Scalar Dissipation Rate based Flamelet Modelling of Turbulent Premixed Flames Hemanth Kolla Department of Engineering University of Cambridge (Trinity Hall) A dissertation submitted for the degree of Doctor of Philosophy November 2009 Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This dissertation contains approximately 35000 words and includes 49 figures. Hemanth Kolla February 1, 2010 To my motherland, for the warmth of her womb and To Nalanda, for the warmth of her embrace Acknowledgements I would like to express my sincere gratitude to Dr. N. Swaminathan for his constant support, advice and encouragement. I have learnt a great deal under his tutelage and benefited immensely from his super- vision. I would also like to thank Professors Stewart Cant and Non- das Mastorakos for their invaluable guidance throughout the course of my PhD. Acknowledgements are due to Cambridge Commonwealth Trust and Cambridge Nehru Trust whose generous financial support helped fund my studies. Conference grants from Cambridge Univer- sity Engineering Department, Trinity Hall and Combustion Institute are also gratefully acknowledged. Acknowledgements are also due to Rolls–Royce plc. and in particular Dr. Ruud Eggels for the technical support. A special note of thanks to Mrs. Kate Graham and Mr. Peter Benie of Hopkinson laboratory without whose help finishing my PhD would have been very arduous. Friends and colleagues have greatly con- tributed to making my stay in Cambridge a memorable experience. I will forever cherish my association with members of Heat Gallery: Dr. Jim Rogerson, Olliver, Mark, Inkyu, Tristan, Tee, Larry, Shokri and Shaohong. In addition, I would like to thank my fellow Indian schol- ars at Cambridge who have helped me on various occasions and in particular Pashmina, Sriram and Sandeep for being such dependable buddies. I’m deeply indebted to my parents and my brother for their continued love and support and their faith in my endeavours. And finally I’d like to thank my wife Nalanda for being a patient companion and making it all worthwhile. Scalar Dissipation Rate based Flamelet Modelling of Turbulent Premixed Flames Hemanth Kolla Lean premixed combustion has potential for reducing emissions from combus- tion devices without compromising fuel efficiency, but it is prone to instabilities which presents design difficulties. From emissions point of view reliable predic- tions of species formation rates in the flame zone are required while from the point of view of thermo–acoustics the prediction of spatial variation of heat re- lease rate is crucial; both tasks are challenging but imperative in CFD based design of combustion systems. In this thesis a computational model for turbu- lent premixed combustion is proposed in the RANS framework and its predictive ability is studied. The model is based on the flamelet concept and employs strained laminar flamelets in reactant–to–product opposed flow configuration. The flamelets are parametrised by scalar dissipation rate of progress variable which is a suitable quantity to describe the flamelet structure since it is governed by convection– diffusion–reaction balance and represents the flame front dynamics. This para- materisation is new. The mean reaction rate and mean species concentrations are obtained by integrating the corresponding flamelets quantity weighted by the joint pdf of the progress variable and its dissipation rate. The marginal pdf of the progress variable is obtained using β–pdf and the pdf of the conditional dis- sipation rate is presumed to be log–normal. The conditional mean dissipation rate is obtained from unconditional mean dissipation rate which is a modelling parameter. An algebraic model for the unconditional mean scalar dissipation rate is proposed based on the relevant physics of reactive scalar mixing in turbulent premixed flames. This algebraic model is validated directly using DNS data. An indirect validation is performed by deriving a turbulent flame speed expression us- ing the Kolmogorov–Petrovskii–Piskunov analysis and comparing its predictions with experimental data from a wide range of flame and flow conditions. The mean reaction rate closure of the strained flamelets model is assessed using RANS calculations of statistically planar one–dimensional flames in corrugated flamelets and thin reaction zones regimes. The flame speeds predicted by this closure were close to experimental data in both the regimes. On the other hand, an unstrained flamelets closure predicts flame speed close to the experimental data in the corrugated flamelets regime, but overpredicts in the thin reaction zones regime indicating an overprediction of the mean reaction rate. The overall predictive ability of the strained flamelets model is assessed via cal- culations of laboratory flames of two different configurations: a rod stabilised V– flame and pilot stabilised Bunsen flames. For the V–flame, whose conditions cor- respond to the corrugated flamelets regime, the strained and unstrained flamelets models yield similar predictions which are in good agreement with experimental measurements. For the Bunsen flames which are in the thin reaction zones regime, the unstrained flamelet model predicts a smaller flame brush while the predic- tions of the strained flamelets model are in good agreement with the experimental data. The major and minor species concentrations are also reasonably well pre- dicted by the strained flamelets model, although the minor species predictions seem sensitive to the product stream composition of the laminar flamelets. The fluid dynamics induced attenuation of the reaction rate is captured by the strained flamelets model enabling it to give better predictions than the un- strained flamelets model in the thin reaction zones regime. The planar flames and laboratory flames calculations illustrate the importance of appropriately ac- counting for fluid dynamic effects on flamelet structure and the scalar dissipation rate based strained flamelet model seems promising in this respect. Furthermore, this model seems to have a wide range of applicability with a fixed set of model parameters. Contents List of Figures viii Nomenclature xiv 1 Introduction 1 2 Background on Premixed Flame Modelling 6 2.1 Laminar Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Turbulent Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 RANS Governing Equations . . . . . . . . . . . . . . . . . 10 2.2.2 Regimes of Turbulent Premixed Combustion . . . . . . . . 13 2.2.3 Bray–Moss–Libby Model . . . . . . . . . . . . . . . . . . . 16 2.2.4 Flame surface density models . . . . . . . . . . . . . . . . 19 2.2.5 G–Equation Level Set Formalism . . . . . . . . . . . . . . 21 2.2.6 PDF transport approach . . . . . . . . . . . . . . . . . . . 22 2.2.7 Presumed PDF approaches . . . . . . . . . . . . . . . . . . 23 2.2.8 Conditional Moment Closure . . . . . . . . . . . . . . . . . 24 2.2.9 Chemistry Tabulation . . . . . . . . . . . . . . . . . . . . 26 3 Scalar Dissipation Rate 28 3.1 Definition and Physical Meaning . . . . . . . . . . . . . . . . . . . 28 3.2 Role in Turbulent Combustion Modelling . . . . . . . . . . . . . . 30 3.2.1 Non–Premixed Combustion . . . . . . . . . . . . . . . . . 32 3.2.2 Premixed Combustion . . . . . . . . . . . . . . . . . . . . 33 3.2.3 Stratified/Partially Premixed Combustion . . . . . . . . . 34 3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 vi CONTENTS 4 Strained Flamelet Formulation for Premixed Flames 36 4.1 Scalar Dissipation Parametrisation . . . . . . . . . . . . . . . . . 38 4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Choice of Flamelet Configuration . . . . . . . . . . . . . . . . . . 40 4.2.1 Reactant–to–Reactant Flamelet . . . . . . . . . . . . . . . 41 4.2.2 Reactant–to–Product Flamelet . . . . . . . . . . . . . . . 42 4.3 Shapes of the PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Conditional Mean Scalar Dissipation Rate . . . . . . . . . . . . . 49 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5 Modelling of Mean Scalar Dissipation Rate 53 5.1 Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Modelling of Dominant Terms . . . . . . . . . . . . . . . . . . . . 55 5.2.1 Dissipation–Dilatation correlation, T2 . . . . . . . . . . . . 55 5.2.2 Turbulence–Scalar interaction, T32 . . . . . . . . . . . . . . 57 5.2.3 Flame front curvature, (T4 −D2) . . . . . . . . . . . . . . 59 5.3 Algebraic Model for ²˜c . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3.1 Realisability . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4.1 Attributes of DNS data . . . . . . . . . . . . . . . . . . . . 62 5.4.2 Comparisons with DNS data . . . . . . . . . . . . . . . . . 64 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 Turbulent Flame Speed 67 6.1 Expression for Turbulent Flame Speed . . . . . . . . . . . . . . . 67 6.1.1 Analysis of the Flame Speed expression . . . . . . . . . . . 69 6.2 Flame Speed definitions in Experiments . . . . . . . . . . . . . . . 72 6.3 KPP analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.1 KPP analysis of a planar one–dimensional flame . . . . . . 75 6.3.2 Applicability to multi–dimensional flames . . . . . . . . . . 76 6.4 Comparisons with experimental data . . . . . . . . . . . . . . . . 78 6.4.1 Flame speed datasets . . . . . . . . . . . . . . . . . . . . . 80 6.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 vii CONTENTS 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7 Application of Strained Flamelet Formulation 96 7.1 Computational methodology . . . . . . . . . . . . . . . . . . . . . 96 7.1.1 Overview of modelling closures . . . . . . . . . . . . . . . 98 7.2 RANS of planar one–dimensional flames . . . . . . . . . . . . . . 99 7.2.1 Computational details . . . . . . . . . . . . . . . . . . . . 99 7.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . . 104 7.3 RANS of laboratory flames . . . . . . . . . . . . . . . . . . . . . . 113 7.3.1 Computational details . . . . . . . . . . . . . . . . . . . . 114 7.3.1.1 V–flame calculations . . . . . . . . . . . . . . . . 114 7.3.1.2 Bunsen flames calculations . . . . . . . . . . . . . 116 7.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . 120 7.3.2.1 V–flame . . . . . . . . . . . . . . . . . . . . . . . 120 7.3.2.2 Bunsen flames . . . . . . . . . . . . . . . . . . . 122 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8 Conclusions and Future work 146 8.1 Recommendations for future work . . . . . . . . . . . . . . . . . . 147 8.1.1 Lewis number effects . . . . . . . . . . . . . . . . . . . . . 148 8.1.2 Sensitivity to turbulence closure . . . . . . . . . . . . . . . 148 8.1.3 Unsteady strain, non–adiabaticity and curvature . . . . . . 149 8.1.4 Stratified combustion . . . . . . . . . . . . . . . . . . . . . 150 8.1.5 Statistically non–planar flames . . . . . . . . . . . . . . . . 150 8.1.6 Modelling for CMC . . . . . . . . . . . . . . . . . . . . . . 150 8.1.7 Modelling for LES . . . . . . . . . . . . . . . . . . . . . . 151 A Relations between Reynolds and Favre averaged quantities 152 B List of Publications 154 References 155 viii List of Figures 2.1 Turbulent combustion regime diagram of Peters (2000). . . . . . . 15 4.1 Schematic of purely strained planar laminar flamelet configura- tions: (a) Reactant–to–Reactant (RtR) and (b) Reactant–to–Product (RtP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Normalised reaction rate, ω˙+, profiles of RtP flames of lean CH4– air mixture (φ = 0.6), at various strain rates, plotted against the distance from the stagnation plane. . . . . . . . . . . . . . . . . 43 4.3 The surface of ω˙+(c,N+c ) from RtP flame calculations of lean CH4– air mixture, φ = 0.6. . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4 The surface of ω˙+(c,N+c ) from RtP flame calculations of lean C3H8– air mixture, φ = 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.5 Representative curves of ω˙+ vs N+c conditioned on the progress variable ζ for three cases: (a) methane–air, φ = 0.6, Le = 0.96, (b) propane–air, φ = 0.8, Le = 1.83 and (c) methane–air, φ = 1.4, Le = 1.17 . The ζ = 0.7 curve for the propane–air case is shown in (d) and its integration is discussed in the text. . . . . . . . . . . . 47 4.6 Profiles of Nc(ζ)/Nc(ζ∗) for three mixtures: (a) CH4–air of φ = 0.6; (b) CH4–air of φ = 1.0 and (c) C3H8–air of φ = 0.8. The unstrained flame is the solid and strained flames are dashed curves. . . . . . 50 5.1 Attributes of the DNS data on the combustion regime diagram. . 63 5.2 Comparisons of ²˜c predicted by various models with DNS data. . . 65 6.1 Typical prediction of Eq. (6.3) for three different values of Λ/δoL with τ = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ix LIST OF FIGURES 6.2 Schematic showing a turbulent flame brush. The thick solid curves denote the flame brush boundaries and the dashed line denotes an iso–surface of c˜ inside the flame brush. The thin solid line denotes an instantaneous c iso–contour with displacement speed sd. The orthogonal curvilinear coordinate system (η, ξ, ζ) is attached locally to the c˜ iso–surface. . . . . . . . . . . . . . . . . . . . . . . 73 6.3 Typical comparisons of turbulent flame speed expressions to the experimental data of Shepherd & Cheng (2001). . . . . . . . . . 79 6.4 Turbulent combustion regime diagram (Peters, 2000) with attributes of various experimental flames: ∗ Smith & Gouldin (1978); ¨ Al- dredge et al. (1998); M Kobayashi et al. (1996); O Savarianandam & Lawn (2006); ◦, • Il’yashenko & Talantov (1966). . . . . . . . . 83 6.5 The predictions of turbulent flame speed expressions are compared to the experimental data of Smith & Gouldin (1978) for various equivalence ratios: (a)–(c) correspond to φ = 0.75; (d)–(f) corre- spond to φ = 0.85 and (g)–(i) correspond to φ = 1.0. The com- parisons on the left, center and right columns correspond to Λ = 0.69mm, 0.95mm and 1.49mm respectively. . . . . . . . . . . . . . 91 6.6 The predictions of turbulent flame speed expressions are compared to the experimental data of Aldredge et al. (1998) for various equiv- alence ratios: (a) φ = 1.0; (b) 1.1; (c) 1.2; (d) 1.3; (e) 1.4; (f) 1.5; (g) 0.8; (h) 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.7 The variation of turbulent flame speed with u′/soL at different pres- sures: (a) 0.1 MPa; (b) 0.5 MPa; (c) 1 MPa; (d) 2 MPa; (e) 3 MPa. The experimental data of Kobayashi et al. (1996) are compared to the flame speed expression. . . . . . . . . . . . . . . . . . . . . . . 93 6.8 Comparisons of turbulent flame speed prediction to the experimen- tal data of Savarianandam & Lawn (2006). . . . . . . . . . . . . 94 6.9 Comparisons of turbulent flame speed prediction to the experimen- tal data of Il’yashenko & Talantov (1966). . . . . . . . . . . . . . 94 6.10 Comparison of turbulent flame speed expressions to the experimen- tal database of Abdel-Gayed et al. (1987) for two values of flame stretch parameter, K: (i) K = 0.053 and (ii) K = 0.15. . . . . . . 95 x LIST OF FIGURES 7.1 The regime diagram with the parameters of the one–dimensional computational flames for two values of flame stretch parameter K: 0.15 (¨) and 1.0 (•). . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.2 The progress of the solution in a typical one–dimensional flame calculation. Starting from an arbitrary initial profile (dashed line), the flame brush travels from right to left. . . . . . . . . . . . . . 102 7.3 History of displacement speed, Sd, in m/s for three c˜ values in two flames: (a) u′/soL = 4, K = 0.15 (b) u′/soL = 12, K = 1.0. The results are shown for the algebraic closure of the mean reaction rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.4 The spatial profiles of c˜ predicted by the various models plotted against the normalised distance for two cases: (a) u′/soL = 4, K = 0.15 (b) u′/soL = 16, K = 1.0. The location of c˜ = 0.5 in each case is denoted by x1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.5 The normalised flame brush thickness, δt/δoL, predicted by various models at different u′/soL, for two values of flame stretch parameter: (a) K = 0.15 and (b) K = 1. . . . . . . . . . . . . . . . . . . . . . 106 7.6 The shape of the density weighted pdf of progress variable, p˜(ζ), at three locations in the flame brush (c˜ = 0.1, 0.5 and 0.9) from the strained flamelet calculations for two cases : (a) u′/soL = 3, K = 0.15 and (b) u′/soL = 24, K = 1. The inset in (b) shows the values of ∫ 0.9 0.1 p˜(ζ; c˜) dζ for the two cases. . . . . . . . . . . . . . . 108 7.7 Comparisons of predicted flame speeds with the experimental data of Abdel-Gayed et al. (1987) for two values of flame stretch param- eter: (a) K = 0.15 and (b) K = 1. . . . . . . . . . . . . . . . . . . 110 7.8 Flame speeds predicted by the strained flamelet closure for two mixtures: stoichiometric methane–air with K = 1.0 (•) and lean propane–air with KLe = 1.0 (♦). The experimental data of Abdel- Gayed et al. (1987) for K = 1.0 are also shown (◦). . . . . . . . . 111 7.9 The comparisons of flame speeds predictions with the predictions of Bradley et al. (1994) for two values of flame stretch parameter: (a) KLe = 0.15 and (b) KLe = 1. . . . . . . . . . . . . . . . . . 112 xi LIST OF FIGURES 7.10 The regime diagram with conditions of the V–flame (•) of Robin et al. (2008) and the Bunsen flames (¨) of Chen et al. (1996). . . 114 7.11 The schematic of the computational domain for the V–flame cal- culation. The ‘O’–grid in the region close to the rod is also shown. 115 7.12 The schematic of the computational domain for the Bunsen flames calculations. The diameter of the nozzle jet is D. . . . . . . . . . 117 7.13 The calculated longitudinal (left) and transverse velocities (right) in m/s using the unstrained flamelets (dashed lines) and strained flamelets (solid lines) models are compared with the experimental data (◦) for the V–flame (Robin et al., 2008) at three downstream locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.14 The calculated mean progress variable using the unstrained flamelets (dashed lines) and strained flamelets (solid lines) models are com- pared with the experimental data (dots) of the V–flame (Robin et al., 2008) at two streamwise locations. . . . . . . . . . . . . . 122 7.15 The contours of ω˙/ρ (s−1) from the V–flame calculations. The contours in x < 0 region are for the strained flamelets while those in x > 0 region are for the unstrained flamelets. . . . . . . . . . . 123 7.16 The normalised mean axial velocity and normalised mean turbulent kinetic energy from non–reacting flow calculations (solid lines) of case F2 are compared with the experimental data (◦) of Chen et al. (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.17 The normalised mean axial velocity and normalised mean turbulent kinetic energy from non–reacting flow calculations (solid lines) of case F1 are compared with the experimental data (◦) of Chen et al. (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.18 The normalised mean axial velocity and normalised mean turbulent kinetic energy from non–reacting flow calculations (solid lines) of case F3 are compared with the experimental data (◦) of Chen et al. (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.19 The normalised mean axial velocity from reacting flow calculations of flames F1 (left), F2 (center) and F3 (right) are compared with the experimental data of Chen et al. (1996). . . . . . . . . . . . . 128 xii LIST OF FIGURES 7.20 The normalised mean turbulent kinetic energy from reacting flow calculations using the strained flamelets model of flames F1 (left), F2 (center) and F3 (right) are compared with the experimental data of Chen et al. (1996). . . . . . . . . . . . . . . . . . . . . . 129 7.21 The contour plot of temperature (K) from the Bunsen flame calcu- lations for case F3. The plot for the entire computational domain (top) and the region close to the nozzle exit (bottom) are shown. 130 7.22 The mean progress variable, c, using the strained flamelets model is compared with the experimental data of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). . . . . . . . . . . . . 133 7.23 The mean CH4 mass fractions from strained flamelets calculations are compared with the experimental data of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). . . . . . . . . . . . . 134 7.24 The root mean square progress variable fluctuations, √ c′2 , from strained flamelets calculations are compared with the experimental data of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.25 The turbulent flame brush thicknesses, δt (mm), from the strained flamelets calculations are compared with the experimental data (•) of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). 136 7.26 The mean progress variable, c, (left) and mean CH4 mass fraction (right) from the unstrained flamelet calculations are compared with the experimental data of Chen et al. (1996) for flame F2. . . . . 137 7.27 The major species mass fractions: O2 (◦, ), CO2 (•, ) and H2O (M, ), from strained flamelets calculations are compared with the experimental data (symbols) of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). . . . . . . . . . . . . 139 7.28 The minor species mass fractions: 10×CO (◦, ) , 100×H2 (•, ) and 75×OH (M, ), from strained flamelets calculations are compared with the experimental data (symbols) of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). . . . . . . 140 xiii LIST OF FIGURES 7.29 The predictions of 10×YCO using strained flamelets model ( ) are compared with predictions of Herrmann (2006) ( ) and Lindst- edt & Vaos (2006) ( ) for flame F3 at x/D = 4.5. The sensitivity of strained flamelets predictions to RtP flamelet boundary condi- tions are shown: (a) equilibrium mixture composition for product stream, (b) product stream comprising of only CO2 and H2O. . . 141 7.30 The predictions of 10×CO from strained flamelet calculations using two definitions of progress variable: cf ( ) and cp ( ) are compared with the experimental data of Chen et al. (1996) for flame F2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 xiv Nomenclature Roman Symbols aT Tangential strain rate c Reaction progress variable C3, C4 Constants in the model for T32 L Markstein length cf Progress variable based on fuel mass fraction Cp Mixture specific heat cp Progress variable based on mass fraction of CO and CO2 Cp,α Specific heat at constant pressure of species α D2, T2, T32, T4 Unclosed leading order terms in ²˜c transport equation Da Damköhler number Dα Equivalent diffusion coefficient of species α Dc Diffusivity of the progress variable Dkα Binary diffusion coefficient of species k and α Dth Thermal diffusivity e Internal energy g Normalised progress variable variance xv LIST OF FIGURES hα Total enthalpy of species α K Flame stretch factor Ka Karlovitz number Kc, K ∗ c Constants in the model for T2 Kf,j Forward rate of reaction j Kr,j Reverse rate of reaction j Leα Lewis number of species α Le Mixture Lewis number m Mass flux eigen value in KPP analysis Nc Instantaneous scalar dissipation rate of c p Hydrostatic pressure Pr Prandtl number Qα Conditional mean of species α qi Component of heat flux vector in direction i Re Turbulent Reynolds number Ru Universal gas constant sc Laminar flame consumption speed Scc Schmidt number of c sd Laminar flame displacement speed SGC Global consumption speed in a turbulent flame brush SLC Local consumption speed in a turbulent flame brush soL Unstrained laminar flame speed xvi LIST OF FIGURES ST Turbulent flame speed T Temperature t time k˜ Favre average turbulent kinetic energy ui Velocity component in direction i; i = 1 .... 