of all species present in thecomposition.2.2 Governing EquationsTo determine the state of the fluid at each point in time and space a number oftransport equations need to be solved on the computational grid: the governingequations. In the derivation of the governing equations the fluid is viewed as acontinuum in space. In order for this to be valid, the smallest characteristic lengthscale ls has to be significantly greater than the mean free path L and with that theKnudsen numberKn≡ Lls(2.1)has to be less than 1.0. In turbulent flow the Kolmogorov length scale [90] η isthe smallest length scale. Employing this convention leads to the following defini-7tion [111]Knη ≡ Lη ≈MaRe1/4(2.2)using Mach number Ma≡ u/a and Reynolds number Re. The latter being definedasRe≡ ρulµ, (2.3)where a is the speed of sound, l a characteristic length of the flow field, u thevelocity, ρ the density and µ the dynamic viscosity.Further assumptions that are commonly made in the derivation of the governingequations are as follows and further details can be found in many text books [69]:• the fluid is at thermal equilibrium• the fluid properties are isotropic• the fluid is an ideal gas and a Newtonian fluid• Stokes’ hypothesis, Fourier’s law of heat conduction and Fick’s law of diffu-sion are appliedThe continuity equation states the conservation of mass as∂ρ∂ t+∂∂xi(ρui) = 0, (2.4)where ui is the velocity in the physical dimension xi and ρ is the density.In addition as chemical reactions occur the composition of the reacting flow changes.Throughout this work, Yα represents the mass fraction of the species α , where α ∈S. One component equation∂∂ t(ρYα)+∂∂xi(ρuiYα)+∂ jα,i∂xi= ρω˙α (2.5)must be solved for each species in the composition that needs tracking. Given thatthe sum of all component equations yields the continuity equation, one of the themcan be ignored. It is most common when observing reacting flows containing airto neglect the inert species N2 and construct its mass fraction from the sum of all8other componentsYN2 = 1−N−1∑1Yα . (2.6)The source term ω˙α on the right-hand side of Eq. 2.5 is the sum of the rates of allreactions in which species α participates (multiplied by its stoichiometric coeffi-cient) per unit volume. On the left-hand side of Eq. 2.5 the molecular diffusive fluxjα,i of species α is unknown as well. The most common way of approximatingthis term is based on Fick’s first law – the law of diffusion. This is a simplificationvalid only for a binary gas mixture without pressure and temperature gradients.Calculating the diffusivity of multicomponent gases is very expensive and the er-ror associated with this simplification is assumed to be small due to the mixturecontaining a large percentage of N2, when using air as oxidizer, whereby the prop-erties of N2 should dominate the flow. With that the diffusivity Dα of species αinto the gas mixture can be calculated from the binary diffusivity Dαβ [41]. Thusthe molecular diffusive flux is defined as∂ jα,i =−ρDα ∂Yα∂x j . (2.7)The momentum equation – also know as the “Navier-Stokes-Equations” – canbe written as∂∂ t(ρui)+∂∂x j(ρuiu j)−∂τi j∂x j+∂ p∂xi= ρgi. (2.8)The left-hand side of the equation represents the following terms from left to right:the local rate of change in time, the convective transport of momentum, the molec-ular transport due to viscosity and the pressure gradient. On the right-hand side isthe gravitational force g in direction i. Assuming a Newtonian fluid, the Stokes’hypothesis can be applied revealing the following solution for the viscous stresstensor [37]τi j = µ(∂ui∂x j+∂u j∂xi)− 23µδi j(∂uk∂xk), (2.9)where µ is the dynamic shear viscosity coefficient and for incompressible flowsthe last term becomes zero.The energy balance can be written in many different forms. Here the enthalpy9equation is chosen and defined as∂∂ t(ρh)+∂∂xi(ρuih)− τi j∂ui∂x j− ∂ p∂ t+∂∂xi(ui p)+∂qi∂xi= ρuigi, (2.10)which also includes the viscous stress tensor τi j . This balance equation contains– similar to the momentum equation – from left to right on the left-hand side: thelocal rate of change with time, the convective transport of enthalpy, the moleculartransport due to viscosity, the local change in pressure, the pressure surface forcesand the heat flux. On the right-hand side is the body force due to gravitation.This system of equations is highly non-linear and its solution can be time de-pendent and chaotic. That is often how turbulence is described.2.2.1 TurbulenceIn an attempt to understand and analyze turbulence the so-called “energy cascade”has been discovered describing how energy is on average transported from largeeddies to smaller eddies. This is due to the vortex stretching phenomenon, thattransfers energy from large eddies to smaller ones through conservation of angularmomentum. Eventually the eddy size is small enough that the turbulent kineticenergy of the eddy is dissipated into heat by molecular viscosity. According toKolmogorov’s first similarity hypothesis for high Reynolds number turbulent flows,the existence of the smallest eddies only depends on the viscosity and the rate ofdissipation of turbulent kinetic energy ε . Hence, the Kolmogorov length η andtime scale τη can be defined as:η ≡(ν3ε)1/4(2.11)τη ≡(νε)1/2(2.12)where ν is the kinematic viscosity. Therefore η and τ become smaller for lowerviscosity fluid and for higher dissipation rate. The dissipation rate is related to theturbulent kinetic energy k of large eddies. In the following description the velocityof an eddy of integral length scale L is uL = O(√23 k)and the time scale is10τL =L /uL. The dissipation rate can be estimated asε = O(u3LL)=O(u2LτL)(2.13)andL = k3/2/ε [35]. In relation to the turbulent Reynolds numberRet ≡ uLLν ∼ O(√kLν), (2.14)the smallest scales can be approximated as [90]lη=Lη∼O(Re3/4t),τLτη∼ O(Re1/2t)(2.15)where L replaces the characteristic length scale l to determine the total numberof cells required to sufficiently represent the computational domain. To resolve thesmallest scales the computational grid would have to contain a minimum ofN ∼O(Re9/4t)(2.16)cells. Furthermore the time step needs to be reduced in accordance with Ret leadingto an increase in computational cost on the order of O(Re3t)[90].In Direct Numerical Simulation (DNS) of turbulent flows, all the turbulentscales are being resolved. DNS is the most accurate but computationally expen-sive method as the grid size of the computational mesh would have to be smallenough to resolve all the turbulent scales. Therefore using this approach on realturbulent flow problems would quickly exceed the capacity of available supercom-puters at this time. However, it is a very useful and necessary tool to study the smallscale phenomena of different flows and with the knowledge gained from these fun-damental studies, models can be developed to then simulate practical applicationsusing different methods in which small scales are filtered out.Depending on the context, the correct time-accurate values of the fluctuationsof variables are not of particular interest; rather what is often needed is a good,time-resolved representation of the mean of the variable. In this case a filter isapplied to derive averaged transport equations. There are different ways to go11about this. For instance the mean can be a time average, a spacial average or anensemble average.Reynolds decomposition [94] splits the instantaneous value of variable ui intoits mean value ui and its fluctuating component u′i such thatui = ui+u′i . (2.17)For Reynolds-averaging u′i becomes zero. A system of equations that is entirelybased on this concept will usually only be applied to inert fluids and given constantdensity [41].When density changes are anticipated due to compressibility, strong tempera-ture gradients or changes in chemical composition, then some variables will com-monly be represented by the density-weighted, or Favre-average instead. Favre(1965) [32] introduced a decomposition into mean and fluctuating components, u˜iand u′′i respectively, which are weighted with the density ρui = u˜i+u′′i , where u˜i =ρuiρ. (2.18)Here u′′i 6= 0, but ρu′′i = 0. This leads to the elimination of many terms in thetransport equations when applying this averaging technique because the term ρu′′iappear frequently.In combustion simulation Reynolds averaging is usually used for density andpressure and Favre averaging for all other variables. When applying the Reynoldsor Favre decomposition to Equations 2.41 - 2.10 then equations for the first mo-ments will evolve from that: the so-called Reynolds-Averaged Navier-Stokes (RANS)equations. For unsteady phenomena the expression Unsteady Reynolds-AveragedNavier-Stokes (URANS) is typically used. For example in the averaged compositionequation∂∂ t(ρY˜α)+∂∂xi(ρ u˜iY˜α +ρ u˜′′i Y ′′α)+∂ jα,i∂xi= ρ ˜˙ωα (2.19)it is now possible to interpret the individual variables as:• first moments, such as: u˜i, ρ , ˜˙ωα12• second moments, such as: u˜′′i Y ′′αDepending on the number of fluctuating variables involved the individual quan-tities are called correlations of respective order. For moments of second order,transport equations can again be derived. These will in turn contain moments ofthird order and so on. The system of equations remains unclosed. This phenom-ena is also referred to as the closure problem of turbulence and ultimately leads toloss of information with each averaging step. Moment closure methods use this ap-proach nevertheless and eventually resort to the application of models for the higherorder moments in terms of lower moments to obtain closure. This has proven towork very well for many applications of turbulent non-reacting flows. As for thechemical source term, closure at the second moment level can not be achieved dueto the highly nonlinear dependence of the reaction rate on turbulence and tempera-ture. How to tackle this particular problem will be explained in Sec. 2.3.2.2.2.2 RANS vs. LESAs previously mentioned, there are different ways of filtering for a mean value.An alternative to RANS would be Large-Eddy Simulation (LES). In LES the spa-tially filtered turbulent flow is captured. In RANS most scales of turbulence aremodelled, while in LES only the subgrid scales are modelled. This is illustrated inFig. 2.1 in the context of the energy cascade as mentioned earlier. The energy con-taining range represents the large scales characterized by the external forces andthe overall flow field. Vortex stretching leads to large scales transferring energy tosmaller eddies in the inertial sub-range, which exhibits a slope of energy contentover length scale size of −5/3 on a log-log plot. Eventually, the dissipation of en-ergy by viscosity becomes more significant in the dissipation range at the smallestturbulent scales.Applying the decomposition as per Eq. 2.18 to the momentum equation,Eq. 2.8, leaves the Reynolds stress tensor ρ u˜′iu′j and the viscous stress tensor τ˜i j isunclosed. The later is often ignored as it is small compared to the former. Commonturbulence models used to close the Reynolds stress tensor belong to one of threecategories: 1) one-equation models and 2) two-equation models and 3) Reynoldsstress models, the last of which involve trying to close and solve transport equations13log(E(k))log(k)−5/3 slopeENERGYCONTAININGRANGE INERTIALRANGEDISSIPATIONRANGE1ηResolvedDNSResolved ModelledLESResolved ModelledRANSFigure 2.1: Illustration of DNS, RANS and LES on the energy cascadefor the Reynolds stresses directly. Depending on the choice additional transportequations have to be solved leading to more unclosed terms that are then modelledfurther. The turbulence model chosen for all numerical simulations using RANSin Chapters 3 and 4 is called the ‘k−ω−SST model’ [77].The LES filter can be applied explicitly, but general practice is to allow thecomputational mesh and spacial discretization scheme to imply the filter size di-rectly. A Subgrid Scale (SGS) model is still required to capture the effects of theviscous (dissipation) range and the smaller scales of the inertial range. Assuminga direct constitutive relation between the turbulence stress and strain components,14first order models known as Eddy Viscosity models (EVM) can be used. For in-stance, the Smagorinsky model [102] serves this purpose, but its parameters areobtained assuming isotropy in the small scales and an infinite turbulent Reynoldsnumber, both assumptions not being universally applicable. A dynamic variationof this model was developed by Germano et al. [42] to eliminate these tunable pa-rameters and has become a popular choice for SGS modelling. For the simulationsperformed with LES in Chapter 5 this model was chosen.LES does not perform as well near walls and so a hybrid of RANS and LES issometimes chosen, depending on the application. A review of these models can befound in Chaouat [22].2.3 CombustionBesides the challenge of resolving and modelling the turbulent structures in a largeindustrially relevant combustor the turbulence-chemistry interaction needs to betaken into account as well. Turbulence will encourage mixing of nonuniformspecies compositions and temperature within the domain. However, this processoccurs at a much slower rate than most chemical reactions causing the system ofequations to become mathematically stiff and difficult to solve. It is necessary tounderstand some fundamental relationships between characteristics found in com-bustion to understand the modelling approaches that have been developed over thepast decades in order to solve the chemical closure problem. Some theoreticalbackground information on combustion in general will be provided in the next sec-tion, Sec. 2.3.1, followed by insight on potential modelling techniques in Sec. 2.3.2and lastly, the theoretical background of the model chosen for this research workin Sec. 2.3.3.2.3.1 Combustion TheoryA common way of differentiating phenomena present in combustion is to discussthem in terms of the combustion types, but it can also be done based on the flametypes. Both shall be presented here.15Combustion TypeCombustion is a high-temperature exothermic reaction caused by oxidizer – air inthis work – and fuel molecules meeting under certain conditions. If all the fuel andoxidizer are mixed before injection, premixed combustion is observed. The flamepropagates as a thin reacting front from burnt product to unburnt reactants. In theabsence of turbulence, the speed at which the unburned gases see the flame frontapproaching is known as the “laminar burning velocity” or “laminar flame speed”,SL, and can be calculated directly given the pressure, temperature and compositionof the mixture [122]. This composition can be described by one variable in the caseof premixed combustion: the equivalence ratio. The fuel-air equivalence ratio φisφ =(Y f uelYoxi)(Yf uelYoxi)s=(X f uelXoxi)(X f uelXoxi)s(2.20)and the air-fuel equivalence ratio λ is defined asλ =1φ, (2.21)where s refers to stoichiometric. In the case of a hydrocarbon fuel (moleculesare made up of H and C atoms only) a mixture is referred to as stoichiometric,when it is such that all fuel and oxidizer can be converted into carbon dioxide andwater (CO2 and H2O) molecules (plus inert species, if present) and still conserve allatoms. The equivalence ratio is then unity. If the mixture is such that there is excessof fuel then it is called rich (φ > 1,λ < 1) and if there is an excess of oxidizer thenthe mixture is lean (φ < 1,λ > 1). So-called lean and rich premixed flammabilitylimits can be determined based on the thermodynamic state of the fluid. Howeverthe kinetic state also needs to be taken into account. If the flow rate of the mixtureuz is higher than SL, a laminar flame would lift off and blow out. It is not assimple as that in a turbulent flame and so it is more difficult to determine theselimitations accurately for industrial applications. In industrial applications leanpremixed combustion is often used for its advantages of being easier to control insome regards as well as leading to lower temperatures and hence lower productionof oxides of nitrogen (NOX ), than would be the case for stoichiometric. Soot is16largely formed by rich combustion due to the excess carbon.Fuel and oxidizer can however be injected separately; leading to non-premixedcombustion. This type of flame is often referred to as a “diffusion flame”. A globalequivalence ratio can be defined for a combustor or a jet with coflow, giving someindication as to what can be expected from the flame. Because mixing is typicallyslow relative to chemical reactions, the overall rate of fuel consumption is limitedby the rate of mixing, which in turn depends on the turbulence – in particularmolecular diffusion and turbulent advection [44]. Under conditions that allow theflame to stabilize in one location, a rich and a lean branch of the flame front canbe found on the fuel and oxidizer side respectively. Mixture fraction is a variablewhich characterizes the reactivity of the composition at different locations. It doesso based on the mixture’s content of the atoms which are important; in the case ofcombustion of hydrocarbons, those atoms are: H, C and O. According to Bilger [7]mixture fraction Z can be defined asZ =2(YC−YC,co f low)/MC +(YH −YH,co f low)/2MH − (YO−YO,co f low)/MO2(YC, jet −YC,co f low)/MC +(YH, jet −YH,co f low)/2MH − (YO, jet −YO,co f low)/MO .(2.22)This definition is also suitable for partially-premixed combustion, which occurswhen part of the oxidizer and fuel are already premixed, but not completely. Inindustrial applications perfectly premixed or purely non-premixed combustion israrely achieved and often not desirable in the first place for reasons of temperaturecontrol or emission reduction.Flame TypeDifferent types of flames have been defined in the literature based on a num-ber of characteristic variables by several authors including Bray [10], Borghi [8],Williams [124] and Peters [85]. One way to summarize turbulent premixed flamesis in a so-called ‘Borghi Diagram’ [8] as shown in Fig. 2.2 . As per the definitionsfound earlier in Sec. 2.2 L represents the largest turbulent length scale or the in-tegral length and lF is the laminar flame thickness. Their ratio with respect to theratio of the turbulence intensity u′ and laminar flame speed SL, can be directly re-lated to the non-dimensional variables Re (describing the turbulence), the Karlovitz1710−1 100 101 102 103 10410−1100101102103L /lFu′/S LLaminarFlamesWell-stirredReactorThinReaction ZonesCorrugated FlameletsWrinkled FlameletsRe = 1Da = 1Ka = 1Figure 2.2: Borghi Diagram showing different combustion regimes of turbu-lent premixed flames (based on Borghi [8])number Ka = τη/τC (ratio of Kolmogorov time scale to chemical time scale) andthe Damko¨hler number Da = τL/τC (ratio of integral time scale to chemical timescale). If the turbulence intensity is not too high and the flame thickness is lessthan the Kolmogorov length, a premixed turbulent flame can be viewed as an en-semble of many premixed laminar flames and the so-called “flamelet” concept canbe applied, as will be discussed in more detail in Sec. 2.3.2. While these character-izations are of great importance when discussing different options for turbulence-chemistry interaction modelling, their connection to industrial applications doesnot appear strikingly obvious. However, in order to correctly predict, for instance,the location at which the flame stabilizes, the implications that follow from the flowand turbulence characteristic strongly affect the chemistry and with that the mod-elling approach must be chosen with care. Particularly for swirl-stabilized flames,18as investigated in research chapters 3 and 4, and for lifted flames in chapter 5 suchmatters are non-trivial. For example, the BRS burner of chapter 4 is operated in theregion marked with the red circle.One more piece of the puzzle is missing to discuss combustion modelling andthat is the chemistry.Chemistry ModellingThe global reaction is considered when determining the stoichiometric compositionas well as for some rough estimations and comparison of different fuel options.In the case of methane (CH4) – the simplest of hydrocarbon fuels – and air, forexample, a stoichiometric mixture would react as follows:CH4+2(O2+3.76N2)−−→ CO2+2H2O+7.52N2 (2.23)If in fact all CH4 molecules present were to react with all the air present, completecombustion would be reached. In the case of complete combustion, the global re-action would be able to correctly predict the energy release in a combustion cham-ber. However in reality there are many more possibilities of how the moleculescan interact and react with each other in which other species can be formed andbecome involved along the way to the final products of the equilibrium state. Theequilibrium state is attainable provided that the mixture has enough time to fullyevolve to it and it again depends on a number of thermodynamic variables suchas pressure and temperature. Since the intermediates are being neglected in theglobal reaction, the reaction progress and the production of pollutants can not bedetermined. Even the position of the flame is often incorrect [41]. Depending onthe purpose of the numerical study using global mechanisms can deliver impor-tant initial insights into an application as long as care is taken when evaluating theresults.To better understand and predict the composition in space and time elementarychemical kinetic mechanisms are developed based on laboratory scale experimentswith highly advanced measurement techniques. These detailed mechanisms areconsidered to be the most complete descriptions of the behavior of the chemistry;they consider a large number of intermediate species and hundreds of possible19reactions to replicate many different pathways to reach equilibrium. Tab. 2.1 showssome examples of detailed mechanisms used in the course of this thesis and thenumber of species and reactions included in them.Table 2.1: List of mechanisms used in the course of this workMechanism Fuel Species Reactions ReferencesGRI-3.0 CH4 53 325 [103]UBC-3.0 CH4 71 379 [51]Wang