3 u˜′′i c′′ Turbulent flux of the progress variable u′ Turbulent r.m.s velocity Vα,i Diffusion velocity of species α in direction i Wα Molecular weight of species α Xα Mole fraction of species α [Xα] Molar concentration of species α xi Spatial coordinate in direction i Ya Mass fraction of species α yα Fluctuation about the conditional mean of species α Greek Symbols β′ Constant in the model for (T4 −D2) α∗, β∗, γ∗ Contributions of the unburnt, fully burnt and burning gases in the BML pdf δ Zeldovich flame thickness ∆hof,α Formation enthalpy of species α δij Kronecker delta; δij = 1 if i = j and 0 otherwise δoL Laminar flame thermal thickness xvii LIST OF FIGURES ε˜ Favre average turbulent kinetic energy dissipation rate ηk Kolmogorov length scale κ Fluid dynamic stretch rate Λ Integral length scale λ Thermal conductivity µ Dynamic viscosity µt Turbulent viscosity ν Kinematic viscosity νt Turbulent diffusivity ω˙α Mass rate of production of species α by chemical reactions ω˙ Reaction rate of the progress variable ω˙o Reaction rate of an unstrained planar flame φ Equivalence ratio ψ Sample space variable for Nc ρ Density Σ Flame surface density σij Stress tensor τ Heat release parameter τc Chemical time scale τij Viscous stress tensor τk Kolmogorov time scale τt Integral time scale xviii LIST OF FIGURES ²˜c Favre averaged scalar dissipation rate of c (η, ξ, ζ) Orthogonal curvilinear coordinate system attached to c˜ iso–surface Ξ Flame wrinkling factor ξ Mixture fraction ζ Sample space variable for c xix Chapter 1 Introduction Combustion is a phenomenon that occurs all around us: a burning candle, a do- mestic boiler, an aircraft engine etc., to name a few instances. Since the dawn of the industrial age, energy derived from combustion of fuels has improved the quality of human life in almost every respect, albeit with consequences for the environment. Until the 1990s the main concerns with fossil fuel combustion were emissions of Oxides of Sulphur and Nitrogen which are known to cause atmo- spheric pollution (smog, acid rain, ozone layer depletion etc.). The scenario, however, has since emerged to be more dire. It is now widely accepted that emission of green house gases (GHG) is having a calamitous impact on our en- vironment. According to IPCC (2007) “many natural systems are being affected by regional climate changes, particularly temperature raise” and “most of the observed increase in global average temperatures since the mid–20th century is very likely due to increase in anthropogenic green house gas concentrations”. This report estimates that combustion of fossil fuels in the domestic, industrial, energy and transportation sectors contributes the most (71%) to total GHG emis- sions of which CO2 constitutes a major portion (57%). While alternative energy sources such as wind, tidal and solar energy are being widely explored, because of economic and technological reasons fossil fuels and alternative hydrocarbon fuels such as bio–fuels are likely to provide a bulk of the growing energy demand for the foreseeable future. This is particularly true for high power density applica- tions, such as transportation. It is hence imperative that combustion devices are designed to be cleaner and more efficient, as recommended by IPCC (2007), to 1 help mitigate the effects of GHG emissions. Lean premixed combustion, which burns a homogeneous mixture of fuel and excess oxidiser, has the potential for significantly reducing emissions from com- bustion devices while improving their efficiency (Correa, 1992; Heywood, 1976; Lefebvre, 1999). Land based gas turbine engines in the energy sector that burn gaseous fuel are already able to operate in this mode (Davis & Washam, 1988). This has not yet been realised for aircraft engines since they burn liquid fuels that require a substantial amount of time to vaporise and mix which can result in operational difficulties such as autoignition or flashback (Correa, 1992; Lefebvre, 1999). Instead, a fuel rich mixture is first burnt in a primary zone followed by the injection of additional air to burn the residual fuel and achieve an overall lean combustion (Mongia, 1998). The reacting mixture in such a scenario is not per- fectly premixed and has compositional inhomogeneities but future technologies in the aviation sector are likely to move towards premixed combustion (Correa, 1998; Rolls-Royce, 2005). Current generation reciprocating engines: spark ig- nition (Takagi, 1998; Zhao et al., 1999) and compression ignition engines (Lee & Lee, 2007; Yao et al., 2009) also burn inhomogeneous mixtures that are lean overall. However, even in combustion of inhomogeneous mixtures, the classi- cal premixed mode is expected to play a major role in determining the overall combustion characteristics (Westbrook et al., 2005). While attractive from the efficiency and emissions point of view, lean premixed combustion is very sus- ceptible to instabilities (Lieuwen et al., 1998; Shih et al., 1996) due to the high sensitivity of heat release rate to mixture composition, and this can contribute to rough combustion, unless sufficient care is taken in the design. Computational simulations are widely used in the design of combustion de- vices as they greatly reduce the time and expense of the design cycle and provide information at a level of detail that is difficult to obtain from experiments. How- ever, most combustion devices operate in the turbulent regime and numerically resolving the entire range of spatial and temporal scales that characterise turbu- lent flames in realistic geometries is not computationally tractable even with the state–of–art computing resources. To circumvent this, modelling strategies are employed whereby the need to resolve all the scales is avoided by mathematical treatments such as averaging, filtering, of the governing equations. As a result of 2 these treatments, terms arise representing the interaction among various physical processes: fluid dynamics, thermodynamics, molecular transport and chemical reactions. These interactions are quite complex in turbulent combustion and not well understood. Consequently, computations of turbulent combustion rely heav- ily on theoretical and mathematical models that provide a description of these underlying physical processes. Hence, the study of turbulent combustion mod- elling is important, not only because it helps us to understand the fundamental physics, but because it is essential in the design process. It is evident that the premixed mode of combustion is likely to play a major role in the drive towards ef- ficient, clean and quiet combustion devices. While significant advances have been made in understanding and modelling turbulent premixed combustion (Borghi, 1988; Bray, 1980; Peters, 2000; Veynante & Vervisch, 2002) a lot remains to be achieved. For instance the prediction of heat release rates and species concentra- tions, important from the noise and emissions perspectives, presents a formidable challenge which is yet to be satisfactorily addressed. The development of a pre- dictive model for turbulent premixed combustion applicable over wide range of conditions encountered in practical devices is an ongoing activity in the com- bustion research community. The present study attempts a contribution to this endeavour. In Direct Numerical Simulations (DNS), which represent one extreme of the computational paradigm, all the scales of the flow and flame are resolved. Al- though this is attractive, the inclusion of detailed chemical kinetics and trans- port processes in DNS tremendously increases the computational expense re- stricting the computational domain size to typically few cm3. At the other ex- treme, Reynolds–Averaged Navier–Stokes (RANS) simulations solve governing equations for average quantities with no attempt to resolve any of the scales. The unclosed terms arising in the RANS governing equations, typically involv- ing correlations of turbulence and thermochemical quantities, require modelling. The most challenging aspect of RANS is the modelling of terms that are dic- tated by the turbulence–chemistry interactions since these interactions are highly non–linear. In Large Eddy Simulations (LES) governing equations of filtered quantities representing the dynamics of energy containing large scale motions of turbulent flow are solved. The influence of smaller scales are represented using 3 models. However, the chemical scales are typically smaller than the smallest tur- bulent scale, the Kolmogorov scale, in turbulent flames of practical interest and the turbulence–chemistry interaction is unresolved and requires modelling even in LES (Pope, 1990). Most LES models are extensions of RANS modelling ideas (Poinsot & Veynante, 2001) and hence turbulent combustion modelling in the RANS paradigm is pivotal. Many concepts/quantities in the realm of combustion modelling embody turbulence– chemistry interactions; one such quantity, the scalar dissipation rate, is the focus of this work. The mean scalar dissipation rate is the average rate of decay of scalar fluctuations and denotes the rate of scalar mixing at the smallest scales. Conventionally, mixture fraction, a conserved scalar, is employed to describe the thermochemical state in non–premixed flames, and its dissipation rate is dictated entirely by the turbulence. The scenario is different for premixed flames (Bilger, 2004b) whose thermochemical description is usually via the reaction progress variable, a reactive scalar, whose dissipation rate is dictated by the turbulence as well as the flame front dynamics. While the modelling of mean dissipation rate of the mixture fraction is well understood, that of the progress variable is not. Nonetheless, scalar dissipation rate is a central parameter in most modelling approaches (Bilger, 1976; Bray, 1980; Klimenko & Bilger, 1999; Pope, 1985) since it is linked to the fundamental quantities in turbulent combustion (Veynante & Vervisch, 2002). The main objectives of this work are: (i) to formulate a RANS computational model for turbulent premixed combus- tion based on scalar dissipation rate, (ii) to study the associated problem of mean scalar dissipation rate modelling in turbulent premixed flames, and (iii) to validate the models using DNS and experimental data over wide range of turbulent flame conditions. The outline of this thesis is as follows. The background on premixed flame modelling is first presented in Chapter 2 giving an overview of the available mod- elling methodologies. The discussion is restricted to laminar flames and RANS modelling of turbulent premixed flames. Chapter 3 focuses on scalar dissipation 4 rate; its physical meaning, significance in turbulent combustion and its modelling challenges. The formulation of a turbulent premixed combustion model based on the scalar dissipation rate is presented in Chapter 4. This model is based on the “flamelet” concept which views turbulent flame as an ensemble of laminar–like re- action layers - flamelets. Strained laminar flames in an appropriate configuration are chosen to represent the flamelets and the choice of scalar dissipation rate to parametrise these flamelets results in a wide range of applicability of the model. The closure of mean scalar dissipation rate, required in the formulation pre- sented in Chapter 4, is studied in Chapter 5 and a new algebraic closure is proposed based on recent findings. This model is validated using DNS data. Validation using experimental data is not possible since direct measurements of this quantity are scarce. However, its indirect validation is performed in Chapter 6 by obtaining an expression for turbulent flame speed from the scalar dissipa- tion rate model and comparing it with experimental data from a wide range of configurations and conditions. The computational model is applied to turbulent flame calculations and re- sults are presented in Chapter 7. The calculations of a test problem: planar one–dimensional flame are discussed first. The results of laboratory flame cal- culations of different configurations are discussed and conclusions are drawn re- garding the regime of applicability of the model. Some limitations of flamelet modelling in predicting pollutants are identified and possible ways to improve this are discussed. Chapter 8 discusses the scope for future work. 5 Chapter 2 Background on Premixed Flame Modelling The classical modes, namely premixed and non–premixed, are logical starting points to study combustion. In non–premixed flames the fuel and oxidiser enter the combustion zone from separate streams and the flame is located at the sto- ichiometric interface. The oxidiser and fuel are supplied to the flame zone via diffusion. In premixed flames the fuel and oxidiser are mixed at the molecular level prior to entering the combustion zone and the flame propagates into the unburnt mixture at a speed, relative to the fluid motion, determined by the rates of reaction and thermal diffusion. The conventional compression ignition engine is a typical example of the former while a spark ignition engine is an example of the latter. However, in practice one mostly encounters combustion situations that lie in between these two classical extremes. As noted in the introduction, many practical devices involve combustion of inhomogeneous reacting mixtures, which is broadly called as partially premixed combustion, which can be further cate- gorised into stratified premixed and premixed/non–premixed combustion (Bilger et al., 2005). In stratified premixed combustion the inhomogeneous mixture is either entirely fuel–lean or fuel–rich and the local flame structures are only of the premixed type whereas in premixed/non–premixed combustion the mixture in- cludes both rich and lean compositions and flame structures of both premixed and non–premixed type are likely to occur. The role of premixed mode in partially premixed combustion is evident and the implications for modelling are obvious. 6 2.1 Laminar Flames Premixed combustion models are often, though not always, an integral part of the modelling strategies for partially premixed combustion (Domingo et al., 2002; Hélie & Trouvé, 2000; Jiménez et al., 2002; Peters, 2000). The focus of this work is premixed flame modelling and in this chapter an overview of the existing models is presented. The term ‘modelling’ is best de- scribed as an effort to provide a mathematical description for physical phenomena. The modelling of the fundamental phenomena in laminar flames is discussed first and then the Reynolds–Averaged–Navier–Stokes (RANS) modelling of turbulent flames is presented. 2.1 Laminar Flames The mathematical framework for laminar combustion is provided by the gov- erning equations of fluid motion: mass and momentum conservation, and the transport of thermochemical quantities: species concentrations and energy. For a multi–component gaseous reacting system with N species and M reactions these equations written in tensorial notation are (Williams, 1985a) : • continuity equation ∂ρ ∂t + ∂ρui ∂xi = 0, (2.1) • momentum conservation ∂ρui ∂t + ∂ρuiuj ∂xj = −∂σij ∂xj + ρ N∑ α=1 Yαfα,i, (2.2) where fα,i is the ith component of the body force acting on species α. The stress tensor is defined in terms of the pressure and viscous stress tensors as σij = pδij − τij where τij = −2 3 µ ∂uk ∂xk δij + µ ( ∂ui ∂xj + ∂uj ∂xi ) ; • conservation of species α ∂ρYα ∂t + ∂ ∂xi [ρ(ui + Vα,i)Yα] = ω˙α, α = 1....N, (2.3) where Vα,i is the molecular diffusion velocity and ω˙α is the total rate of mass production by chemical reactions. 7 2.1 Laminar Flames • conservation of energy ∂ρe ∂t + ∂ρuie ∂xi = − ∂qi ∂xi − σij ∂ui ∂xj + ρ N∑ α=1 Yαfα,iVα,i + Q˙, (2.4) where qi is the energy flux in direction i due to thermal conduction and Q˙ in- cludes effects such as ignition source, radiative flux etc. The energy equation is often recast in terms of enthalpy or temperature using the identity e =∑ α Yαhα − p/ρ and the caloric equation of state hα = ∆hof,α + ∫ T T o Cp,αdT . The pressure is obtained from the equation of state p = ρRuT N∑ α=1 Yα Wα . (2.5) For multi–component mixtures simplifications are often made for the molecular diffusive flux terms qi and Vα,i. Typically Dufour effects are neglected and the heat flux vector is expressed as qi = −λ(∂T/∂xi) + ρ ∑ α hαYαVα,i. The expression for species diffusive velocity, after neglecting Soret effects, is very complicated and usually the approximation of Curtiss & Hirschfelder (1949) is employed which yields the Fickian form Vα,i = −Dα 1 Xα ∂Xα ∂xi = −Dα 1 Yα ∂Yα ∂xi where the equivalent diffusivity, Dα, is given by Dα = (1 − Yα)/( ∑ k 6=αXi/Dkα) and Dkα is the binary diffusivity of species k and α. This expression is based on the approximation that all the species are in trace quantities relative to one major species, which is reasonable since in most cases air is used as the oxidiser and the Nitrogen concentration is much larger compared to the rest of the species. The relative magnitude of the thermal and mass diffusivities is given by the Lewis number of a species defined as Leα ≡ Dth/Dα where the thermal diffusivity is defined as Dth ≡ λ/(ρCp) and Cp is the mixture specific heat. The Prandtl number is the ratio of the momentum and thermal diffusivities and is defined as Pr ≡ ν/Dth where the kinematic viscosity is ν ≡ µ/ρ. 8 2.1 Laminar Flames If the system of M reversible reactions is expressed using the stoichiometric coefficients as ∑ α ν ′ αjMα ­ ∑ α ν ′′ αjMα for j = 1, ...M where Mα is the chemical symbol for species α, then the total mass production rate of species α is given by ω˙α = Wα M∑ j=1 (ν ′′αj − ν ′αj) [ Kf,j N∏ α=1 [Xα] ν′αj −Kr,j N∏ α=1 [Xα] ν′′αj ] . (2.6) The forward reaction rates are usually expressed in the Arrhenius form, Kf,j = AjT βjexp(−Ta,j/RuT ) where Aj and βj are constants and Ta,j is the activation temperature for reaction j. The reverse reaction rate is obtained from Kr,j = Kf,j/Keq,j where Keq,j is the equilibrium constant. The above mathematical framework is not specific to a combustion mode and the differences between the premixed and non–premixed modes emerge only through the boundary conditions. In principle the above system of equations can be solved numerically if appropriate initial and boundary conditions are spec- ified. In practice the detailed chemical mechanisms for typical hydrocarbons involve tens of species and hundreds of reactions which means the dimension of the system of equations (2.1)–(2.4), equal to (N + 5), is large. Furthermore the system of equations is stiff due to the exponential dependence of reaction rates on temperature and a wide range of length and time scales are involved. This requires special numerical techniques to obtain a solution. Nonetheless compu- tational codes for canonical flame configurations have been developed (Bradley et al., 1996; Kee et al., 1985; Lutz et al., 1997; Somers, 1994) and are widely used. Canonical flames are usually one–dimensional (1D) or quasi–one–dimensional and their spatial scales are relatively small; the thickness of a freely propagating planar unstrained flame, which is a canonical flame, is typically less than a millimetre. Multi–dimensional flames are seldom computed with the detailed thermochem- istry due to the enormous computational cost involved. Instead the thermochem- istry is tabulated using a set of reduced number of representative scalars (Maas & Pope, 1994; van Oijen & de Goey, 2000) and only transport equations of these scalars are solved along with equations describing the fluid flow. Such a strategy is also widely used in turbulent flame computations which are discussed next. 9 2.2 Turbulent Flames 2.2 Turbulent Flames For turbulent flames the computational expense is further compounded because they are inherently unsteady and are comprised of cascade of eddies that span a wide range of length and time scales. Direct numerical simulations solve the full set of equations described in Sec. 2.1 and resolve all the scales1 involved. Hence they are prohibitively expensive and are restricted to computational domains that are orders of magnitude smaller than engineering devices. However, for engineering applications one is mostly interested in the statistical moments rather than the instantaneous values of temperature, velocity etc. These statistical moments are obtained economically by solving the Reynolds–Averaged Navier– Stokes (RANS) equations. 2.2.1 RANS Governing Equations By averaging Eqs. (2.1) and (2.2) one obtains, respectively, the continuity equa- tion ∂ρ ∂t + ∂ρu˜i ∂xi = 0 (2.7) and the momentum equation ∂ρu˜i ∂t + ∂ρu˜iu˜j ∂xj = − ∂p ∂xi + ∂ ∂xj ( τ ij − ρu˜′′i u′′j ) , (2.8) for the mean motion, when the body forces are assumed to be negligible. The over bar denotes Reynolds averaging and tilde denotes density weighted (Favre) averaging: u˜i ≡ ρui/ρ. The Favre fluctuation is denoted by a double prime i.e. u′′i = ui − u˜i. In the above equation the Reynolds stresses, u˜′′i u′′j , are unclosed. One could solve transport equations for the six components of the symmetric Reynolds stress tensor, but these equations contain further unclosed terms and at some point modelling closures will be required2. Instead a two–equation approach 1for reacting flows the numerical resolution requirement is more stringent compared to non– reacting flows since the scales of chemical reactions are often smaller than the turbulence scales. 2the transport equations for u˜′′i u′′j contain the triple correlations u˜′′i u′′j u′′k . In general trans- port equations of correlations of fluctuating quantities will contain higher order correlations. 10 2.2 Turbulent Flames (Jones & Launder, 1972) is widely followed whereby the Reynolds stresses are closed using the Boussinesq approximation ρu˜′′i u ′′ j = −µt ( ∂u˜i ∂xj + ∂u˜j ∂xi − 2 3 ∂u˜k ∂xk δij ) + 2 3 ρk˜δij , (2.9) and the turbulent viscosity is modelled as µt = ρCµk˜2/ε˜. Modelled transport equations are solved for the Favre averaged turbulent kinetic energy, k˜ (Bray et al., 1992) ∂ρk˜ ∂t + ∂ρu˜ik˜ ∂xi = ∂ ∂xi [( µ+ µt Sck ) ∂k˜ ∂xi ] + Pk + p′ ∂u′′k ∂xk − ρε˜ , (2.10) and its dissipation rate, ε˜, ∂ρε˜ ∂t + ∂ρu˜iε˜ ∂xi = ∂ ∂xi [( µ+ µt Sc ε ) ∂ε˜ ∂xi ] + Cε1Pk ε˜ k˜ − Cε2ρε˜ 2 k˜ . (2.11) The production term in the above equations is Pk = −ρu˜′′i u′′j (∂u˜i/∂xj)−u′′i (∂p/∂xi). The values for the model constants in Eqs. (2.9)-(2.11) are: Cµ = 0.09, Sck = 1.0, Scε = 1.3, Cε1 = 1.44 and Cε2 = 1.92. This two–equation approach, known as k˜–ε˜ approach, was originally proposed (Jones & Launder, 1972) for non–reacting turbulent flows but it is retained in many turbulent flame calculations due to its simplicity and low computational expense. However, the pressure related terms are usually neglected for non–reacting flows. As noted earlier, computations of turbulent flames can be made tractable by describing the thermochemistry via few representative scalars instead of solving for all the reactive species. For premixed flames the most commonly used scalar for this purpose is the reaction progress variable, c. The progress variable is typ- ically defined as either normalised fuel mass fraction or normalised temperature3 such that it takes a value of zero in unburnt mixture and 1 in the burnt mix- ture. Alternative definitions of progress variable have also been proposed (Bilger, 2004a; Bilger et al., 1991). Here we use the temperature based definition, c ≡ T − Tu Tb − Tu , (2.12) 3The two definitions are equivalent strictly for the case of a single step irreversible reaction and when the Lewis number of the fuel is equal to 1 (Poinsot & Veynante, 2001). 11 2.2 Turbulent Flames where Tu is the unburnt mixture temperature and Tb is the adiabatic flame tem- perature. The transport equation for the instantaneous progress variable is ∂ρc ∂t + ∂ρuic ∂xi = ∂ ∂xj ( ρDc ∂c ∂xj ) + ω˙ , (2.13) which can be obtained from Eq. (2.4) (Poinsot & Veynante, 2001) by setting the body forces and Q˙ to zero and assuming that the work done by pressure and viscous stresses are negligibly small, which is reasonable for flames in low Mach number flows. The diffusivity and the reaction rate of the progress variable are Dc and ω˙ respectively. If the specific heats of all the species are assumed to be equal to the mixture specific heat, Cp, then Dc is equal to the thermal diffusivity Dth. The reaction rate can be written from Eq. (2.6) as ω˙ = − N∑ α=1 hαω˙α Cp(Tb − Tu) . (2.14) The transport equation for the Favre averaged progress variable, c˜, is ∂ρc˜ ∂t + ∂ρu˜ic˜ ∂xi = ∂ ∂xi ( ρDc ∂c˜ ∂xi − ρu˜′′i c′′ ) + ω˙. (2.15) The two terms, the turbulent scalar flux, u˜′′i c′′, and the mean reaction rate, ω˙, on the right hand side in Eq. (2.15) are unclosed. The turbulent scalar flux can be closed similar to the Reynolds stresses using a gradient transport hypothesis as ρu˜′′i c′′ = − µt Scc ( ∂c˜ ∂xi ) . (2.16) However this closure contradicts the phenomenon of counter–gradient transport (Bray et al., 1981; Moss, 1980) which is known to occur in turbulent premixed flames and will be discussed in Section 2.2.3. A closure for mean reaction rate, ω˙, is more challenging and is the primary concern of premixed flame modelling. The difficulty in closing ω˙ can be illustrated using a simple case involving single irreversible reaction between fuel (F) and oxi- diser (O) giving rise to a product (P). The mean reaction rate in this case is related to the mean fuel consumption rate via ω˙ = Qω˙F/Cp(Tb−Tu), where Q is the heat 12 2.2 Turbulent Flames released per unit mass of fuel consumption. Using an Arrhenius form for the rate constant the fuel consumption rate is written as ω˙F = Aρ2T βYFYOexp(−Ta/T ). The mean of this quantity, and hence ω˙, cannot be expressed as a simple function of Y˜F , Y˜O and T˜ mainly because of the non–linear exponential term. Using a Tay- lor series expansion for the mean fuel consumption rate gives (Libby & Williams, 1980; Poinsot & Veynante, 2001) ω˙F = Aρ 2T˜ βY˜F Y˜Oexp ( −Ta T˜ )[ 1 + ( Ta T˜ )2 T˜ ′′2 T˜ 2 + Y˜ ′′F Y ′′ O Y˜F Y˜O + ..... ] . (2.17) The second term in the square brackets is not small compared to the first term since Ta is usually large, and it can only be neglected for unrealistically small tem- perature fluctuations (Libby & Williams, 1980). The higher order terms in such an expansion involve powers of (Ta/T˜ ) and neglecting these terms results in large truncation errors. The third term involving Y˜ ′′F Y ′′O is also not negligible and its closure is difficult. Furthermore, for a general case involving multi–step chemistry this expression becomes more complicated and difficult to estimate. Thus, the closure of ω˙ using the moments is difficult and alternative methods are employed. Modelling of this quantity is usually based on physical ideas of the interaction between the turbulent flow and the chemical reactions; the turbulence–chemistry interaction. A laminar premixed flame has a well defined velocity and length scale while turbulent flows are characterised by a range of length and time scales. The influence of turbulent flow on a premixed flame can be qualitatively described by the relative magnitude of the flame and the turbulent flow scales. This gives rise to the concept of turbulent combustion regimes which forms a first basis for model development. 