C2H4 52 367 [120]It is from these mechanisms that one can derive the chemical source term –the reaction rate of a species – needed to solve the component transport equation(Eq. 2.5), but in order to do that all species and reactions included in the mecha-nism need to be considered at the same time leading to a vast range of time scalesand with that a stiffness in the system of equations, which is only manageable un-der extremely high computational costs. Fig. 2.3 illustrates the range of time scalespresent in a detailed chemical mechanism in comparison to the time scales of tur-bulence in a flow field such as is relevant for a turbulent flame in a combustor forexample.chemistryflowphysics100 sslowe.g. NOX formation10−2 s 10−4 sintermediate10−6 s 10−8 sfaststeady statepartial equilibriumtime-scales of flow, transport and turbulenceFigure 2.3: Turbulent time scales in comparison with chemical time scales(adopted from Warnatz et al. [122])There are ways of lessening the efforts by a) reducing the mechanism or b) em-ploying a manifold technique to tabulate the chemistry. Reduced mechanisms aregenerally targeted at specific operating conditions and must be chosen with care,but if an educated choice is made, they can provide good results. Tabulation tech-niques allow the user to determine the chemistry a priori and tabulate the resultsin a clever way to reduce the computational cost and decouple the fluid mechanics20from the chemical kinetics – alleviating the numerical stiffness issues caused by thelarge range of time scales present in combustion. Only certain types of combus-tion models have the ability to use these types of chemistry modelling successfully,some of which will be discussed in Sec. 2.3.2. Examples of manifold techniquesare given here, but the ability to utilize them to replicate turbulence-chemistry in-teraction well depends entirely on the combustion model of choice:1. Intrinsic Low-Dimensional Manifold (ILDM) [75] was one of the firstmanifold techniques ever suggested. Mass and Pope used a Jacobian analy-sis of the governing equations of reactive scalars and found an intrinsic low-dimensional manifold. This means that reaction progress practically takesplace only in a small fraction of the highly multidimensional space whichrepresents all possible species compositions.2. In-Situ Adaptive Tabulation (ISAT) [89] was developed under the assump-tion that the fluid state remains within a relatively small area of thermo-chemical space. This allows for the table to be filled “in-situ” – meaning onthe fly during runtime of the CFD simulation. The idea can be employed byany manifold technique which is generated locally.3. Flamelet-Generated Manifold (FGM) [114] is based off the same prin-ciple as ILDM. Individual flamelet solutions are generated using a one-dimensional laminar premixed flame. The trajectories of these flameletsbuild the manifold. This method is suitable for premixed combustion only.4. Trajectory-Generated Low-Dimensional Manifold (TGLDM) [52, 88, 91]can be used for non-premixed combustion by generating individual mani-folds for different mixture fractions and interpolating between them. The as-sumption for each manifold is that the S-dimensional space can be reduced toa two-dimensional space made up of the most important products (or productcombinations) because the trajectory of each mixture would quickly evolveto this space and progress from there.Besides the challenges involved in solving the problem of stiffness in the sys-tem of equations, there are different phenomena occurring on micro and macro21scales of the combustion chemistry that are still not fully understood – let alonecorrectly modelled – in current chemical kinetic mechanisms. For instance, theformation of NOX can have different origins. Three formation mechanisms havebeen identified each with its own characteristics [112]: (1) thermal NOX ; (2) fuelNOX ; and (3) prompt NOX . Thermal NOX is determined by the molar concen-tration of nitrogen and oxygen and the local temperature. At temperatures below1300 K the rate of formation of thermal NOX is essentially negligible. Fuel NOXresults from the oxidation of nitrogen already present in the fuel and is thereforeonly relevant for fuels containing a significant amount of nitrogen (e.g. coal). Thelast type, prompt NOX , is formed from molecular nitrogen in the air oxidizingwhen combined with fuel and therefore the probability for prompt NOX formationis much higher in fuel-rich regions. In combustors the chemical and physical pro-cesses determining the rate of NOX formation are: (i) atomization and vaporizationof liquid fuel, (ii) turbulent mixing of fuel, air and combustion products, (iii) chem-ical kinetics of fuel oxidation, and (iv) chemical kinetics of NO formation [119].Turbulent mixing rates typically have a much greater influence on NOX emissionsthan do spay characteristics in large industrial combustors [112]. Mixing rates arevery important in thermal NO since they determine the residence time at a certaintemperature. For thermal NOX formation the chemical kinetics of fuel oxidationare not important but they can be for prompt NOX and fuel NOX formation [112].Combustion, prompt NOX and fuel NOX all occur on about the same time scale;thermal NO is much slower. There has been much disagreement in the literatureamong different authors on the rate of controlling reactions for thermal NO andthe abundance of prompt NOX [79]. Due to the complexity and incomplete under-standing of the processes involved, prediction of NOX emissions is difficult.2.3.2 Combustion ModellingFor the purposes of aiming to simulate combustion for the development of indus-trial applications, the focus is on turbulent flames and the goal is to replicate theturbulence-chemistry interaction with the combustion model. As will become obvi-ous in the following subsections, some of the models use averaged mean quantitiesto calculate ˜˙ωα and solve the chemical closure problem. To account for the turbu-22lence, these models use simplified algebraic formulations that are usually special-ized and tuned to work well for specific types of flames and combustion regimes.They are known as algebraic models. The relations between source terms, ther-modynamic quantities and species mass fractions are highly non-linear and hardto capture this way such that they would be universally applicable. Neverthelessthese models are widely used and particularly common in industry, which is whysome examples will be further described below.Another popular approach used in academia and sometimes industry is the cat-egory of laminar flamelet models [84]. They are based on the assumption thatthe chemical length and time scales are smaller than the smallest scales associatedwith turbulence. Hence, the chemical reaction can be assumed to occur in a regionwhere the flow is locally laminar. Therefore their validity is limited to the flameletregime; in other words: high Da and small reaction zone thickness relative to theKolmogorov scale.Other types of models begin their assumptions by noting that˜˙ωα (ρ,T,Y1,Y2, . . . ,YN−1) 6= ω˙α (ρ, T˜ ,Y˜1,Y˜2, . . . ,Y˜N−1) (2.24)in the context of turbulent flames. The two major categories that can be foundfor these types of models are: Probability Density Function (PDF)-transport mod-els [87] and Conditional Moment Closure (CMC) models [63].While discussing the different modelling approaches in this section not only theacademic value of the technique but also its suitability with regards to the Criteriamentioned in Sec. 1.2 will be a focal point and a summary of this evaluation canbe found in Sec. 2.3.2.Algebraic modelsSome examples of algebraic models readily available in most open-source andcommercial CFD software are the Eddy-Breakup-Model [104] for premixed com-bustion or the Eddy-Dissipation Concept (EDC) [76], which can be viewed as anextension for non-premixed combustion. In premixed flames fuel and oxidizer arepresent in fine-structures (eddies) encountering regions of hot combustion prod-ucts and if very fast chemistry is being assumed, then the break-up process of the23eddies dictates the combustion progress and chemical reaction rates become irrel-evant [41]. Filtered reaction rates are calculated based on the filtered compositionand turbulence variables (for instance turbulent kinetic energy k and its dissipa-tion rate ε) [76]. Similarly in diffusion flames the fluid transport becomes the ratelimiting process, so that the mixing on a molecular level (dissipation) dictates thecombustion progress. The EDC model can be used with finite-rate chemistry aswell. In this case the filtered reaction rates ω˙α are represented as weighted aver-ages of the reaction rates in the fine-structures ω˙∗α and their surroundings ω˙0α sothatω˙α = κω˙∗α +(1−κ)ω˙0α , (2.25)in which κ is the reacting volume fraction and the reaction rates in the fine-structuresand the surroundings are calculated using the respective mean thermochemicalstates [33]ω˙∗α = (ρ,Y∗α ,T∗) and ω˙0α = (ρ,Y0α ,T0). (2.26)The Partially Stirred Reactor (PaSR) model [95] is another of the standard com-bustion models implemented in OpenFOAM [83]. The underlying assumption is thesame as in the EDC model except that the reacting volume fraction κ is typicallyestimated based on theoretical knowledge or DNS data [2, 107]. A chemical timescale τC ≈ lF/SL representing the universal combustion reaction and a time scaleτ∗ related to turbulence become the weights by which the reacting volume fractionκ is calculated:κ ≈ τCτC +Cmixτ∗, (2.27)where Cmix is a tunable parameter to scale the turbulence mixing time scale.Many algebraic combustion models are based on the assumption of very fastor infinitely fast chemistry; making it impossible to reproduce phenomena such asautoignition, extinction and re-ignition. Nevertheless depending on the purpose ofthe numerical study using algebraic models with global reactions can neverthelessbe employed to provide useful information about the flow field and temperaturedistribution in the chamber as well as material temperatures of the combustor andprediction of thermal power outputs.24Flamelet approachNon-premixed laminar Flamelet models are based on Peters [84, 85] and his as-sumption that the flame is thin compared to the length scales of the flow – mean-ing the thickness of the flame is less than the smallest length scale of turbulencepresent: the Kolmogorov length scale η . The idea is that chemistry and flow canbe decoupled due to the separation of scales and a library can be built a priori froman ensemble of numerous laminar one-dimensional flames. This allows for a largedegree of freedom in terms of complexity of fuels and chemical mechanisms usingthis method at low computational cost. Species transport equations are being re-placed by other scalars based on the type of combustion that is being considered.The thermochemical state of the fluid is obtained from the flamelet table basedon the local value of the mixture fraction [9]. The influence of the fluid dynam-ics on the flame is usually represented by the stretch rate or the scalar dissipationrate which is then used for tabulation along with the mixture fraction. A more re-cent approach is to use a progress variable instead when simulating non-premixedcombustion with a so-called Flamelet/progress variable (FPV) approach [86].In premixed flames the assumption is that chemical reaction takes place in thinpropagating surfaces that locally resemble laminar flames [47]. Many differentapproaches have been developed within the laminar flamelet framework resultingin a large variety of modelling approaches such as the Bray-Moss-Libby formalism[11], the Flame Surface Density (FSD) model [47], the G-equation [60] and manymore.Over time many different flamelet models have been developed to provide so-lutions for many different types of combustion problems, however regardless of thecombustion type this model is always limited by its base assumptions: it is onlystrictly valid in the flamelet regime. This needs to be considered when applyinga flamelet model to an industrial application as it can greatly affect the validityand accuracy of the results. For instance when a one dimensional problem can nolonger be assumed – for example, when exhaust gases are being recirculated [41].In this case the chemistry would need to be adjusted.25PDF-transport modelsAs the name suggests, a transport equation is being solved for the Probability Den-sity Function (PDF) to describe the evolution of the fluid transport and composi-tion progress. The chemical source term for species transport in that equation isclosed, however new unclosed terms appear that need to be modelled such as thediffusion terms. Furthermore an integral-differential equation with more than fourindependent variables is obtained making it near-impossible to solve the systemof equations in a conventional Eulerian framework on the computational grid. Tomake the PDF transport models feasible with today’s standard of computationalpower several sub-classes of this modelling approach have evolved. Amongst themost popular ones are the Lagrangian PDF model [62] and the Stochastic Fieldsmodel [13, 58]. The former allows for efficient handling of the high dimension-ality of turbulent combustion by coupling the transport of velocity and pressurefields in a conventional Eulerian framework to the evolution of a large number ofparticles representing the transported PDF on a Lagrangian particle-mesh. Thisapproach has become particularly popular for low-Mach number reacting turbu-lent flows and is also suitable for high-Mach numbers [25]. The Stochastic Fieldsmodel on the other hand is based on a strictly Eulerian formulation to form a sys-tem of stochastic partial differential equations that is stochastically identical to themodelled PDF evolution equation [96, 113]. In practice, a large set of stochasticfields is evolved and statistics are recovered through weighted sums upon thesedifferent realizations. [96] A main advantage is that the solutions give rise to fieldsthat are continuous and differentiable in space and which are thus free of spatiallyvarying stochastic errors. [13] In all cases the Monte Carlo method is employed togenerate the stochastic fields used in both Lagrangian PDF and Stochastic Fieldsmethods.It has been proven that PDF-transport models provide a great opportunity fordetailed investigations of the physical processes and it is possible to achieve ex-cellent emission predictions [58, 117, 126]. Another major advantage compared toall other modelling approaches is that with this type of model any chemical kineticmechanisms can be used and complete flexibility with regards to the combustiontype is implied [48]. The utilization of ISAT [55] and stochastic fields [57] and26more recently the application of sparse particle methods enabled by the multiplemapping conditioning model [61, 127] have brought down the cost significantly.Despite these improvements the computational cost of PDF transport models re-mains very high in comparison to other modelling approaches and much know-howis required to implement such a model.Conditional Moment Closure modelsThe CMC method [63] provides a middle ground between the extremes of lami-nar flamelet models and PDF-transport models; leading to good predictions withconsiderably less computational cost than the PDF transport models. Based on thefundamental hypothesis that most fluctuations in the reactive scalars of interest areassociated with fluctuations in certain characteristic quantities, one or multiple ofthese quantities are chosen to be conditioning variable(s) γi. In most implemen-tations of CMC only one conditioning variable γ is used and in the case of non-premixed combustion, the obvious choice is mixture fraction. For instance, whenanalyzing the Sandia measurements [3] for flame D at one location downstreamregarding the correlation between mixture fraction and water mass fraction YH2O,a clear trend is noticeable amongst the scattered data in Fig. 2.4. The curve plot-ted through the data is then the function of the mean conditional averaging of thewater mass fraction (YH2O|Z). It is apparent that the departure of the line is muchsmaller in magnitude than the overall fluctuation for measurements represented bythe collective of all the dots in Fig. 2.4. A similar plot could be generated for othermajor species mass fractions or thermodynamic properties such as temperature.Conditional transport equations are then derived using the assumption that theconditional fluctuations of the scalar quantities are small and can therefore be ne-glected. Under this assumption, the conditional mean of the chemical source term(˜˙ωα | γ) can be calculated if the conditional state of the fluid is known:(˜˙ωα | γ)(ρ,T,Y1, . . . ,YN−1) = ω˙α((ρ|γ),(T˜ |γ),(Y˜1|γ), . . . ,( ˜YN−1 | γ)). (2.28)Mixture fraction is a conserved scalar and so its balance equation has no chemicalsource term and therefore no additional closure problem is introduced. The meanand variance of mixture fraction can successfully be transported using moment27(YH2O|Z)Y H2OFigure 2.4: Sandia D data for YH2O with respect to mixture fraction Z at z =30djet and r = 12mm; modification of image in Hendra [49]closure methods.To evaluate an unconditional chemical source term and use it in the transportof the filtered species, a PDF P(γ) has to be presumed. The inner product of theconditional source term with P(γ) gives the filtered source termω˙α =∫ 10(ω˙α |γ) ·P(γ)dγ. (2.29)Over the past decades CMC has become a well-established approach suitable forsimulations of complex geometries, parameter studies and optimizations in funda-mental research [6, 14, 81]. A major challenge in conventional CMC implemen-tations is that transport equations include the conditioning variable as independentvariables to the system. Naturally this increases the computational expense signif-icantly and therefore CMC has, for the most part, been limited to using only oneconditioning variable [21, 68, 97]. Therefore the use of one version of the CMCmethod which would be universally applicable to all types of combustion has notbeen feasible. Furthermore, the implementation into an existing CFD environmentis non-trivial and challenging for potential industry partners and the computational28cost remains quite high compared to laminar flamelet or algebraic models. Never-theless the results that can be achieved with this method are of higher accuracy thanthose of flamelet models, most specifically for flames not in the flamelet regime.Evaluation summaryThe criteria to be satisfied by a combustion model to deem it suitable for use inindustry as a predictive method to develop new technology were listed in Sec. 1.2.In short they are:1. Computational cost (low preferred)2. Ease of integration (high preferred)3. Versatility with regards to combustion types (high preferred)4. Flexibility towards fuel types (high preferred)5. Accuracy of emission predictions (high preferred)Based on the literature review of the state-of-the-art combustion modellingtechniques Tab. 2.2 gives a summary of their evaluation – highlighting again theadvantages and disadvantages of each modelling approach.The conclusion is that none of them seem fit to fulfill all five criteria. Twospecializations of the CMC method are conceivably able to satisfy all criteria: 1)the Conditional Source-term Estimation (CSE) method, which has been developedand tested for some time already, but only on laboratory scale flames, and 2) theUniform Conditional State (UCS) [49, 50] method, which has only recently beenproposed.Bushe and Steiner [16] developed the CSE method as a computationally advan-tageous extension to CMC. The fundamental difference is the idea of evaluating theconditional averages of the reactive scalars from the unconditional mean values inthe computational domain rather than solving the transport equations for the condi-tional averages. Mathematically, the unconditional mean value of a reactive scalarY˜i is defined by the conditional mean value (Yα |γ) and the PDF P of the condition-ing variable γY˜α =∫ 10P(γ) · (Yα |γ)dγ, (2.30)29Table 2.2: State-of-the-Art combustion model evaluationCriteria AlgebraicLaminarflameletCMCPDFtransport#1ComputationalCostlow low medium very high#2Ease ofIntegrationhigh medium low low#3VersatilityregardingCombustionTypemedium low medium high#4Flexibilityregarding FuelTypemedium high medium high#5 Accuracy low medium high very highwhere γ is the sample space variable and its range is from 0 to 1.0. The proposalof Bushe and Steiner [17] meant that the conditional averages of the two majorspecies mass fractions (YCO2 |γ) and (YH2O|γ) could be obtained through inversionof the integral equation 2.30 and these could then in turn be used to look up allconditional source terms from a low-dimensional manifold. In particular TGLDMand FGM are used for non-premixed and premixed combustion respectively. Forthe inversion, a PDF has to be presumed and then later on used again to evaluate theunconditional source terms for species transport. This leads to a strong dependenceon the PDF and its accurate representation of the behavior of the conditioningvariable. Furthermore, when inverting an integral equation, one typically findsit necessary to provide regularization to obtain a physically meaningful solution.This can become a non-trivial issue.CSE has been successfully applied in non-premixed [121] [46] [70] and pre-mixed flames [98] [27] and a two-condition version, Doubly-Conditional Source-term Estimation (DCSE), has been developed by Dovizio [28] to simulate partially-premixed combustion. Major challenges were encountered when attempting toapply this model to swirl-stabilized flames leading to divergence of the solutionand also when trying to use the low-dimensional manifold method, TGLDM, on30fuels other than methane. These results significantly reduce the versatility of theCSE (and DCSE) approach in its current state.The theory behind the remaining viable option, UCS, suggests exceptionalqualities to suit the needs of