2.2.2 Regimes of Turbulent Premixed Combustion Damköhler (1940) was among the first to describe turbulent flame behaviour based on combustion regimes. He envisaged a regime where all the scales of the turbulent eddies are much larger than the flame scales and inner structure of the flame front is virtually undisturbed in the “flamelet” limit. This occurs when the Damköhler number, Da, is much greater than 1. The Damköhler number is the 13 2.2 Turbulent Flames ratio of the integral turbulent time scale to the chemical time scale Da ≡ τt τc = Λ/u′ δ/soL (2.18) where Λ is the integral length scale, u′ is the turbulent root mean square (RMS) velocity and soL is the laminar flame speed. The Zeldovich flame thickness is defined as δ ≡ Dthu/soL where the thermal diffusivity of the unburnt mixture is Dthu . In the other extreme, Da < 1, all the turbulent time scales are smaller than the chemical time scale and the species are mixed by the turbulent motions at a rate faster than the chemical reactions. This limit is called “perfectly stirred” limit (Libby & Williams, 1980). In the flamelet regime the turbulence wrinkles and contorts the flame front which increases its area and results in greater reactant consumption per unit time. The flame brush propagates at a greater speed, ST , than the laminar flame speed, soL. A simple expression proposed by Damköhler (1940) gives ST/soL ' AT/Ac = 1+ (u′/soL), where AT is the area of the wrinkled turbulent flame and Ac is its projected area. Thus, the turbulent flame speed, ST , incorporates the influences of turbulence as well as chemistry and has been the subject of many theoretical and experimental studies. This quantity is studied in more detail in Chapter 6. The Klimov–Williams criterion (Klimov, 1963; Williams, 1976) introduces the notion that even when Da > 1, the smallest eddies which are of the Kol- mogorov scale, ηk, can penetrate and disturb the flame front. This occurs when the Karlovitz number, Ka, defined as (Peters, 1986) Ka ≡ τc τk = ( δ ηk )2 (2.19) is larger than unity implying that the Kolmogorov eddies are smaller than the flame thickness. The Kolmogorov time scale is τk. Peters (2000) refines this idea by noting that the laminar flame has three distinct layers: a preheat layer, an inner reaction layer and an oxidation layer. The reaction layer is typically one– tenth of the flame thickness and hence the Kolmogorov eddies have to be much smaller to penetrate this layer. The regime diagram of Peters (2000) is shown in Fig. 2.1. The turbulent Reynolds number is defined as 14 2.2 Turbulent Flames 0.1 1 10 100 1000 0.1 1 10 100 1000 u′/soL Λ/δ ........................................................................................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . ................................................................................................................................................................................................................................. ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ....... ......... ......... ......... ......... ......... ......... ......... Re = 1 Ka = 100 Da = 1 Ka = 1 Wrinkled Corrugated Thin reaction zones Distributed reaction zones Figure 2.1: Turbulent combustion regime diagram of Peters (2000). Re = u′Λ νu , (2.20) and for a unity Prandtl number, Re = (u′Λ)/(soLδ). The region bounded by the Ka = 1 and Ka = 100 lines is the “thin reaction zones” regime where the Kolmogorov eddies are smaller than the flame thickness and enter the preheat layer but are unable to penetrate the reaction layer. For Ka > 100 the entire flame structure is disturbed by the turbulent eddies and reactions occur over a distributed region in space. Below the Ka = 1 line lie the “corrugated” and “wrinkled flamelets” regimes which are separated by u′/soL = 1 line. In the former the flame front is likely to be corrugated with possibility of isolated pockets of burning mixture while in the latter the flame front is merely wrinkled with its layer like structure intact. It must be stressed that the regimes in Fig. 2.1 are based on qualitative ideas and make many simplifying assumptions. Modified combustion diagrams based on experimental observations have been proposed by Chen & Bilger (2001b). Poinsot et al. (1991) construct a regime diagram 15 2.2 Turbulent Flames using rigorous analysis of flame/vortex interactions from DNS data. Nonetheless, such qualitative ideas are widely used in model formulation and as guidelines to estimate the regimes of applicability of various models. Interesting modelling simplifications can be made in the classical limits of combustion regimes. In the perfectly stirred limit the turbulent time scales are faster than the chemical processes and fluctuations in the thermochemical quan- tities are small since they are quickly mixed by the turbulent motions. The mean reaction rate in Eq. (2.17) can then be reasonably approximated to that evaluated using only the mean quantities (Libby & Williams, 1980). On the other hand, in the flamelet limit it can be assumed that the flame front structure is laminar– like; the “laminar flamelet” approximation. The mean reaction rate can then be deduced from the reaction rate in a laminar flame - typically a canonical flame - and the statistics of the flamelets at a given location. The perfectly stirred limit is unlikely to be attained in practical devices and in most cases the scenario is somewhat closer to the flamelet regime (Peters, 1986). Many turbulent premixed (and non–premixed) combustion models are based on the flamelet concept. These and other models, specifically for premixed flames, are discussed below. 2.2.3 Bray–Moss–Libby Model The Bray–Moss–Libby (BML) model (Bray, 1980; Bray & Libby, 1976; Bray & Moss, 1977) provides a thermochemical closure centred on the progress variable pdf. This pdf is expressed as the sum of the probabilities of finding unburnt, fully burnt and the burning gases at a location by p(c = ζ;x, t) = α∗(x, t)δ(ζ)︸ ︷︷ ︸ unburnt gases + β∗(x, t)δ(1− ζ)︸ ︷︷ ︸ burnt gases + γ∗(x, t)f(ζ)︸ ︷︷ ︸ burning gases , (2.21) where δ(ζ) and δ(1− ζ) are the Dirac delta functions and ζ is the sample space variable for c. Normalisation of the pdf yields the conditions: α∗ + β∗ + γ∗ = 1, and ∫ 1 0 f(ζ) dζ = 1. (2.22) The mean value of a thermochemical quantity, Φα, can be written as Φα = ∫ 1 0 Φα(ζ) p(ζ) dζ , (2.23) 16 2.2 Turbulent Flames which upon substituting Eq. (2.21) gives Φα = α ∗Φα(0) + β∗Φα(1) + γ∗(x, t) ∫ 1 0 Φα(ζ) f(ζ) dζ. (2.24) For flames with large Reynolds and Damköhler numbers that are well within the flamelet regime, the flame front is thin and the probability of finding the burning gases is small compared to the probabilities of unburnt or burnt gases. The burning gas component γ∗ ∼ O(1/Da), is negligible compared to α∗ and β∗ and the mean quantity Φα has contributions mostly from the unburnt and fully burnt gases when Da is large. Simple relations can be obtained by noting that the Favre averaged progress variable c˜ is ρc˜ ≡ ρc = ∫ 1 0 ρ(ζ) ζ p(ζ) dζ = ρbβ ∗ (2.25) where ρb is the density of the burnt gases. If the combustion occurs in low Mach number flows then the density can be expressed as ρ = ρuTu/T = ρu/(1 + τζ), where ρu is the unburnt density and the heat release parameter is defined as τ ≡ Tb/Tu − 1 = ρu/ρb − 1. This yields (Bray, 1980) ρ = ρu ∫ 1 0 1 1 + τζ p(ζ) dζ and ρc˜ = ρu ∫ 1 0 ζ 1 + τζ p(ζ) dζ , (2.26) which can be combined to give ρ = ρu 1 + τ c˜ , (2.27) and β∗ = (1 + τ)c˜ 1 + τ c˜ and α∗ = 1− c˜ 1 + τ c˜ . (2.28) Thus, the thermochemical state of the mixture is completely determined by c˜ which can be obtained from its transport equation, Eq. (2.15), provided the tur- bulent scalar flux and the mean reaction rate are modelled. The turbulent scalar flux in Eq. (2.15) is sometimes closed via a gradient transport hypothesis as noted in Section 2.2.1 which assumes that, by analogy to molecular transport, the turbulent transport is positive in the direction of negative gradient of c˜. However, counter–gradient transport, where the scalar flux is positive along positive c˜ gradient, has been observed in experiments (Bray 17 2.2 Turbulent Flames et al., 1981). This is explained in BML framework by noting that the scalar flux can be expanded as u˜′′i c′′ = (u˜ic− u˜ic˜) = c˜(1− c˜)(ubi − uui ), (2.29) where ubi and uui are the mean velocities conditioned in the burnt and the un- burnt gases respectively. Since the burnt gases have lower density they are pref- erentially accelerated by the pressure gradient compared to unburnt gases i.e., ubi > u u i . Hence u˜′′i c′′ is positive in the direction going from the unburnt to the burnt side; the direction of positive mean pressure gradient as well as positive c˜ gradient. Thus the BML model provides a theoretical basis for the correct be- haviour of turbulent scalar flux, however, the conditional mean velocities need to be modelled. The main modelling challenge, the closure of ω˙, remains unresolved in the BML approach. This is because the mean reaction rate, when evaluated using Eq. (2.24), will be O(γ∗), which is assumed to be negligible. Alternative ap- proaches have been developed to close the mean reaction rate. Bray (1979) analyses the transport equations of c˜ and c˜′′2 and deduces that, under the BML assumptions, the mean reaction rate is proportional to the mean scalar dissipation rate, ²˜c, as ω˙ = 2 2Cm − 1ρ²˜c (2.30) where Cm ≡ ∫ 1 0 ζ ω˙(ζ) p(ζ) dζ∫ 1 0 ω˙(ζ) p(ζ) dζ , (2.31) and the value of Cm is typically 0.7. The mean scalar dissipation rate is defined as ρ²˜c = ρDc ∂c′′ ∂xi ∂c′′ ∂xi . (2.32) Unfortunately, the mean scalar dissipation rate is an unclosed quantity and its closure for premixed flames is challenging. One could use classical models de- veloped for passive scalars in non–reacting turbulence which give ρ²˜c = ρc˜′′2/τt. Under the BML assumptions c˜′′2 ≈ c˜(1 − c˜), and by estimating the turbulence 18 2.2 Turbulent Flames time scale as τt ' k˜/ε˜, one recovers the Eddy–Break–Up (EBU) model of Mason & Spalding (1973) ω˙ ≈ CEBU ρ τt c˜′′2 ≈ CEBU ρ ε˜ k˜ c˜(1− c˜). (2.33) An obvious limitation of the EBU model is that it contains no information about the chemistry. In the above framework this stems from the modelling of ²˜c based on passive scalar mixing concepts and further details on this will be discussed in Chapter 5. Another approach is to estimate the mean reaction rate from the number of times the flame front crosses a given point (Bray et al., 1984). Mathematically, this is expressed as ω˙ = ω˙fνf (2.34) where νf is the flame crossing frequency and ω˙f is the reaction rate per flame crossing. If the flame front is infinitely thin the progress variable signal at a location resembles a telegraph signal whose analysis yields νf = 2 c(1− c) Tˆ (2.35) where the mean period of the telegraph signal, Tˆ , is typically set equal to τt. The reaction rate per flame crossing is expressed as ω˙f = ρus o L |σf | (2.36) where |σf | is a flamelet orientation factor (Bray et al., 1984). 2.2.4 Flame surface density models Perhaps the most widely employed approach, which is complementary to the BML methodology, is to express the mean reaction rate per unit volume as a product of reaction rate per unit flame area times flame surface density, Σ (Marble & Broadwell, 1977) ω˙ = ρu〈sc〉sΣ , (2.37) where sc is the flame consumption speed and the notation 〈 〉s denotes averaging over the flame surface. The flame surface density is the mean flame surface area 19 2.2 Turbulent Flames per unit volume. Algebraic expressions for the flame surface density have been proposed based on the flame crossing frequency analysis (Bray et al., 1989) and fractal theories (Gouldin et al., 1989). Pope (1988) presents a definition for the surface density of c = ζ iso–surface as, Σ = 〈|∇c|c = ζ〉 p(ζ) , (2.38) where 〈|∇c|c = ζ〉 is the average of |∇c| conditioned on c = ζ, and p(ζ) is the marginal pdf. A balance equation for Σ is given by (Candel & Poinsot, 1990; Pope, 1988; Trouvé & Poinsot, 1994) ∂Σ ∂t + ∂〈ui〉sΣ ∂xi + ∂ ∂xi [〈sdni〉sΣ] = 〈aT + sd∂ni ∂xi 〉sΣ, (2.39) where sd is the flame surface displacement speed, aT is the tangential strain rate and ni is the ith component of the unit vector normal to the flame surface. These quantities are defined as (Poinsot & Veynante, 2001) aT = (δij − ninj)∂ui ∂xj , sd = ( 1 |∇c| Dc Dt c = ζ ) . (2.40) In typical implementation of this equation in turbulent flame calculations the last term on the left hand side (LHS), the normal propagation term, is neglected. The convective term is written as 〈ui〉s = u˜i + 〈u′′i 〉s, and the tangential strain rate term is split into terms corresponding to the mean flow field and turbulent fluctuations. The last term in the right hand side (RHS), the curvature term, is treated as a destruction term which is modelled such that it prevents the transport equation from predicting an infinite growth of flame surface area. Numerous modelling closures for the tangential strain rate and the destruction terms have been proposed and these are summarised by Poinsot & Veynante (2001) and Veynante & Vervisch (2002). If the flame front is assumed to be planar unstrained laminar flame then the consumption speed in Eq. (2.37), sc, is close to soL. However the turbulence eddies are likely to wrinkle and stretch the flame front and alter its consumption speed. Bray & Cant (1991) account for the stretch rate, κ, via 〈sc〉s = IosoL, where the “stretch factor”, defined as Io = 1 soL ∫ sc(κ) p(κ) dκ , (2.41) 20 2.2 Turbulent Flames is the ratio of the mean flamelet consumption speed to unstrained laminar flame speed. 2.2.5 G–Equation Level Set Formalism Peters (2000) developed a level set formalism based on the well known G–equation (Williams, 1985b) that describes the kinematics of flame front propagation. The scalar G is defined such that its value is Go at the flame front, but away from the flame front its value is not uniquely defined. To ascribe a meaningful field value in a turbulent flame, the fluctuating scalar G is interpreted as the distance between the mean and the instantaneous flame front measured in the direction normal to the mean turbulent flame. The mean flame front location is given by G˜ = Go. The variance G˜′′2 thus becomes a measure of the flame brush thickness. The transport equation for G˜ is (Peters, 1999) ρ ∂G˜ ∂t + ρu˜i ∂G˜ ∂xi = ρST ∣∣∣∣∣∂G˜∂xi ∣∣∣∣∣− µtK ∣∣∣∣∣∂G˜∂xi ∣∣∣∣∣ . (2.42) The first term on the RHS represents the normal propagation of the turbulent flame front while the second is a curvature term. The turbulent flame speed, ST , needs to be modelled and it is obtained as ST = soL(1 + σ˜t) where σ˜t is the ratio of the increase in turbulent flame surface area to that of the laminar flame. A balance equation for σ˜t, valid in the corrugated and thin reaction zones regimes, was derived by Peters (1999) leading to algebraic expressions for the turbulent flame speed, ST . To describe the thermochemical state of the mixture, a rescaled quantity xn is derived from the scalar G as xn = G−Go |∇G| , (2.43) where xn is the normal distance from the flame front. One–dimensional laminar flamelet equations are solved to yield Φα as a function of xn which, along with presumed pdfs are used to obtain the mean quantities, Φα, in the turbulent flame (Herrmann, 2006). 21 2.2 Turbulent Flames 2.2.6 PDF transport approach The motivation behind the pdf transport approach is that an exact closure for the mean reaction rate can be obtained if the joint pdf of all the thermochemical scalars is known. Revisiting the case of the single step reaction in Section 2.2.1, the mean fuel consumption rate can be closed exactly as ω˙F = ∫ ∫ ∫ ω˙F (YF , YO, T ) p(YF , YO, T ) dYF dYO dT , (2.44) where p(YF , YO, T ) is the joint pdf of the species mass fractions and temperature. Thus, for a multi–component system with N species, the one–point joint compo- sition pdf, p(Φ), provides a closed form expression for the reaction rates, where the vector Φ = (Y1, Y2, ....YN , T ). An evolution equation for the mass–weighted joint pdf, p˜(Φ), is (O’Brien, 1980; Pope, 1985) ∂ ρ p˜(Φ) ∂t + ∂ ∂xi [ρu˜i p˜(Φ)] + N∑ α=1 ∂ ∂Φα [ 1 ρ ω˙α(Φ) ρ p˜(Φ) ] = − ∂ ∂xi [〈u′′i |Φ〉 ρ p˜(Φ)] + N∑ α=1 ∂ ∂Φα (〈 1 ρ ∂Ji,α ∂xi Φ 〉 ρ p˜(Φ) ) , (2.45) where the molecular diffusion flux is Ji,α = Vi,αYα. The three terms on the LHS, including the reaction source term, are closed. The first term on the RHS denotes the turbulent transport in the physical space and it is usually modelled using the gradient transport hypothesis. However, the need to model this term, as well as the unclosed terms in the momentum equation, can be avoided if the evolution of the joint velocity–composition pdf is solved. The second term on the RHS denotes molecular mixing in composition space and poses the main modelling challenge for this approach (Pope, 1990). The joint pdf at each location has a large dimension, (N+1), and conventional numerical techniques such as finite differences are unsuitable for solving the pdf equation. Instead, Monte–Carlo methods are employed whereby a large num- ber of notional fluid particles are introduced whose compositional distribution approximates the joint pdf. The advantage of this method is that the computa- tional expense increases only linearly with the pdf dimension (Pope, 1985) making 22 2.2 Turbulent Flames the approach computationally tractable for any arbitrarily large chemical mech- anism. However, in practice, the implementation of pdf transport approach is complicated and expensive compared to other approaches discussed earlier. 2.2.7 Presumed PDF approaches The two main difficulties associated with the pdf transport approach: numerical solution of the pdf equation and the large dimension of the joint pdf, can be simplified by representing the thermochemistry using fewer scalars and by pre- suming a shape for the scalar pdf. The BML model and the G–equation approach described earlier are examples of this where the progress variable and the scalar G, respectively, determine the thermochemical state. Presuming a pdf shape also allows separating the complex thermochemistry calculation from the flow field calculation. The mean thermochemical quantities, including mean density and reaction rates, can be calculated a priori and tabulated as functions of scalar moments and looked up during the turbulent flame calculations. The number of scalar moments required for the tabulation is dictated by the presumed pdf shape. For instance, the BML pdf requires only the first moment, c˜, whereas the beta–pdf, which is widely employed (Bradley et al., 1994; Poinsot & Veynante, 2001; Schneider et al., 2005), requires the first two moments: c˜ and c˜′′2 . These need to be obtained from their transport equations. Fundamental differences between various presumed pdf approaches arise de- pending on the number and definition of the representative scalars employed, and their relationship to the thermochemistry. In the laminar flamelet models, which are based on the flamelet approximation, the relationship is constructed from canonical laminar flame calculations. For example, the mean quantities can be evaluated from Eq. (2.23) by presuming p(ζ) and taking Φα(ζ) to correspond to a laminar flame. Similarly, the mean reaction rate can be obtained from laminar flame values, ω˙(ζ), as ω˙ = ∫ ω˙(ζ) p(ζ) dζ. (2.46) This is the basis of the laminar flamelet approaches proposed by Bradley et al. (1994), van Oijen & de Goey (2000) and Gicquel et al. (2000). However, other approaches to construct the mapping exist, as will be discussed in Section 2.2.9. 23 2.2 Turbulent Flames 2.2.8 Conditional Moment Closure Conditional Moment Closure (CMC) is based on the hypothesis that in turbu- lent flames the fluctuations of all thermochemical quantities are well correlated with the fluctuations of just one scalar: mixture fraction in non–premixed and progress variable in premixed flames. Hence, while the fluctuations about the un- conditional mean can be significant, the fluctuations about a mean conditioned on such a scalar are small. The unconditional mean is related to the conditional mean as Φα = ∫ 〈Φα|c = ζ〉 p(ζ) dζ . (2.47) Note that the above equation, which is mathematically exact, is very similar to Eq. (2.23) which is an approximation. In CMC the conditional mean scalar values, Qα ≡ 〈Φα|c = ζ〉, are obtained from their respective transport equations. For the case of premixed flames this equation is written as (Klimenko & Bilger, 1999) 〈ρ|ζ〉∂Qα ∂t + 〈ρui|ζ〉∂Qα ∂xi = 〈ρ|ζ〉〈Nc|ζ〉∂ 2Qα ∂ζ2 + 〈ω˙α|ζ〉 − 〈ω˙|ζ〉∂Qα ∂ζ + eQα + eyα , (2.48) where the reaction rate terms, ω˙α and ω˙, are as in Eqs. (2.3) and (2.13) respec- tively and the instantaneous scalar dissipation rate is Nc ≡ Dc ∂c ∂xi ∂c ∂xi (2.49) Equation (2.48) has been arrived at independently by Klimenko (1990) using the joint pdf equation, and by Bilger (1993) using a decomposition about the conditional mean as Yα(x, t) = Qα(ζ,x, t) + yα(x, t), where yα is the conditional fluctuation. The term eQα is (Swaminathan & Bilger, 2001a) eQα ≡ ∂ ∂xi ( ρDα ∂Qα ∂xi ) + 〈 ∂Qα ∂ζ ∂ ∂xi ( [1− Leα]ρDα ∂c ∂xi ) + ρDα ∂c ∂xi ∂ ∂xi ( ∂Qα ∂ζ )∣∣∣∣ζ〉 (2.50) and it represents the contribution of molecular diffusion which becomes negligible when Re is large. However, the second term in the above equation arises from 24 2.2 Turbulent Flames differential diffusion effects and might have to be cautiously considered (Swami- nathan & Bilger, 2001a). The last term on the RHS in Eq. (2.48) is eyα ≡ − 〈 ρ ∂yα ∂t + ρui ∂yα ∂xi − ∂ ∂xi ( ρDα ∂yα ∂xi )∣∣∣∣ζ〉 (2.51) and it can be modelled as (Klimenko & Bilger, 1999) eyα ≈ 1 p(ζ) ∂ ∂xi (〈ρu′′i yα|ζ〉p(ζ)) (2.52) where u′′i is the conditional fluctuation of velocity ui. Modelling of the conditional velocity, 〈ρui|ζ〉, was investigated by Swami- nathan & Bilger (2001a) and some simple closures seem promising. On the other hand, modelling of the conditional mean scalar dissipation rate, 〈Nc|ζ〉, is yet to be addressed satisfactorily (Swaminathan & Bilger, 2001b)4 and it appears to be inextricably linked to the unconditional mean dissipation rate, ²˜c. For the con- ditional reaction rates a first order closure can be obtained by writing a Taylor series expansion of 〈ω˙α|ζ〉, similar to Eq. (2.17), and noting that the conditional fluctuations are negligible which gives (Klimenko & Bilger, 1999) 〈ω˙α(Y1, ....YN , T )|ζ〉 ≈ 〈ω˙α(Q1, ....QN , QT )|ζ〉 = ω˙α(Q1, ....QN , QT ). (2.53) This first order closure has been found to be remarkably good for the conditional reaction rates of major species and progress variable, but not so for minor species since their conditional fluctuations are not small (Swaminathan & Bilger, 2001a). For such species, conditioning on more than one scalar might be necessary. Apart from modelling of the conditional scalar dissipation rate, the main difficulty in implementing CMC for premixed flames seems to be with the suitable definition of the progress variable (Klimenko & Bilger, 1999). It is worth noting that CMC is considerably more expensive than the approaches discussed earlier, except possibly the transported pdf approach. The number of transport equations to be solved for the thermochemistry is equal to (N + 1) times the number of 4For non–premixed flames the conditioning variable, the mixture fraction, is a conserved scalar and its pdf equation leads to an expression for the conditional scalar dissipation rate under certain conditions (Klimenko & Bilger, 1999). This is considerably more difficult when the conditioning variable is a reactive scalar, such as c for premixed flames. 25 2.2 Turbulent Flames points in the conditioning variable space. However, the rigorous mathematical framework of the CMC is likely to provide better predictions of pollutants and their formation rates compared to other approaches that are based on physical or phenomenological arguments. 2.2.9 Chemistry Tabulation Tabulating the thermochemistry is a crucial aspect of most of the modelling approaches described in the previous sections, with the exception of CMC5. In the pdf transport approach an on–the–fly computation of the reaction rates in the (N + 1)–dimensional composition space is time consuming unless a reduced chemical mechanism is considered. In presumed pdf approaches (G–equation and flamelet) chemistry tabulation is an integral part of their formulation. Hence strategies have been developed to simplify the detailed chemical kinetics and tabulate the chemistry as functions of few scalars. The Intrinsic Low Dimensional Manifold (ILDM) approach identifies that for every reacting system of N species, there exists subspaces of much smaller di- mension, ns, in which reactions proceed according to slow time scales (Maas & Pope, 1992b). At each point in this subspace there is local equilibrium with re- spect to (N − ns) fastest time scales and any point away from this subspace is attracted to it via these faster processes. Hence the dynamic evolution of the reacting system can be parametrised in terms of only ns variables and these com- pletely determine the thermochemical state of the mixture. The physical and mathematical foundation of this approach and its implementation are discussed in (Maas & Pope, 1992a,b, 1994). ILDM performs a simplification of the detailed chemical mechanism as opposed to reducing it. In reduced chemical mechanisms certain fast chemical reactions are identified and the assumption of partial equi- librium for these reactions is invoked at the start of the reduction process. These assumptions provide poor approximations beyond certain composition range. In ILDM, the chemical reactions corresponding to the fast time scales are identified locally at each point in the subspace and different initial compositions optimally approach this manifold. 