application in industry and academia alike. UCSis described in detail in Sec. 2.3.3 and the results of its applications as a predictivemethod to three test cases can be found in Chapters 3 through 5. This will allowfor a detailed qualitative and quantitative evaluation of UCS with regards to the fiveCriteria similar to Tab. 2.2.2.3.3 Uniform Conditional State (UCS) ModelThe UCS model [49, 50] is a variation and at the same time a specialization ofthe Conditional Moment Closure (CMC) method described in the previous section.In addition to the CMC chemical closure hypothesis, UCS is built upon the as-sumption that the conditional state of the gas is statistically independent from thephysical location within the burner, in which case the thermodynamic state can bedefined entirely by the conditional state. Considering only one conditioning vari-able as universally representative of the turbulence-chemistry interaction might bereasonable in very simple burner geometries, for instance, at low Reynolds num-bers and when the combustion is purely non-premixed or perfectly premixed. Inindustrial applications regions of partial premixedness are far more likely. A re-cent study [15] showed excellent results representing the conditional state of allflames in the Sandia series [3] using two conditioning variables - mixture fractionZ and progress variable c based on CO2 mass fraction, while completely neglectingthe location of the measurement point in the flame. A deviation of less than 5%from the mean value of any species mass fraction and the temperature suggests thatthe UCS hypothesis is valid – at least in the case of the Sandia series. Applyingthis assumption to the CMC equations leads to the UCS equations – the transportequations of the conditionally-filtered species mass fractions (˜Yα |γi)∂ (˜Yα |γi)∂ t+nc∑j=1˜(ω˙c j |γi)∂ (˜Yα |γi)∂γ j = (˜ω˙α |γi)+nc∑l,m=1∂ 2(˜Yα |γi)∂γl ∂γm(χlm|γi), (2.31)31where the conditionally-filtered reaction rate of species α is (˜ω˙α |γi) and theconditionally-filtered scalar dissipation is (χlm|γi). The index i indicates the re-spective independent non-random conditioning variables where {γi} = {ζ ,γ} fortwo conditioning variables. ˜(ω˙c j |γi)is the conditionally-filtered chemical sourceterm of the conditioning variables {c j(x, t)}. Fickian diffusion #»j α = −ρD #»∇Yαand a single global diffusivity D were presumed for the derivation and a detaileddiscussion on this can be found in Kuo [69]. More details on the derivation of theUCS equations are given by Hendra and Bushe [50] and Hendra [49].It should be mentioned at this point that even though the derivation is en-tirely different the equation that evolved is identical to that of the multidimensionalflamelet-generated manifold (MFM) method [82]. Despite the fact that the equa-tions take on the same form, UCS remains free of the limitations associated withMFM due to the differences in simplifications and assumptions which were ap-plied during the derivation of the UCS equations compared to those applied forMFM. Therefore detailed chemistry can be solved in the conditional space withoutneglecting the effects of diffusion and dissipation.In summary, the most important assumptions which have been made in thederivation are as follows:1. First Conditional Moment Closure approximation: fluctuations about theconditional state {γi} are sufficiently small:(ω˙α |γi)≈ ω˙α((ρ|γi),(T |γi),(Ys 6=α |γi))(2.32)2. Uniform Conditional State assumption: conditionally-filtered thermodynamicproperties are spacially uniform.3. Fick’s first law of diffusion from 1855 and single global diffusivity D [69]:the molar flux due to diffusion is proportional to the concentration gradient.#»j α =−ρD #»∇Yα (2.33)4. Uniform Conditional Scalar Dissipation assumption: the conditional aver-age of the scalar dissipation is also assumed to be independent of #»x .325. Filter definition: the filter kernel has spatial extent only, i.e. a filtered field attime t0 has no dependence on an un-filtered field at time t1 6= t0.The conditioning variables were chosen to be the same as they had been inprevious studies [15, 50]: mixture fraction Z and progress variable c. Mixturefraction Z is the obvious choice for the conditioning variable of non-premixedflames and represents the degree of mixedness between oxidizer and fuel. Usinga progress variable c as one of the conditioning variables is a common approachfor premixed flames and is widely used both in flamelet [47] as well as some CMCapproaches [115]. It reflects the current state of the fluid with respect to the com-bustion progress and can be defined based on temperature or a major species massfraction or a sum of multiple species mass fractions. Using both Z and c as condi-tioning variables leads to a chemistry solution which is theoretically applicable onthe whole spectrum of premixed, non-premixed and partially-premixed combus-tion.In accordance with Bilger [7] Z is defined as per Eq. 2.22. For the case ofnon-premixed combustion this results in Z ranging from 0 to 1 - referring to pureoxidizer and pure fuel respectively. As for the case of partially-premixed combus-tion Z = 1.0 is associated with the jet stream and Z = 0.0 with the coflow stream.Traditionally c is defined based on the mass fraction of major species CO2c =YCO2YCO2,equi(Z)or c =YCO2YCO2,cc(Z), (2.34)where the subscript equi refers to the equilibium condition and cc to complete com-bustion according to the Burke-Schumann definition [125] as the reference value.Both options will be explored in the research chapters. Furthermore the effect ofthe definition of c was investigated by considering a second option of defining itbased on the sum of YCO2 and YCOc∗ =YCO2 +YCOYCO2,equi(Z)+YCO,equi(Z). (2.35)The UCS equations (Eq. 2.31) are fully closed with the exception of the con-ditional scalar dissipations (χii|γi). Three different approaches are conceivable formodelling this term leading to three different options for implementing the UCS33solver into the CFD solver:1. Fully coupled: Chemistry is being solved on the conditional mesh itera-tively at every time step while the conditional scalar dissipation is extractedfrom the physical mesh of the CFD simulation using an inversion of the inte-gral equation similar to the CSE approach mentioned at the end of Sec. 2.3.2.2. Partially coupled: Chemistry is being solved on the conditional mesh it-eratively after a certain number of time steps while the conditional scalardissipation is extracted from the physical mesh of the CFD simulation usingan inversion of the integral equation similar to the CSE approach mentionedat the end of Sec. 2.3.2.3. No coupling: Chemistry is being solved on the conditional mesh iterativelya priori using a presumed conditional scalar dissipation function leading tocomplete separation of conditional and physical space.For the purposes of this thesis the option of least computational expense - No cou-pling - was chosen as the first implementation to explore.Hendra and Bushe [50] carried out an investigation on the Sandia flames C-F [3]showing only a small dependency of the result on the value of the conditional scalardissipation rate in that particular case. In all the simulations carried out in this workthe conditional scalar dissipation of mixture fraction (χZZ|ζ ,γ) has been modelledusing the inverse of the complementary error function:(χZZ|ζ ,γ)≈ kZ exp[−2(erfc−1(2ζ ))2] . (2.36)As for the conditional scalar dissipation of the progress variable, the choice is notas obvious and so a constant function was applied initially(χcc|ζ ,γ)≈ kc (Option A) (2.37)and later on began to be questioned and was therefore part of an investigation pre-sented in Chapter 5. A similar definition to that of the conditional scalar dissipation34of mixture fraction was explored:(χcc|ζ ,γ)≈ b(Z) exp[−2(erfc−1(2ccequi))2](Option B) (2.38)whereb(Z) = kc exp[−2(erfc−1(2ZZs))2](2.39)and the subscript s refers to stoichiometric. This has previously been tested byNguyen et al. [82] in a study of the MFM chemistry; however they only exploredhow the chemistry table is affected and did not use the result in a CFD simulationof a burner.Fig. 2.5 illustrates the procedure of using the UCS solution to solve for thereactive flow in a CFD simulation as per the No coupling-approach. Theyxphysical space conditional spacec˜, c˜′′2, Z˜, Z˜ ′′2+ P˜(ζ ,γ)UCS TableZc˜˙ωc, T˜ , ( ˜˙ωi, Y˜j)Figure 2.5: Flow chart schematic for using UCS as a predictive method in aCFD simulationa priori simulation of the chemistry in conditional space is carried out on a cus-tomized solver, UCSFoam, which is embedded in the OpenFOAM-framework. Af-35ter processing the conditional reactive scalars with the appropriate assumed PDFsfor the conditioning variables, the chemistry is stored in the UCS Table shownin Fig. 2.5. The implementation of UCS as a predictive model for turbulence-chemistry interaction was carried out using an existing framework for combustionmodelling in OpenFOAM, the rhoReactingFoam solver [83, 123]. One advan-tage of the resulting custom solver, rhoFlameFoam, is that it does not depend onthe choice of turbulence modelling and can be used in an LES context as well asRANS without modifications.In the CFD simulation of the combustor in question perfect gas was assumedusing the ideal gas law [20]pV = mRT, (2.40)where V is the volume and R the specific gas constant, and all thermodynamicproperties are evaluated using the JANAF thermodynamic tables. The fluid stateis then computed by solving the governing equations (see also Sec. 2.2) in theirfiltered form. While the continuity equation∂ρ∂ t+∂∂xi(ρ u˜i) = 0, (2.41)ensures overall mass conservation, momentum is conserved by solving the filter-averaged Navier-Stokes equations∂∂ t(ρ u˜i)+∂∂x j(ρ u˜iu˜ j−ρ u˜′′i u′′j)=∂ τ˜i j∂x j− ∂ p∂xi+ρgi. (2.42)Turbulence modelling is required to close this equation and ought to be chosenbased on the filtering approach for the governing equations. Assuming a Newto-nian fluid, the viscous stress tensor in its filtered form becomes:τ˜i j = µ(∂ u˜i∂x j+∂ u˜ j∂xi)− 23µδi j(∂ u˜k∂xk). (2.43)The nature of NOX emissions, for instance, requires its direct transport in phys-ical space as it is slow forming and does not reach equilibrium state in the combus-36tor. To track such certain species of particular interest their transport equations∂∂ t(ρY˜α)+∂∂xi(ρ u˜iY˜α +ρ u˜′′i Y ′′α)+∂ jα,i∂xi= ρ ˜˙ωα (2.44)are solved as well and the source-terms ω˙α are obtained from the UCS table, whichcontains the unconditional values of all reactive scalars and thermophysical proper-ties of interest. The temperature field of the physical space too is obtained from theUCS table in the same manner as the chemical source-terms. These unconditionalvalues are obtained by computing the inner product of the conditional source-termsof the UCS chemistry solution with a presumed joint PDF (JPDF) model P˜(ζ ,γ)which in turn is a function of the mean and variance of the conditioning variables:ω˙α =∫ζ∫γ(ω˙α |ζ ,γ) · P˜(ζ ,γ) dγdζ . (2.45)Unconditional values of reactive scalars and source-terms can thereafter be lookedup in the UCS table and used to transport all species of interest in physical spacealong with the conditioning variables and their variances, which are necessary forthe look-up. In this work, since neither the UCS solution nor the JPDF presumedfunction are changing with time, significant computational cost-savings are real-ized by pre-computing the inner product at fixed mean and variance values Z˜, Z˜′′2,c˜ and c˜′′2 and tabulating the results for subsequent lookup during the CFD calcula-tion. It is being assumed that the two marginal PDFs P˜(ζ ) and P˜(γ) are statisticallyindependent and therefore the joint PDF P˜(ζ ,γ) can be constructed from the outerproduct of the two marginal PDFs. More details on the PDF approaches chosen forthis work can be found in Sec. 2.3.3. The UCS tables used in this work contain theunconditional values and thus source terms for species transport and for the trans-port of mean and variance of the conditioning variable c as well as temperature andunconditional species mass fractions. These values can be extracted from the tablevia a look-up and interpolation algorithm described in Appendix A .To perform this look-up, four additional transport equations need to be solvedby the custom solver. Those are the transport equations of the mean and variance37of the conditioning variables Z and c:∂∂ t(ρZ˜)+∂∂xi(ρ u˜iZ˜)=∂∂xi(ρD∂ Z˜∂xi)(2.46)∂∂ t(ρZ˜′′2)+∂∂xi(ρ u˜iZ˜′′2)=∂∂xi(ρD∂ Z˜′′2∂xi)+2ρD∂ Z˜∂xi∂ Z˜∂xi−ρχZZ (2.47)∂∂ t(ρ c˜)+∂∂xi(ρ u˜ic˜) =∂∂xi(ρD∂ c˜∂xi)+ρ ˜˙ωc (2.48)∂∂ t(ρ c˜′′2)+∂∂xi(ρ u˜ic˜′′2)=∂∂xi(ρD∂ c˜′′2∂xi)+2ρD∂ c˜∂xi∂ c˜∂xi−ρχcc+2c′′ω˙c.(2.49)The last term on the right-hand side of Eq. 2.49 is unclosed in the CFD contextand requires an additional entry in the chemistry table, where it can then be lookedup. It is being approximated asc′′ω˙c = (c− c˜)ω˙c ≈∫ζ∫γ∫ 10(γ∗− c˜) · ˜(ω˙c|ζ ,γ)(γ∗) · P˜(ζ ,γ,γ∗) dγ∗dγdζ , (2.50)where γ∗ is the control variable.PDF ModellingMany types of combustion models resort to the use of statistical methods to ac-count for the chaotic nature of turbulence and its interaction with the chemistry asdescribed in Sec. 2.3.2. Laminar flamelet models [84], conditional moment clo-sure models [63] (including CSE [17] and DCSE [28]) and now UCS do not relyon the transport of the PDF but utilize knowledge about its shape, based on whicha simple function of the lower statistical moments can be defined. For modellingprobability density behavior with respect to mixture fraction in particular, the β -PDF has been greatly successful in combination with various combustion models[71, 86, 121] and therefore it was used as the presumed PDF for mixture fraction in38this work. As for the progress variable, the correct distribution is only guaranteedat the extremes of variance being maximum (c˜′′2 = c˜(1− c˜)) or zero [56], whileeverywhere else the reaction progress is being either over- or under-predicted bythe β -PDF [12, 99]. Often this model is nevertheless used for premixed flames[34, 64] for simplicity and so it was initially used for the non-premixed test casestudied in this work in chapter 3. More recently the Modified Laminar Flamelet(MLF) PDF [56] has become a viable alternative and has shown great success whenapplied to laboratory-scale premixed flames [28, 100]. This approach was tested inchapter 4 for investigation of premixed flames and was furthermore used in chap-ter 5 for the simulation of a partially-premixed lifted jet flame.To build the JPDF, statistical independence of the two conditioning variables isassumed following previous studies [26, 116] and so P˜(ζ ,γ) can be constructed asP˜(ζ ,γ) = P˜(ζ )P˜(γ). (2.51)Regardless of the model choice and the grid size in conditional space, a bin-ning scheme is to be defined for all dimensions of the PDF. The number of binsdetermines the overall size of the chemistry table. In this work the binning is150×50×50×50 for Z˜× Z˜′′2× c˜× c˜′′2. The values of the PDF for the appropriatebin are then defined asP[a≤ γi ≤ b] =∫ baf (γi)dγi, (2.52)where a and b are the boundaries of the bin and f (γi) is the function to be definedby one of the assumed PDF models in the following sections to represent the distri-bution of probability over the range of the conditioning variable γi given a certainmean and variance value.β -PDFThe β -PDF is based on the Gamma functionΓ(t)≡∫ ∞0xγi−1e−xdx, (2.53)39which is a steady function considering γi ∈ [0,1] for the purposes here. The βfunction distribution is given byf (γi)≡ Γ(α+β )Γ(α)Γ(β ) γˆiα−1 (1− γˆi)β−1 (2.54)and requires the calculation of the two parametersα = γ i [Segi−1] and β = (1− γ i) [Segi−1] (2.55)based on the mean and segregation of the conditioning variable, γ i and Segi respec-tively. Segregation is defined asSegi =γ ′′2iγi(1− γi) . (2.56)For maximum variance σ = γ(1− γ) the Γ function has a singularity and thereforethe β -PDF is defined by two Dirac delta functionsf (γi) = (1− γ i)δ (γˆi)+ γ iδ (γˆi−1). (2.57)Some examples of a β -distribution for various values of mean and segregation areplotted in Fig. 2.6 for γi = Z.MLF-PDFThe original version of laminar flamelet-based PDFs proposed by Bray et al. [12]was only applicable for relatively high variance values. The distribution of theBray PDF is defined byf (γi) = Aδ (γi)+Bg(γi)+Cδ (1− γi) (2.58)using the function g(γi) obtained from the solution of an unstrained laminar pre-mixed flameg(γi) =1∇γi. (2.59)40Figure 2.6: Examples of β -distribution for various values of mean and segre-gation values for γi = ZIn this work the proposition of Jin et al. [56] of modifying the Bray function wasfollowed to achieve higher accuracy and more reliable results for the whole rangeof mean and variance values. Four cases are being differentiated according tothe mean γ˜i and segregation values Segγi , clipping the premixed flame in spaceas needed to obtain the Modified Laminar Flamelet (MLF) PDF [56, 99]. Fig. 2.7visualizes which cases demand elimination of one or both delta functions and trun-cation of the function g(c) in the case of the conditioning variable being c.Other works have shown success with this approach in recent years [28, 100].2.4 SummaryHaving now presented the background behind the turbulence-chemistry interactionmodelling approach used in this thesis, the results of three different test cases willbe presented, starting with a non-premixed swirl-stabilized flame. Throughout thenext three chapters the application of the UCS model to the whole spectrum of non-premixed, premixed and partially-premixed combustion will be tested and evalu-ated. Based on this information a baseline for the limitations and current range ofapplicability of UCS will be established with regards to the criteria mentioned inTab. 2.2.41c˜ [−]Seg c[−]Figure 2.7: Four cases of Modified laminar flamelet PDF; Image: Salehi etal. [100]42Chapter 3Non-Premixed Combustion –Swirl-Stabilized FlameThe Fuel Injection Research for Sustainable Transport (FIRST) combustor is amodel aero engine combustor designed by the DLR (German Aerospace ResearchCenter) to investigate soot formation. That however is not the focus of this work.It is rather the ideal vehicle to test the limitations and the performance of the newUCS modelling approach in its current state against the modelling approach of in-dustry partner MTU Aero Engines AG directly on a burner designed with realisticindustrial combustor features. First the experimental and the numerical setup arebeing outlined in Sec. 3.1 and 3.2 respectively. Then the evaluation of the UCSsolution for the chemistry in conditional space will be discussed for ethylene incomparison to methane in Sec. 3.3. Followed by the results of the numerical studybeing presented in comparison to other published works in Sec. 3.4 and a summaryof concluding thoughts on the results in Sec. 3.5.3.1 Experimental SetupA schematic of the test rig in Fig. 3.1 shows the geometry of the combustion cham-ber and the apparatus enclosing it. The fuel distribution was designed such that itcould mimic the atomizing lip between swirled air flows as used for spray flamesin aero engine combustors. Three concentric feeds make up the injector system.43Figure 3.1: Schematic of geometry of FIRST burner [38]The outermost annular nozzle (ri = 14.4mm; ro = 19.8mm) and the central noz-zle supply the flame with air at room temperature, which is drawn from separatepleni and pass through radial swirlers consisting of 12 and 8 channels respectively.In between the two are 60 straight channels forming a concentric ring. From theinjector exit plane the combustor measures a total height of 120mm containing a68 × 68mm2 square section with beveled edges. In addition there are four oxida-tion tubes with a diameter of 5mm to reduce the gas temperature and to mix theremaining fuel with air and complete the combustion of all the fuel before cham-ber exit. They are located at a height of 80mm from the injector. More detailedinformation on the experimental setup can be found in Geigle et al. [38]. Differentboundary conditions were investigated in the FIRST series, but only one is beingexamined in this work. For that the chamber pressure was slightly elevated withp = 3bar and the inflow conditions of the different streams are described in Ta-ble 3.1; leading to a fuel-air equivalence ratio of 1.2 in the primary section of thecombustor and a global equivalence ratio of 0.86 considering the secondary air.44Table 3.1: Inflow conditions for FIRST burnerInflow Mass flow [g / s] Temperature [K]Inner air 3.03 293Outer air 7.08 293Cooling air 4.04 293Fuel 0.83 2973.2 Numerical SetupThe simulation was carried out using a URANS approach and an unstructuredmesh of 2.8 million cells. This same mesh had been used in an earlier publica-tion by Konle et al. [67] – hereafter referred to as the “Konle simulation” – andwith that allows for a back-to-back comparison of the results with the two differentapproaches. As shown in Fig. 3.2, the computational domain includes the injectorsand swirlers as well as the secondary air ducts. In Sec. 3.4 a direct comparisonbetween the two approaches is being presented as it provides a reliable basis forcomparison. It is important to note that the Konle simulation took heat transferto the wall into account while also capturing the production of soot and radiation.However the turbulence-chemistry-interaction is based on a Partially-Stirred Reac-tor (PaSR) approach using simple 1-step chemistry. In contrast, the “UCS simula-tion” was carried out under the assumption of adiabatic walls while using detailedchemistry in the form of the UCS table as described in Sec. 3.3. The turbulencemodel k−ω−SST [77] was used in both the “UCS simulation” and the previouslypublished “Konle simulation” with a standard choice for tunable parameters.To ground this work further in the literature, the results of other publishedsimulations have been studied for further comparison. An overview of all the sim-ulations, which will later on be referred back to in Sec. 3.4, is given in Tab. 3.2to highlight the different modelling approaches. The simulation results presentedby Eberle et al. [29, 30] were obtained using URANS and applying two differ-ent approaches in both publications: (1) an assumed probability density function(APDF) [40, 43] and (2) finite-rate chemistry (FRC), where chemistry-turbulenceinteraction is neglected. On the other hand Franzelli et al. [36] carried out theirsimulations using LES and a full tabulation technique as well as a hybrid method,45xyz(a) computational domain (b) numerical mesh [67]Figure 3.2: Computational domain and mesh for numerical simulation ofFIRST burnerTable 3.2: Comparison of simulations published for FIRST burnerpublication CFDTurbulence-chemistry-interactionChamber wallboundary conditionDe Graaf et al.,2020 [23]URANSUCS with Wangmechanism [120]adiabaticKonle et al.,2017 [67]URANSPaSR with 1-stepchemistryT=1100KFranzelli et al.,2015 [36]LES FPI-TTC adiabaticEberle et al.,2015 [30]URANS(1) Assumed PDF;(2) FRC;T=900KEberle et al.,2014 [29]URANS(1) Assumed PDF;(2) FRC;T=900K46coupling a reduced finite-rate formulation with a look-up table. For full tabulationthe FPI-TTC [118] technique was used, while the hybrid method was originallysuggested by Lecocq et al. [73].3.3 Ethylene ChemistryDue to the a priori processing of the chemistry when using UCS, some insightscan already be gained from the solution in conditional space. The most interest-ing aspect of the use of ethylene in model engine combustors is the high rate ofproduction of soot. It can be a useful tool to evaluate a combustor regarding it’spotential to encourage pockets of soot formation. Despite the fact that soot is notthe topic of our discussion in this work, it is a significant step forward to be ableto simulate a fuel other than methane with UCS and generate a table to be used ina CFD simulation. An investigation was carried out comparing the ethylene chem-istry to the chemistry of methane – as the simplest hydrocarbon and the fuel mostcommonly used for validation purposes – while also varying some other bound-ary conditions. Ethylene is still a simple fuel, however, chemically-speaking it issignificantly different from methane.The following analysis compares 4 different UCS solutions obtained based onthe properties and boundary conditions defined in Table 3.3. All four cases areTable 3.3: Boundary conditions of investigated chemistriesFuel mechanism Case p[bar]Tfuel[K]Tair[K]CH4 GRI 3.0A 1 300 300B 10 300 650C2H4 WangC 1 300 300D 3 300 300initialized according to the following recipe:1. Setting values at the boundaries:• south: c = 0 → air-fuel mixture• west: Z = 0 → pure air47• north: c = 1.0 → complete combustion• east: Z = 1.0 → pure fuel2. Interpolate between north and south boundary for all cellsThe east and west boundary can be collapsed onto a single point, however it waschosen to be expanded into a fixed boundary to maintain a structured mesh.In Sec. 2.3.3 various parameters were identified that can be altered for differ-ent simulations including the definition of the progress variable and the functionsand constants of the scalar dissipation model in conditional space. To develop auniversally applicable version of the model, these parameters will be documentedin each research chapter and evaluated at the end. As for the simulation discussedin this chapter the parameters are summarized in Tab. 3.4.Table 3.4: UCS parameters for this investigationSimulationprogressvariabledefinitionkZχccmodelkc PDF of Z PDF of cFIRSTburnerc=YCO2YCO2 ,cc(Z)25 constant 35 β -PDF β -PDFWith that the UCS equations Eq. 2.31 are solved to steady-state assuming adi-abatic conditions using a L-stable, stiffly-accurate embedded Rosenbrock ODEsolver of order (2)3 for the chemistry.After solving these equations in conditional space with their respective bound-ary conditions, all species mass fractions and reaction rates are available to betabulated along with the chemical source term of the progress variable ω˙c, the tem-perature T and the heat release rate dhdt . The chemical source term of the progressvariable ω˙c for example for Case D can then be plotted in conditioning space asdemonstrated in Fig. 3.3.Fig. 3.4 shows plots of some of these conditional variables at the stoichiomet-ric mixture fraction Zstoi plotted over the progress variable c. The variables chosenhere are the ones, which would be tabulated and later on used in a CFD simulationalongside any other species mass fractions of interest with regards to the particular48Figure 3.3: Chemical source-term of progress variable in conditional spacecombustor to be investigated. Slow forming species such as NO do not get any-where near equilibrium as the reverse reaction rates are always small. Thereforethe direct calculation of NO mass fraction with UCS would inevitably fail. NOis still being created at a relatively high rate as the CO2 mass fraction asymptotestowards equilibrium and indeed is still greater one for c = 1.0. For that reasonNO needs to be transported in the CFD simulation using the reaction rate retrievedfrom the UCS solution. Please note that the reaction rate of NO mass fraction ω˙NOis only available for the cases A and B – simulated with the GRI-3.0 mechanism[103]. The Wang mechanism for C2H4 [120] does not include NOX -chemistry.The overall trend of the reaction rates of progress variable and species is verysimilar for all cases with the exception of Case A. In Case A, the maximum isreached at c = 0.81, which is much closer to the equilibrium value of the progressvariable for this case: cequi = 0.9; whereas all others have a predominant peaklocated near c = 0.3. Case B displays a much higher magnitude for all reactionrates compared to the other cases as shown in Fig. 3.4(a), 3.4(b), 3.4(e) and 3.4(f).This is due to the elevated temperature and pressure.The maximum value of the heat release rate shown in Fig. 3.4(b) is located490 0.5 1-505101520 104(a) ω˙c [ kgm3s ]0 0.5 1-505101520 1010Case ACase BCase CCase D(b) dhdt [Jkgs ]0 0.5 100.020.040.060.080.1(c) YCO [ ]0 0.5 105001000150020002500(d) T [K]0 0.5 1progress variable c [ ]-202468Case ACase B(e) ω˙NO [kgm3s ]0 0.5 1progress variable c [ ]10 -410 -210 010 2(f) ω˙NO [kgm3s ] on log-scaleFigure 3.4: UCS solution for cases A-D (as per Tab. 3.3) at their respectivestoichiometric mixture fraction50around c = 0.25/0.3 for Cases C and D, whereas for Case A it is much closer toequilibrium at c = 0.75. The latter indicates a delayed onset of combustion andwith that the heat release, while the maximum magnitude itself is an indicator forpotential energy that can be released burning this fuel at these conditions. Thebehavior of the CO mass fraction YCO in Fig. 5.9 reflects the same trend, where themaximum in Case A is reached near c = 0.75.In terms of the overall reactivity of the mixture, the effect of switching frommethane to ethylene do not seem to be the most determining factor in the overallperformance of the chemistry. The elevated pressure and/or temperature evoke amuch greater influence. Increasing the temperature significantly – as in Case B –causes the heat release rate to go up by an order of magnitude, while the magni-tude of the maximum CO mass fraction is reduced. Naturally the production ofCO is generally much greater for the C2H4 chemistry than for the CH4 chemistrysince there is more carbon available. Furthermore the overall mass fraction of CObecomes smaller with increased pressure for both methane and ethylene chem-istry. All of these effects are in line with the “Le Chaˆtelier’s principle” [72] whichstates that equilibrium will always be displaced in such a way that any changesimposed from outside the system are minimized [105]. In the case of pressure, theCO oxidation equation would shift towards a reduced number of moles which andtherefore a reduction of CO compared to CO2.Fig. 3.4(f) as well as Fig. 3.4(e) show the reaction rate of NO ω˙NO where the lat-ter is on a logarithmic scale and therefore only displays positive values. It becomesevident here that the reaction rate for NO remains positive beyond equilibrium asc approaches complete combustion. Increasing the pressure and temperature sig-nificantly increases the NOX production. Between Case A and B the difference inmaximum magnitude is more than two orders of magnitude. The reaction rates arebeing compared here rather than the mass fractions as NO has to be transported tobe able to capture its production accurately.For the test case to be investigated in this Chapter, the chemistry of Case D isthe relevant one and was used to produce the simulation results of Section 3.4.513.4 Results and DiscussionThe simulation of the FIRST burner using the UCS model as a predictive toolmatches the experimental global equivalence ratio of 0.86 [39] with an error ofless than 3% and the mean mixture fraction is conserved. Furthermore the globalthermal power of the combustor of 38.6kW [39] is being matched very closely bythe simulation. An energy balance on the system using the time-averaged scalarsshowed a prediction of 38.8kW.The computational cost of UCS proved to be extremely low with a time stepof about 2.22 ·10−07 to 2.24 ·10−07 s to maintain a maximum CFL number of 0.4,it took 7 days on 96 processors (≈ 16,100 CPU hours with a processor clock rateof 2.1GHz) to simulate 220ms of real time, which is equivalent to over 20 flowthrough times. For comparison, the simulation performed by Eberle et al. [30]took 338,000 CPU hours on Intel Xeon X5570 quad-core processors with a clockrate of 2.93GHz.3.4.1 Flow FieldsA comparison of the results of the FIRST combustor simulation using UCS as apredictive method with the results published by Konle et al. [67] show that theflow field is captured reasonably well. The contour plots of Fig. 3.5 indicates verysimilar location of both inner and outer recirculation zones as well as the magnitudeof velocities.There is an asymmetry noticeable in both simulations in the downstream sec-tion of the combustor around z= 80mm despite the fact that they are time-averagedand that the flow is statistically stationary. The source of this phenomena is likelyattributable to the interaction of the swirling flows with the cooling air stream lead-ing to an intrinsic unsteadiness. In Fig. 3.6 the profiles of the axial velocity atz = 6mm, z = 12mm, z = 18mm and z = 24mm are displayed as examples com-paring the UCS simulation results to the experimental data.Overall the maximum value of the velocity is captured fairly well but the dis-tribution profiles are different. In particular looking at the values near the centerof the combustor (x≈ 0mm) the negative values of axial velocity at z = 6mm andz = 18mm and z = 24mm are not in agreement with the experimental data and in52(a) Konle et al. [67] (b) UCSFigure 3.5: Contour plot of the magnitude of velocityconclusion the exact location of the inner recirculation zone is not being capturedwell in this work. As the temperature and velocity field are highly dependent oneach other due to the strong interaction of chemistry and turbulence it is unclearwhat is the root cause of the discrepancy here. Further downstream after mixingwith the secondary air, the velocity profile predicted by UCS seems in very goodagreement with the experimental data in Fig. 3.6(e) near the center of the combus-tor.3.4.2 Flame Structure and TemperatureOverall the temperature predicted by UCS lies within the error bars of the experi-mental data reported but the location and magnitude of the maximum temperatureare not in agreement. This becomes apparent in Fig. 3.7, where the temperaturealong the centerline is being plotted against the experimental data and other sim-ulations. According to the simulation results published by Eberle et al. [30] andFranzelli et al. [36] rate of increase in temperature is much greater than that pre-dicted by the UCS simulation. However they seem to predict for the flame tostabilize closer to the inlet judging by their over-prediction of the temperature at53(a) z = 6mm (b) z = 12mm(c) z = 18mm (d) z = 24mm(e) z = 100mmFigure 3.6: Axial velocity profiles of UCS simulation (—) and experiment(· · ·) at different locations downstream from the inlet54Figure 3.7: Temperature along the centerline: Comparison with differentsimulations from the literature [30, 36, 67] and experimental data [39]z = 1mm and z = 3mm.The Konle simulation distinguishes itself in that it has a stronger incline oftemperature and further downstream also a stronger decline of temperature than allother simulations. With that it seems to capture the trend of both the temperatureat z = 3mm as well as the peak temperature. With UCS, an overall lower maxi-mum temperature is being predicted and it indicates for the temperature to level offaround z= 12mm rather than spike up in the same manner as in the experiment andother simulations showed. Yet the good overlap of the UCS results with the exper-imental data at z = 3mm in Fig. 3.7 suggests that the UCS simulation is capturingthe lift-off height of the flame or flame location more accurately than the Eberle orFranzelli simulations with further confirmation provided below in the plot of theradial temperature profile in Fig. 3.8(a).A closer look at the radial temperature profiles at different downstream loca-tions in Fig. 3.8 strengthens the previous observations: the general trend of theflame shape and location is captured well – however the maximum temperaturelocation is not. In particular at z = 12mm the UCS simulation is barely within theerror bars at x = 0mm in Fig. 3.8(b), while Konle and Eberle seem to predict anaveraged scalar field that is well in line with the mean temperature measurement.Only intermittently around z = 24mm downstream does the UCS simulation agreewell with all the experimental data points; see Fig. 3.8(c).Looking at the temperature contour plot as a whole in Fig. 3.9(b) the flame55(a) z = 3mm (b) z = 12mm(c) z = 24mmFigure 3.8: Temperature comparison of UCS simulation with the mean exper-imental data Tmean and the mode temperature Tmp with it’s uncertaintyas well as the simulation results available from publications of Eberle etal. [30] and Konle et al. [67] at different locations z downstreamstretches far into the exit of the combustor and it is apparent that the combustionzone continues into the second half of the combustor where the fresh cooling airmeets the unburnt fuel from the rich primary combustion region.When comparing to the contour plot in Eberle et al. [29] the asymmetry thatwas previously discussed is even more noticeable in Fig. 3.10. The flame shape inboth these simulations is quite similar. Both reaching far into the second half of thecombustor, where the fresh cooling air is injected, while the biggest discrepancy inboth comparisons Figs. 3.9 and 3.10 is the wall temperature and the exit tempera-56(a) Konle et al. [67] (b) UCSFigure 3.9: Contour plot of temperature on x− z−planeture. In the UCS simulation, the fluid near the walls heats up as heat transfer withthe wall cannot be accounted for in the current state of the modelling approach.Hence, this discrepancy from the results of the Konle and Eberle simulations inFigs. 3.9 and 3.10 is to be expected.Looking back at the centerline temperature comparison in Fig. 3.7, the exittemperature predicted by Franzelli is quite close to and even higher than the UCSresult. The contour plot comparison in Fig. 3.11 shows similarities in terms of thenear wall temperatures and exit temperature profile.As Franzelli too does not include heat transfer to the walls, while distinguishingitself in the same way from the other two simulations (Konle and Eberle) regardingthe near wall temperatures, this is a strong indication of how future simulationswith UCS can most likely be improved by adding the capacity to incorporate theseeffects. Adding a third conditioning variable to account for heat loss would accom-plish this goal – Bushe [15] has suggested total enthalpy for this purpose. In spiteof this current limitation, the contour plot comparisons further emphasize that theUCS simulation is capturing the lift-off height more accurately than the Eberle orFranzelli simulations.57Figure 3.10: Comparison of UCS simulation results (right-hand side) againstthe results published by Eberle et al. [29] (left-hand side) for tempera-ture3.4.3 Species AnalysisUnfortunately there is no experimental data available for minor and major species.However a comparison between UCS and Eberle et al. [29] can be made for time-averaged fields of O2 and CO2 as shown in Fig. 3.12.In addition, a qualitative comparison for OH distribution in the time-averagedfield is presented in Fig. 3.13.Overall the comparison of the species mass fractions is encouraging from aqualitative and quantitative standpoint. The most significant differences are foundin the bottom corners which would again be related to the heat transfer modellingalong the walls.58Figure 3.11: Comparison of UCS simulation results (left-hand side) againstthe results published by Franzelli et al. [36] (center and right-handside)3.5 Concluding ThoughtsThe attempt of simulating a non-premixed flame in an industrially-relevant burneroperated on ethylene fuel with the UCS method for turbulence-chemistry interac-tion modelling was a qualified success in that it provided some insight into thestatus of the methodology. The overall flow field was predicted reasonably wellcomparing to the PIV data of the experiment at various downstream locations andthe contour plot of the simulation published by Konle et al. [67]. As for the tem-perature, the comparison with other publications and the CARS measurements re-vealed some of the limitations of the current UCS implementation and potentialareas of improvements. The location and magnitude of the maximum tempera-ture in different regions was not well-captured. A potential source of error is theuse of RANS and the tendency of turbulent fluctuations smearing out the peaksin the temperature and lowering the magnitude overall. Furthermore the wall andexit temperatures were overpredicted which is a consequence of applying adiabaticwalls as boundary conditions. In order to capture the heat transfer at the wall well,59(a) YCO2(b) YO2Figure 3.12: Comparison of UCS simulation results (right-hand side) againstthe results published by Eberle et al. [29] (left-hand side)60Figure 3.13: Qualitative comparison of YOH of UCS simulation results (leftside) with the results published by Eberle et al. [29] (right side)one would want to include another conditioning variable to represent the heat lostto the wall – Bushe [15] suggested that using total enthalpy might accomplish thisgoal. This would lead to a substantially higher computational cost and also intro-duces challenges with modelling the JPDF of mixture fraction, reaction progressvariable and total enthalpy.The temperature along the centerline clearly shows that the location of theflame and its anchor point is more accurately predicted by UCS than by Eberle etal. [30] and Franzelli et al. [36]. Their simulations suggest the flame stabilizingvery close to the central nozzle. The results suggest that UCS has the potential tocorrectly predict lift-off height. Major advantages of UCS should become promi-nent when simulating a test case where emission mass fractions can be compared toexperimental data. The possibility of comparing different fuels for the same config-uration and evaluating the NOX or CO emissions under fixed boundary conditionsis attractive for investigations of alternative as well as conventional fuels.The FIRST burner provided a good starting point to test the applicability andcapability of the new modelling approach. This simulation has shown that thecomputational cost remains low – even compared to simpler models – while pro-viding additional detailed information about emission production. Before furtherdeveloping the methodology significantly, an assessment of a premixed flame and61a partially-premixed flame remains a priority in order to establish how universallyapplicable the model truly is. Furthermore, the ability of UCS to predict emissionand temperature trends is of interest as well as the accuracy of quantitative NOemission predictions. These will be topic of the next two chapters.62Chapter 4Premixed Combustion –Swirl-Stabilized FlameThrough the work presented in this chapter, the analysis of the performance of UCSon swirl-stabilized flames in industrially relevant combustors will be expanded topremixed flames. The focus is primarily to demonstrate the versatility of the modelwith regards to the combustion type and the chemical kinetic mechanism, whichis why the study was carried out with both the GRI-3.0 mechanism [103] and theUBC-3.0 mechanism [51].Different equivalence ratios, φ , were investigated, keeping the thermal poweroutput constant at 50kW; allowing for an evaluation of the predictive quality of themodel for temperature and different emissions, where NO is of particular interest.Two of the equivalence ratios were tested experimentally- φ = 0.67 and φ = 0.77.The experimental setup is described in Sec. 4.1 and the numerical investigationis outlined in Sec. 4.2. In Sec. 4.3 more details can be found on the chemistrymodelling with UCS for this study and finally the results are presented in Sec. 4.4.4.1 Experimental SetupThe Beschaufelter RingSpalt (BRS) burner [66] was investigated at the TechnicalUniversity of Munich for two different swirler locations (front and back), threedifferent fuel-to-air equivalence ratios (φ = 0.67/0.77/0.91) as well as three dif-63ferent thermal power ratings (P = 30/50/70kW) under atmospheric conditions(294K; 1bar). In particular the front-swirler configurations producing 50kW ofthermal power with φ = 0.67 and φ = 0.77 were selected for this study. Fig. 4.1illustrates the methane operated test rig in that set-up. PIV measurements wereFigure 4.1: Schematic of the experimental test rig of the BRS burner [66]including geometric dimensions in [mm]taken to extract the velocity field on the central cross plane x− z using a Nd:YLFLaser with a wavelength of 527nm and a temporal separation of 20µs. Measure-ments of