5The BML model also does not require the tabulation since it does not attempt to resolve the chemical structure of the flame front. 26 2.2 Turbulent Flames Gicquel et al. (2000) note that at low temperatures the chemical time scales are not disparate and hence ILDM provides a poor approximation in such re- gions. This is remedied by a Flame Prolongation of ILDM (FPI) (Gicquel et al., 2000) which is conceptually similar to the Flamelet–Generated Manifolds (FGM) approach of van Oijen & de Goey (2000). Both involve generating a manifold from freely propagating unstrained planar laminar flame calculations, which is used to tabulate the thermochemistry in terms of a progress variable. Physically, when applied to turbulent flame calculations, FPI and FGM can be viewed as laminar flamelet models where the turbulent flame front has the same structure as that of an unstrained laminar flame. Bradley et al. (1994) propose a stretched laminar flamelet model that accounts for the influence of fluid–dynamic stretch rate on the reaction rate. While the focus of their approach was on the closure of the mean reaction rate, ω˙, it could be extended to provide a closure for the mean species concentrations also. In Chapter 4, an alternative flamelet approach based on the scalar dissipation rate will be presented. This approach is motivated by physical considerations. But first it is worth emphasising the role played by scalar dissipation rate in turbulent premixed flame modelling, and this is discussed in the next chapter. 27 Chapter 3 Scalar Dissipation Rate This chapter focuses on the scalar dissipation rate which is unclosed yet cen- tral in many modelling approaches. The mathematical definition and physical meaning of this quantity will be discussed first. Simple relations between chemi- cal reaction rates and the scalar dissipation rate in premixed and non–premixed flames will be presented. The role of mean scalar dissipation rate in turbulent combustion modelling will then be illustrated. Although the main interest of this study is lean premixed flames, non–premixed flames are discussed for the sake of completeness and to emphasise the central role of the scalar dissipation rate in turbulent combustion in general. 3.1 Definition and Physical Meaning It is worthwhile to examine the balance equation for the scalar variance, c˜′′2 . By substituting c = c˜ + c′′ in Eq. (2.13), multiplying by c′′ and then averaging the result leads to the exact equation (Bray, 1980; Veynante & Vervisch, 2002) ∂ρc˜′′2 ∂t + ∂ρu˜ic˜′′ 2 ∂xi + ∂ρu˜′′i c′′ 2 ∂xi = ∂ ∂xj ( ρDc ∂c′′2 ∂xj ) + 2c′′ ∂ ∂xj ( ρDc ∂c˜ ∂xj ) − 2ρu˜′′i c′′ ∂c˜ ∂xi︸ ︷︷ ︸ production + 2ω˙′′c′′︸ ︷︷ ︸ reaction − 2ρDc∂c ′′ ∂xi ∂c′′ ∂xi︸ ︷︷ ︸ dissipation . (3.1) Modelled form of the above equation is particularly relevant to presumed pdf approaches since the pdf shapes requiring the first two moments are popular 28 3.1 Definition and Physical Meaning choice among modellers. The three terms in the LHS denote the unsteady, mean convection and the turbulent transport terms. The first two terms in the RHS denote the influence of molecular transport, and are negligible for large Reynolds numbers. The third term denotes a production of scalar fluctuations due to gradients in the mean, if the scalar flux, ρu˜′′i c′′, is gradient type. The fourth term denotes the reaction rate contribution and needs to be modelled. This term is related to the mean reaction rate (Bray, 1979) by ω˙′′c′′ = ω˙c− ω˙c˜ = (Cm − c˜)ω˙ . (3.2) The last term in the RHS of Eq. (3.1) is the mean scalar dissipation rate, ²˜c, defined earlier by Eq. (2.32) and it is unclosed. This term has a negative sign in Eq. (3.1) and for a simplified homogeneous case with no c˜ gradients and no reaction, it denotes one half of the rate of decay of scalar fluctuations. Since the absence of fluctuations implies a well mixed fluid, the mean scalar dissipation rate can be physically interpreted as the rate at which turbulent mixing occurs at the molecular level (Cant & Mastorakos, 2008). Mathematically, it is the average of the instantaneous dissipation rate ρNc = ρDc ∂c˜ ∂xi ∂c˜ ∂xi + 2ρDc ∂c′′ ∂xi ∂c˜ ∂xi + ρDc ∂c′′ ∂xi ∂c′′ ∂xi , (3.3) since the contribution of the gradients of mean progress variable is negligible compared to the gradients of fluctuations. Hence ρNc ≈ ρ²˜c. The EBU model of Mason & Spalding (1973), mentioned in Section 2.2.3, is based on the physical idea that in the limit of fast chemistry, the mean reaction rate is limited by the rate at which the reactants are brought into contact with hot products by the turbulent mixing. Bilger (1976) provided a mathematical basis for this physical argument by considering the mixture fraction formulation in non–premixed flames. Considering a two–feed problem, assuming all species have equal diffusivities, Bilger (1976) defines the mixture fraction as ξ = Zi − Zi2 Zi1 − Zi2 , (3.4) where Zi is the mass fraction of any element i, and the subscripts 1 and 2 denote the two feeds. The mixture fraction is a conserved scalar and satisfies L(ξ) = 0 29 3.2 Role in Turbulent Combustion Modelling where the linear operator L ≡ ρ ∂ ∂t + ρui ∂ ∂xi − ∂ ∂xi ( ρD ∂ ∂xi ) . (3.5) The conservation equation for species α can be written as L(Yα) = ω˙α. If the species concentrations are assumed to be functions of mixture fraction alone, then through simple transformation one obtains (Bilger, 1976) ω˙α = ∂Yα ∂ξ L(ξ)− ∂ 2Yα ∂ξ2 ( ρD ∂ξ ∂xi ∂ξ ∂xi ) = −∂ 2Yα ∂ξ2 ρNξ (3.6) and the mean reaction rate is directly proportional to the mean scalar dissipation rate i.e. ω˙α ∼ ²˜ξ. The definitions of Nξ and ²˜ξ are analogous to Nc and ²˜c respec- tively. For premixed flames, one can similarly work with the progress variable and, noting that L(c) = ω˙ from Eq. (2.13), obtain ω˙α = ∂Yα ∂c ω˙ − ∂ 2Yα ∂c2 ρNc . (3.7) Again the dependence of mean species reaction rates on the mean scalar dissipa- tion rate, ²˜c, is apparent. However, this analysis leads nowhere useful for premixed flame modelling since, in the first instance, closure of ω˙ is itself unknown and, further, the modelling of ²˜c is challenging. In contrast, modelling of ²˜ξ is relatively easier. Nonetheless, these simple results suggest that the mean scalar dissipation rate fundamentally represents the turbulence–chemistry interaction both in pre- mixed and non–premixed scenarios. Inevitably, it appears directly or indirectly in most combustion modelling approaches, as we shall see in the next section. 3.2 Role in Turbulent Combustion Modelling Veynante & Vervisch (2002) analyse the various mathematical tools of turbulent combustion and conclude that the scalar dissipation rate is a key parameter and often a stumbling block in modelling. In presumed pdf approaches, which are conceptually the simplest, the mean scalar dissipation rate is required in the variance equation, Eq. (3.1). Provided the turbulent scalar flux and reaction terms are closed for the c˜ equation, the scalar dissipation rate presents the only additional closure problem. In CMC the conditional mean scalar dissipation rate 30 3.2 Role in Turbulent Combustion Modelling appears as an unclosed term in Eq. (2.48) and it is related to the mean scalar dissipation rate as (Swaminathan & Bilger, 2001b) ²˜c = ∫ 1 0 〈Nc|ζ〉 p˜(ζ) dζ. (3.8) For transported pdf approaches, as discussed in Section 2.2.6, the only un- closed term is the molecular mixing term which is the last term in the RHS of Eq. (2.45). An elementary implementation of this approach would involve solv- ing the transport equation of the pdf of a single scalar. The molecular mixing term for such a case becomes equal to (Klimenko & Bilger, 1999; O’Brien, 1980; Swaminathan & Bilger, 2001b; Veynante & Vervisch, 2002) − 1〈ρ|ζ〉 ∂ ∂ζ [〈∇ · (ρDc∇c) | ζ〉 p(ζ)] ≈ − ∂ 2 ∂ζ2 [〈ρ Nc|ζ〉 p(ζ)] (3.9) involving only the conditional mean dissipation rate, when the Reynolds number is large. On the other hand, in solving transport equations of multi–component pdfs, stochastic mixing models are used to mimic the molecular mixing process. A key ingredient of these models is the scalar mixing frequency, τ−1c , which is the inverse of the decay time scale of scalar fluctuations and is related to the scalar dissipation rate by τ−1c ≡ ²˜c c˜′′2 . (3.10) The mixing frequency is usually estimated from the decay time scale for turbulent velocity fluctuations, τt, via τ−1c = Cφτ −1 t = Cφ ε˜ k˜ , (3.11) and for a passive scalar Cφ is about 1.0 (Spalding, 1971). Bilger et al. (2005) point out that such mixing models based on passive scalar dynamics involve no explicit coupling between reaction and mixing, although pdf simulations have been suc- cessful in predicting complex phenomena such as local extinction in non–premixed turbulent flames. However, the need to include the coupling is acknowledged by Subramaniam & Pope (1998) and Mitarai et al. (2005). 31 3.2 Role in Turbulent Combustion Modelling 3.2.1 Non–Premixed Combustion Passive scalar mixing models have been quite useful in non–premixed flame mod- elling. In the class of laminar flamelet models, spawned by the work of Peters (1984), the dissipation rate of mixture fraction assumes a special role. Attaching a coordinate system to the stoichiometric mixture fraction surface, Peters (1984) performs a Crocco–type transformation from (x1, x2, x3, t) system to (ξ, Z2, Z3, t1) system where, by definition, ξ is locally normal and Z2 and Z3 are locally tangen- tial to the stoichiometric surface. This leads to the flamelet equations in terms of the mixture fraction ρ ∂Yα ∂t1 = ρNξ ∂2Yα ∂ξ2 + ω˙α −R(Yα) . (3.12) The derivatives with respect to Z2 and Z3, denoted by the operator R, and the unsteady term are of lower order in the inner reaction zone (except close to extinction) (Peters, 1984). Under such conditions the structure of a burning flamelet that defines Yα(ξ) is parametrised only by the dissipation rate, Nξ. This is identical to Bilger’s result, Eq. (3.6), except for the unsteady term. Thus the statistics of the mixture fraction and its dissipation rate are sufficient to describe the thermochemistry of a turbulent flame via Φα = ∫ Nξ,q 0 ∫ 1 0 Φα(ξ,Nξ) p(ξ,Nξ) dξ dNξ , (3.13) where Nξ,q is the dissipation rate at extinction. The mean dissipation rate, ²˜ξ, required for the statistics of Nξ can be obtained from a transport equation but it is usually closed algebraically using a simple model ²˜ξ ≈ Cξ ε˜ k˜ ξ˜′′2 , (3.14) where Cξ = 1.01 and it is obtained from Eq. (3.11) . 1the definition of scalar dissipation rate in (Peters, 1984) includes a factor of 2, i.e., Nξ = 2D(∂ξ/∂xi)2. Accordingly the constant Cξ takes a value of 2.0 in that work. 32 3.2 Role in Turbulent Combustion Modelling 3.2.2 Premixed Combustion Bilger (2004b) suggests that the heat release is unlikely to influence the fine–scale structure of turbulence strongly in non–premixed flames since the local dilatation at the flame front is much smaller than the turbulence strain rate. However, in premixed flames, the strong local dilatation will be subdued only under intense turbulence conditions. Hence, models based on passive scalar ideas might be reasonable for the former but are unjustifiable for the latter. Furthermore, the parametrising scalars for premixed flame modelling are invariably reactive. In flamelet combustion the gradients of reactive scalars are steepened by chemical reactions which is not incorporated in the current mixing models (Subramaniam & Pope, 1998). Consequently, attempts at tailoring the stochastic mixing models for premixed flame calculations have been few. Pope & Anand (1984) solve the transport equation of p(ζ) for a single step chemistry, and consider the closure of the mixing term for the two extreme conditions: distributed reaction zones and wrinkled flamelets regimes. For the former regime passive scalar type mixing models were retained. For the latter, the mixing term was not explicitly modelled. Instead, invoking the flamelet assumption, the mixing term combined with the reaction term was expressed as a local function of the progress variable and was obtained from planar laminar flame results. Lindstedt & Vaos (2006) report premixed flame calculations for conditions that are far from the classical extremes, with a mixing model using modified model constant Cφ which includes a chemical time scale. Despite its modelling difficulties, the scalar dissipation rate is mathemati- cally linked to the fundamental quantities in premixed combustion (Veynante & Vervisch, 2002). The direct relation between the mean reaction rate, ω˙, and the mean scalar dissipation rate, ²˜c, is already shown in Eq. (2.30). When combined with Eq. (3.7) it shows that the mean reaction rate of species α, ω˙α, is propor- tional to ²˜c, similar to the case of non–premixed flames. It is also related to the flame surface density from Eq. (2.38) as Σ = 〈√ (Nc/Dc) ∣∣∣c = ζ〉 p(ζ) . (3.15) 33 3.2 Role in Turbulent Combustion Modelling 3.2.3 Stratified/Partially Premixed Combustion Recent modelling strategies for partially premixed combustion have employed both reactive and passive scalars: progress variable to track the reaction zones and mixture fraction to denote the local stoichiometry. Bray et al. (2005) sug- gest that an appropriately defined progress variable can be a suitable scalar to represent the thermochemistry in partially premixed flames. One possible def- inition, suggested therein, is as a normalised fuel mass fraction with a mixture fraction dependence. The reaction source term for such a progress variable then introduces a dependence on the mixture fraction dissipation rate, Nξ, and an additional parameter; the cross scalar dissipation rate2, Nξc ≡ D ∂ξ ∂xi ∂c ∂xi . (3.16) Libby & Williams (2000) and subsequently Ribert et al. (2004) have formulated a presumed pdf approach for partially premixed systems where the joint pdf, p(ξ, c), consists of two Dirac delta functions in the (ξ, c) space. The strength and the location of the delta functions are determined by, in the least, ξ˜, ξ˜′′2 , c˜ and c˜′′2 for which transport equations are solved. The shape presumed for the pdf constrains the cross correlation to satisfy ξ˜′′c′′ = √ ξ˜′′2 c˜′′2 (3.17) which is not necessarily physical. Robin et al. (2006) attempt to relax this con- straint by presuming a pdf shape with four Dirac delta functions and solving a transport equation for the cross correlation, ξ˜′′c′′, which requires the closure of the mean cross scalar dissipation, ρNξc. Thus, it appears that models of partially premixed flames need to address the closure of the mean cross scalar dissipation, apart from the mean dissipation rates of individual scalars. The cross scalar dissipation rate can be expressed as (Bray et al., 2005) Nξc = (eˆξ · eˆc) √ NξNc (3.18) where eˆξ and eˆc are the unit vectors normal to the ξ and c iso–surfaces. The dot product (eˆξ · eˆc) could assume a value between -1 and 1 in a turbulent flame and 2for the simple case of all species having unity Lewis numbers Dξ = Dc = D. 34 3.3 Summary it is possible that the mean value is negligible (Bray et al., 2005). However, this is only speculative and further study is required to justify any assumptions of the magnitude of the cross scalar dissipation on physical grounds. 3.3 Summary The role of scalar dissipation rate in turbulent combustion was studied in this chapter. Simple mathematical relations that link the scalar dissipation rate to the fundamental quantities in turbulent combustion were presented. From modelling point of view, conditional and unconditional mean scalar dissipation rates are central to most modelling methodologies. For premixed flames, closures for the reaction rates of chemical species and the progress variable are of interest and these are seen to be proportional to the progress variable dissipation rate, Nc. It is instructive to study a balance equation for Nc. Swaminathan & Bray (2005) present the equation ∂ρNc ∂t + ∂ρuiNc ∂xi = ∂ ∂xj ( ρDc ∂Nc ∂xj ) − 2ρ ( Dc ∂2c ∂xj∂xk )2 −ρDc ∂c ∂xj [ ∂uj ∂xk + ∂uk ∂xj ] ∂c ∂xk + 2ρNc ∂ui ∂xi + 2Dc ∂c ∂xk ∂ω˙ ∂xk , (3.19) where the dependence of Dc on temperature is assumed to be negligible. In the above equation the first and second terms in the RHS denote the diffusive flux and dissipation respectively. The third term is governed by the interaction of the fluid–dynamic strain rate with the gradients of progress variable. The fourth term arises due to density jump across the flame front and the last term represents the correlation between progress variable gradient and chemical reaction. This equation suggests that the balance of Nc is intricately linked to the fundamental processes that affect the flame front dynamics. The observations on the central role of Nc and its close coupling to the physical processes influencing the dynamics of the flame front provide motivation for formulating a flamelet model based on the scalar dissipation rate, which is discussed in the next chapter. 35 Chapter 4 Strained Flamelet Formulation for Premixed Flames In this chapter the formulation of a mathematical model for turbulent premixed flame computations is developed using the flamelet concept. The objective is to describe how the mutual influence of turbulence and flamelets at the scales rele- vant to combustion can be captured, with a specific interest on the mean reaction rate modelling, within the framework of the laminar flamelets approach. The ex- isting models based on flamelet ideas will be briefly summarised before presenting a new formulation which is based on a scalar dissipation rate parametrisation. Closures for the unknown terms, the joint pdf of progress variable and scalar dissipation rate and the requisite moments, will be discussed. The choice of a canonical laminar flamelet configuration will then be discussed which results in the final closure. In laminar flamelet modelling the turbulent flame is viewed as an ensemble of thin locally one–dimensional laminar structures called flamelets. Experimental studies (Chen et al., 1996; Dunn et al., 2007, 2009) indicate that even at very high turbulence intensities, while the flame front is disrupted by the turbulence, flamelet–like structures can still be discerned. Numerical studies (Poinsot et al., 1991) have demonstrated that flamelet type combustion is likely to extend to regimes well beyond those suggested by classical ideas. However, the success of this modelling method depends on the approach used to ascribe the influence of turbulence on flamelets. The flame front is contorted and strained by the eddies but if the flame scales are much smaller than the typical small (Kolmogorov) 36 scales of turbulence, such as in the wrinkled flamelets regime, then the effects of turbulent straining can be ignored. Some existing models such as FGM (van Oijen & de Goey, 2000), FPI (Gicquel et al., 2000) and the flamelet model of Bradley et al. (1988) are based on such an implicit assumption; the turbulent flamelet structure corresponds to a freely propagating planar unstrained laminar flame. They provide a closure for the mean reaction rate as ω˙ = ∫ 1 0 ω˙o(ζ) p(ζ) dζ, (4.1) where ω˙o(ζ) is the reaction rate in an unstrained planar laminar flame. At higher turbulence intensities, such as in the thin reaction zones regime, the influence of turbulence straining is unlikely to be negligible. It is well known from asymptotic theories (Clavin, 1985) that the consumption speed, sc, depends on the fluid dynamic stretch rate, κ, as sc(κ) = s o L − Lκ, (4.2) where L is the Markstein length. The stretch rate can be written as (Candel & Poinsot, 1990) κ = aT + sd/r (4.3) and it incorporates the effects of both the strain rate, aT , and the flame front curvature, r. The displacement speed, sd, denotes the speed of advancement of the flame relative to unburnt mixture while the consumption speed, sc, denotes the rate of reactants consumption by the flame. Bradley (1992) and co–workers (Bradley et al., 1994, 1998, 2005) propose a flamelet model that accounts for the stretch rate effects as ω˙ = ∫ ∫ ω˙(ζ, κ) p(ζ, κ) dζ dκ . (4.4) They assume that ζ and κ are statistically independent which allows the joint pdf to be decomposed as p(ζ, κ) = p(ζ) p(κ). Further, noting that there is a correspondence between consumption speeds and reaction rate profiles, they write, by analogy to Eq. (4.2), ω˙(ζ, κ) = f̂(κ) ω˙o(ζ) . (4.5) 37 4.1 Scalar Dissipation Parametrisation Empirical correlations for f̂(κ) in terms of L were obtained from numerical calcu- lations of spherical laminar flames (Bradley et al., 1996, 1998). Their final closure for the mean reaction rate can be expressed as ω˙ = Pb ∫ ω˙o(ζ) p(ζ) dζ, (4.6) and the “burning rate factor”, Pb = ∫ κ+q κ−q f̂(κ) p(κ) dκ, (4.7) incorporates the stretch effects and is analogous to the factor Io in Eq. (2.41). The positive and negative extinction stretch rates are denoted by κ+q and κ−q respectively. 4.1 Scalar Dissipation Parametrisation The stretch rate dependence of the consumption speed given by Eq. (4.2) is strictly valid for low stretch rates (Poinsot & Veynante, 2001). Accordingly, at high stretch rates the validity of the analogous relationship, Eq. (4.5), as well as Eq. (2.41) might be questionable. Bradley et al. (1998) indicate that the Mark- stein number correlations proposed for f̂(κ) are likely to be valid only for weakly curved corrugated flames. On the other hand, a recent DNS study (Hawkes & Chen, 2006) showed significant burning in turbulent flames at strain rates much larger than the extinction strain rates of symmetric counterflow laminar flames. Therefore, it is proposed to use a scalar dissipation rate parametrisation, instead of the stretch rate, in the form of ω˙ = ∫ ∫ ω˙(ζ, ψ) p(ζ, ψ) dψ dζ , (4.8) where ψ is the sample space variable for the scalar dissipation rate Nc. Physically, this implies that the influence of turbulence dynamics on the flame front is tracked using the local scalar dissipation rate. 38 4.1 Scalar Dissipation Parametrisation 4.1.1 Motivation There are many reasons for choosing the scalar dissipation rate to parametrise the flamelets. To emphasise a few of these: • On a physical basis, the mixing of cold and hot fluid parcels on the flame surface is required to sustain combustion and the rate of this mixing is given by the scalar dissipation rate. • As discussed in Section 3.3, scalar dissipation rate is governed by the convection–diffusion–reaction balance and represents the flame front dy- namics whereas the stretch rate represents flame front kinematics. • The progress variable gradients, and hence Nc, are mainly produced by chemical reactions. It can be expected, as is also evident from Eq. (3.7), that locally the reaction rate is strongly correlated with the dissipation rate. • It can be inferred from the asymptotic analyses of Libby & Williams (1982) and Libby et al. (1983) that the flamelet structure is closely related to gradient of c, and hence Nc, both at low and high stretch rates. Furthermore, in the implementation of the stretch rate parametrisation, when one considers the distribution of stretch rates acting on a turbulent flame front at a given spatial location, it is not distinguished whether the stretch rate corre- sponds to a location on the burnt side, the unburnt side or within the flame front. When embedding the laminar flamelets, however, the stretch rate characterising a flamelet is defined at a location on the unburnt side only according to Bradley et al. (1996). On the other hand, the use of a scalar dissipation rate parametrisa- tion automatically allows such a distinction to be accounted for. This is because the scalar dissipation rate is defined at every location in the laminar flamelet. One could also define a stretch rate at every location in the laminar flamelet but existing models do not follow such an approach. The scalar dissipation rate parametrisation proposed here is some what similar to the non–premixed laminar flamelets formulation of Peters (1984), Eq. (3.13). However, in Eq. (3.13), the flamelet structure is characterised by a single dissi- pation rate value, that at the stoichiometric mixture fraction, whereas here the 39 4.2 Choice of Flamelet Configuration dissipation rate at every location in the flamelet is considered. Also, statistical independence between the mixture fraction and its dissipation rate is typically assumed which is strictly valid when the mixture fraction pdf is Gaussian (Gao & O’Brien, 1991). Such an assumption might be drastic for the progress variable since it is a reactive scalar and its pdf is unlikely to resemble a Gaussian. The assumption of statistical independence is eschewed and the joint pdf is written using the Bayes theorem as p(ζ, ψ) = p(ζ) p(ψ|ζ) . (4.9) The mean reaction rate in Eq. (4.8) now becomes ω˙ = ∫ [∫ Nmax 0 ω˙(ζ, ψ) p(ψ|ζ) dψ ] p(ζ) dζ . (4.10) One could solve for the pdfs from the transport equations but a simpler framework to test the scalar dissipation rate parametrisation is followed by presuming these pdfs which will be discussed in Section 4.3. The quantity ω˙(ζ, ψ) is to be obtained from strained laminar flames which is the focus of the next section. 