OH∗ chemiluminescence were carried out using an interference filter fora wavelength of 307.1 +3/-0nm with a half power bandwidth of 10±2nm and atransmission of 15% to separate the signal from other emissions [66]. The signalswere recorded using an intensified CMOS camera. It should be noted as well thatthe walls of the chamber were cooled, however the wall temperatures achieved inthe experiment were not measured, nor were the properties of cooling air or the heatflux. More details on the experimental investigation can be found in Komarek etal. [65, 66].4.2 Numerical SetupFor the numerical study, the walls were assumed to be adiabatic and the swirler wasincluded in the numerical domain as shown in Fig. 4.2. Here the flame is visual-ized using the magnitude of the sensible enthalpy source term in three dimensionalspace. A hybrid mesh was used for the simulations, where three prism layers wereapplied along the side walls of the chamber while the rest of the domain contains64Figure 4.2: Simulation domain for CFD simulation using UCS as a predictivemethodunstructured tetrahedral cells; amounting to a total of 2.4 million cells. In the flameregion the cell size is around 0.6mm. A URANS approach was chosen and the tur-bulence is modelled with the k−ω−SST model [77] with standard coefficients.Initially the two experimental operating points were simulated with the sameparameters as presented in Sec. 3.3 for the FIRST burner and this evaluation leadto an adjustment of some parameters as well as the investigation of two differentapproaches for modelling the PDF of the progress variable: (1) the β -PDF; (2)the MLF-PDF. The findings of this investigation are presented in Sec. 4.4.1 andconsequentially the NO-trend study was then carried out with the MLF-PDF. Tendifferent cases were studied with the fuel-air equivalence ratio φ ranging from 0.6to 1.2. An overview of the equivalence ratios chosen for the trend study can befound in Table 4.1, where the two experimental cases are depicted in highlightedfont.65Table4.1:EquivalenceratioseriesforvalidationstudyPTpφλZUGRImechanismUBCmechanism[kW][K][bar][-][-][-][m/s]withJPDF:withJPDF:5029410.61.670.033814.30β-MLF1β−MLF10.671.50.037612.88(1)β-β;(2)β-MLF1(1)β-β;(2)β-MLF10.71.430.039212.36β-MLF1720.771.30.042911.30(1)β-β;(2)β-MLF1(1)β-β;(2)β-MLF10.81.250.044510.90β-MLF1720.91.110.04989.77(1)β-β;(2)β-MLF1(1)β-β;(2)β-MLF1110.05518.86β-MLF1β-MLF11.0520.950.05788.90β-MLF1721.10.910.06028.93β-MLF1β-MLF11.20.830.06549.00β-MLF1β-MLF131 bothvariationsoff YCO2andf Tweresimulated;seedefinitionsandexplanationsinSec.4.32 notsimulated3 thissimulationdiverged66The equivalence ratios φ are defined asφ ≡ m˙fuel/m˙oxi(m˙fuel/m˙oxi)stoiand λ =1φ, (4.1)which leads to a mass flow rate ofm˙lean =PLHVCH4Z; m˙rich =PLHVCH4Zφ ; (4.2)to maintain the thermal power output P of 50kW considering the lower heatingvalue LHV of CH4.As for the UBC mechanism, not all equivalence ratios were repeated since thesealready provided a good mean of comparison for the trend study. One simulationfailed due to divergence: the one using the UBC mechanism at an equivalence ratioof φ = 1.2. Three flames were also simulated using the Partially-Stirred Reactor(PaSR) approach – the standard combustion model implemented in the OpenFOAMsolver rhoReactingFoam – with the same detailed GRI-3.0 mechanism as usedin the UCS simulation and the same approach for turbulence modelling. The equiv-alence ratios chosen are: (1) φ = 0.77 (lean); (2) φ = 1.0 (stoichiometric); (3)φ = 1.2 (rich). Results obtained from these simulations are used to give an addi-tional means to judge the results of the UCS simulations – in particular the NOemissions – in Sec. 4.4.2.4.3 ChemistryTo generate the UCS tables for the CFD simulations, the chemistry of both theGRI-3.0 mechanism [103] and the UBC-3.0 mechanism [51] were solved on theconditional mesh assuming adiabatic conditions. Both simulations are initializedin the same way as was done in the case of the FIRST burner (see Sec. 3.3) whileusing a mesh is made up of 300 cells in the mixture fraction dimension – unevenlyspaced with higher resolution near stoichiometric – and 100 cells in progress vari-able dimension. The UCS equations (Eq. 2.31) were then solved using the sameL-stable, stiffly-accurate embedded Rosenbrock ODE solver of order (2)3 [92] asbefore. As for the UCS parameters chosen for this test case, they are listed in67Tab. 4.2 in direct comparison with those previously used in Chapter 3. Initially theTable 4.2: UCS parameters of BRS burner simulations compared to FIRSTburner simulationSimulationprogressvariabledefinitionkZχccmodelkc PDF of Z PDF of cFIRSTburnerc=YCO2YCO2 ,cc(Z)25 constant 35 β -PDF β -PDFBRSburnerc=YCO2YCO2 ,cc(Z)10 constant 15 β -PDF β -PDF /MLF-PDFsame constants for kZ and kc were tested as had been used in the FIRST burner sim-ulation but the results quickly showed that the scalar dissipation was set too highfor a premixed case. In particular, kZ for the scalar dissipation of mixture fractionwould be expected to be close to zero in a premixed burner.The use of the MLF-PDF invokes a problem for generating the joint PDF andwith that the UCS table. The progress variable c is defined to become unity at astate of complete combustion (cc). However the MLF-PDF is defined to be unityat equilibrium (equi) as it is based on the laminar flamelet solution. Therefore theMLF-PDF had to be rescaled to align with the range of the UCS solution. Thiswas done using the equilibrium solution obtained from Cantera: once using theCO2 mass fraction (YCO2,equi) and later on using the temperature (Tequi). Both ofthese approaches result in a clipping of the UCS solution based on the ratio of theequilibrium (equi) value to the complete combustion cc value:fYCO2 (Z) =YCO2,equi(Z)YCO2,cc(Z)and fT (Z) =Tequi(Z)Tcc(Z)(4.3)Fig. 4.3 shows all four functions necessary for the operation. Evidently the dif-ference between equi and cc for lean mixtures is minute while for rich mixturesthey deviate significantly for both T and YCO2 , but more so the latter. The factthat YCO2,equi would exceed YCO2,cc is unphysical with regards to conservation ofelements. Therefore the function YCO2,equi was corrected to take on a linear shapebeyond Z = 0.3. Since, for all cases considered, the burner was operated far from680 0.2 0.4 0.6 0.8 100.020.040.060.080.10.120.140.16Z [−]Y CO2[−]5001,0001,5002,0002,500T[K]YCO2 at ccYCO2 at equiYCO2 at equi correctedT at ccT at equiφ=1.0Z=0.3Figure 4.3: Complete combustion according to Burke-Schuhmann [125] andequilibrium computed with Cantera for YCO2 and T plotted over Zthis limit, it does not affect the calculations to be discussed. It should be noted thatuse of two conditioning variables is not necessary in the case of a premixed flame,however it is a convenient way to use a single table for all flames regardless of theirequivalence ratio and in a general sense it ensures a procedure that would in futureallow one to use the same table for premixed, non-premixed and partially-premixedflames of the same boundary conditions.Fig. 4.4 shows the UCS solution for temperature in conditional space using(a) the GRI-3.0 mechanism and (b) the UBC-3.0 mechanism. The two options forclipping the UCS solution to build the joint PDF with the MLF-PDF are includedin the contour plot. All simulations using the MLF-PDF which will be discussedin Sec. 4.4 were performed with both clipping options to study their effects.69Z [−]c[−]fYCO2fTφ=1.0(a) GRI mechanismZ [−]c[−]fYCO2fTφ=1.0(b) UBC mechanismFigure 4.4: UCS solution for temperature with clipping functions based on(a) GRI-3.0 Mechanism [103] and (b) UBC-3.0 mechanism [51]70A first comparison of the two different UCS solutions with the two mechanismscan be made based on the chemistry solution in conditional space alone. The be-havior of a limited set of variables over progress variable c is presented in Fig. 4.5.All of them show more rapid combustion using the UBC mechanism in comparisonto the GRI mechanism with a tendency to grow further apart with increasing equiv-alence ratio. The overall characteristics of the prediction are qualitatively similarbut quantitatively significantly different. For instance the temperature in both so-lutions in Fig. 4.5(a) shows a rapid increase and then a steady but slow increasetowards equilibrium. From there the second steep slope is needed to reach com-plete combustion temperatures. The UBC mechanism however leads to a muchhigher temperature closer to c = 0.5 than does the GRI mechanism and also thevalues for the reaction rate of the progress variable in Fig. 4.5(f) is much highercloser to c= 0.5. In particular the latter is what inevitably embodies the more rapidprogression when applying the UBC mechanism. This mechanism was originallydesigned for auto-ignition problems at high pressure and so it will be interestingto see its performance in this application. It is to be expected that the shapes ofthe flames predicted with these two mechanisms would differ and it remains to beevaluated in the following sections which one performs better.As for the minor species mass fractions such as CO and H2, the magnitudes ofmaximum values are very similar, however the locations of their peaks are differentas shown in Fig. 4.5(b) and 4.5(c). The magnitudes do differ significantly forthe source terms of various species such as NO and NO2 as well as the progressvariable rate of reaction ω˙c in Fig. 4.5(d) - 4.5(f). Particularly interesting is alsothe double peak in Fig. 4.5(d) for φ = 1.0 that both chemical mechanisms producein conditional space. The production rate ω˙NO2 in the GRI solution is about 40%larger than that of the UBC solution.4.4 Results and DiscussionIn this section many different aspects of the modelling approach will be discussedand compared and evaluated. To maintain structure the two approaches for PDFmodelling shall be investigated in Sec. 4.4.1, followed by the equivalence ratiostudy of the prediction of trends of temperature and emissions and the effect of the710.670.771.0(a) T [K]0.670.771.0(b) CO [− ]0.670.771.0(c) H2 [− ]0.670.771.0(d) ω˙NO [1/s]0.670.771.0(e) ω˙NO2 [1/s]0.670.771.0(f) ω˙c [1/s]Figure 4.5: UCS chemistry solution for various equivalence ratios φ as la-beled in the figures for both chemical mechanisms: red lines withoutsymbols representing the results using the GRI mechanism, while bluelines with symbols represent the results using the UBC mechanism.72clipping method for the MFL PDF approach in Sec. 4.4.2. Simultaneously, somecomparisons to the works of Tay-Wo-Chong et al. [108–110] will be made to placethis new modelling approach into a more general context. Different approaches forcombustion modelling were employed in the different publications. Once URANSwas used in combination with the enhanced Schmid model [109]. Furthermore,LES has been used – once with the thickened Flame Model [108] and then againwith the extended Turbulent Flame Closure (TFC) model [110]. All results ofthe simulations by Tay-Wo-Chong et al. displayed in the following sections werechosen as the ones matching the experiments best, whenever there were multiplevariations published in the paper. The various modelling approaches include acombined effect of strain and heat losses. They all took into account wall heattransfer and assumed an isothermal wall condition of 600K. Lastly in Sec. 4.4.3further investigation will be made into the prediction of the flame shape and thetemperature.4.4.1 PDF StudyThe experimental measurements that can be used to compare to are limited to thevelocity field. Therefore the evaluation of the results of the simulations with ei-ther of the two PDF approaches is being made based on their contour plots of thevelocity and their velocity profiles – beginning with the contour plots in Fig. 4.6.The streamlines on the x-z-plane of the simulations using the β -PDF and theMLF-PDF for the progress variable (in Fig. 4.6(a) and Fig. 4.6(c) respectively)were extracted and superposed onto the post-processed experimental PIV image ofthe axial velocity and are illustrated in Fig. 4.6(b) and in Fig. 4.6(d) . It should benoted here that the simulation results using the MLF-PDF were essentially identicalfor both the use of fYCO2 and fT and therefore the discussion of the more suitableclipping function will be found in the next section, when looking at the whole rangeof equivalence ratios. For now, the results presented here were obtained with thefT clipping method.In the experiment the outer recirculation zone is observed to reach from aboutz = 10mm to z = 25mm downstream, while the inner recirculation zone coversthe full length of the image generated from the PIV measurements – well beyond73(a) UCS with β -β -JPDF (b) Experiment with streamlines of (a)(c) UCS with β -MLF-JPDF (d) Experiment with streamlines of (c)Figure 4.6: Contour plots of the axial velocity field prediction of UCS in [m/s]for the case of φ = 0.77 using (a) the β -PDF and (c) the MLF-PDF forprogress variable compared to (b) and (d) with the experimental imagesuperposed by the corresponding streamlines of (a) and (c) respectivelyz = 100mm – and stretches from about x = −25mm to x = 25mm. Judging bythe streamlines in Fig. 4.6(d) the overall size of the inner recirculation zone isbeing captured fairly well in the UCS simulation when using the MLF-PDF. Thepredictions using UCS with the β -PDF produced a much shorter inner recirculation74zone as shown in Fig. 4.6(a) and Fig. 4.8(b) than what is being observed in theexperiment.Using the MLF-PDF for progress variable presents itself to be significantly su-perior to using the β -PDF in terms of predicting the flow field correctly. Lookingmore closely at the line graphs of the axial velocity Uz in Fig. 4.7 strengthens thishypothesis. In Fig. 4.7(a) the UCS results with the β -β approach are clearly sug-gesting a much larger magnitude of velocity in negative z direction – both in theinner and outer recirculation zone – of close to 20m/s. With the β -MLF approachthe prediction is not quite as far off from the experimental value, however the ve-locity is still being overpredicted for most downstream locations. At z = 40mmdownstream in Fig.4.7(b) the magnitude of the velocity is about twice of that of theexperiment at x = 0mm. This shows that the details of the inner recirculation zoneare not captured by UCS even with the better results of the β -MLF approach. Theresults of Tay-Wo-Chong et al. [108–110] do not predict this location either; in-deed, their velocity profiles do suggest a trend similar to that of the UCS prediction.In particular in Fig. 4.7(c) and 4.7(d) the prediction of Tay-Wo-Chong et al. [109]is almost identical to that of UCS with the β -β -JPDF. For most downstream lo-cations the gradients of the profiles seem to be captured sufficiently by the UCSsimulation with the β -MLF-JPDF. A quantitative accuracy of the velocity can notbe expected here based on the fact that the heat transfer to the walls is not beingtaken into account and with that the velocity will inevitably be overpredicted in theregions affected by this assumption.By contrast, in the case of φ = 0.67 as per Fig. 4.8 the two UCS simula-tions – using the β -PDF approach in Fig. 4.8(a) and the MLF-PDF approach inFig. 4.8(c) – showed very similar results. Here again, the experimental image inFig. 4.8(b) is superposed by the streamlines of the simulation with the β -MLF-JPDF approach. As noted earlier in Sec. 4.3, the progress of reaction shows greatersimilarity in magnitude for φ = 0.67 when comparing the two different predictions,than it does for higher equivalence ratios. Such behavior can be the potential causefor the similarity in the overall result for the velocity field. Comparing all contourplots in Fig. 4.8 to one another shows that the inner recirculation zone is capturedwell by the UCS simulations in size both in axial as well as radial direction. Anotable difference between simulations and experiment is present for the location75(a) z = 20mm (b) z = 40mm(c) z = 60mm (d) z = 80mm(e) z = 100mmFigure 4.7: Line plot of the axial velocity field [m/s] prediction of UCSusing the two different JPDF approaches compared to the experi-mental measurements and other simulations published by Tay-Wo-Chong et al. [108–110] for φ = 0.77 at different downstream locations76(a) UCS with β -β -PDF (b) Experiment (c) UCS with β -MLF-JPDFFigure 4.8: Contour plots of the axial velocity field prediction of UCS in [m/s]for the case of φ = 0.67 using (a) the β -PDF and (c) the MLF-PDF forprogress variable compared to (b) the experimental imaging superposedby the streamlines of (c)of the outer recirculation which is predicted to be further downstream in the UCSsimulations than shown by the PIV data of the experiment.The study of trends predicted by UCS over different equivalence ratios beyondthe two experimental cases should provide some further insights into the validityof the results with this approach and in particular the NO emissions.4.4.2 Equivalence Ratio VariationFig. 4.9 illustrates the adiabatic flame temperature over mixture fraction Z in com-parison with the function obtained for complete combustion cc. It can be noticedhere that the highest temperature predicted for the adiabatic flame is at φ = 1.052and not stoichiometric. This is why φ = 1.052 was included in the study. It isreassuring to see that the Cantera function for equilibrium is in exact agreementwith the adiabatic flame temperature behavior, since the boundary conditions ofthe UCS chemistry simulation were determined using Cantera [45]. This obser-vation further validates the legitimacy of the fT clipping function. Furthermore thePaSR model predicts the average outlet temperature to be exactly the same as the77φ=0.77φ=1.0φ=1.21,0001,2001,4001,6001,8002,0002,2002,400Tadiabatic flame [69]ccCantera equi [45]UCS GRI with fYCO2UCS UBC with fYCO2UCS GRI with fTUCS UBC with fTPaSR with GRIFigure 4.9: Average temperature on exit plane for all simulated cases andvariationsadiabatic flame temperature. The predictions obtained with the UCS method usingonly the β -MLF-JPDF approach are included in Fig. 4.9 as it was found to be su-perior in the evaluation of Sec. 4.4.1. For lean mixtures the results mostly overlapfor both chemical kinetic mechanisms and both clipping approaches, however forstoichiometric and rich flames differences of up to 160K are observed. A deviationof the predicted maximum temperatures from expected equilibrium values can befound in other works as well – such as [82] using the MFM method and [31] usinga Doubly-Conditional Moment Closure (DCMC) approach. Indeed it is not to beexpected that the temperature would reach the adiabatic flame temperature as scalardissipation and diffusion effects drive down the temperature. One example on theexperimental side would be the Cabra flame [18] – a premixed CH4/air jet flame invitiated coflow – where temperatures measured were significantly below the adia-batic flame temperature [19]. The only series that fully reproduces the trends of theadiabatic flame is the one using the GRI mechanism and fT . The highest tempera-78ture is being reached at φ = 1.052 and the gradient of decline is similar to that ofthe adiabatic flame temperature. In comparison using the same mechanism – and,with that, the same UCS solution but a different clipping function ( fYCO2 ) – leads toa rapid decline of the temperature starting at φ = 0.9. Looking at the two functionsin Fig. 4.4(a), it can again be noticed that the decline of fYCO2 in the rich regionis much steeper than that of fT and therefore this effect can be attributed to theclipping function. The last data point of φ = 1.2 can be discounted due to the factthat the flame did not stabilize near the bluff body and instead stretched far into thechamber – resulting in a notable discrepancy in predicted thermal power Pthermal ofonly 26kW. As illustrated in Fig. 4.10, the thermal power output of all cases in thelean combustion region is within 2% of the experimental power output. In the richregion, and for the stoichiometric mixture, some cases deviate more than others.φ=0.77φ=1.0φ=1.220304050P thermal[kW]UCS GRI with fYCO2UCS UBC with fYCO2UCS GRI with fTUCS UBC with fTPaSR with GRIFigure 4.10: Overall thermal power produced in the combustor for all simu-lated cases using the UCS approach as per Table 4.1One reason why the full power output cannot be reached in some cases is thatc does not reach unity as Fig. 4.11 shows. It could be argued that the temperatureof all graphs – and with that also “UCS GRI Mech with fT ” – is limited by theclipping function. However, looking at Fig. 4.11 it is clear that the limiting factorhere is the prediction of incomplete combustion in the sense that the residence time79of the gas in the chamber is not long enough to reach equilibrium for rich mixtures.