4.2 Choice of Flamelet Configuration To put the current reaction rate closure in perspective it is helpful to write Eq. (4.10) as ω˙ = ∫ 1 0 〈ω˙|ζ〉 P (ζ) dζ . (4.11) The difference in various modelling methodologies can be illustrated in the way the conditional mean reaction rate, 〈ω˙|ζ〉, is evaluated1. In the unstrained flamelets closure given by Eq. (4.1), 〈ω˙|ζ〉 = ω˙o(ζ), whereas the stretch rate based closure of Bradley and co–workers is equivalent to 〈ω˙|ζ〉 = Pb ω˙o(ζ). But, in the current closure 〈ω˙|ζ〉 = ∫ N2 N1 ω˙(ζ, ψ) P (ψ|ζ) dψ . (4.12) 1As noted in Section 2.2.8, in CMC a closure for 〈ω˙|ζ〉 is obtained by evaluating the Arrhenius type reaction rate expression using the conditional mean species mass fractions and temperature. 40 4.2 Choice of Flamelet Configuration The value of ω˙(ζ, ψ) and the limits N1 and N2 will be dictated by the choice of the flamelet configuration as different configurations exhibit different responses to stretching (Law, 2006). Since stretched premixed flamelets can be established in a variety of configurations such as spherical flames, planar flames in Reactant–to– Reactant or Reactant–to–Product opposed flows etc., it is important to choose a configuration that reasonably represents the turbulent flame characteristics (Pe- ters, 1984). Two configurations that are widely considered (Dixon-Lewis, 1990) in the context of flamelet modelling are the purely strained steady planar flames; the opposed flow symmetric Reactant–to–Reactant (RtR) and asymmetric Reactant– to–Product (RtP) flamelets shown schematically in Fig. 4.1. In these flamelets, the stretch rate is equal to the strain rate as there is no curvature. It is worth dis- cussing the salient characteristics of these flamelets and their response to straining before identifying the configuration of choice. (a) (b) Figure 4.1: Schematic of purely strained planar laminar flamelet configurations: (a) Reactant–to–Reactant (RtR) and (b) Reactant–to–Product (RtP). 4.2.1 Reactant–to–Reactant Flamelet In the RtR configuration the reaction zones move towards the stagnation plane as the strain rate is increased. The response of the flame structure - the reaction rate or species concentration profiles versus a flame coordinate - to strain rate depends on the mixture Lewis number, Le. The mixture Lewis number is defined as the ratio of thermal diffusivity of unburnt mixture, Dthu , to the diffusivity of 41 4.2 Choice of Flamelet Configuration the deficient species. For equidiffusive mixtures (Le = 1) the flame structure is insensitive to strain rate changes (Dixon-Lewis, 1990; Law et al., 1994) whereas for non–equidiffusive mixtures the burnt temperature, Tb, and the reaction rate, ω˙, decrease (increase) with strain rate for mixtures with Le greater (less) than unity (Sung et al., 1996). When the strain rate is high enough the reaction zone reaches the stagnation plane and is progressively truncated leading to an abrupt extinction caused by the depletion of the radical pool by the diverging streamlines (Dixon-Lewis, 1990). 4.2.2 Reactant–to–Product Flamelet In RtP flames at low strain rates the reaction zone is on the reactants side far from the stagnation plane and is characterised by reactive–diffusive balance with convection of lower order. As the strain rate is increased the reaction zone pro- gressively moves closer to the stagnation plane and at sufficiently high strain rates it crosses over and locates on the products side. Under such conditions the convective flux of reactants is away from the reaction zone but the diffusion is able to overcome the negative convection and ensure a net positive flux of reac- tants enabling an attenuated chemical reaction (Libby & Williams, 1982). This is evident in Fig. 4.2 which shows ω˙+ profiles of lean methane–air flames in RtP configuration at various strain rates as a function of distance from the stagnation plane located at x = 0. The equivalence ratio, φ, of the mixture is 0.6. The su- perscript + denotes a quantity normalised appropriately using ρu, soL and δoL. The thermal thickness is defined based on the temperature gradient in the unstrained laminar flame as (Poinsot & Veynante, 2001) δoL = Tb − Tu (dT/dx)max (4.13) The reactants stream is in the region x < 0 and the products stream is in the region x > 0. The strain rate, a, corresponding to each profile is noted in the figure and its value, as per convention, is equal to the maximum velocity gradient on the unburnt side of the flame. These profiles are calculated using the OPPDIF code (Lutz et al., 1997) and the GRI–3.0 chemical mechanism. The product 42 4.2 Choice of Flamelet Configuration 0.0 0.5 1.0 1.5 2.0 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 ω˙+ x (mm) .................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ......................................................................................................................................................................................................................................................................................................................................................................................................................................... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .................................................................................................................................................................................................................... ...... .... .... .... .... .... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ......................................................................................................................................................................................................... ... .... .... ... .... .... .... .... .... .... .... .... .... .... ................................................................... ............ 90/s 276 468 3363 4874 Figure 4.2: Normalised reaction rate, ω˙+, profiles of RtP flames of lean CH4–air mixture (φ = 0.6), at various strain rates, plotted against the distance from the stagnation plane. stream boundary conditions i.e. equilibrium compositions and temperature are obtained using the STANJAN code (Reynolds, 1986). In RtP flames there is no abrupt extinction as in RtR flames but an extinction regime can be identified when the reaction rate becomes negligible and there is no net additional product creation (Libby & Williams, 1982). The approach to this regime is smooth for adiabatic equidiffusive flames, although, abrupt extinction could occur if the Lewis number is sufficiently far from 1 or under non–adiabatic conditions (Libby et al., 1983). In this work however, only adiabatic flames are considered. Another distinction is that RtP flames have a negative displacement speed when the reaction zone is located on the products side whereas negative displacement speed is not possible in RtR flames. Hawkes & Chen (2006) used DNS data to examine the suitability of the RtR and RtP configurations for flamelet modelling and found that the RtP configu- ration reproduced the overall turbulent flame characteristics well over a range of conditions. The trends of displacement and consumption speeds versus the strain rate in turbulent flames were in reasonable agreement with those in the RtP 43 4.2 Choice of Flamelet Configuration flames, whereas, the RtR configuration showed trends that were qualitatively op- posite to the DNS data. Furthermore, the turbulent flames were observed to burn at strain rates much larger than the extinction value of RtR flames and burning at such high strain rates was better explained by the RtP flames. This too is evident from Fig. 4.2 which shows considerable reaction rate even at a = 3363/s whereas the extinction strain rate for the lean methane–air flame (φ = 0.6) in RtR configuration is about 200/s (Law et al., 1986). Also, negative displacement speeds observed in DNS studies by Gran et al. (1996) and Hawkes & Chen (2006) can be explained by the RtP but not the RtR configuration. Based on the above considerations the RtP configuration is chosen in the present formulation. The quantity ω˙(c = ζ,Nc = ψ), required to evaluate ω˙, is obtained from a series of RtP flame calculations i.e. ω˙(ζ, ψ) = ω˙RP (ζ, ψ) in Eq. (4.12), where the subscript RP denotes RtP flamelets. It is clear from Eq. (4.8) that ω˙ is equal to the volume under the surface defined by the product of ω˙RP (ζ, ψ) with the pdf p(ζ, ψ). The surface of ω˙+(c,N+c ) obtained from RtP flamelets is shown in Fig. 4.3 for the lean methane–air mixture and in Fig. 4.4 for lean propane–air mixture. For lean methane–air mixture having Lewis number, Le, less than 1, the surface has a simply connected topology since the value of Nc increases monotonically with strain rate for all c. However, for the lean propane–air mixture having Le > 1, as the strain rate increases Nc decreases first as noted by Sung et al. (1996) and then increases. Hence, the surface in Fig. 4.4 is folded but it is still integrable. The response of ω˙(c,Nc) to the strain rate is better understood when these surfaces are viewed as a series of ω˙ vs Nc curves for different values of the sample space variable ζ as shown in Fig. 4.5. Data from the laminar flame calculations are used to plot representative curves in this figure. The area under each curve weighted by the conditional pdf, p(ψ|ζ), is equal to 〈ω˙|ζ〉 in Eq. (4.12). It is also worth noting that the limits N1 and N2 are in general different for different ζ. The curves shown in Fig. 4.5a are for Le < 1 case and those in 4.5b and 4.5c are for Le > 1 cases. The unstrained flame value, No, gives N1 and the extinction limit gives N2 for the Le < 1 case. However, for the Le > 1 case the upper limit N2 corresponds to the extinction limit but the lower limit N1 corresponds to an intermediately strained value since No lies between the two. This results in the reaction rate being multivalued for a certain range of the dissipation rate. 44 4.2 Choice of Flamelet Configuration Figure 4.3: The surface of ω˙+(c,N+c ) from RtP flame calculations of lean CH4–air mixture, φ = 0.6. Physically, this implies that in the domain ψ ∈ [N1, No], there exist two states: a higher and a lower burning state. Mathematically, the integration of such a curve, shown schematically in Fig. 4.5d, can be performed in two ways: (i) The total integral can be written as I1 + I2. The area of the “C” portion, bounded by the solid curve and the dashed line, is I1 and I2 is the area under the solid curve in the domain ψ > No. The dashed line is defined by ψ = No. (ii) Either the upper or lower branches of the “C” portion can be neglected which yields a single–valued curve in the domain ψ ∈ [N1, No]. The integral of this single–valued curve will be I1 and I2 is the same as before. It is more likely that the higher burning state occurs in turbulent flames of large Damkohler number and hence neglecting the lower branch is probably more physical. Preliminary calculations show that the two approximations above yield mean 45 4.3 Shapes of the PDFs Figure 4.4: The surface of ω˙+(c,N+c ) from RtP flame calculations of lean C3H8–air mixture, φ = 0.8. reaction rate values very close to each other for Le > 1 mixture. This will be ev- ident from calculations of statistically planar one–dimensional flames in Chapter 7. It must be noted that the characteristics of a ω˙(c,Nc) surface are governed by the mixture Lewis number and not its equivalence ratio. Rich methane–air mix- tures with Le > 1 have the same characteristics as the lean propane–air mixture and this is evident from Figs. 4.5b and 4.5c. The proposed modelling formulation is equally applicable for both rich and lean mixtures. We now turn the attention to the other unknown quantities in Eq. (4.11); the pdfs p(ζ) and p(ψ|ζ). 4.3 Shapes of the PDFs As noted in Section 2.2.3, a BML type shape for p(ζ) has the drawback that it neglects the contributions from the reacting part of the flamelet, 0 < ζ < 1 yielding a zero mean reaction rate value. One approach to overcome this is to relate the burning mode, f(ζ) in Eq. (2.21), to the inverse of the progress variable 46 4.3 Shapes of the PDFs 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 (a) ω˙+ ............................................................................................................ ............................................................................................................. ................................................................ .... .... .... .... .... .... .... .... .... .... .... .... ........................................................................................................... ◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦ ◦◦◦◦ ◦ ◦ N1 ..... ..... ...N2 ζ = 0.8 .... .... .. 0.7 0.9 0.0 0.5 1.0 1.5 2.0 0 2 4 6 (b) ................................................................................................................................................ ............................................................................................................................................................... ............................................................................................................................................................................................. ◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦ ◦ ◦ ◦◦◦ ◦ ◦◦◦◦◦ ◦ ◦ ..... N1 ..... ..N2 ζ = 0.7 0.6 0.5 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 (c) ω˙+ N+c ....................................................................................................................................................................... ..................................................................................................................... ................................. ............................................................................................................................ ◦ ◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦ ◦ ◦ ζ = 0.8 0.7 0.9 0.0 0.5 1.0 1.5 2.0 0 2 4 6 (d) N+c ................................................................................................................................................................................................ .... .. .... .. .... .. .... .. .... .. .... .. .... . .... .. .... .. .... .. .... .. .... .. .... .. .... .. .... .. .... . .... .. . ..... N1 ..... ... N2 .......... No ◦ ◦ ◦ Figure 4.5: Representative curves of ω˙+ vs N+c conditioned on the progress vari- able ζ for three cases: (a) methane–air, φ = 0.6, Le = 0.96, (b) propane–air, φ = 0.8, Le = 1.83 and (c) methane–air, φ = 1.4, Le = 1.17 . The ζ = 0.7 curve for the propane–air case is shown in (d) and its integration is discussed in the text. 47 4.3 Shapes of the PDFs gradient in an unstrained laminar flame (Bray, 1980; Bray et al., 2006) f(ζ) = [ dζ d(x/δ) ]−1 . (4.14) Another alternative is to reasonably presume p(ζ) to be a beta–pdf (Poinsot & Veynante, 2001) as has been done in many earlier works (Bradley et al., 1994; Schneider et al., 2005). The BML–pdf is valid in the limit of large Da and in this limit the normalised variance, g ≡ c˜′′2/c˜(1 − c˜), becomes nearly equal to 1. However, experimental studies of Chen & Bilger (2001a, 2002) show that for flames with moderate Da, even though the pdf resembles the BML model with peaks at 0 and 1, the contribution of the burning mode is non–negligible. Furthermore, the peak values of g in the flame brush were as low as 0.4 and the beta–pdf reasonably represents the experimental pdfs (Chen & Bilger, 2002). Accordingly, the beta shape is presumed for the density weighted pdf p˜(ζ) = Γ(a+ b) Γ(a) Γ(b) ζa−1 (1− ζ)b−1 , (4.15) where the parameters are a = c˜ ( c˜(1− c˜) c˜′′2 − 1 ) , and, b = a ( 1 c˜ − 1 ) . (4.16) The gamma function is defined as Γ(x) = ∫ +∞ 0 e−t tx−1 dt. (4.17) Note that p(ζ) is related to the density weighted pdf via (Swaminathan et al., 1997) p(ζ) = ρ p˜(ζ) ρ = 1 + τζ 1 + τ c˜ p˜(ζ) . (4.18) Since the progress variable gradient is produced by chemical reactions and the flamelets are spatially intermittent inside the flame–brush, one can take the conditional dissipation rate pdf, p(ψ|ζ), to be log-normal. Experimental (Chen & Bilger, 2002) and DNS (Swaminathan & Bilger, 2001b) studies show that the scalar dissipation is log–normally distributed. However, this is an open issue and 48 4.4 Conditional Mean Scalar Dissipation Rate further experimental and DNS studies examining this pdf are required. Presum- ing the log–normal shape one can write p(ψ|ζ) = 1 (ψ|ζ) σNc √ 2pi exp { − 1 2σ2Nc [ln(ψ|ζ)− µNc ]2 } , (4.19) where the mean and variance of natural logarithm of the conditional dissipation rate, ln(ψ|ζ), are respectively denoted by µNc and σ2Nc . These two quantities are related to the conditional mean, 〈Nc|ζ〉, and conditional variance, G2Nc , of the dissipation rate respectively through the relations (Peters, 1984) 〈Nc|ζ〉 = exp(µNc + 0.5σ2Nc) and G2Nc = 〈Nc|ζ〉2(expσ2Nc − 1) . (4.20) A nominal value of σNc is taken to be 0.3 and this value is found (Rogerson et al., 2007) by analysing the DNS data of Nada et al. (2004). The sensitivity of the model predictions to the value of σNc is a subject for future studies. Now, it is clear that only the conditional mean dissipation rate 〈Nc|ζ〉 is required to get the pdf using Eq. (4.19). 4.4 Conditional Mean Scalar Dissipation Rate Klimenko & Bilger (1999) consider the closure of the conditional mean scalar dissipation rate in the context of CMC and prescribe an approach based on the pdf transport equation. The equation for p(ζ) can be written as (Veynante & Vervisch, 2002) ∂ ∂t [ρ p(ζ)] + ∂ ∂xi [ρ 〈ui|ζ〉 p(ζ)] + ∂ ∂ζ [ω˙ p(ζ)] = − ∂ 2 ∂ζ2 [ρ 〈Nc|ζ〉 p(ζ)] . (4.21) The terms in the LHS can be integrated twice to yield closed form expression for [〈Nc|ζ〉 p(ζ)], subject to the appropriate pdf boundary conditions. The resulting closure for 〈Nc|ζ〉 will be consistent with the pdf equation. This approach was originally developed for the mixture fraction pdf, p(ξ), whose transport equation has no reaction term. Although more rigorous, this is difficult to adopt for the case of a progress variable due to the presence of the reaction term. However, it is worth exploring in future studies. 49 4.4 Conditional Mean Scalar Dissipation Rate 0.0 0.4 0.8 1.2 1.6 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0 (a) (b) (c) Nc(ζ) Nc(ζ∗) ζ ζ ζ ................. ........ ...... .... .... ..... .... ..... .... ..... ..... ..... ..... ..... ..... .... .... .... ..... ..... ..... ..... .... .... .... .... ..... ..... .... .... ..... ..... .... ............................................................................................................................................................... .... .... .... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . .... . .... ..... .... . .. .. .... .... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . .... . . . . ..... ..... ..... ...... .... .... .... .... .... .... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . .... . ... . . . ..... ...... ...... ...... . .... .... . ... .... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... .... .... . ..... .. . . . ...... ...... ...... .. .. .... .... .... ..... . ..... . ..... . ..... . ..... .. . ... .... ...... .. ... .. .. .. .. ... ... ... .. ............ ..... .... ..... ..... ..... ..... ..... ..... .... ..... ..... .... .... ..... ..... ..... .... .... .... .... ..... ..... ..... .... ..... ..... .... .... ..... .... .... ..... ..... ..... .................................................................................................................................................................................. . . ... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . .... . .... ..... ...... ...... ...... ...... .. .... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . .... ..... .... . ...... ....... ... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . .. . . . . ..... . .... ... .. .. ...... ...... ...... ...... ...... ...... ...... ...... ..... ...... ...... ................ ....... ..... ..... ..... .... .... ... ... ... ..... ..... ..... .... .... .... .... .... .... ..... .... .... ...... ........................................................................................................................ .... .... .... .... .... .... ..... .. ..... .... . .... . .... .... ...... .. ...... ...... ...... ...... ...... ...... ...... ...... ......... .... .... .... .... .... .... . ..... . ..... . ..... .... .... .... ...... .... ..... ...... ...... .. .. .... .... .... .... .... ..... . .... .... ... ..... .... ..... ...... . . Figure 4.6: Profiles of Nc(ζ)/Nc(ζ∗) for three mixtures: (a) CH4–air of φ = 0.6; (b) CH4–air of φ = 1.0 and (c) C3H8–air of φ = 0.8. The unstrained flame is the solid and strained flames are dashed curves. Alternatively, a closure based on physical arguments can be developed. In the context of strained flamelet approach, since the turbulent flame is treated as an ensemble of strained laminar flamelets, the conditional dissipation rate can be taken to be 〈Nc|ζ〉 ≈ ∫ Nc(ζ, a) p(a) da , (4.22) where Nc(ζ, a) is the scalar dissipation rate in a flamelet subject to a strain rate a. The asymptotic studies of RtP flames (Libby &Williams, 1982; Libby et al., 1983) showed that the strain rate primarily influences the progress variable gradient at the inner reaction zone, denoted by ζ∗ in the discussion below, which in turn dictates the gradient at other locations. Thus, one may expect Nc(ζ, a) ≈ Nc(ζ∗, a)f(ζ) . (4.23) This approximation is evaluated in Fig 4.6 using the results of laminar flames calculations for different values of a and for three mixtures; lean (φ = 0.6) and stoichiometric (φ = 1.0) methane–air and lean (φ = 0.8) propane–air mixtures. Figure 4.6a and 4.6b are for the methane–air flames and Fig. 4.6c is for the propane–air flame. These curves are obtained from calculations using the PRE- MIX (Kee et al., 1985) and OPPDIF codes and detailed chemical mechanisms. The mechanism of Sung et al. (1998) for propane–air mixtures is used. The solid curve in Fig. 4.6 corresponds to the unstrained laminar flame while the dashed 50 4.4 Conditional Mean Scalar Dissipation Rate curves correspond to increasing strain rates upto a point where the reaction zone just reaches the stagnation plane in an RtP opposed flow configuration. This strain rate value is 500/s for lean CH4–air, 1800/s for stoichiometric CH4–air and 1200/s for lean C3H8–air respectively. The inner reaction zone corresponds to the location of maximum fuel consumption rate and it remains reasonably constant at ζ∗ ≈ 0.8, 0.7 and 0.62 respectively, over the range of strain rates considered in these flames. Although the collapse of the curves in the thermal regions is not ideal, it is reasonable for the most part of flames where chemical reactions are expected to be intense, providing good support for the approximation in Eq. (4.23). Substituting the approximation (4.23) into Eq. (4.22) results in 〈Nc|ζ〉 ≈ 〈Nc|ζ∗〉f(ζ), which, when combined with Eq. (3.8), gives ²˜c ≈ ∫ 1 0 〈Nc|ζ∗〉 f(ζ) p˜(ζ) dζ (4.24) leading to 〈Nc|ζ〉 ≈ ²˜c f(ζ)∫ 1 0 f(ζ) p˜(ζ) dζ . (4.25) Any of the curves in Fig. (4.6) can be taken to approximate f(ζ) and hence we choose the curve corresponding to the unstrained flame. The Eqs. (4.11), (4.12), (4.18), (4.19) and (4.25) constitute a closure for the mean reaction rate ω˙ provided a model for the mean scalar dissipation rate, ²˜c, is available. The mean reaction rate can be calculated a priori and tabulated as a function of c˜, c˜′′2 and ²˜c. The progress variable mean, c˜, and variance, c˜′′2 , can be obtained from solving their transport equations (2.15) and (3.1) respectively. The mean scalar dissipation rate, ²˜c, can also be obtained from a modelled transport equation or an algebraic model. However, as noted in Chapter 3, existing models are based on passive scalar ideas whereas c is a reactive scalar. The next chapter is devoted to the challenging but important task of providing a closure for ²˜c. A consistent closure can also be written for the mean species mass fraction as Y α = ∫ 1 0 〈Yα|ζ〉 p(ζ) dζ, where, 〈Yα|ζ〉 = ∫ N2 N1 YαRP (ζ, ψ) p(ψ|ζ) dψ. (4.26) 51 4.5 Summary The upper limit, N2, is determined using the extinction limit given by (Libby & Williams, 1982) ∫ 1 0 ω˙RP (ζ) dζ∫ 1 0 ω˙o(ζ) dζ ≤ 0.1 . (4.27) This criterion is a practical limit and not a theoretical one. The upper limit, N2, can be infinity in principle but for ψ > N2 there is negligible chemical reaction. Also, the limits N1 and N2 do not cover the negative stretch rate regime and this is a limitation of the RtP configuration. 4.5 Summary A strained flamelet model for turbulent premixed combustion is proposed. The flamelets are parametrised using the scalar dissipation rate, Nc, since the stretch rate parametrisation is not valid for high values of stretch. The scalar dissipation rate is defined at every location in the flame front and asymptotic studies (Libby & Williams, 1982) have shown that Nc is closely related to the flamelet structure both at low and high stretch rates which motivated its choice as a parameter. Strained laminar flames in Reactant–to–Product (RTP) opposed flow configura- tion are chosen to represent the flamelets since recent DNS studies (Hawkes & Chen, 2006) have shown that they reproduce the turbulent flame characteristics well. A closure for the mean reaction rate, ω˙, of the progress variable, c, is ob- tained using Eq. (4.8) and presuming shapes for the pdfs of the progress variable and the conditional scalar dissipation rate. A β–pdf is used for the marginal pdf p˜(ζ) and a log–normal pdf is used for p(Nc|ζ). The required conditional mean scalar dissipation rate, 〈Nc|ζ〉, is to be obtained from the mean scalar dissipation rate, ²˜c, which needs to be modelled. The modelling of ²˜c is studied in the next chapter. 