φ=0.77φ=1.0φ=1.20.70.80.91c exit[−]UCS GRI with fTFigure 4.11: Progress variable c at the exit plane plotted over mixture fractionZThe latter observation also feeds into the evaluation of the NO emissions. Gen-erally the trend to be expected would be for NO to increase with φ for lean mix-tures, to peak around stoichiometric and then decrease again. This is partially thecase for either of the series with the GRI or UBC mechanisms as per Fig. 4.12,but not exactly. All cases show the correct trend of rising average NO emissionson the exit plane with an increase of φ for all lean mixtures. The predictions aregenerally slightly higher with the GRI mechanism then with the UBC mechanism.Furthermore the average ppm are higher when using the clipping function fYCO2opposed to fT . The leveling off in the case of “UCS GRI Mech with fT ” could beexplained by the fact that the progress is not fast enough to reach equilibrium inthe chamber and therefore more NO is being produced. With respect to the resultsof the three cases simulated with the PaSR, the UCS predictions are much higher;in particular for the stoichiometric and for the rich mixture, there is an order ofmagnitude difference between “PaSR with GRI Mech” and “UCS GRI Mech withfT ”.Overall it can be concluded that the modelling of the PDF for progress variable80φ=0.77φ=1.0φ=1.2101102103104NO[ppm]UCS GRI with fYCO2UCS UBC with fYCO2UCS GRI with fTUCS UBC with fTPaSR with GRIFigure 4.12: Average NO in ppm by mass on the exit plane for all BRS seriessimulationswith the MLF-PDF and fT allows for a wider range of application of the resultingtable – in particular near stoichiometric and rich mixtures it is superior to the clip-ping with fYCO2 . In the following sections the UCS results presented were obtainedwith the β -MLF approach using the fT clipping function. To evaluate the two dif-ferent mechanisms further, again, the focus will lie on the two experimental casesof φ = 0.67 and φ = 0.77, when comparing the UCS predictions for velocity fields,the flame shape and temperature to experiments and other works.4.4.3 Flame ShapeIn the line plots in Fig. 4.13, the differences in the UCS results with the UBCmechanism and the UCS results with the GRI mechanism are compared to the ex-periments and the results of Tay-Wo-Chong et al. [108–110]. Fig. 4.13(a) clearlyillustrates the location of the recirculation zones of the simulation with the UBC81mechanism to be much further upstream than is the case in the experiment. Fur-thermore they are over-predicted in magnitude. For instance, the peak velocity atz = 40mm is about 15m/s upstream in the center, while the experiment suggestsabout 5m/s upstream. The peak velocity locations cannot be matched for the down-stream locations of z = 20 to z = 60mm. These factors lead to the conclusion thatthe flame shape would be significantly different from the experiment. When com-paring to the simulation results of Tay-Wo-Chong et al. [108] using the thickenedFlame Model, the prediction of UCS with the UBC mechanism is very similar atz = 40 and z = 60mm. However overall the simulation results obtained by usingthe GRI mechanism follow the trends in the experiment much better and so it mustbe concluded that while the comparison of both mechanisms provided some in-teresting insights, the mechanism of choice would be the GRI mechanism henceforth. Similar to Fig. 4.7 the velocity is overall being overpredicted for all down-stream locations. This can be attributed to the higher temperatures that are presentin the UCS simulation based on the assumption of adiabatic walls and significantlyimpact the density of the fluid.As had been observed earlier in Sec. 4.3 in connection with Fig. 4.5 the reactionrates were much higher for the UBC mechanism at lower progress variable and thetemperature much greater. How that affects the flame shape, shall neverthelessbe discussed here for the sake of completeness, regardless of the unsatisfactoryperformance of this mechanism.A comparison of the heat release rates dhsdt on the x− z-plane is made in Fig.4.14 to results obtained by Tay-Wo-Chong et al. [110] with an LES approach andthe Turbulent Flame Closure (TFC) combustion model proposed by Zimont andLipatnikov [128]. It becomes apparent that the flames predicted in all UCS simu-lations are an M-shaped flame. This is to be expected considering the fact that heatloss is not being taken into account by this version of the model. The simplificationof adiabatic walls in the corners of the combustor lead to a stabilization of the flamein this region causing a similar magnitude of heat release in the outer shear layercompared to the inner shear layer which would not be expected to take place if thewalls were cool as they were in the experiment. Similarly the Standard TFC modeldoes not include heat loss effects and is therefore a good candidate for comparisonhere to evaluate the results. The extended version of TFC however does account82(a) z = 20mm (b) z = 40mm(c) z = 60mm (d) z = 80mm(e) z = 100mmFigure 4.13: Line plot of the axial velocity field [m/s] prediction of UCS us-ing the UBC mechanism and the GRI mechanism compared to the ex-perimental measurements and other simulations published by Tay-Wo-Chong et al. [108–110] for φ = 0.77 at different downstream locations83(a) UCS with GRI mech using β -β JPDF(b) UCS with GRI mech using β -MLF JPDF (c) UCS with UBC mech using β -MLF JPDF(d) Tay-Wo-Chong et al. [110] with standardTFC(e) Tay-Wo-Chong et al. [110] with extendedTFCFigure 4.14: Contour plots of the enthalpy source term predicted by UCSwith the GRI mechanism using (a) the β -PDF approach and (b) theMLF-PDF approach for the PDF of progress variable and (c) UCSwith the UBC mechanism compared to the LES simulation results ofTay-Wo-Chong et al. [110] using (d) the standard TFC model and (e)their extended TFC modelfor the heat loss to some degree alongside incorporating stretching phenomena andis therefore able to predict the V-shaped flame as quenching occurs in the outershear layer. In general, however the flame in Fig. 4.14(b) matches the shape and84penetration length of the V-shaped flame obtained with the extended TFC, whilethe two flames in Fig. 4.14(a) and Fig. 4.14(c) are much shorter and exhibit veryhigh levels of heat release in the tip of the flame. In the case of UCS with the UBCmechanism, the shortness of the flame can be explained by the accelerated progressof combustion that had been noticed already upon analysis of the UCS chemistrysolution in Fig. 4.5.A direct comparison can also be made between the UCS results and the workperformed by Tay-Wo-Chong et al. [109] using a RANS approach and the reactionrate model of Schmid [101]. The contour of the temperature fields in Fig. 4.15shows an agreement in magnitude however the shape of the flame appears to bedifferent, which is to be expected given the fact that Tay-Wo-Chong et al. didaccount for the heat transfer to the wall by assuming a constant wall temperatureof 600K. In the UCS simulation the walls were assumed to be adiabatic and so the(a) UCS usingβ -β -JPDF withGRI(b) UCS usingβ -MLF-JPDFwith GRI(c) UCS usingβ -MLF-JPDFwith UBC(d) originalSchmid [109](e) enhancedSchmid [109]Figure 4.15: Contour plots of the temperature for φ = 0.77 predicted by UCSwith the GRI mechanism using (a) the β -β PDF approach and (b) theβ -MLF PDF approach and also (c) the prediction of UCS with theUBC mechanism using the β -MLF PDF approach compared to theLES simulation results of Tay-Wo-Chong et al. [109] using (d) theoriginal Schmid model and (e) their enhanced Schmid modelheat transfer that is visible in the simulation results in Fig. 4.15(d) and Fig. 4.15(e)85cannot be reproduced with this current version of UCS. Certainly the flow field isaffected by the temperature and so the differences in temperature in the corner ofthe combustor could explain the deviations of the flow field predicted by UCS andthe experiments in particular for the outer recirculation zone.4.5 Concluding ThoughtsBased on the limited experimental data available for the BRS burner, it is diffi-cult to draw decisive conclusions about the validity and the accuracy of the results,however some important statements can be made. The overall ability of the UCSmodel to simulate both a premixed and a non-premixed flame with essentially allidentical parameters is encouraging. Some modifications were made to improvethe quality of the results and this was justified using the PIV data available for thevelocity field as well as direct comparisons to other modelling approaches. A majorimprovement was using the MLF-PDF in place of the more commonly used β -PDFto model the behavior of the progress variable. With that approach, the UCS modelprovided a realistic representation of the turbulence-chemistry interaction with re-spect to flame shape and flow field, when using the GRI-3.0 kinetic mechanism forgenerating the chemistry solution in conditional space. The UBC-3.0 mechanismappears to be less suitable for this particular application, which is not entirely sur-prising as it was developed for autoignition problems at elevated pressure. It wasfound that the reactivity is predicted to be much greater with the UBC mechanismthan with the GRI mechanism. In turn this leads to a shorter flame and a differentflow field dominated by a large outer recirculation zone.The nature of reactivity and the behavior of temperature, heat release and emis-sion production purely based on changes in equivalence ratio were studied andshow similarities to other cases in the literature [19, 59]. It was observed that theaverage outlet temperature falls below the values of the adiabatic flame temperatureacross all equivalence ratios and that this difference becomes greater near stoichio-metric and in the rich region; the reason being the chemistry-turbulence interaction.The NO emission prediction is reasonable as it follows the trend of the temperatureacross the equivalence ratio range examined. These results are encouraging withregards to the validity of the UCS chemistry solution. However, to come to a def-86inite verdict on the accuracy of the modelling approach presented here, validationdata in the form of comparison with more experimental measurements would berequired – in particular for species composition and temperature.Comparisons to both RANS and LES results of this burner from the archivalliterature help to put the model in perspective with state-of-the-art approaches. Itseems that the difference associated with choosing RANS over LES does not havea big impact for the study that was conducted in this particular test case. Thechoice of turbulence-chemistry interaction model however does have an impactand more so the fact that in the UCS approach the heat transfer with the wall wasnot taken into account. In particular for the heat release rate, the results presentedhere highlight further the need to include a third conditioning variable to representthe effects of heat transfer with the wall.The investigations of this chapter demonstrate the applicability of the UCSmodel to premixed combustion in addition to the previous study of non-premixedcombustion in the FIRST burner (Chapter 3). With this, a significant step is takenin evaluating the approach as a universal tool for use in industry anywhere on thepremixed - non-premixed continuum. It remains to be seen how the model per-forms in the case of partially-premixed combustion. In the case of non-premixedcombustion the choice of the PDF of progress variable seems not as important andso one might consider using the β -PDF over the MLF-PDF in such a case as itrequires less effort a priori. However if the goal is to reduce the number of vari-able parameters of the modelling approach all together it might be worthwhile toestablish the MLF-PDF as the standard approach to modelling the progress vari-able behavior. Further investigation of a partially-premixed combustor might helpin establishing the preferred avenue here.Even though the heat transfer is without a doubt an important issue that needsto be addressed, at this stage of the evaluation of the UCS method in other aspectsappeared to be more pressing, at the forefront being the quantitative analysis ofthe NO emission prediction. Therefore it made sense to move away from technicalapplications slightly in order to gain access to high quality NO emission measure-ments and compare to them directly; and so the test case chosen for the next inves-tigation is the Cabra flame [19], a methane jet flame without enclosure. No heattransfer with a wall is relevant here and the flame is partially-premixed, enabling87yet another step towards achieving the goal of universal applicability.88Chapter 5Partially-Premixed Combustion –Jet FlameThe test case investigated in this chapter is the Cabra burner [18, 19]. Over theyears the Cabra flames [18, 19] have been studied by many groups with variousmodelling approaches [26, 78, 80, 106] and this test case supplies ample data ofthe emissions of interest without the additional complexity of a swirl-stabilizedflame. It therefore serves as an ideal candidate to explore the limitations of theUCS method and to establish a baseline for the ability of UCS to provide qualitativeand quantitative predictions of temperature and emissions of interest as well as theprediction of lift-off height. In Sec. 5.1 the test case is outlined and in Sec. 5.2details of the numerical setup are described. The results will be presented anddiscussed in Sec. 5.4.5.1 Experimental SetupThe Cabra burner [18, 19] has been tested in different configurations to investi-gate the effects of temperature and equivalance ratio on the lift-off height. In allcases the vitiated coflow originates from a cluster of 2200 small hydrogen flameswhile the central premixed jet is generated either from air with H2 [18] or fromair and CH4 [19]. The latter is chosen for this investigation as it is advantageousto validate the approach for methane chemistry further before moving on to differ-89ent fuels. Fig. 5.1 illustrates the configuration, in which multi-scalar multi-pointmeasurements were taken during the experiment.(a) Burner schematic [19] (b) PhotoFigure 5.1: (a) Burner schematic [19] and (b) a photograph of the experimentfor the lifted methane-air flame in vitiated coflowMajor species concentrations and temperature were measured using a single-point Raman-Rayleigh scattering technique. In addition laser-induced fluorescence(LIF) was used to measure the OH radical and CO mass fractions. It was re-ported that for CO the LIF results are more accurate than the results of the Raman-Rayleigh scattering technique. Both measurement sets will be used below. Theexperiment was carried out under ambient conditions and the boundary conditionsof the two streams are described in Tab. 5.1. For this configuration the lift-offheight was determined to be 35d, where d is the diameter of the jet (4.57mm).5.2 Numerical SetupThe simulation domain covers 210mm in diameter and 125.3d (= 70mm+110d)in length, where 70mm is the length of the methane-air jet pipe from the vitiatedcoflow flames to the inlet as illustrated in Fig. 5.2. A structured O-grid mesh was90Table 5.1: Cabra flame boundary conditions from experimentJet CoflowRe [−] 28000 23300d [mm] 4.57 210Uz [m/s] 100 5.4T [K] 320 1350Global φ [-] 0.4Global Zs [-] 0.177JetCoflow110d70 mmadiabatic wallsxzFigure 5.2: Numerical domain of Cabra burner91constructed using 540 cells in axial, 126 cells in radial and 128 cells in circumfer-ential direction to obtain a total of 7.88 million cells, where 6 prism layers wereused on the walls. The Smagorinsky model [102] was chosen to model the subgrid-scale turbulence in the LES approach.Time-averaged fields were generated over a total of five flow-through timesafter the first full flow-through time based on the velocity of the coflow of 5.4m/s.5.3 ChemistryThe GRI-3.0 mechanism [103] was used for this study as it had been more suc-cessful than the UBC-3.0 mechanism [51] in the previous study of the BRS burner.Furthermore the MLF-PDF had proven superior to the β -PDF for progress variableand was therefore used here as well. To eliminate the need for a clipping function(see Sec. 4.3) the conditional space was defined such that c = 1.0 represents theequilibrium state equi. Apart from this change, the initialization procedure is thesame here as in the previous chapters and the mesh is made up of 300 cells inthe mixture fraction dimension – unevenly spaced with higher resolution near sto-ichiometric – and 100 cells in progress variable dimension. The UCS equations(Eq. 2.31) were then solved under adiabatic conditions using the same L-stable,stiffly-accurate embedded Rosenbrock ODE solver of order (2)3 [92] as before.A number of UCS parameters have been identified, but not much have been var-ied thus far. In this study a number of different approaches were explored for theseparameters as per Tab. 5.2 building on the table of the previous chapter, Tab. 4.2.With that a total of four simulations were performed using four different chemistrytables, but otherwise the numerical setup was identical. Throughout this chapterthe simulations will be referred to as “Simulation #X” rather than “Cabra #X”.Simulation #1 is the baseline UCS approach. Simulation #2 is an improved versionusing a set of conditional scalar dissipation coefficients better suited for this testcase. The values of kZ and kc were chosen based on an extensive study of otherworks regarding the optimization of these constants [31, 82] and the desire to con-verge towards an optimal set of values to be used in a universally applicable tool.Simulation #3 serves the purpose of studying the effect of modelling conditionalscalar dissipation with a function described in Eq. 2.38 . Simulation #4 explores the92Table 5.2: UCS parameters of Cabra flame simulations compared to FIRSTand BRS burner simulationSimulationprogress variabledefinitionkZχccmodelkcPDF ofZPDF of cFIRSTburnerc =YCO2YCO2,cc(Z)25 constant 35 β -PDF β -PDFBRSburnerc =YCO2YCO2,cc(Z)10 constant 15 β -PDF β -PDF /MLF-PDFCabra #1 c =YCO2YCO2 ,equi(Z)25 constant 35 β -PDF MLF-PDFCabra #2 c =YCO2YCO2 ,equi(Z)16 constant 20 β -PDF MLF-PDFCabra #3 c =YCO2YCO2 ,equi(Z)16refer toEq. 2.3820 β -PDF MLF-PDFCabra #4c∗ =YCO2+YCOYCO2 ,equi(Z)+YCO,equi(Z)16refer toEq. 2.3820 β -PDF MLF-PDFoption of using c∗, which is based on the sum of CO2 and CO, as progress variablerather than using c, which is based on CO2 alone. CO is known to have a signifi-cant impact on the energy released during oxidation and therefore its influence onthe progress of combustion may be important.5.4 Results and Discussion5.4.1 Lift-off HeightIn the experiment the lift-off height, as determined using visible chemilumines-cence as flamefront indicator, was reported to be at 35d [19]. There are differentways of defining the lift-off height of a flame. In experiments it is usually donebased on the chemiluminescence originating from species such as OH* and others.As this specie in particular is not being tracked, the lift-off height was determinedbased on both the time-averaged temperature T˜ exceeding the temperature of thecoflow Tco f low = 1350 and the time-averaged OH mass fraction Y˜OH exceeding thethreshold of 1× 10−4. All four variations simulated are compared and evaluatedwith regards to the predicted lift-off height and the flame shape.Figs. 5.3 through 5.6 include a contour plot of (a) the instantaneous Tempera-931d15d30d40d50d70d(a) T at t = 0.5s (b) T˜ (left) and Y˜OHFigure 5.3: Contour plots of Simulation #1 (baseline) with (a) instantaneoustemperature and (b) mean temperature (left) and mean OH mass fraction(right)ture on the central x− z-plane and (b) a mean temperature and OH mass fraction,which are time-averaged over a total of 350ms – 70 flow through times based onthe jet velocity of 100m/s and almost 4 flow through times based on the coflowvelocity of 5.4m/s. The baseline simulation (#1) predicts a lift-off height of 22d asper Fig. 5.3(b) and distinguishes itself from the other simulations through a widerbase of the flame anchor. The instantaneous temperature contour suggests that theflame otherwise does not look significantly different from the one in Simulation#4 – apart from the lift-off height that is – however the time averaged tempera-941d15d30d40d50d70d(a) T at t = 0.5s (b) T˜ (left) and Y˜OH(right)Figure 5.4: Contour plots of Simulation #2 with (a) instantaneous tempera-ture and (b) mean temperature (left) and mean OH mass fraction (right)ture shows a wider base than all other simulations. Therefore it is not the flamebase itself that is wider but the anchor point that is not as stable as in the othersimulations.Changing the scalar dissipation parameters as per Table 5.2 leads to an interest-ing change in the behavior of the flame near the base. As per Fig. 5.4(a) the flameseems to stretch and become thinner around the core of the jet forming a lengthybase upon which the flame stabilizes. With that the lift-off height shifts to about18d downstream. By introducing the alternative definition of the scalar dissipationof progress variable using the error function as per Eq. 2.38 the lift-off height and951d15d30d40d50d70d(a) T at t = 0.5s (b) T˜ (left) and Y˜OH(right)Figure 5.5: Contour plots of Simulation #3 with (a) instantaneous tempera-ture and (b) mean temperature (left) and mean OH mass fraction (right)some of the character of the flame shape remains, but the flame base becomes thin-ner once again. Fig. 5.5(a) suggests that the core of the jet starts opening up slightlyaround 6d downstream, but ignition doesn’t occur until about 17d downstream.By far the best result was achieved with the Simulation #4 where the lift-offheight was predicted correctly at 35d. This flame does not show a stretched outflame base as becomes clear from comparing Fig. 5.6(a) to Fig. 5.4(a) and 5.5(a).Both ways of evaluating the lift-off height are equally valid and interchange-able in this case. Comparing all contour plots (b) in the Figs. 5.3 through 5.6 bearsevidence of that. Since the OH production is highly dependent on the tempera-961d15d30d40d50d70d(a) T at t = 0.5s (b) T˜ (left) and Y˜OH(right)Figure 5.6: Contour plots of Simulation #4 with (a) instantaneous tempera-ture and (b) mean temperature (left) and mean OH mass fraction (right)ture this conclusion is to be expected. Furthermore the behavior of OH is mostlyidentical for all four simulations. In regions of highest temperature, maximum OHvalues can be found and beyond the maximum, both temperature and OH massfraction begin to decline, which can be observed near the top of the computationaldomain.5.4.2 Temperature and Emission PredictionsThe plots in Fig. 5.7 show predictions of the temperature as well as the CO and NOmass fractions along the centerline for all four variations. In the case of YCO, both97sets of experimental data are used for comparison.The results of Simulation #4 seem to slightly outperform those of the base-line and the other two variations. Simulation #4 leads to a later onset in increasedheat release and with that the lift-off height is further downstream, but also thetemperature measurements near z = 50d are matched very well as can be seen inFig. 5.8(e) and the peak temperature level is the same as that of the experiment,however slightly further downstream (see Fig. 5.7(a)). This suggest that the flamepredicted by the UCS simulation is longer than what was observed in the experi-ment. The most notable difference between the Simulation #4 and the experimen-tal data in Fig. 5.7(a) is that the smaller hump that is noticeable in the experimentbetween z = 10d and z = 30d – and with that before the flame base – was notreproduced by the simulation. In fact, none of the 4 simulations suggest this pre-mature increase in temperature prior to the ignition onset. This could have to dowith the turbulence modelling as it is clear from the contour plot of the velocitythat little mixing occurs in the center of the jet until the combustion process sets in.Therefore the modelling approach as it stands can not predict a preheating in thisupstream location, when there is insufficient mixing. This is, of course, assumingthat the experimental profile is correct.Since there is no velocity data available it is not possible to confirm whetherthe flow field produced by the LES simulations is correct. A closer inspectionof the mass fraction of CO along the centerline extracted from the simulations inFig. 5.7(b) shows that the predictions are in the correct order of magnitude butthey do not exactly match up with the experiments. However it appears that withthe modifications made through changing the modelling function for conditionalscalar dissipation and the progress variable definition, the predictions improved toa significant degree. Firstly, the location of the peak mass fraction of CO is moreclosely matched by Simulation #3 and #4 and so is the magnitude of the peak.Looking at Fig. 5.7(c), which shows the same results normalized to the maximumvalue of CO in the data set, this trend is confirmed. Secondly, it also illustratesthat the gradients for the increase as well as the decrease in CO value is capturedcorrectly by Simulations #2 through #4.As for the NO prediction, Fig. 5.7(d) shows the mass fraction of NO predictedagainst the measured data and Fig. 5.7(e) shows the same data but on a logarithmic98(a) T [K](b) YCO [−] (c) YCOYCO,max(d) YNO [−] (e) YNO [−] on log-scaleFigure 5.7: UCS predictions for different variables along the centerline with:Simulation #1 (—), Simulation #2 (- · -), Simulation #3 (−−) and Simu-lation #4 (· · ·); experimental data (∗); normalized experimental data (•)99scale. This illustrates clearly that the NO emissions are mostly predicted to be inthe correct order of magnitude between about z = 25d and z = 60d, but furtherdownstream is dramatically overpredicted. The NO being present further upstreammostly a result of mixing from the coflow stream. In the simulations, the in-flowinggas does not contain NO as it had been recorded to be less than 1ppm and so it wasdecided not to take inflow NO into account in the simulation, therefore it is to beexpected that the NO can not appear in the UCS simulation this early on.Fig. 5.8 shows the temperature prediction at different downstream locationsand from the comparison with the experimental data an idea can be formed on thedifferences in flame shape.The exact match of the temperature profile at z = 1d is merely a confirmationof the correctly set inlet conditions. On the whole, for all other downstream lo-cations the same observation can be made as earlier – the temperature along thecenterline is only captured correctly in few places and those places vary for the dif-ferent simulations. Again Simulation #4 shows the most promising overall resultsas it captures the trends correctly for at least the locations downstream of z = 40d.Looking in particular at Figs. 5.8(c) and 5.8(d) the temperatures of Simulation #4are underpredicted in the center and tend to be slightly overpredicted at a radiusnear 10mm. Nevertheless the shape of the temperature profiles seems to be moreclosely matched with Simulation #4 and from that can be concluded that the flameshape in thickness and flame base characteristics is best represented by Simula-tion #4. Simulation #3 differs in a number of ways – not only in this set of graphsbut in general. Is had previously been observed in Sec. 5.4.1 that the flame is thin-ner and the base stretching up much higher than in the other simulations, whichcauses the profile at z = 70d downstream to have a dip in the center. This does notagree with the profile seen in the experiment. Furthermore it is to be noted thatthe fluctuations in the LES simulation over 300ms were compared to the fluctua-tion in the experimental data and it was found that they are of the same order ofmagnitude.When considering the comparison of CO mass fraction to experimental mea-surements the downstream locations of 40d, 50d and 70d are of particular interestas the measurements upstream are of very low magnitude. The plots in Fig. 5.9show the time-averaged mass fractions in the top row and again the same data100(a) z = 1d (b) z = 15d(c) z = 30d (d) z = 40d(e) z = 50d (f) z = 70dFigure 5.8: Temperature predictions at different downstream locations forSimulation #1 (—), Simulation #2 (- · -), Simulation #3 (−−) and Sim-ulation #4 (· · ·) in comparison to experimental data (∗)101(a) z = 40d (b) z = 40d(c) z = 50d (d) z = 50d(e) z = 70d (f) z = 70dFigure 5.9: First column of (a), (c) and (e): CO mass fraction predictionsat different downstream locations for Simulation #1 (—), Simulation#2 (- · -), Simulation #3 (−−) and Simulation #4 (· · ·) in comparisonwith experimental data from Raman-Rayleigh scattering (∗) and LIF(o); Second column of (b), (d) and (f): CO mass fraction predictionsnormalized by the maximum value in each respective data set comparedto normalized experimental data (•)102but normalized to the maximum value in the respective data set to better observethe trends. The overprediction of the CO mass fraction is quite significant in allsimulations and again Simulation #4 performs best overall. It is also remarkableto see such a big difference between Raman-Rayleigh scattering and LIF resultsand shows how difficult it is to measure this species correctly. From the plots inFigs. 5.9(b) and 5.9(d) it is also obvious that the gradients are not captured cor-rectly near the center of the jet flame. Only at z = 70d is a very nice agreementachieved both qualitatively and quantitatively.Turning back to the OH prediction discussed earlier, the downstream locationsof z= 40d, z= 50d and z= 70d are being studied in some more detail. All plots inFig. 5.10 show the trends being captured quite well by all simulations – in particularwhen looking at the qualitative graphs in Figs. 5.10(b) to 5.10(f). The peak locationand magnitude are overall best matched by Simulation #4 for both the quantitativeset of graphs and the qualitative set.Lastly the NO emission prediction is being discussed with the help of log scaledplots in the first row of Fig. 5.11 and qualitative plots in the second row. Whilesome of the results are within the correct order of magnitude, the overall impressionis that the simulations can not predict the NO levels correctly – in particular theybecome worse further downstream, which is not ideal when considering the overallpurpose of using this model to predict NO emissions at the exit of a combustor.This may in part be due to inaccuracies in the NOx chemistry in GRI-3.0 or due tothe simulations neglecting the effects of radiation heat transfer on the temperature.5.5 Concluding ThoughtsAmongst the most important criteria for a user-friendly industrially-relevant toolis the applicability of the model to all types of combustion from premixed to non-premixed including partially-premixed combustion. This present study illustratedthat partially-premixed combustion of a lifted flame is possible with UCS afteralready having demonstrated success in the previous chapters in the context of pre-mixed and non-premixed swirl-stabilized flames. Having a tool that can fulfill thiscriterion is of tremendous importance to enable the development of future combus-tion technology and therefore the results presented here are exciting. Perhaps the103(a) z = 40d (b) z = 40d(c) z = 50d (d) z = 50d(e) z = 70d (f) z = 70dFigure 5.10: First column of (a), (c) and (e): OH mass fraction predictionsat different downstream locations for Simulation #1 (—), Simula-tion #2 (- · -), Simulation #3 (−−) and Simulation #4 (· · ·) in com-parison with experimental data (∗); Second row of (b), (d) and (f): OHmass fraction predictions normalized by the maximum value in eachrespective data set compared to normalized experimental data (•)104(a) z = 40d (b) z = 40d(c) z = 50d (d) z = 50d(e) z = 70d (f) z = 70dFigure 5.11: First column of (a), (c) and (e) : NO mass fraction predictionsat different downstream locations on a logarithmic scale for Simula-tion #1 (—), Simulation #2 (- · -), Simulation #3 (−−) and Simula-tion #4 (· · ·) in comparison with experimental data (∗); Second columnof (b), (d) and (f): NO mass fraction predictions normalized by themaximum value in each respective data set compared to normalizedexperimental data (•)105most obvious criterion and the most difficult to validate and achieve is the qualita-tive and quantitative accuracy of emission predictions, in particular NO emissions.This was the focus of the present chapter.A number of parameters of the UCS method had thus far been set based onweakly justified assumptions: (1) the modelling of the conditional scalar dissi-pation rate of the progress variable with a constant value; (2) CO2 as the singlespecies upon which the definition of the progress variable should depend. In thischapter four different variations of the aforementioned parameters were tested andcompared. The most promising results were ultimately produced using the sumof CO2 and CO mass fraction to define the progress variable and a function forthe conditional scalar dissipation based on the inverse of the complementary errorfunction for both mixture fraction and progress variable.Particularly encouraging is the fact that the lift-off height of the flame wascorrectly predicted by this improved version of the UCS model. Overall the tem-perature trends were well captured from the flame base downstream and at thesame time species mass fractions were predicted with some success if not quan-titatively, then at least qualitatively. However, it became clear that the model stillrequires further development to be able to reliably predict emissions – in particularCO and NO – with high accuracy and further investigations will be necessary toreach the ultimate goal of using this model to develop real combustors. A majorchallenge is the lack of high quality experimental data for applications of technicalrelevance. The dramatic differences in CO values extracted from LIF vs. Raman-Rayleigh scattering data is a good indicator for just how difficult it is to measurethese species accurately.106Chapter 6Conclusions and Future WorkHaving tested the UCS modelling approach as predictive method for turbulence-chemistry interaction on three different cases allows for an overall assessment of(1) the requirements for industry application stated in the five Criteria established inSec. 1.2 and (2) the limitations of the model as per its current state of development.These aspects are discussed in Sec. 6.1 and Sec. 6.2 respectively and suggestionsfor future work to address remaining issues are presented in Sec. 6.3.6.1 Evaluation based on Industry CriteriaA qualitative analysis of the UCS method with regards to the five criteria – as sum-marized in Tab. 2.2 for other combustion modelling techniques – is necessary butnot sufficient. The following Tab. 6.1 gives an overview of the status based on theanalysis of this thesis and provides a measure of success with regards to the ulti-mate goal of being able to use this modelling approach in industry for combustordevelopment. More detailed descriptions of the conclusions drawn from this studywith regards to the individual criteria is given in the following sections.6.1.1 Computational CostIn its current state the UCS model is able to deliver emission predictions for verylow computational costs. The study of the FIRST burner revealed that in com-parison with the performance of the URANS simulation of Eberle et al. [30] the107Table 6.1: Quantitative analysis of UCS as per current status and ultimategoals with regards to the five Criteria for industry applicationCriteria current status ultimate goal#1LowComputationalCost20% of computationalcost of Eberle et al. [30]X#2 Ease of Integrationimplementation intoexisting softwareenvironment ofOpenFOAMX#3Versatility withregards toCombustion Typespremixed, non-premixedand partially-premixedcombustionX#4Flexibility withregards to FuelTypes2 different CX HY fuels; 3different mechanismsJet A surrogatecapabilityrequired#5Accuracy ofEmissionPredictionsNO: 50-150% error;reasonable trends;NO: 25% errorand correcttrendscomputational cost using UCS is estimated to be about 5 times lower. As for theBRS study, three test points were simulated with both the UCS approach and thePaSR model using detailed chemistry of the GRI-3.0 mechanism. Using the PaSRmodel took approximately 30 times longer to simulate the same flow time. ThePaSR represents a simple readily available model able to simulate NO chemistrywhen combined with detailed mechanisms, but it is rarely used in this way due tothe enormous computational cost involved with solving transport equations for allbut one species included in the mechanism.In addition to the cost of the CFD simulation itself, one needs to considerthe preprocessing costs and efforts involved in using UCS. The simulation of thechemistry in conditional space can – depending on the mechanism – take betweenseven and ten days on 96 CPUs (with a processor clock rate of 2.1GHz). If a gridconvergence study on the conditional mesh is required, then this might increase thecomputational cost. Thus far the grid size of 300× 100 for mixture fraction andprogress variable has proven sufficient for all test cases given that the spacing in108the dimension of mixture fraction resolves the region near stoichiometry well.6.1.2 Ease of IntegrationThe accessibility and compatibility is inherently satisfied as the custom solvers,UCSFoam and rhoFlameFoam, are implemented in the open-source softwareenvironment of OpenFOAM, which is available to industry partners.In addition to the accessibility of the model itself through the solvers, an im-portant aspect is the time requirement involved in setting up a new test case witheither a new fuel or different boundary conditions. This includes the preparationand initialization of the UCS simulation in conditional space, the generation of anassumed PDF and the processing of the resulting chemistry manifold to a table ofunconditional values of reactive scalars.In order to eliminate some of the difficulties around this aspect of the accessi-bility of the model, a user-friendly tutorial with detailed documentation was createdand can be found in Appendix B. With that it should be possible to set-up a newtest case with a new fuel in two days and generate the chemistry table using themanifold in about 24 hours. In the case of using a fuel that had previously beensetup already, where only the boundary conditions (pressure, temperature) havechanged, the setup should take no more than 2 hours.6.1.3 Versatility with regards to Combustion TypesAmongst the most important criteria is the applicability of the model to all types ofcombustion from premixed to non-premixed including partially-premixed combus-tion. This present study illustrated that UCS satisfies this criterion by simulating anon-premixed swirl-stabilized flame, various premixed swirl-stabilized flames anda partially-premixed jet flame. Having a tool that can fulfill this criterion is oftremendous importance to enable the development of future combustion technol-ogy and therefore the results presented in Chapters 3 through 5 are exciting.Over the course of these various studies a number of parameters of the UCSmethod were adjusted and their effects analyzed; in particular: (1) the modellingof the conditional scalar dissipation rate of the progress variable; (2) the speciesupon which the definition of the progress variable depends; and (3) the model109for the marginal PDF of progress variable. Tab. 6.2 shows the different optionsavailable thus far. While it is acceptable to maintain some adjustable parametersgoing forward, suggestions as to which options should become common practicefor future work are given in Sec. 6.3.Table 6.2: UCS parameters tested in this studyOption 1 Option 2 Option 3progress variabledefinitionc =YCO2YCO2 ,cc(Z)c =YCO2YCO2 ,equi(Z)c∗ = YCO2+YCOYCO2 ,equi(Z)+YCO,equi(Z)χcc model constantrefer toEq. 2.38PDF of Z β -PDFPDF of c β -PDF MLF-PDF6.1.4 Flexibility towards fuel typesIn order to accommodate future developments using alternative fuels, the versatilityof the model with regards to fuel types and chemical kinetic mechanisms is veryimportant. Two different fuels and three different mechanisms have been studiedwith UCS thus far, but further investigations, in particular with more complex fuels,will be needed in the future to demonstrate the necessary flexibility of the model.As a next step, simulation using a Jet A surrogate mechanism is essential.6.1.5 Accuracy of Emission PredictionsThe perhaps most obvious criteria and the most difficult to validate and achieveis the qualitative and quantitative accuracy of emission predictions, in particularNO emissions. From discussions with industry it has been established that an errormargin of 25% would, in fact, be acceptable at the current state, keeping in mindthe difficulties involved in measuring the concentration of NO correctly as well asthe difficulty to account for all aspects involved in NOX chemistry due to its strongdependency on the temperature. The qualitative analysis of the NO production inthe case of the BRS burner in Sec. 4.4.2 suggested a reasonable prediction of thetrends both with respect to the equivalence ratios and the temperature, however110since there was no experimental data to compare to it is not sufficient to evaluatethis criteria. In this study only the Cabra flame provided experimental measure-ments to compare the NO emission predictions of UCS directly. An error between50 and 150% had to be noted here and suggests the need for further investigationsof other burners with similarly high quality measurements preferably of industrialrelevance to develop the UCS model towards more accurate emission predictions.Successes can be noted in predicting the lift-off height of the Cabra flamecorrectly with improved modelling parameters and also the flame location in theFIRST burner test case. In the Cabra test case the overall temperature trends werewell captured from the flame base downstream and at the same time the trends forthe species mass fractions were predicted qualitatively and in the case of OH alsoquantitatively.6.2 Current LimitationsThe first two test cases presented, the FIRST burner and the BRS burner, indicatethe need for further development of the UCS method to include effects of heattransfer with sources inside and outside of the numerical domain of the combustorand in particular for the interaction with cooled walls or heated surfaces. In or-der to accomplish that a third conditioning variable would need to be implementedto extend the conditional space into 3D. Major challenges involved in that imple-mentation are: (1) choice of a suitable conditioning variable – total enthalpy hasrecently been suggested for this purpose [15]; (2) transport of such a variable inphysical space; (3) a PDF representation of the conditioning variable – differentmethods to model the PDF of total enthalpy are currently being investigated; (4)with each conditioning variable the table size grows by a factor of n ·m, where n isthe number of bins chosen for the mean and m the number of bins chosen for thevariance of the additional conditioning variable in the chemistry table. The latteraspect is currently the main bottleneck as it severely influences the practicality ofthe model. One potential method to avoid this and future explosions in table sizewould be to employ artificial neural networks to represent the multidimensionaltable. Currently this approach is a work in progress and will require further devel-opment [5]. Even though the necessity for such an extension can not be emphasized111enough, it will need to be addressed in future work.Another highly restrictive limitation at the moment is the intensive RAM stor-age requirement inflicted by the code due to a combination of OpenFOAM codestructure and data processing routines and the large size of the chemistry table. Atthe moment each processor involved in the simulation needs to provide a minimumof 6GB of storage to handle the solver operations. Being able to include methodssuch as artificial neural networks would therefore not only be a good solutions forthe extension of the conditional space to 3 dimensions (mixture fraction, progressvariable, total enthalpy), but could also eliminate the large storage requirementsof the current solver and chemistry table. Similarly hyperoctree [1, 93] could beused to recursively adapt the number of cells in this multi-dimensional space ofconditioning variables.6.3 Suggestions for Future WorkThe first priority in future development ought to be the implementation of totalenthalpy into the UCS solver, UCSFoam, to ensure that the model can accountfor effects of heat transfer. Related to this advance, the development of a marginalPDF for total enthalpy and the construction of a joint PDF for the three conditions isessential to using a third condition. Furthermore, the option of employing artificialneural networks to generate a smaller multidimensional table goes hand in handwith this development and would resolve the major limitations UCS is currentlyfacing.In addition, simulations of more complex fuels such as Jet A should be carriedout and evaluated.To optimize UCS further, a preferred set of parameters should be chosen fromthe options developed in this work (see Table 6.2) and applied to all previous testcases in the same manner in order to determine whether the matrix of parameterscan be converged into one universally applicable set of parameters. Based on thethree test cases studied in this work, the ideal universal parameters would be:• progress variable definition based on CO2 and CO mass fraction (Option 3)• definition of the conditional scalar dissipation rate of progress variable using112the erfc function as per Eq. 2.38 (Option 2)• β -PDF for mixture fraction (Option 1)• MLF-PDF for progress variable (Option 2)In light of the insufficient accuracy of the model in its current state, furtherinvestigations of other burners should be chosen with care to ensure high qualitymeasurements preferably of industrial relevance to develop the UCS model towardsmore accurate emission predictions.113Bibliography[1] S. Aluru and F. E. Sevilgen. Dynamic compressed hyperoctrees withapplication to the n-body problem. International Conference onFoundations of Software Technology and Theoretical Computer Science,pages 22–33, 2000. → page 112[2] R. S. Barlow. 