52 Chapter 5 Modelling of Mean Scalar Dissipation Rate The modelling of unconditional mean dissipation rate of the progress variable is addressed in this chapter. This quantity is a central modelling parameter in the flamelet formulation described in the previous chapter, as well as other modelling approaches. An exact transport equation for this quantity, which was derived in a recent work (Swaminathan & Bray, 2005), will be presented first. Modelling of various leading order terms of this transport equation will be discussed leading to an algebraic closure for the mean dissipation rate. The algebraic model will be validated by making comparisons with DNS data. It is worthwhile to make a few observations on the common practices of mean scalar dissipation rate modelling. By far the most popular closure is the classical passive scalar model given in Eq. (3.11), with the constant Cφ = 1.0. The quantity Cφ ≡ (²˜c/c˜′′2)/(ε˜/k˜) is the ratio of turbulence–to–scalar time scales and a constant value is based on the notion that the scalar fluctuations decay rate is proportional to the velocity fluctuations decay rate. Hence this model is sometimes referred to as “linear relaxation model”. Based on an initial survey of experimental data, Pope (1985) questions the universality of value of Cφ, especially for scalar fields in decaying turbulence. However, the DNS study of Eswaran & Pope (1988) later confirmed that, even for a wide variation in the initial value of scalar integral length scale, Cφ tends to a value of about 1.0 after some early transients. Although this model is applied in a variety of scenarios, it is strictly valid for a passive scalar transport in incompressible turbulence. However, it has often been used to 53 5.1 Transport Equation model the mean and the cross dissipation rates involving reactive scalars without sufficient justification. Chomiak et al. (1991) prescribe a modification to Cφ with a dependence on the Prandtl number, Pr, and the Reynolds number, Re, but suggest that the reactiveness of a scalar has no influence. It will be shown later that when the physics of reactive scalars are appropriately accounted, a chemical time scale appears in the scalar dissipation rate model naturally. 5.1 Transport Equation An exact transport equation for ²˜c was derived by Swaminathan & Bray (2005) : ∂ρ²˜c ∂t + ∂ρu˜j ²˜c ∂xj = ∂ ∂xj ( ρDc ∂²˜c ∂xj ) ︸ ︷︷ ︸ D1 − 2ρ ( Dc ∂2c′′ ∂xj ∂xk )2 ︸ ︷︷ ︸ D2 +T1 + T2 + T3 + T4 (5.1) T1 ≡ − ∂ρu′′j ²c ∂xj − 2ρDc ( ˜ u′′j ∂c′′ ∂xk ) ∂2 c˜ ∂xj ∂xk T2 ≡ 2ρ²c∂ui ∂xi T3 ≡ −2ρDc ∂c˜ ∂xj  ∂˜c′′ ∂xk ∂u′′j ∂xk  ︸ ︷︷ ︸ T31 −2ρDc  ˜∂c′′ ∂xj ∂u′′j ∂xk ∂c′′ ∂xk  ︸ ︷︷ ︸ T32 −2ρDc ( ∂˜c′′ ∂xj ∂c′′ ∂xk ) ∂u˜j ∂xk︸ ︷︷ ︸ T33 T4 ≡ 2 ( Dc ∂c′′ ∂xk ∂ω˙′′ ∂xk ) , when the diffusivity Dc has a weak dependence on temperature. A similar equa- tion was also derived earlier but with a constant density approximation by Borghi and co-workers (Borghi, 1990; Mantel & Borghi, 1994; Mura & Borghi, 2003). The various terms in Eq. (5.1) have the following meaning. The left hand side represents temporal and convective changes of ²˜c. The molecular diffusion and dissipation are denoted by D1 and D2 respectively. The effects of dilatation rate is represented by T2 which results from the density change across the flame front and hence this term is absent in the equation derived by Mantel & Borghi (1994). This term is generally positive representing a source contribution to ²˜c evolution. 54 5.2 Modelling of Dominant Terms The turbulence–scalar interaction is represented by T3 and the predominant con- tribution to this term comes from T32 (Chakraborty & Swaminathan, 2007a; Swaminathan & Grout, 2006) which is traditionally considered to be positive. The influence of chemical reaction is denoted by T4. For large Reynolds and Damköhler numbers, the order of magnitude analysis (Swaminathan & Bray, 2005) showed the following balance at leading order: T2 + T32 + T4 −D2 ' 0. (5.2) All of these terms are unclosed and need to be modelled. Swaminathan & Bray (2005) discussed the past modelling studies underpinned by passive scalar tur- bulence physics. It has been recognised recently (Chakraborty & Swaminathan, 2007a; Swaminathan & Grout, 2006) that the heat release strongly influences turbulence scalar interaction and models based on passive scalar ideas need to be revised. Accounting for these findings led to precise modelling of these terms by Chakraborty et al. (2008). Our interest is on a simple algebraic model for the mean scalar dissipation rate which can be obtained from the leading order balance in Eq. (5.2). Modelling of the terms in this equation follows the study of Chakraborty et al. (2008) in a broad sense and the details of these models are discussed in the next section. 5.2 Modelling of Dominant Terms The modelling of each of the terms in Eq. (5.2) is considered in this section. 5.2.1 Dissipation–Dilatation correlation, T2 The term T2 denotes the correlation between dissipation and dilatation rates and a model for this term can be obtained by adopting the BML–type pdf from Eq. (2.21). Since the dilatation rate is zero everywhere except inside the flame front, T2 must be proportional to γ∗ as T2 = 2γ ∗ ∫ ρNc∇ · u f(ζ) dζ . (5.3) 55 5.2 Modelling of Dominant Terms In the limit of large Da the internal structure of the flamelet is less likely to be disturbed by the turbulence and hence, making use of Eq. (2.30), Swaminathan & Bray (2005) propose: T2 = 4Kc (2Cm − 1) ( soL δoL ) ρ²˜c , (5.4) where Kc ≡ ( δoL soL ) ∫ {ρNc(∇ · u)}oL f(ζ) dζ∫ ω˙o f(ζ) dζ . (5.5) The notation {}oL denotes quantities corresponding to an unstrained laminar flame. It is clear that the values of Kc cannot be chosen arbitrarily once the internal flamelet structure is specified. The values of Kc for methane–air and propane–air mixtures at various equivalence ratios are given in Table 5.1. These are calculated from planar laminar flame results. The values for stoichiometric methane–air flames at elevated pressures are also given in Table 5.1. Table 5.1: T2 Model constants Kc and K∗c . CH4–air, p = 1 atm CH4–air, φ = 1 C3H8–air, p = 1 atm φ τ Kc/τ K ∗ c /τ p (atm) τ Kc/τ K∗c /τ φ τ Kc/τ K∗c /τ 0.6 4.58 0.24 0.82 1 6.48 0.18 0.87 0.6 4.74 0.27 0.85 0.7 5.17 0.21 0.83 1.5 6.51 0.18 0.86 0.7 5.36 0.23 0.86 0.8 5.70 0.19 0.84 2 6.52 0.19 0.85 0.8 5.91 0.20 0.88 0.9 6.17 0.18 0.85 3 6.54 0.20 0.85 0.9 6.36 0.19 0.90 1.0 6.48 0.18 0.87 5 6.56 0.21 0.85 1.0 6.62 0.18 0.92 1.1 6.42 0.19 0.88 10 6.59 0.23 0.84 1.1 6.59 0.20 0.94 1.2 6.17 0.23 0.89 20 6.62 0.24 0.84 1.2 6.36 0.24 0.96 1.4 5.64 0.29 0.91 1.4 5.83 0.31 0.99 It was shown in earlier studies (Chakraborty et al., 2008; Rogerson & Swami- nathan, 2007) that for large Da flames the weight of the burning mode can be expressed as γ∗ = ρ²c/ ∫ ρN oLf(ζ) dζ. Using this γ∗ one obtains T2 = 2 ( soL δoL ) K∗c ρ ²˜c, (5.6) where K∗c = ( δoL soL ) ∫ {ρN(∇ · u)}oL f(ζ) dζ∫ {ρN}oL f(ζ) dζ . (5.7) 56 5.2 Modelling of Dominant Terms The model in Eq. (5.6) is similar to that in Eq. (5.4) but with a distinctly different model constant. The values of K∗c for various cases are also listed in Table 5.1 and it is evident that the variation in K∗c /τ with equivalence ratio, φ, is less than 10%, while the variation in Kc/τ is nearly 60% in methane–air flames. A similar trend is also noted for propane–air flames. Even for elevated pressures, Kc/τ varies substantially whereas K∗c /τ remains almost constant. The sensitivity of Kc/τ to the equivalence ratio is because of N oL sensitivity to the flamelet structure. This high sensitivity is not desirable from the modelling point of view and thus the model in Eq. (5.6) is preferred. The comparison of this model with DNS data was observed to be very good in earlier studies (Kolla et al., 2009; Rogerson & Swaminathan, 2007). 5.2.2 Turbulence–Scalar interaction, T32 The tensor scalar product of turbulent strain rate and scalar gradient represents the turbulence–scalar interaction denoted by T32 in Eq. (5.1). This term can be written as T32 = −2ρ²c(α cos θα + β cos θβ + γ cos θγ) , (5.8) using the eigenvalue decomposition, where α, β and γ are the eigen values of the turbulent strain tensor ∂u′′i /∂xj. The eigen values are ranked as α > β > γ, α being the most extensive principal strain rate and γ being the most compressive principal strain rate. The relative orientation of the scalar gradient vector with respect to the eigen vector corresponding to α strain rate is denoted by θα. Thus it is clear that the source or sink nature of T32 is dictated by the predominant alignment of the scalar gradient with the eigen vectors. It is well known (Ashurst et al., 1987; Batchelor, 1952) that the scalar gradient preferentially aligns with the most compressive principal strain rate in turbulent flows yielding a source contri- bution from T32. Recent studies (Chakraborty & Swaminathan, 2007a; Hartung et al., 2008; Swaminathan & Grout, 2006) clearly showed that the scalar gradient aligns preferentially with the most extensive principal strain rate in regions of intense heat release. This is because of the competition between dilatation rate and turbulent strain rate (Swaminathan & Grout, 2006). 57 5.2 Modelling of Dominant Terms The dilatation rate can be scaled as ∼ τsoL/δoL while the turbulent strain rate can be scaled as ∼ ε˜/k˜. When the dilatation rate overcomes the turbulence strain rate, T32 becomes negative and it dissipates the scalar gradient. To capture this behaviour, a model for T32 is proposed as below following recent studies (Chakraborty & Swaminathan, 2007b; Chakraborty et al., 2008). The turbulent strain rate effect is modelled as C3ρ²˜c(ε˜/k˜) while the strain rate due to heat release is modelled as −τC4ρ²˜c(soL/δoL), where C3 and C4 are model constants. Combining these two contributions, one gets: T32 = [C3 − τC4DaL] ( ε˜ k˜ ) ρ ²˜c, (5.9) where DaL ≡ (s o L/δ o L) (ε˜/k˜) (5.10) is the local Damköhler number. In a similar expression proposed earlier (Chakraborty & Swaminathan, 2007b; Chakraborty et al., 2008), a density weighted Damköhler number Da∗L = (ρuDaL/ρ) was used instead of DaL. The model constants C3 = 1.5 and C4 = 1.1 (1 + Ka) −0.4 (5.11) are suggested by Chakraborty & Swaminathan (2007b). These forms were pro- posed so that Eq. (5.9) reduces to the classical expression signifying a source, when DaL → 0 and/or Ka → ∞. Also, the classical form is recovered when τ = 0. The classical model for ²˜c from Eq. (3.11), which is strictly for a passive scalar, can be written as ²˜c = Cφ(ε˜/k˜)c˜′′2 , where, as noted earlier, Cφ is the ratio of integral time scales of turbulent velocity fluctuation to scalar fluctuation. This model can also be obtained by balancing T32 and D2 in Eq. (5.2) when the scalar is passive (Mantel & Borghi, 1994). Hence C3 in Eq. (5.9) is directly related to the time scale ratio, Cφ. For a passive scalar the time scale ratio is typically about 1 but in premixed flames its variation is large (Mantel & Bilger, 1995; Swaminathan & Bray, 2005). These analyses showed that Cφ ≈ 0.05 within the flame–brush. The classical model with Cφ ≈ 1 will over predict the scalar dissipation rate and it is also evident that this model is strictly not valid for turbulent premixed flames. 58 5.3 Algebraic Model for ²˜c In order to satisfy this change in the time scale ratio the model constant C3 is modified as C3 = 1.5 (1 + Ka−m) (5.12) so that C3 ≈ 1.5 when Ka is large and C3 would decrease as Ka decreases. For the present work m is taken to be 0.5. 5.2.3 Flame front curvature, (T4 −D2) It was shown by Mantel & Borghi (1994), Mura & Borghi (2003) and Swami- nathan & Bray (2005) that (T4 − D2 + D1) is related to the flame front cur- vature contribution and accordingly an algebraic model was proposed for this using flamelet theories. However, D1 is a closed term and the order of magnitude analysis by Swaminathan & Bray (2005) showed that it is not of leading order when the Damköhler and Reynolds numbers are large. Thus, after excluding D1 a revised model was proposed by Chakraborty et al. (2008) based on DNS data analysis. This model is (T4 −D2) = −β′ ρ ²˜c 2 c˜(1− c˜) , (5.13) where β′ = 6.7, is a model constant. Since c˜′′2 ≈ c˜(1 − c˜) for high Da flames, Eq. (5.13) can be rewritten as (T4 −D2) = −β′ ρ ²˜c 2 c˜′′2 . (5.14) The recent analysis by Chakraborty et al. (2008) showed that dissipative na- ture of D2 overwhelms contribution from T4 thereby making (T4 −D2) negative throughout the flame brush, which justifies the negative sign in Eq. (5.13). 5.3 Algebraic Model for ²˜c Apart from the classical model, algebraic models for ²˜c have been proposed in the past. The recent models are briefly discussed first before deriving and analysing a 59 5.3 Algebraic Model for ²˜c new model. Combining the relation ω˙ ≈ ρusoLΣ (Bray et al., 1989) with Eq. (2.30), Vervisch et al. (2004) obtained ρ²˜c = ( 2Cm − 1 2 ) ρus o LΞ|∇c| ( c′2 c(1− c) ) , (5.15) where Ξ ≡ Σ/|∇c| is the flame wrinkling factor commonly used in the flame surface density modelling. However, to evaluate ²˜c from Eq. (5.15) one requires to model Ξ which is beyond the scope of this study. Furthermore, such modelling attempts can be viewed as relating two unknowns, ²˜c and Σ, via a third unknown ω˙. The last factor containing c′2 is introduced intuitively to ensure that ²˜c goes to zero when c′2 goes to zero. An interesting comparison of this model to DNS data is shown by Vervisch et al. (2004). Fiorina et al. (2005) employ this model for RANS calculations with a nominal value for the unknown quantity Ξ. Borghi and his co-workers (Borghi, 1990; Mantel & Borghi, 1994; Mura & Borghi, 2003) proposed a model: ²˜c ' ( 1 + 2C²cs o L 3 √ k˜ )( CD ε˜ k˜ ) c˜′′2, C²c = 0.1, CD = 0.21, (5.16) when (√ k˜/soL ) > 0.067, based on the analysis of ²˜c transport equation but with constant density assumption. This limitation is remedied by Swaminathan & Bray (2005) in the following model by allowing the density to vary across the flame front: ²˜c ' ( 1 + 2 3 C²c soL√ k˜ )( CDc soL δoL + CD ε˜ k˜ ) c˜′′2, CDc = 0.24. (5.17) The relaxation of the constant density assumption allowed a chemical time scale to appear naturally and explicitly in the model. If the laminar flame scales are used to normalise Eq. (5.17), it follows that ²˜c scales as (soL/δoL) for large Da flames (Bray & Swaminathan, 2006). Such explicit scaling is not apparent in the other models for the scalar dissipation rate. The model constant CDc is proportional to Kc in Eq. (5.4) and hence the scalar dissipation rate model given by Eq. (5.17) is sensitive to the flamelet struc- ture. Furthermore, the classical model for T32 based on the alignment of scalar 60 5.3 Algebraic Model for ²˜c gradient with the most compressive eigen vector of turbulent strain rate, is used to deduce Eq. (5.17). Thus this model does not incorporate the change in the alignment characteristics noted earlier in subsection 5.2.2. To account for the correct physics, a new model is derived by using Eqs. (5.6), (5.9) and (5.13) in Eq. (5.2): ²˜c ' 1 β′ ( 2K∗c soL δoL + [C3 − τC4DaL] ε˜ k˜ ) c˜′′2 . (5.18) It should be noted here that this model is strictly valid for large Damköhler number and it is straightforward to see that ²˜c scales as (soL/δoL) in this limit. It is worth mentioning that Eq. (5.18) includes the chemical time scale, which is important (Swaminathan & Bray, 2005), and it incorporates the correct physics of turbulence-scalar interaction which is absent in Eq. (5.17). Furthermore, the heat release parameter τ appears explicitly in Eq. (5.18) unlike in the earlier models. These models predictions will be compared to DNS data in Section 5.4 while our immediate attention is focussed on realisability of the model given by Eq. (5.18). 5.3.1 Realisability Duclos et al. (1993) noted that ∫ ω˙ dv given by various flame surface density models has to be finite for realisability. This condition from Eq. (2.30) implies that ²˜c must be bounded within the flame brush. Furthermore, Eq. (2.30) suggests that ²˜c should be zero at the flame brush boundary. Also, ²˜c → 0 when c′2 → 0 as noted by Vervisch et al. (2004) which is equivalent to the condition given by Duclos et al. (1993). This was the reason for adding the last factor proportional to c′2 in Eq. (5.15). It is straightforward to see that Eq. (5.18) satisfies all of these criteria. The remaining concern is on simple physical realisability, i.e., the semi-positive definiteness (²˜c ≥ 0) of the scalar dissipation rate. Equation (5.17) automatically satisfies this criterion but it is not obvious for Eq. (5.18). This equation is rearranged as ²˜c ' 1 β′ ( [2K∗c − τC4] soL δoL + C3 ε˜ k˜ ) c˜′′2 . (5.19) 61 5.4 Validation It is clear that a sufficient condition for the physical realisability is 2K∗c τ − C4 ≥ 0 . (5.20) The constant C4 is 1.1(1 + Ka)−0.4 and thus C4 ≤ 1.1 for any value of Ka. On the other hand K∗c /τ > 0.8 for methane-air and propane–air mixtures as in Ta- ble 5.11. For hydrogen-air mixtures, the laminar flame calculations yield K∗c /τ ' 0.65 (Rogerson & Swaminathan, 2007). Hence, although K∗c is weakly depen- dent on fuel, the sufficient condition (2K∗c /τ − C4) ≥ 0 will always be satisfied thereby making the current model unconditionally realisable. It is worth noting that while the models for T2 and T32 along with their respective constants were proposed independently to capture the relevant physics, the algebraic model for ²˜c is automatically unconditionally realisable. Furthermore, if a density weighted Damköhler number is used instead of DaL in Eqs. (5.9) or (5.18), then the suffi- cient condition becomes [2K∗c /τ − (ρu/ρ)C4] ≥ 0 which is not guaranteed always since ρu/ρ = (1 + τ c˜). The unconditional realisability of Eq. (5.18) noted above is an indication that correct physics is captured by the model. However, DNS comparisons presented in the next section provide more rigorous validation for the model. 5.4 Validation The algebraic scalar dissipation rate model derived in the previous section is validated using DNS data in this section. Before discussing these results, the attributes of the DNS data are briefly discussed first. 5.4.1 Attributes of DNS data Five different DNS datasets are considered in this study. The main attributes of these data sets are given in Table 5.2 and are also shown in the regime diagram in Fig. 5.1. All these cases directly simulated the propagation of a premixed flame in three dimensional homogeneous turbulence. Turbulence inflow and outflow 1It is believed that even for higher hydrocarbons including paraffins and aliphatic compounds,K∗c /τ ' 0.8 to 0.9. This needs to be verified with either flame calculations or experiments. 62 5.4 Validation boundary conditions were applied in the direction of mean flame propagation and periodic boundary conditions were used in the other two directions. It is evident from Table 5.2 and Fig. 5.1 that the flames cover a range of combustion conditions, viz. the wrinkled flamelets, the corrugated flamelets and the thin reaction zones regime, all with Da>1. Table 5.2: Initial parameters of DNS data used. Flame u′/soL Λ/δ Re Da Ka R1 1.41 28.3 56.7 20.1 0.37 R2a 0.85 78.0 106.8 91.8 0.2 R2b 1.70 39.0 106.8 22.9 0.8 R2c 3.40 19.5 106.2 5.7 3.2 R2d 3.38 41.5 201.7 12.3 1.5 0.1 1 10 100 1000 0.1 1 10 100 1000 u′/soL Λ/δ ........................................................................................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . ................................................................................................................................................................................................................................. ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ....... ......... ......... ......... ......... ......... ......... ......... Re = 1 Ka = 100 Da = 1 Ka = 1 Wrinkled Corrugated Thin reaction zones Distributed reaction zones ¨ ¨ ¨ ¨ ¨ R1 R2a R2b R2c R2d Figure 5.1: Attributes of the DNS data on the combustion regime diagram. The case R1 was simulated by Rutland & Cant (1994) using a single ir- reversible reaction with large activation energy. In R2 cases a stoichiometric 63 5.4 Validation hydrogen-air flame was simulated (Nada et al., 2004) with detailed kinetic mech- anism involving 27 reactions and 12 reactive species. Reactant mixtures preheated to 700 K were used. The DNS data at about 4.4 initial eddy turnover time from the case R1, and at about 2.5 initial eddy turnover time from the set of R2 cases are considered for the present analysis. Full details of these DNS can be found elsewhere (Nada et al., 2004; Rutland & Cant, 1994). 5.4.2 Comparisons with DNS data It is to be noted that the models discussed in Sections 5.2 and 5.3 were developed using DNS data which are completely different from the DNS data used here. The predictions of various scalar dissipation rate models given in Section 5.3 are compared to the DNS results in Fig. 5.2. The superscript ‘+’ denotes that ²˜c is normalised using soL and δoL appropriately. The progress variable c1 is the local temperature increment (T − Tu) normalised by the maximum temperature increment observed in the respective DNS. Since model of Ξ is kept beyond the scope of the present study, model prediction of Eq. (5.15) is not shown. One– fourth of the values predicted by the classical model, Eq. (3.11), are also shown in Fig. 5.2. The values of model constants used are those given in Section 5.3. Since the R2 DNS considered hydrogen-air mixture, K∗c /τ = 0.65 is used but for R1 case K∗c /τ = 0.85. It is evident from Fig. 5.2 that Eq. (5.18) satisfactorily predicts the scalar dissipation rate demonstrating the model capability across the combustion regimes. While there is a slight over prediction for the R1 and an under prediction for the R2b and R2d cases, the overall agreement might be considered reasonable. The predictions of Eq. (5.17) also appear reasonable although an over prediction is observed for the R2a case. However, this model does not incorporate the correct scalar gradient alignment physics as noted in Section 5.2. Figure 5.2 also shows that Eq. (5.16) under predicts the scalar dissipation rate while the classical model over predicts the values by nearly a factor of four. It was observed that the Ka dependence of C3 in Eq. (5.9) is required for correct prediction of ²˜c by the model in Eq. (5.18). This modification to C3 allows for a slow change of T32 effects across the flame–brush. 64 5.4 Validation 0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.2 0.4 0.6 0.8 1.0 (R1) ²˜c + c˜1 ◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦......... ..... .... .... .... ..... ....... ..... ....... ..... ..... ..... ......... ...... ..... ..... ......... ...... ..... ...... ...... ..... ..... ............................................................................................................................................................................... ................ ......... ................ ......... ................................................................................... .... .... .... .... .... .... .... .... .... ... ... ... .... ... ... ... .... .... .... .. ..... .. .................... ... .... ..... ..... ... ....... ...... ........ ....... ............................................................ 0.00 0.01 0.02 0.03 0.0 0.2 0.4 0.6 0.8 1.0 (R2a) c˜1 DNS Eq. (3.11)/4 Eq. (5.16) Eq. (5.17) Eq. (5.18) ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ...... .... .... ..... ..... ..... ....... ..... ..... ....... ..... ..... ..... ...... ................ .......................................................................................................................... ................ ................ .............................................................................................. .... .... .... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ... .... ... ... ... ... ................................................................................................................... ....... ....... .... ............................................... 0.00 0.01 0.02 0.03 0.0 0.2 0.4 0.6 0.8 1.0 (R2b) ²˜c + c˜1 ◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦......... ..... ..... .... ........ .... ..... ..... ..... ..... ..... ....... ............ ................................................................................................................................... ................ ............... ................................................................................................ .... ... ... ... ... ... .... ... ... .... .... .... .... .... ... .... .... ......................................................................... .......... .............................................................. 0.00 0.01 0.02 0.03 0.04 0.0 0.2 0.4 0.6 0.8 1.0 (R2c) c˜1 ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦.......... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ........ ..... ....... ......... ..... ..... ..... ..... ...................................................................................................................................................................................................... ...... ..... ............ ..................... ...................................................................................... ... .... .... .... .... ... ... ... ... .... .... ....... ........................................................... ........ ...... ...... ....... .......... .......... .......................................... 0.00 0.01 0.02 0.03 0.0 0.2 0.4 0.6 0.8 1.0 (R2d) ²˜c + c˜1 ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦.......... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ....... ........ ... .... ..... ..... .... .... ...................... ........................................................................................................................................... ........... ......... ............... ............ ..................................................................................... .... .... .... ... ... ... .. .... ... . .... ..... .............. .............................................................. ....... ......... ....... ................................................................................ Figure 5.2: Comparisons of ²˜c predicted by various models with DNS data. 65 5.5 Summary 5.5 Summary An algebraic model for the Favre averaged scalar dissipation rate, ²˜c, is derived for high Damköhler number turbulent premixed flames. This model is based on the leading order balance of an exact transport equation for ²˜c, which includes the effects of dilatation, interaction of turbulence and scalar gradients, chemical reaction and dissipation processes. The model for ²˜c is examined for its physical realisability and it is shown that the proposed model is always unconditionally realisable. Predictions of this new algebraic model are compared to DNS data of turbulent premixed combustion in different regimes and the agreement is found to be satisfactory. Direct validation of the ²˜c model with experimental data could not be per- formed since extensive measurements of this quantity are currently scarce. Accu- rate measurement of ²˜c is made difficult by the stringent resolution requirements for measuring the steep scalar gradients (Bilger, 2004b). However, an indirect ex- perimental validation is possible via the turbulent flame speed; a quantity which has been widely measured in a variety of flame configurations and over a wide range of conditions. Such an indirect validation is performed in the next chapter. 66 Chapter 6 Turbulent Flame Speed The aim of this chapter is to provide an indirect experimental validation for the algebraic mean scalar dissipation rate model via turbulent flame speed. The an- alytical framework that establishes a mathematical link between these two quan- tities is presented first that yields an algebraic expression for the latter from the model for the former. The characteristics of this flame speed expression will be examined in relation to some well known theories. Attention is then paid to the various definitions of turbulent flame speed adopted in experiments so as to iden- tify the definition that corresponds to the algebraic expression being considered. Experimental data are accordingly chosen for validation and comparisons, which will be discussed. 6.1 Expression for Turbulent Flame Speed Turbulent flame speed, ST , is a fundamental quantity in turbulent premixed flames. It signifies the intensity of burning and it is influenced by the dynamics of both large and small scales of turbulence, and turbulence–chemistry interaction. It has been the subject of a large number of theoretical and experimental studies (see for example Bradley (2002); Driscoll (2008)) and it is known to be central in many premixed flame modelling methods such as the Flame Surface Density (FSD) models (Veynante & Vervisch, 2002), the Turbulent Flame speed Closure (TFC) (Zimont, 2000) and the G–equation approach (Peters, 1999). Many of the existing flame speed expressions are based on semi–empirical scaling laws (Gülder, 67 6.1 Expression for Turbulent Flame Speed 1990) while others have used the physics of passive scalar dynamics (Kerstein & Ashurst, 1992; Ronney & Yakhot, 1992; Yakhot, 1988). As noted in Chapter 3 the fine scale mixing rate strongly influences the burning rate in turbulent flames and hence the flame speed. Consequently, the reactive scalar physics inherent in the ²˜c model discussed in the previous chapter will have implications on ST as well. This provides the motivation to test the ²˜c model by making turbulent flame speed predictions. This is made possible by the Kolmogorov–Petrovskii–Piskunov (KPP) analysis (Duclos et al., 1993; Lipatnikov & Chomiak, 2002) which yields: ST = 2 √ νt (ρu Scc) ( ∂ω˙ ∂c˜ ) c˜→0 , (6.1) where νt ≡ µt/ρ is the turbulent diffusivity. From Eq. (2.30) one can write( ∂ω˙ ∂c˜ ) c˜→0 = 2 2Cm − 1 ρu ( ∂²˜c ∂c˜ ) c˜→0 (6.2) for large Damköhler number flames. Substituting for the ²˜c model from Eq. (5.18) in the above two equations, and noting that for large Da, c˜′′2 ≈ c˜(1 − c˜), yields after some algebra( ST soL ) = { 18Cµ (2Cm − 1)β′ [ [2K∗c − τC4] ( u′Λ soLδ o L ) + 2C3 3 ( u′ soL )2]}1/2 . (6.3) In obtaining the above equation the standard closure for turbulent diffusivity νt = Cµk˜ 2/ε˜ (see Eq. (2.9)) is employed and the simple scaling laws; k˜ = 3u′2/2, ε˜ = u′3/Λ are used. The value of Scc is taken to be 1. Apart from the standard modelling constants Cµ and Cm, all the constants in Eq. (6.3) are from the ²˜c model. To reiterate, each of these constants represents an important aspect of scalar mixing physics: K∗c denotes the effects of dilatation rate, C3 and C4 repre- sent the interaction between turbulence strain and the scalar gradient, β′ denotes flamelet curvature contribution. The KPP analysis will be closely examined in Section 6.3 in light of the various definitions of turbulent flame speed in experi- ments. But first we turn our focus on the characteristics of the algebraic flame speed expression 68 6.1 Expression for Turbulent Flame Speed 6.1.1 Analysis of the Flame Speed expression In Eq. (6.3) one can see the dependence of ST on u′ and Λ coming out naturally from the modelling of ²˜c. The role of turbulence length scale on the flame prop- agation is well established (Abdel-Gayed et al., 1987; Bray, 1990; Peters, 1999). Dependence on τ is also seen in Eq. (6.3) which is not shown by many existing expressions. Although Eq. (6.3) seems to involve many parameters, a close study shows that only three parameters: u′, Λ and the fuel type can be varied indepen- dently. From an experimental point of view this corresponds to two independent conditions, the turbulence (with two independent parameters u′ and Λ) and the fuel mixture (characterising the thermo–chemistry with soL, δoL, τ and K∗c ). For a scalar propagating passively, Eq. (6.3) reduces to (ST/soL) ∼ (u′/soL) sinceK∗c = C4 = 0. A similar scaling is seen in the classical expression (Lipatnikov & Chomiak, 2002): (ST/soL) = 1 + C(u′/soL)n when (u′/soL)À1. Hence Eq. (6.3) can be seen as a special case of the classical form with the constant C dependent on the length scale ratio (Λ/δoL). In his seminal work, Damköhler (1940) proposed that for low turbulence intensities (ST/soL) ' (AT/Ac) where AT is the area of the wrinkled flame surface and Ac is its projected area. This yields the linear dependence: ST soL = 1 + u′ soL . (6.4) For high turbulence intensity cases, Damköhler postulated that ST soL ∼ ( νt Dthu )0.5 ∼ ( u′Λ soLδ )0.5 . (6.5) The similarity of this result to the first term on the right hand side of Eq. (6.3) is to be noted. The second term can be seen to correspond to the linear depen- dence noted above. Kerstein & Ashurst (1992) consider the flame as a passive interface and study its dynamics in turbulent flow. Their analysis yields a scaling (ST/s o L) ∼ (u′/soL)4/3 in the limit u′ << soL, and in the limit u′ >> soL the scaling ST ∼ u′ multiplied by a factor ln(u′/soL)−3/4 was reported. To further examine the characteristics of Eq. (6.3) in the two limiting cases, i.e. small and large u′/soL, let us consider Λ/δoL to be fixed. For a given experiment this is a reasonable assumption since Λ is usually a function of geometrical parameters 69 6.1 Expression for Turbulent Flame Speed and δoL is constant for a given mixture. For a fixed Λ/δoL, Eq. (6.3) becomes ST/s o L ∼ √ (A+B)(u′/soL)2 where A = (2K ∗ c − τC4)Da represents the influence of thermo–chemistry and B = 2C3/3 represents the influence of turbulence. The group of constants 18Cµ/(2Cm− 1)β′ is of order unity and it does not depend on turbulence characteristics or thermo–chemistry. Thus it is excluded in A and B above. • When u′/soL is small the Karlovitz number, Ka, is also small in high Da flames. This gives C4 ≈ 1 and C3 ∼ √ Ka. Since 2K∗c ∼ τ , in this limit A is one order smaller than Da and one can expect O(B) ≈ O(A). Thus one recovers the linear dependence (ST/soL) ∼ (u′/soL), which is consistent with Damköhler’s result. Physically, it implies that the influence of thermo– chemistry is comparable to the effects of turbulence. • In the other limit of large u′/soL, Ka is large. Thus C4 ≈ 0 and C3 ≈ 1.5. Hence A ∼ τDa and B ∼ 1 which implies that ST soL ∼ √ O(τ) u′Λ soLδ o L + O(1) ( u′ soL )2 . While Damköhler’s result in this limit shows a dependence only on the first term inside the square root, Eq. (6.3) suggests that the second term is also comparable. Mathematically, this scaling is similar to the result of Kerstein & Ashurst (1992). The well known “bending” phenomena at high turbulence intensities, reported in many works, is consistent with Damköhler’s square–root dependence. Following the analysis of Peters (1999), the behaviour of Eq. (6.3) for a fixed Λ/δoL value is examined to study if it predicts the bending. The predictions of Eq. (6.3) for τ = 5, for three values of Λ/δoL, are shown in Fig. 6.1. The similarity of this figure to Fig. 5 of Peters (1999), and the associated discussion therein, indicates that Eq. (6.3) does predict bending. One conclusion often drawn in literature is that the bending phenomena indicates that at high enough u′/soL values, the ST/s o L curve should level off. This trend however is not apparent for Eq. (6.3) from Fig. 6.1. 70 6.1 Expression for Turbulent Flame Speed 0 20 40 60 80 100 120 140 160 0 10 20 30 40 50 ST soL u′/soL Λ/δoL = 10 Λ/δoL = 40 Λ/δoL = 100 ....... ....... ....... ........ ....... ......... ......... ......... ......... ......... ......... ......... .......... .......... .......... .......... .......... .. ..... ... ..... ... ..... ...... ..... ........ ....... ..... ........ ........ ...... ....... ........ ...... ....... ........ ......... ....... ....... ......... ......... ......... .......... .......... ........ .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ...... ....... ....... ....... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ......... ......... ......... .... Figure 6.1: Typical prediction of Eq. (6.3) for three different values of Λ/δoL with τ = 5. It is interesting to note some qualitative agreement of Eq. (6.3) with the classical analyses (Damköhler, 1940; Kerstein & Ashurst, 1992). However, it must be noted that when u′/soL = 0, these analyses recover the laminar flame speed i.e. ST = soL. This is not so for Eq. (6.3) and for other KPP expressions derived by Duclos et al. (1993) which lead Kolla et al. (2009) to surmise that the KPP expression is inapplicable in the limit u′/soL = 0. However, the reason for this apparent inapplicability is clarified by Libby (1989) who points out that such a behaviour is expected since the molecular diffusivity is assumed to be negligible in the KPP analysis. If the molecular diffusivity is included and the total diffusivity is taken to be (νt+ν), then for u′/soL = 0 an estimate for (ST/soL) from Eq. (6.3) is ( ST soL ) = {[ 2K∗c − τC4 (2Cm − 1)β′ ] 8ν soLδ o L }1/2 ∼ O(1) . However, our interest is on turbulent flames with large Re for which the assump- tion νt >> ν is reasonable and hence we exclude the molecular diffusivity from the total diffusivity. Equation (6.3) will be compared with experimental data in 71 6.2 Flame Speed definitions in Experiments Section 6.4. However, it is essential to note that the definition of turbulent flame speed is integral to this task. 6.2 Flame Speed definitions in Experiments A precise definition of turbulent flame speed is crucial while making comparisons with experimental data since the measured values can be based on quite different definitions. Here we adopt the definitions of Driscoll (2008) : • Global Consumption Speed, SGC : It is defined as the total mass consump- tion rate of reactants expressed as a velocity and it is given by SGC = ∫∫∫ ω˙ dV ρu Aref , (6.6) where dV is an elemental volume of the flame brush and Aref is the area of a reference iso–surface of c˜. • Local Consumption Speed, SLC : It is defined as the local mass consumption rate per unit area of the flame brush and it is given by SLC = ∫ ∞ −∞ ω˙ dxn ρu , (6.7) where xn is the local normal coordinate to the flame brush. • Leading Edge Displacement Speed, ST : It is defined as the speed of prop- agation of the flame brush leading edge (c˜ ≈ 0 iso–surface) relative to the unburnt mixture velocity. This quantity is referred to as turbulent flame speed in this study and is illustrated schematically in Fig. 6.2. It is to be noted that the quantity sd used in the flame surface density modelling is also commonly referred to as displacement speed and it corresponds to the advancement speed of an instantaneous iso–contour of c. This quantity is defined by combining the two terms on the r.h.s of Eq. (2.13); the diffusive flux and the reaction rate terms: ∂ ∂xj ( ρDc ∂c ∂xj ) + ω˙ = ρsd|∇c| (6.8) 72 6.2 Flame Speed definitions in Experiments Hence sd pertains to the instantaneous c iso–surface while ST pertains to the iso– surface of c˜, as shown in Fig. 6.2. In this study, the leading edge displacement speed is called as turbulent flame speed and it is denoted as ST . The three quantities: SGC , SLC and ST , are equal only for the case of a statistically stationary planar one–dimensional flame brush. The c˜ equation for this case is ρu˜ ∂c˜ ∂x = − ∂ ∂x (ρu′′c′′) + ω˙. (6.9) Integrating from the reactant (x = −∞) to the product side (x = +∞) and noting that ρu˜ = ρuST = constant, by mass conservation, yields ρu˜ = ∫∞ −∞ ω˙ dx. Thus ST = 1 ρu ∫ ∞ −∞ ω˙ dx (6.10) which by definition is equal to SLC . In this case SLC is the same as SGC since Aref does not depend on the choice of c˜ iso–surface. Figure 6.2: Schematic showing a turbulent flame brush. The thick solid curves denote the flame brush boundaries and the dashed line denotes an iso–surface of c˜ inside the flame brush. The thin solid line denotes an instantaneous c iso– contour with displacement speed sd. The orthogonal curvilinear coordinate sys- tem (η, ξ, ζ) is attached locally to the c˜ iso–surface. 73 6.2 Flame Speed definitions in Experiments In real turbulent flames the scenario is different from the statistically planar one–dimensional case. In general SGC is not equal to SLC . The thickness of a turbulent flame brush is considerable and area of different c˜ iso–surfaces can be quite different. This implies that the value of SGC depends on the choice of the c˜ iso–surface that defines Aref as pointed out by Lipatnikov & Chomiak (2002). Driscoll (2008) suggests that c˜ = 0.5 iso–surface can be adopted as a convention. Kim & Bilger (2007) showed that even the local consumption speed, SLC , can vary with locations on the flame brush in multi–dimensional flames. Furthermore, if the mean flow is strained on the unburned side, the diverging streamlines result in a reactant flux in the transverse direction within the flame brush. In such cases the normal mass flux of the reactants at the flame brush leading edge can be greater than the reactant mass consumption rate per unit area in the flame brush and therefore ST > SLC . Shepherd & Kostiuk (1994) illustrate this clearly using the example of a stagnation point flame. Also, it can be envisaged from Fig. 6.2 that in multi–dimensional flames, ST is likely to vary from one location to another even at the leading edge which is less likely in a statistically planar one–dimensional flame. The leading edge displacement speed, ST , is relatively easy to measure in experiments. In statistically stationary flames, ST is the component of local mean flow velocity in the direction normal to the flame brush at the leading edge. Measurements of the mean flow velocity and the flame brush orientation are sufficient to obtain ST at a given location on the leading edge. On the other hand, due to the reasons described above, measurements of SLC and SGC have to account for factors such as flame geometry, flow divergence and the various turbulent scalar fluxes involved. In envelope flames, such as a Bunsen flame or the slot flame of Filatyev et al. (2005), the flame geometry ensures that all the reactants are consumed by the flame and SGC = m˙R/(ρuAref ), where m˙R is the mass flow rate of reactants into the burner which is known. Shepherd and co–workers (Shepherd & Cheng, 2001; Shepherd & Kostiuk, 1994; Shepherd et al., 1998) measured the reactant mass fluxes over a control volume along the local flame brush normal direction in axisymmetric flames and the integral of these fluxes gives ∫ ω˙ dxn and hence the local consumption speed, SLC . Another approach to obtain SLC is from measurements of the mean flame surface density 74 6.3 KPP analysis Σ (Shepherd & Cheng, 2001) via SLC = soLIo ∫ Σdxn since the mean reaction rate can be modelled as ω˙ = ρusoLIoΣ (see Eq. 2.41). Gouldin (1996) also obtained the burning rate integral ∫ ω˙ dxn by appropriately integrating the measured convective and diffusive fluxes in a V–flame. The understanding of these subtleties are important for interpreting whether a particular experimental study gives ST , SGC or SLC . Furthermore, since the focus here is on a flame speed expression which is derived from the KPP analysis, it is equally important to ascertain which of these three quantities the KPP expression corresponds to. This is examined in the next section. 6.3 KPP analysis Equation (6.3) was derived using the KPP analysis. The application of this analysis to a laminar flame (Zeldovich et al., 1985) gives a flame speed expression as a solution to an eigenvalue problem and shows that the flame speed depends on conditions at the leading edge. An analogous turbulent case is the statistically planar one–dimensional flame for which the KPP analysis has been applied in earlier works (Corvellec et al., 2000; Duclos et al., 1993; Lipatnikov & Chomiak, 2002). As noted earlier, for this flame ST = SLC = SGC and there is a single flame speed which is an eigenvalue of the flame propagation problem. However, this is not the case in a multi–dimensional flame. Thus, it is important to examine the applicability of the KPP analysis and to understand whether the resulting expression gives ST , SLC or SGC . To distinguish from the multi–dimensional case, we will refer to the eigenvalue in the planar one–dimensional case as burning velocity. 6.3.1 KPP analysis of a planar one–dimensional flame It is well known that Eq. (6.9) can be reduced to (see for example Lipatnikov & Chomiak (2002)) dy dc˜ = my − Ω y , (6.11) 75 6.3 KPP analysis by taking the turbulent scalar flux to be of gradient type and defining it as y ≡ −ρu′′c′′ = ρ νt Scc ∂c˜ ∂x . (6.12) The quantity Ω is defined as Ω ≡ ρνtω˙/Scc and the mass flux m = ρuST is an eigenvalue of the problem. The well known analysis by Zeldovich et al. (1985) gives m = 2 √ Ω′(0), which yields Eq. (6.1) for the burning velocity. The prime here denotes a derivative with respect to c˜. When the turbulent scalar flux is counter–gradient, the turbulent diffusivity changes sign at some location c˜ = c∗ in the flame brush. Corvellec et al. (2000) point out that the burning velocity may be dictated by Ω′(1) rather than by Ω′(0) when the turbulent scalar flux is counter–gradient. On the other hand, Bray (1990) points out that earlier studies (for instance Libby (1989)) accounting for counter–gradient transport arrive at burning velocity expressions which are equivalent to the KPP expression. 6.3.2 Applicability to multi–dimensional flames For statistically stationary, multi–dimensional turbulent flame, the transient terms are zero in the Favre averaged continuity equation, Eq. (2.7), and the c˜ equation, Eq. (2.15). Furthermore for large Reynolds numbers the molecular diffusion term in Eq. (2.15) is negligible compared to the other terms. Consider a curvilinear or- thogonal coordinate system (η, ξ, ζ) attached to the flame brush, in which η is the coordinate locally normal to an iso–surface of c˜ and ξ, ζ are the local tangential coordinates as shown in Fig. 6.2. By definition c˜ varies only in the η direction. In such an orthogonal curvilinear coordinate system the above continuity and c˜ equations transform to (Gouldin, 1996) 1 hηhξhζ [ ∂ ∂η hξhζρu˜η + ∂ ∂ξ hζhηρu˜ξ + ∂ ∂ζ hηhξρu˜ζ ] = 0 (6.13) and ρu˜η 1 hη ∂c˜ ∂η + 1 hηhξhζ [ ∂ ∂η hξhζρuη′′c′′ + ∂ ∂ξ hζhηρuξ′′c′′ + ∂ ∂ζ hηhξρuζ′′c′′ ] = ω˙ (6.14) respectively, where hη, hξ, hζ are the corresponding scale factors. The scale factors are defined by the partial derivative of the position vector ~r(x1, x2, x3). Note 76 6.3 KPP analysis that Eq. (6.13) and the conditions (∂c˜/∂ξ) = (∂c˜/∂ζ) = 0 are made use of in obtaining Eq. (6.14). Gouldin (1996) examines the conditions under which Eq. (6.14) poses an eigenvalue problem and notes that the sufficient condition for this arises when Eq. (6.13) reduces to ρu˜η = constant. This is satisfied only for a planar one–dimensional flame (Gouldin, 1996) while for a multi–dimensional flame this condition is usually not met. The studies of Shepherd and co–workers (Shepherd & Cheng, 2001; Shepherd & Kostiuk, 1994; Shepherd et al., 1998) indicate that the tangential turbulent fluxes, ρuξ′′c′′ and ρuζ′′c′′, are negligibly small compared to the convective flux and the normal turbulent flux ρuη′′c′′. This plausible assumption seems to be supported by the experimental measurements of Gouldin (1996). However, more experimental studies on this issue would be enlightening. Here, this experimental observation is used to reduce Eq. (6.14) as ρu˜η 1 hη ∂c˜ ∂η + 1 hη ∂ ∂η ρuη′′c′′ + ρuη′′c′′ 1 hηhξhζ ∂hξhζ ∂η = ω˙. (6.15) If eˆi is the unit vector in direction i then 1 hηhξhζ ∂hξhζ ∂η = ∇ · eˆη = 1 R where R is the curvature of the c˜ iso–surface 1. This reduces Eq. (6.15) to a quasi–1D form ρu˜η 1 hη ∂c˜ ∂η + ρuη′′c′′ R = − 1 hη ∂ ∂η (ρuη′′c′′) + ω˙. (6.16) If the turbulent scalar flux is of gradient type and one defines y as before: y ≡ ρ(νt/Scc)(∂c˜/hη∂η) and uses the transformation 1 hη ∂ ∂η = 1 hη ∂c˜ ∂η d dc˜ = y Scc ρνt d dc˜ then the following equation results dy dc˜ = mˆy − Ω y . 1The vector eˆη points in the direction of increasing c˜. 77 6.4 Comparisons with experimental data where mˆ = ρu˜η − (ρνt/RScc) and Ω is defined as before. It is to be noted that mˆ 6= constant and thus it is not an eigenvalue of the problem. Also mˆ, y and Ω depend on the ξ and ζ coordinates as well, but for the moment it is enough to focus on the dependence on η alone. By writing the Taylor’s series expansions for these quantities about c˜ = 0, following Hakberg & Gosman (1984), one obtains s2 − mˆ(0)s+ Ω′(0) = 0 where s ≡ y′(0). Note that Ω(0) = 0. Now, the condition for real roots be- comes mˆ2(0) ≥ 4Ω′(0) and the minimum of which gives ρuST − (ρνt/R Scc)c˜=0 = 2 √ Ω′(0). Thus ST = 2 √ νt (ρu Scc) ( ∂ω˙ ∂c˜ ) ec→0 + ( νt R Scc ) c˜=0 . (6.17) If the mean curvature is negligible, one readily recovers Eq. (6.1) which indicates that the KPP expression corresponds to the leading edge displacement speed, ST . It is worth mentioning that the above analysis can also be performed when the turbulent flux is counter–gradient, following Corvellec et al. (2000). It is interesting to note from Eq. (6.17) that the leading edge displacement speed is enhanced by positive curvature. However, in the present study we focus on the case of zero curvature for which Eq. (6.3) is directly applicable. Comparisons of this expression with experimental data are considered next. 6.4 Comparisons with experimental data Many flame speed expressions have been proposed earlier and they are inves- tigated in detail in the review by Lipatnikov & Chomiak (2002). While some expressions correspond to the leading edge displacement speed, other expressions correspond to consumption speed. Their analysis of large number of experimental data revealed qualitative trends that were reproduced by only few flame speed expressions. The typical comparison of two such expressions with the experimen- tal data of Shepherd & Cheng (2001) is shown in Fig. 6.3. The comparisons of Eq. (6.3) are also shown in the same figure. The data corresponds to ST of lean methane–air flames of equivalence ratio φ = 0.7 stabilised on a low–swirl burner. 78 6.4 Comparisons with experimental data 0 4 8 12 16 0 2 4 6 8 ST soL u′/soL Expt. Eq. (6.18) Eq. (6.19) Eq. (6.3) M M M MM ... ... ... ... ... ... ... ... ... ... ... .... ..... .... ..... .... ..... .... ..... .... ...... ...... ...... ...... ...... .. ..... ... ..... ... .... ... .... ... .... ... .... ... .... .... .... ..... ... ..... .... .... ..... .... .. . . .. . .. .