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Scarpato, and W. Polifke. LES combustion modelwith stretch and heat loss effects for prediction of premix flamecharacteristics and dynamics. ASME Turbo Expo, (GT2017-63357), 2017.→ pages xiv, 73, 75, 76, 81, 82, 83, 84[111] H. Tennekes and J. Lumley. A First Course in Turbulence. The MIT Press,1972. → page 8[112] J. L. Toof. A Model for the Prediction of Thermal, Prompt, and Fuel NOXEmissions From Combustion Turbines. Journal of Engineering GasTurbines Power, 108(2):340–347, 1986. → page 22[113] L. Valin˜o. A Field Monte Carlo Formulation for Calculating the ProbabilityDensity Function of a Single Scalar in a Turbulent Flow. Flow, Turbulenceand Combustion, 60:157–172, 1998. → page 26124[114] J. A. van Oijen and L. P. H. de Goey. Modelling of premixed laminarflames using flamelet-generated manifolds. Combustion Science andTechnology, 161:113–137, 2000. → page 21[115] C. A. Velez, S. M. Martin, and S. S. Vasu. Reacting unsteadyreynolds-averaged navierstokes with the tabulated premixed conditionalmoment closure method. Journal of Propulsion and Power, 33(3), 2017.→ page 33[116] L. Vervisch, R. Hauguel, P. Domingo, and M. Rullaud. Three facets ofturbulent combustionmodelling: DNS of premixed v-flame, LES of liftednonpremixed flame and RANS of jet-flame. Journal of Turbulence, (5):1–36, 2004. → page 39[117] W. Vicente, M. Salinas, E. Barrios, and C. Dopazo. PDF modeling of COand NO formation in lean premixed methane flames. Combustion Scienceand Technology, 176:585–601, 2004. → page 26[118] R. Vicquelin, B. Fiorina, S. Payet, N. Darabiha, and O. Gicquel. Couplingtabulated chemistry with compressible cfd solvers. Proc. Combust. Inst., 1:1481–1488, 2011. → page 47[119] A. Vranos. Turbulent mixing and NOX formation in gas turbinecombustors. Combustion and Flame, 22(2):253–258, April 1974. → page22[120] H. Wang and A. Laskin. A Comprehensive Kinetic Model of Ethylene andAcetylene Oxidation at High Temperatures. Progress report, University ofDelaware, 1999. → pages 20, 46, 49[121] M. Wang, J. Huang, and W. K. Bushe. Simulation of a turbulentnon-premixed flame using Conditional Source-term Estimation withTrajectory Generated Low-Dimensional Manifold. Combustion Theory andModelling, 31:1701–1709, 2007. → pages 30, 38[122] J. Warnatz, U. Maas, and R. Dibble. Combustion: Physical and ChemicalFundamentals, Modeling and Simulation, Experiments, PollutantFormation. Number 3rd Edition. Springer, 2001. → pages xii, 16, 20[123] H. G. Weller, G. Tabor, H. Jasak, and C. Fureby. A tensorial approach tocomputational continuum mechanics using object-oriented techniques.Computers in Physics, 12(6):620–631, 1998. → page 36125[124] F. A. Williams. Turbulent combustion. In J. D. Buckmaster, editor,Mathematics of Combustion, pages 97–132, 1985. → page 17[125] F. A. Williams. Combustion Theory. CRC Press, 2018. → pages xiii, 33, 69[126] J. Xu and S. Pope. PDF calculations of turbulent nonpremixed flames withlocal extinction. Combustion and Flame, 123(3):281–307, 2000. → page26[127] M. C. Z. Huang and H. Zhang. Application of the sparse-lagrangianmultiple mapping conditioning approach to a model supersonic combustor.Physics of Fluids, 32(10), 2020. 105120. → page 27[128] V. Zimont and A. Lipatnikov. A numerical model of premixed turbulentcombustion of gases. Chemical Physics Reports, 14(7):993–1025, 1995. →page 82126Appendix AInterpolation in 4DAs mentioned above, the UCS table spans over four dimensions – Z˜× Z˜′′2× c˜× c˜′′2– and with that it is not entirely trivial to interpolate to a single look-up value inthis space. A three dimensional look-up can easily be imaged as a cube in which tointerpolate and so to visualize the look-up in four dimensions two separate cubesare used to illustrate the 4th dimension. Fig. A.1 shows them side by side withevery one of the 16 corners representing one value of the UCS table, all of whichare used to calculate the interpolated value.Initially placeholder values u∗, v∗, w∗ and x∗ are defined to represent the po-sition in the individual dimension with respect to the length of the bin in which itcan be foundu∗ =u−u0u1−u0 , v∗ =v− v0v1− v0 , w∗ =w−w0w1−w0 , x∗ =x− x0x1− x0 . (A.1)For instance, u0 and u1, would be the bin boundaries of the bin in which u wouldbe found. Then the interpolation can be carried out reducing the dimensions one127D0100D0110D0010D0000D1100D1110D1010D1000D0101D0111D0011D0001D1101D1111D1011D1001C010 C011C001C000C110 C111C101C100B10 B11B01B00A0 A1Figure A.1: Schematic for UCS table lookup interpolationby one as per the following system of equations:C000 = D0000(1−u∗)+D1000 u∗,C001 = D0001(1−u∗)+D1001 u∗,C100 = D0100(1−u∗)+D1100 u∗,C101 = D0101(1−u∗)+D1101 u∗,C010 = D0010(1−u∗)+D1010 u∗,C011 = D0011(1−u∗)+D1011 u∗,C110 = D0110(1−u∗)+D1110 u∗,C111 = D0111(1−u∗)+D1111 u∗,(A.2)B00 =C000(1− v∗)+C100 v∗,B01 =C001(1− v∗)+C101 v∗,B10 =C010(1− v∗)+C110 v∗,B11 =C011(1− v∗)+C111 v∗,(A.3)A0 = B00(1−w∗)+B10 w∗,A1 = B01(1−w∗)+B11 w∗,(A.4)S = A0(1− x∗)+A1 x∗. (A.5)128Appendix BUser Guide for UCS softwareThis tutorial is designed for users who have standard OpenFOAM-3.0.1, OpenFOAM-4.0 orOpenFOAM-5.0 installed and have received and integrated the UCS-softwarepackage and successfully compiled it.B.1 Step-by-step: How to simulate the chemistry with UCS?In this section it is assumed that the test case to be prepared is located in the fol-lowing directory:” myDirectory / run / tes tChemis t ry / ”Within this directory there are subdirectories specifically for the conditional grid:” myDirectory / run / tes tChemis t ry / 0 / cnd2D / ”” myDirectory / run / tes tChemis t ry / constant / cnd2D / ”” myDirectory / run / tes tChemis t ry / system / cnd2D / ”1. Design conditional grid:(a) Find Zstoichiometric for your combustion application.(b) Define two regions of mesh in the ".../system/cnd2D/blockMeshDict" fileusing for example the simpleGrading method as per this example:/∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗− C++ −∗−−−−−−−−−−−−−−−−−−−−−−−−−−−∗\| ========= | || \\ / F i e l d | OpenFOAM: The Open Source CFD Toolbox |129| \\ / O pera t i on | Version : 1.5 || \\ / A nd | Web: h t t p : / / www.OpenFOAM. org || \\ / M a n i p u l a t i o n | |\∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗/FoamFile{vers ion 2 . 0 ;format a s c i i ;c lass d i c t i o n a r y ;ob jec t blockMeshDict ;}/ / ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ / /v e r t i c e s((0 0 0.01) (0 0 −0.01)(0 1 0.01) (0 1 −0.01)(1 0 0.01) (1 0 −0.01)(1 1 0.01) (1 1 −0.01)) ;b locks(hex (1 5 7 3 0 4 6 2) (300 100 1)simpleGrading (((0 .1 0.12 0 .7 )(0 .9 0.88 2.05))11)) ;edges() ;patches(patch COFLOW((0 2 3 1))130patch JET((5 7 6 4))patch UNREACTED((0 1 5 4))patch REACTED((2 6 7 3))empty SIDES((0 4 6 2)(1 3 7 5))) ;/ / ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ / /(c) Tips for designing the mesh:i. To better understand and ensure a smooth transition between thetwo regions, one can use the following online tool to support thedesign of the mesh:https://openfoamwiki.net/index.php/Scripts/blockMesh grading calculationii. As a rule of thumb, strive to have a growth rate of less than 5%within each region as well as between the two neighboring cellsconnecting the two regions.iii. Previous simulations have shown that a total number of cells of300×100 in mixture fraction and progress variable dimension re-spectively are sufficient.(d) Generate the mesh with the command:blockMesh −reg ion cnd2D2. Generating an initialization for the new mesh OR for an old mesh:131(a) If you are ...i. using a new chemistry:A. Place files of the chemical mechanism in the directory:” myDirectory / run / tes tChemis t ry / constant / cnd2D / ”B. Edit the file thermophysicalProperties to include thecorrect location and name of the chemistry files needed andspecify the boundary conditions of your applicationC. Choose whether you would like to use equilibrium or com-plete combustion for the definition and upper boundary of theprogress variable by setting cMaxIsEquilibrium eithertrue or false respectively.ii. using the same chemistry as before but new boundary conditions:A. Edit the file thermophysicalProperties to specify theboundary conditions of your applicationB. Choose whether you would like to use equilibrium or com-plete combustion for the definition and upper boundary of theprogress variable by setting cMaxIsEquilibrium eithertrue or false respectively.(b) If you are using a new grid or you are using new boundary conditionsor new chemistry on an old grid, you will need to rerun a initializationscript, which is customized for your chemistry. At the moment youhave the choice between the following:in i t ia l izeUCS2D C2H4ini t ia l izeUCS2D CH4in i t ia l izeUCS2D JetA Aachenin i t ia l i zeUCS2D Cabrain i t ia l izeUCS2D Cabra copco2The latter is a special case for using c∗ = (YCO2 +YCO)/(YCO2,equi +YCO,equi) as progress variable, where equi represents equilibrium. Inthe case of Jet A every mechanism uses a different composition of anumber of long chain hydrocarbons to represent this fuel and thereforethe initialization script needs to be tailored towards that specific mech-anisms. If the chemistry you are using is not amongst this list, adapt132one of the existing scripts to your needs. The source code can be foundin the directory:” myUCSSoftwareDirectory / a p p l i c a t i o n s / u t i l i t i e s / ”3. Eventhough the CFD simulation on the physical mesh is not being carriedout when running UCS, the output of the UCSFoam simulation and with thatthe content of the resulting chemistry table depends on just a few parametersset in the file thermophysicalProperties in the directory:” myDirectory / run / tes tChemis t ry / constant / ”For all species listed in species, the reaction rate will be part of eachoutputted solution generated during the UCSFoam simulation.4. After running the new initialization a file called “ maxCO2s.dat” will ap-pear in the main case directory. This file contains the values of mixturefraction Z for the bin centers of each cell along the upper boundary ANDlower bin boundary of first cell and upper bin boundary of last cell (aka themin and max values of Z) and the values for YCO2 corresponding to thoseZ values at the upper boundary of the progress variable dimension. Thesetwo sets of values ( Z 302( ... ) and maxCO2 302( ... );) need to be copied to the file:” myDirectory / run / tes tChemis t ry / constant / combust ionProper t ies ”5. Set UCS parameters in that same file combustionProperties. Here isan example:/∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗− C++ −∗−−−−−−−−−−−−−−−−−−−−−−−−∗\| ========= | || \\ / F i e l d | OpenFOAM: The Open Source CFD Toolbox || \\ / O pera t i on | Version : 2 .3 .0 || \\ / A nd | Web: www.OpenFOAM. org || \\ / M a n i p u l a t i o n | |\∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗/FoamFile{vers ion 2 . 0 ;format a s c i i ;c lass d i c t i o n a r y ;133l o c a t i o n ” constant ” ;ob jec t combust ionProper t ies ;}/ / ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ / /combustionModel UCS<psiThermoCombustion>;a c t i v e t rue ;caseLabel burnerX ; / / re ferenced i n l a b e l sUCSCoeffs{k ch i ZZ 16; / / u n i t s 1 / s − equal to peak ch ik c h i c c 1.55e−5; / / e f f e c t i v e l y the value o f D,/ / the d i f f u s i o n c o e f f i c i e n t ;/ / t h i s i s on ly used f o r the sqrGradC/ / model o f ch i cc/ / D e f i n i t i o n o f ch i ccch i cc model constant ; / / o ther op t ions : i n v e r r o r ; sqrGradC ;ch i cc min 20;ch i cc 20;i nvEr ro rCoe f f s{b ch i cc 20;Zs to i 0 .177;upperBouEqm t rue ;Z 302 ( 0 0.00250327 0.00745924 . . . 0.990494 0.996825 1 ) ;eqmCO2 302 ( 0 0.00163766 0.00487719 . . . 0.057715 0.0581052 ) ;maxCO2 302 ( 0 0.00163766 0.00487719 . . . 0.057715 0.0581052 ) ;}/ / Should a l l terms i n a l l c o n d i t i o n a l t r a n s p o r t equat ions be p r i n t e d ?pr in tTranspor tTerms no ;/ / CFL number f o r c o n d i t i o n a l d i f f u s i o n134dCFL 0 . 2 ;/ / Cond i t i on ing Var iab lescnd 2 (Z c ) ;/ / Maximum CO2 given ZmaxCO2 {Z 302 ( 0 0.00250327 0.00745924 . . . 0.990494 0.996825 1 ) ;maxCO2 302 ( 0 0.00163766 0.00487719 . . . 0.057715 0.0581052 ) ;}/ / Fluxes to i n v e r t fo r , i n a d d i t i o n to f l u x o f Z and QextraSpeciesFluxes 1(CO2) ;/ / Note : Every species s p e c i f i e d i n thermophys ica lProper t ies/ / ( save i n t e r t S p e c i e ) w i l l be t ranspor ted/ / Transpor t equat ions f o r cnd and varCndt ranspo r tCoe f f s {Z {Sc t 1 . 0 ; / / not used i n premixed}varZ {Sc 1 0 . 7 ; / / Used i n var iance t r a n s p o r t eqnSc 2 0 . 7 ; / / Used i n var iance t r a n s p o r t eqn}c {Sc t 1 . 0 ; / / not used i n premixed}varc {Sc 1 0 . 7 ; / / Used i n var iance t r a n s p o r t eqnSc 2 0 . 7 ; / / Used i n var iance t r a n s p o r t eqn135}}isoSDRModels {Z{type l i n e a r R e l a x a t i o n ;C D 1 . 0 ;}c{type l i n e a r R e l a x a t i o n ; / / INAPPROPRIATE FOR CO2 ! !C D 1 . 0 ;}}/ / PDFPDFModel beta ;betaPDFCoeffs {segregat ions 50 ( 0.0000000000e+00 . . . 9.9999999999e−01 ) ;}biBetaPDFCoeffs {means 100 ( 0 0.0101010101 . . . 1 ) ;segregat ions 50 ( 0.0000000000e+00 . . . 9.9999999999e−01 ) ;}/ / PDFlabel $ : caseLabel ;}/ / ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ / /6. Run the simulation with the command:UCSFoamOnce convergence is reached you can no longer see a change in the contourplot of T or wdotc or any other variable of critical importance for yourapplication.136B.2 Step-by-step: How to use UCS in a CFD simulation?1. Before beginning to set-up the CFD simulation one will need to generatethe UCS table, which entails generating the inner product of the UCSChemistry solution and a JPDF of the users choice. Technically this can bedone using any customized software, but in the package discussed here thereare a number of options provided. If this approach should be chosen then thefollowing steps apply:(a) Create a new directory:” myDirectory / run / testChemist ry 150x50 / ”Copy over necessary files and generate a new mesh (as before) but nowwith 150×50 cells.(b) Interpolate the solution obtained in Sec. B.1 using the openfoam appli-cation mapFields via the command:mapFields −cons i s ten t −sourceRegion cnd2D −ta rgetRegion . . .cnd2D −sourceTime la tes tT ime . . / tes tChemis t ry / .(c) Open the last solution (which you just interpolated onto the coarsergrid) with paraFoam or paraview and save the solution as a .csvfile.(d) Using this solution one of the matlab scripts can be applied either di-rectly (if the mesh is the same) or modified to match the new mesh.The matlab scripts available are:MakeUCSxPDF forCandZ brs .mMakeUCSxPDF forCandZ Cabra .mMakeUCSxPDF forCandZ first .mMakeUCSxPDF beta MLF cabra noClipping .mMakeUCSxPDF beta MLF brs clipUsingCO2 .mMakeUCSxPDF beta MLF brs clipUsingTemp .mwhere the first three use beta PDFs for both mixture fraction and progress vari-able and the latter three use beta PDF for mixture fraction but MLFPDF for progress variable.(e) It should be noted that in the work thus far, one large table,UCSxPDFData xx would result from this matlab script, however a137smaller version was extracted from it to use on the fly during CFD sim-ulation in parallel to reduce RAM usage. This table ought to containno more than 6 variables in total.2. Given that the mesh for the CFD simulation is already prepared for the phys-ical space, one can proceed by moving the chemistry table into a convenientlocation and naming the smaller table dualUCSData xx and the largetable dualUCSData xx full. The directories created for this purposemight be:myDirectory / run / tes tBurner /myDirectory / run / tab les /Note one might choose to keep all simulations in a single directory (both theUCSFoam and the rhoFlameFoam simulation). The mesh of the physicalspace does not matter for the chemistry simulation and so they can bothbe located in the same folder, however this might lead to the use of largeamounts of storage and furthermore one ought to be aware of some files thatneed to be changed when switching between those simulations.3. Prepare your initialization including the following files:myDirectory / run / tes tBurner / 0 / ZmyDirectory / run / tes tBurner / 0 / cmyDirectory / run / tes tBurner / 0 / varZmyDirectory / run / tes tBurner / 0 / varc( myDirectory / run / tes tBurner / 0 / Yde fau l t )As for the internal fields, one can choose to run rhoFlameFloam withoutcombustion first to generate an internal field for the oxidizer-fuel mixture inthe correct flow field.4. Furthermore add those four variables (Z, c, varZ and varc) to the fvSchemesand the fvSolution file.5. Set all remaining parameters for the CFD simulation as you normally would.As for UCS the following would be your standard choice for thecombustionProperties file:138/∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗− C++ −∗−−−−−−−−−−−−−−−−−−−−−−−−∗\| ========= | || \\ / F i e l d | OpenFOAM: The Open Source CFD Toolbox || \\ / O pera t i on | Version : 4.0 || \\ / A nd | Web: www.OpenFOAM. org || \\ / M a n i p u l a t i o n | |\∗−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−∗/FoamFile{vers ion 2 . 0 ;format a s c i i ;c lass d i c t i o n a r y ;l o c a t i o n ” constant ” ;ob jec t combust ionProper t ies ;}/ / ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ / /combustionModel CSEforUCS<rhoThermoCombustion>;a c t i v e t rue ;caseLabel tes tBurner / / re ferenced i n l a b e l sSc t 0 . 8 ; / / used i n Yeqn regard less o f whether combustion/ / i s a c t i v eCSEforUCSCoeffs{/ / Se t t i ngsi n v e r t f a l s e ; / / t r ue ;timeAverage f a l s e ; / / not i n i t i a l l yresetTimeAverage f a l s e ;/ / Chemistry Back−EndchemistryBackEnd dualUCS ;dualUCSParams {l a b e l $caseLabel ;}/ / Transpor t equat ions f o r cnd and varCndt ranspo r tCoe f f s {139c { / / Used when t r a n s p o r t i n g Z and varZSc t 0 . 8 ; / / Used to evaluate s u b f i l t e r f l u x/ / o f Z i n t r a n s p o r t equat ions f o r Z and varZSc t va r 0 . 8 ; / / Used to evaluate s u b f i l t e r f l u x/ / o f varZ i n t r a n s p o r t equat ion f o r varZSc t sd r 0 . 8 ;C D 1 . 0 ;sca la rD iss ipa t ionMode l l i n e a r R e l a x a t i o n ;/ / Used to eva luate unresolved sca la r d i s s i p a t i o n/ / o f Z i n t r a n s p o r t equat ion f o r varZl i nea rRe laxa t i onCoe f f s{Sc t 0 . 8 ; / / used i n LES}}Z { / / Used when t r a n s p o r t i n g Z and varZSc t 0 . 8 ; / / Used to evaluate s u b f i l t e r f l u x/ / o f Z i n t r a n s p o r t equat ions f o r Z and varZSc t va r 0 . 8 ; / / Used to evaluate s u b f i l t e r f l u x/ / o f varZ i n t r a n s p o r t equat ion f o r varZSc t sd r 0 . 8 ;C D 1 . 0 ;sca la rD iss ipa t ionMode l l i n e a r R e l a x a t i o n ;/ / Used to eva luate unresolved sca la r d i s s i p a t i o n/ / o f Z i n t r a n s p o r t equat ion f o r varZl i nea rRe laxa t i onCoe f f s{Sc t 0 . 8 ; / / used i n LES}}}/ / Ensemble D iv ide rs/ / −c u r r e n t l y not being used−/ / −i n flameFoam solver−ensembleDividers (z {140type distanceFromPlane ;po in t (0 0 0 ) ;normal (0 0 1 ) ;re la t i veTh icknesses 1 {1};neighbourWeight 1 ;}) ;/ / Load BalancingloadBalancing {type none ;}}/ / ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ / /Note that the caseLabel specifies which tables is being used for the chem-istry look-up.6. In the thermophysicalProperties file two lines need to be addedand specified according to the user’s preferences apart from the usual speciesproperties which are required for standard rhoReactingFoam simula-tions:species ( NO N2 ) ;t r a n s p o r t ( 1 0 ) ;What this means is that only NO is being transported in this simulation. N2is also the inert species, which means it takes the place of all species whoare not being transported for the time being and will be looked up later on.7. Run the CFD simulation with the following command:rhoFlameFoam8. Once convergence is reached and enough flow-through times have been sim-ulated, make a copy of the last solution tlast and assigning it for exampletnext = tlast + 0.001s. Then use function funkySetFields to overwrite141the instantaneous fields of Z, c, varZ and varc with the mean fields(ZMean, cMean, varZMean and varcMean) for the9. Modify the caseLabel in file combustionProperties to betestBurner full.10. Modify the species in file thermophysicalProperties to includeall the species whose mass fraction you wish to know in the last solutionand who will therefore be looked up in the next step. Assign them thetransport label “0” to indicate that they are not being transported butlooked-up.11. Make sure to add additional species to the fvSchemes and fvSolutionsfile if necessary.12. Rerun the simulation for only one time step using the same command asbefore:rhoFlameFoam142
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Evaluation of the Uniform Conditional State model for gas turbine combustor design
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