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .. Figure 6.3: Typical comparisons of turbulent flame speed expressions to the ex- perimental data of Shepherd & Cheng (2001). Shepherd & Cheng (2001) compare their experimental data with the expressions of Peters (1999) and Zimont & Lipatnikov (1995). The expression given by Peters (1999) is ST soL = 1− 0.78 2b1 Λ δ + [( 0.78 2b1 Λ δ )2 + 0.78 u′Λ soLδ ]0.5 , (6.18) where b1 is a model constant. The expression of Zimont & Lipatnikov (1995) is ST soL = 1 + κ u′ soL Da0.25, (6.19) where κ is a model constant. The values of these constants as proposed by the original authors are b1 = 2.0 (Peters, 1999) and κ = 0.51 (Zimont & Lipatnikov, 1995). The expression of Gülder (1990) is very similar to Eq. (6.19). Note that the values of the model constants had to be adjusted to b1 = 3.4 and κ = 0.85 to get good agreement with experimental data by Shepherd & Cheng (2001). However, none of the constants in Eq. (6.3) have been altered to obtain the result in Fig. 6.3 and the agreement is acceptable. Comparisons of Eq. (6.3) with a range of experimental data will be discussed next without changing these model 79 6.4 Comparisons with experimental data constants. Comparisons with other flame speed expressions are not attempted since the values of their model constants may have to be adjusted from one experiment to the other. As noted earlier, the turbulent flame speed ST in this study is the propaga- tion speed of the leading edge of the mean flame brush and the flame speeds measured in experiments can sometimes correspond to SLC or SGC . Accordingly experimental studies which reported ST have been chosen after careful scrutiny for comparison. The objective is to compare the flame speed values obtained us- ing Eq. (6.3), while keeping the model constants fixed, with data representing a wide range of experimental conditions, burner geometry, flame configuration and measurement technique. These considerations guide the choice of experimental data sets, which are briefly discussed below with the relevant details. 6.4.1 Flame speed datasets Table 6.1: Salient characteristics of various experimental ST data sets. Flame Burner configuration Mixture φ Pressure Measurement location Source V–shape Rod stabiliser at nozzle exit CH4–Air 0.75–1.0 Ambient c = 1/τ Smith & Gouldin (1978) Planar Taylor- Couette apparatus CH4–Air 0.8–1.5 Ambient Leading edge Aldredge et al. (1998) Conical Nozzle burner in high pressure CH4–Air 0.9 0.1–3.0 MPa c = 0.5 Kobayashi et al. (1996) Planar Diverging duct CH4–Air 0.75–0.95 Ambient c = 0.5 Savarianandam & Lawn (2006) Right Cone Parallel plates at square channel exit City gas–Air - Ambient Leading edge Il’yashenko & Talantov (1966) The configuration and conditions of various experimental data sets considered in the present study are listed in Table 6.1. As one can see, these experiments 80 6.4 Comparisons with experimental data correspond to a variety of flame configurations and burner geometry. The relevant attributes of these flames are given in Table 6.2. Table 6.2: Attributes of various experimental ST data. Source soL (m/s) δoL (mm) u′ (m/s) Λ (mm) Re Da Smith & Gouldin (1978) 0.25–0.40 0.45–0.67 0.095–1.13 0.69–1.49 3–75 3-39 Aldredge et al. (1998) 0.17–0.48 0.37–0.92 0.2–1.06 5.5 50–265 5-219 Kobayashi et al. (1996) 0.06–0.34 0.095–0.5 0.053–0.83 1.0–1.5 7–830 6-97 Savarianandam & Lawn (2006) 0.19–0.32 0.51–0.79 0.01–0.26 3–36 7-497 142-4535 Il’yashenko & Talantov (1966) 0.2 0.7 0.22–10.5 6.8–11.2 114–3230 1.2-92 The study of Smith & Gouldin (1978) considered V–shaped rod stabilised methane–air flames of three equivalence ratios: φ = 0.75, 0.85 and 1. They measured the orientation of the T = 2 Tu isotherm, which corresponds to the c = 1/τ iso–surface in the flame brush. This surface is not strictly the flame brush leading edge and this is to be noted while comparing the turbulent flame speed. Smith & Gouldin (1978) reported the component of local mean velocity normal to the above iso–surface as the turbulent flame speed for each φ and three different values of turbulence integral length scale, Λ: 0.69, 0.95 and 1.49mm. As established in Section 6.3, this normal mean velocity corresponds to the turbulent flame speed. Smith & Gouldin (1978) identify that ST varies locally on the iso– surface and they clearly note the importance of correlating measurements of ST at a location where u′ and Λ are also measured. Aldredge et al. (1998) measured both laminar and turbulent flame speeds of methane–air mixtures with equivalence ratio ranging from 0.8 to 1.5, varied in 81 6.4 Comparisons with experimental data steps of 0.1, in a Taylor–Couette (TC) apparatus. This apparatus consisted of two vertical concentric counter–rotating cylinders with an annular gap of 1.1cm. The planar flame, ignited at the open end, propagates into the nearly stationary turbulent mixture in the annulus. The rate of displacement of the propagating flame, which was calculated from video record, was reported as the flame speed. Clearly, this speed is the turbulent flame speed, ST . Kobayashi et al. (1996) measured the turbulent flame speed for lean methane– air mixture with φ = 0.9 at pressures ranging from 0.1 to 3 MPa. The turbulence measurements under these conditions were reported in another study (Kobayashi et al., 1997). A Hydrogen pilot flame was used to stabilise the turbulent flame at the exit of a nozzle in high pressure environment. Unlike the combustion bombs and SI engines in which the pressure change is unsteady and brief, the apparatus of Kobayashi et al. (1996) was designed to maintain high pressures for long durations. From instantaneous tomographic images of flame, the mean flame cone was identified and the flame speed was calculated as the component of the mean flow velocity normal to the mean flame cone, which corresponds to ST as noted in sub–section 6.3.2. Savarianandam & Lawn (2006) focussed on the weakly wrinkled flame regime (u′/soL < 1) where the Darrieus–Landau (DL) instabilities have dominant influ- ence on ST . While the analysis of Damköhler suggests that ST/soL should be relatively closer to unity in this regime, other studies (Cambray & Joulin, 1992; Paul & Bray, 1996) indicate that Darrieus-Landau hydrodynamic instabilities result in ST/soL value significantly greater than unity. Flame speeds of nearly planar flames stabilised in a wide angled diffuser at very low turbulence intensi- ties, reported by Savarianandam & Lawn (2006), provide evidence for this. Three methane–air mixtures of φ = 0.75, 0.85 and 0.95 were used and values of u′/soL ranged from 0.05 to 1.37. The flame speed was estimated to be the mean flow velocity at the mean flame height location. As noted by Savarianandam & Lawn (2006), this velocity is the leading edge displacement velocity ST . Il’yashenko & Talantov (1966) measured turbulent flame speeds in a flame configuration which was obtained in the region confined between two parallel plates, at the exit of a square channel. Ignition flames were used on two opposite sides of the channel exit and the other two sides were extended as walls. This 82 6.4 Comparisons with experimental data resulted in a symmetric configuration of two flame surfaces at an angle to the flow and intersecting at a downstream location. The experiments were done with both a smooth channel and a channel with turbulising grid at the exit, which allowed very high turbulence level to be achieved. They used city gas (a mixture of natural gas and coke gas) as the fuel and measured flame speeds up to u′/soL ∼ 50. The flame speed was estimated from the mean flow velocity and inclination angle of the flame surfaces. As established in Section 6.3.2, this normal velocity corresponds to the turbulent flame speed ST . 0.1 1 10 100 1000 0.1 1 10 100 1000 u′/soL Λ/δ ........................................................................................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . ................................................................................................................................................................................................................................. ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ....... ......... ......... ......... ......... ......... ......... ......... ∗∗∗∗∗∗ ∗∗∗∗∗ ∗ ∗ ∗∗∗∗∗∗ ∗ ∗∗∗∗∗ ∗∗∗∗∗∗∗∗ ∗ ∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗ ∗∗∗∗∗ ∗∗∗∗∗ ¨¨¨ ¨¨ ¨¨ ¨ ¨¨¨ ¨¨ ¨¨ ¨ ¨¨¨ ¨¨ ¨¨ ¨ ¨¨¨ ¨¨ ¨¨ ¨ ¨¨¨ ¨¨ ¨¨ ¨ MM M M M M M M MM M M MMMM M M MM M M OO OOO O OOO OOOOOO O OOOO OOOO O ◦◦ ◦◦ ◦◦◦ ◦◦◦◦ ◦• •• • Re = 1 Ka = 100 Da = 1 Ka = 1 Wrinkled Corrugated Thin reaction zones Distributed reaction zones Figure 6.4: Turbulent combustion regime diagram (Peters, 2000) with attributes of various experimental flames: ∗ Smith & Gouldin (1978); ¨ Aldredge et al. (1998); M Kobayashi et al. (1996); O Savarianandam & Lawn (2006); ◦, • Il’yashenko & Talantov (1966). Figure 6.4 shows the above five data sets in the combustion regime diagram, which is adapted from Peters (2000). Although most of the experimental flames are within the corrugated and the wrinkled flamelets regimes, there are some cases spanning the thin reaction zones regime. More experimental data in this 83 6.4 Comparisons with experimental data regime would be very useful for model validation. Comparisons between Eq. (6.3) and the above data sets are discussed next. 6.4.2 Results Equation (6.3) is strictly valid for a flame brush with zero curvature and it is clear from the foregoing discussion that the experimental flames in Table 6.1 are free from effects of mean curvature. The flame speed values obtained using Eq. (6.3) are compared to the measured values in these experiments in Figs. 6.5 - 6.9. It is worth emphasising that the constants in Eq. (6.3) remain the same in all the comparisons shown in Figs. 6.5 - 6.9. The thermal thickness of laminar flame is obtained using (Poinsot & Veynante, 2001) δoL δ = 2(1 + τ)0.7 , (6.20) where the Zeldovich thickness, δ, is obtained using the thermal diffusivity (' 2.2×10−5m2/s for atmospheric methane–air flames) and the laminar flame speed measured in the respective experimental work. If soL is not measured, for example in the work of Smith & Gouldin (1978), then the values reported by Law (1993) are used. The Karlovitz number required for the constants C3 and C4 in Eq. (6.3) is evaluated as (Swaminathan & Bray, 2005) Ka = { (u′/soL) 3 (Λ/δ) }0.5 . (6.21) The V–flame data of Smith & Gouldin (1978), shown in Fig. 6.5, are from measurements at T = 2Tu isotherm which corresponds to c = 1/τ iso-surface as noted earlier. V–flames are characterised by increasing flame brush thickness with downstream distance, which implies that different c iso-surfaces have different inclinations. However, since τ ∼ 5.5, the c = 1/τ iso-surface is reasonably close to, but not exactly at, the leading edge of the flame brush. As evident from Fig. 6.5, Eq. (6.3) compares well with the experimental data. The flame speed predicted by Eq. (6.3) is compared to the data of Aldredge et al. (1998) in Fig. 6.6 and it is clear that Eq. (6.3) compares well with the experimental data. While the quantitative agreement is good for rich equivalence ratios, the predictions are slightly higher for lean mixtures as in Figs. 6.6(g) and 84 6.4 Comparisons with experimental data 6.6(h). The laminar flame speed values, soL, measured by Aldredge et al. (1998) for lean mixtures are much lower than those measured in other studies (Andrews & Bradley, 1972; Vagelopoulos et al., 1994). Hence for comparisons of lean cases shown in Figs. 6.6(g) and 6.6(h), the planar laminar flame speed values reported by Law (1993) are used. For all other cases the values of soL measured by Aldredge et al. (1998) are used. If one uses the measured soL for the lean cases, then an underprediction of ∼ 30% (maximum) is observed. The effect of pressure on the turbulent flame speed was attributed by Kobayashi and his co–workers (Kobayashi & Kawazoe, 2000; Kobayashi et al., 1996, 1997, 2002) to laminar flame instabilities. These studies observed that increasing pres- sure resulted in a decrease of the characteristic scale of flame instability, li, while the integral length scale Λ was not affected much. At elevated pressures, li < Λ and hence flame instabilities are the predominant cause of the increase in ST/soL for a given u′/soL (Kobayashi & Kawazoe, 2000). The characteristic flame insta- bility scale li is dictated by both the Darrieus–Landau hydrodynamic instabilites and the thermal–diffusive effects and hence it is a function of δ, soL, τ and the Lewis number Le. The effects of non–unity Lewis number were not considered in (Kolla et al., 2009) and hence Le does not appear explicitly in Eq. (6.3). Nonethe- less, for a given mixture, while Le does not depend on pressure, the other relevant parameters (δ, soL, τ and Λ) appear in Eq. (6.3). Furthermore, Le is close to unity for the mixtures considered by Kobayashi and co–workers and hence it is expected that the pressure effects on ST/soL reported by them are likely to be captured by Eq. (6.3). This is evident from the comparisons shown in Fig. 6.7. Using the TFC closure and empirical correlations given by this experiment for the flame wrinkling ratio, Muppala et al. (2005) show an interesting comparison with this data. Kobayashi et al. (1996) note that the mean flame cone corresponds to the c = 0.5 iso-surface and ST estimated from the inclination of flame brush leading edge might be higher because of the difference in the inclination angles. Nonethe- less, it is encouraging to note that the effect of pressure on ST is well captured by Eq. (6.3). However, the comparison for 1 MPa is not so good and the reason for this is unclear at this time. In the high pressure studies, the laminar flame instabilities were predominant in u′/soL > 1 regime, due to a decrease in li at elevated pressures. On the other 85 6.4 Comparisons with experimental data hand, Savarianandam & Lawn (2006) studied the effects of instabilities at atmo- spheric pressure in u′/soL < 1 regime, by studying weakly turbulent flames with very large turbulence integral length scale, Λ. For the planar flames stabilised in a wide angled diffuser (Savarianandam & Lawn, 2006) the turbulent flame speed was taken to be the flow velocity at the mean flame height location. This data is compared to the predictions of Eq. (6.3) in Fig. 6.8. The large scatter in the experimental data is because of large variation in the integral length scale. By- chkov (2000) notes that theories that ignore the effects of dilatation due to heat release, would not predict the influence of flame instabilities. Savarianandam & Lawn (2006) found good agreement of their experimental data with the theory of Akkerman & Bychkov (2005) which explicitly includes the density ratio, (τ + 1). Figure 6.8 shows that Eq. (6.3) compares well with the experimental data even though the instability effects were not explicitly considered in the analysis. In particular it is interesting to note that Eq. (6.3) is able to capture the sensitivity to the turbulence integral length scale observed in the experimental data. As mentioned earlier, the KPP result will not yield ST = soL when u′/soL = 0 unless the molecular diffusivity is included in the total diffusivity. However, the compar- ison with this data at small u′/soL will not be erroneous here since the Re values are sufficiently large as can be seen from Fig. 6.4. The values of u′/soL in the study of Il’yashenko & Talantov (1966) are much higher than in most other investigations of turbulent flame speed. This data is shown in Fig. 6.9, which corresponds to a mixture with soL = 20cm/s and the width of the channel is 40mm. The smooth channel data are denoted by open circles and the data with turbulence generating grid are denoted by closed circles. Because of this difference in the experimental arrangements, one can expect the integral length scales to be different. Unfortunately the values of the length scales are not reported by Il’yashenko & Talantov (1966). Here, the approach of Abdel- Gayed & Bradley (1981) is followed to obtain Λ. For the smooth channel case the turbulence Reynolds number is related to a Reynolds number, ReP, which is based on the mean flow velocity and hydraulic diameter (Abdel-Gayed & Bradley, 1981) via the correlation u′Λ/ν = 0.0153(ReP)0.911. (6.22) 86 6.4 Comparisons with experimental data The values of ReP are reported by Il’yashenko & Talantov (1966) and thus for the given values of u′ and ν the turbulence integral length scale can be obtained from Eq. (6.22). For the case with turbulence generating grid the following correlation suggested by Abdel-Gayed & Bradley (1981), is used: Λ/d = 0.155(x/d)0.452, (6.23) where d is the diameter of the rod in the turbulence generating grid and x is the measurement location downstream of the channel exit. The experimental data is compared to the predictions of Eq. (6.3) in Fig. 6.9. The experimental results are shown as two different sets to signify the difference in the experimental set– up and thus in the turbulence integral length scale. Accordingly, the theoretical curves (solid lines) are broken around u′/soL ∼ 20. The following observations can be made from Fig. 6.9. (i) The prediction of Eq. (6.3) shown in Fig. 6.9 is very good. The dotted lines extended from the solid lines in Fig. 6.9 are the predictions by Eq. (6.3) if one were to use the same integral length scale over the whole range of u′/soL. The upper dotted line is for Λ/δoL obtained from Eq. (6.22), while the lower dotted line is for Λ/δoL obtained from Eq. (6.23). This clearly shows the sensitivity of ST/soL predicted by Eq. (6.3) to the turbulence integral length scale. (ii) The expression of Klimov (1983): ST/soL = 3.5(u′/soL)0.7, is developed specif- ically for this experiment and thus the prediction is close to the experimental data. (iii) Figure 6.9 supports the linear dependence of ST/soL on u′/soL for large values of u′/soL noted earlier in subsection 6.1.1. The levelling off of the ST/soL curve, as suggested by the bending phenomenon, is not seen in Fig. 6.9. If one considers the data only in the region of u′/soL < 20 then there is some semblance of bending. But upon further increase of u′/soL, the data shows a persistent increase in ST/soL with no sign of bending. It would appear that more experimental ST measurements at such high u′/soL values are needed to provide conclusive evidence (or lack thereof) of the bending phenomenon. 87 6.4 Comparisons with experimental data Furthermore, this phenomenon is seen to depend on whether the velocity ratio or the length scale ratio is held constant as noted by Bray (1990). The comparison between predictions of Eq. (6.3) and various experimental data is consistently good. While only few data sets amongst large number avail- able are considered in the present work, these data sets are representative to show the influence of various factors affecting turbulent premixed flame speed and its behaviour. However, it is worth considering the experimental database of Abdel- Gayed et al. (1987). This compilation is based on the dimensionless parameter K, known as the “flame stretch factor”, which seems to be a useful basis for tur- bulent flame speed comparison. The flame stretch parameter quantifies the effect of stretch on the flame propagation and is defined as (Abdel-Gayed et al., 1987) K ≡ u ′δ λsoL = 0.157 ( u′ soL )2 Re−0.5 , (6.24) where λ is the Taylor microscale. The vast experimental data of turbulent flame speed was correlated with u′/soL for different values of the stretch parameter by Abdel-Gayed et al. (1987). This compilation is shown in Fig. 6.10 for K = 0.053 and 0.15. Figure 6.10 also compares the prediction of Eqs. (6.3), (6.18) and (6.19) with the experimental data. For this comparison, the values of constants used in Eqs. (6.18) and (6.19) are b1 = 2.0 and κ = 0.51 as originally proposed. For small values of K the collapse of experimental data of Abdel-Gayed et al. (1987) seems reasonable, but the scatter in the experimental data seems large for high K values. However, it must be noted that the experimental data span a narrow range of K values: 0.045 ≤ K ≤ 0.06 in Fig. 6.10(i) and 0.12 ≤ K ≤ 0.17 in Fig. 6.10(ii). Furthermore, Driscoll (2008) points out that the scatter is possibly because the database includes measurements of both displacement and consumption speeds. This comparison suggests that Eq. (6.3) predicts the experimental data well for low K but the flame speed seems to be overpredicted for high K. This is expected since Eq. (2.30), which is central to the derivation of Eq. (6.3), is strictly valid for low stretch (when γ∗ is small). As the stretch parameter increases validity of Eq. (2.30) progressively diminishes since the flame front thickens. Nonetheless, Eq. (6.3) seems to be showing the correct physical behaviour and parametric dependence observed in a number of experiments and the predicted 88 6.5 Summary ST compares well with those experimental data. Driscoll (2008) suggests that the turbulent flame speed, ST , is not just a function of the local fluid–dynamic and thermochemical parameters alone but burner geometry has a significant influence since the wrinkling processes in turbulent flames are geometry–dependent. Hence models that are obtained from differential equations that simulate the flame wrin- kling are likely to perform better than empirical models. Our approach is similar since the scalar dissipation rate is related to the flame surface density and in the present work, the flame speed expression is derived from a model for scalar dissi- pation rate that accounts for the dominant terms in its transport. Furthermore Driscoll (2008) prescribes that flame speed databases should be categorised based on the flame geometry and ST expressions should include an appropriate geomet- ric parameter to account for geometry–dependent wrinkling. However the com- parisons with various experiments shown here have demonstrated that Eq. (6.3) gives good predictions for flames with different geometry. This is because the small–scale mixing rate and thus the chemical reaction rate are independent of large scales in turbulent flows with sufficient scale separation. The success of Eq. (6.3) underscores the importance of the scalar dissipation rate and confirms the hypothesis that a deep understanding of its physics can help in predicting the turbulent flame speed reliably. 6.5 Summary An expression for turbulent flame speed, ST , is studied as a means of provid- ing an indirect experimental validation for the mean scalar dissipation rate, ²˜c, model derived in Chapter 5. Unlike many of the earlier turbulent flame speed expressions, this expression has no adjustable parameters and the model con- stants carry specific physical significance and they are related to the physics of scalar dissipation rate. The turbulent flame speed expression explicitly includes the influence of the heat release parameter, τ , the fuel type, the velocity ratio u′/soL and the length scale ratio Λ/δoL. These parameters signify the influences of thermo–chemical and turbulence processes, which can be altered independently in experiments. The current expression agrees qualitatively with the classical flame speed theories such as those of Damköhler (1940) and Kerstein & Ashurst 89 6.5 Summary (1992). In the wrinkled flamelets regime i.e. for low turbulence level, the nor- malised turbulent flame speed, ST/soL, varies almost linearly with the normalised turbulence RMS velocity, u′/soL, for a given value of the length scale ratio Λ/δoL. However Damköhler’s result gives a square–root dependence for ST/soL on u′/soL for high levels of turbulence while the current expression suggests a combination of linear and square–root dependence similar to the result of Kerstein & Ashurst (1992). The flame speed expression is derived using the KPP theorem which has hith- erto been applied for the case of a statistically planar one–dimensional flame. Analysis of multidimensional flame brush in a local curvilinear orthogonal coor- dinate system attached to a c˜ iso–surface suggests that a quasi one–dimensional structure can be observed when the turbulent scalar flux in the tangential direc- tions of the c˜ iso–surface is negligible. The analysis of the quasi one–dimensional equation gives an expression for ST , which is similar to the KPP result if the mean curvature of the flame brush is negligible. Including the mean flame brush curva- ture in the analysis shows explicitly that increasing positive curvature causes an almost linear increase of the flame speed. Since the flame brush curvature was ex- cluded in the development of Eq. (6.3), which is analysed here, the experimental data sets are chosen such that they are meaningful for its validation. The ST/soL values predicted by the current expression, with fixed model con- stants, compare well with a wide range of experimental data from various flow configurations and conditions. Notably, the comparisons are good with data upto u′/soL ' 50. The increase in ST due to laminar flame instabilities for unity Lewis number mixtures, which are expected to be significant in the weakly wrinkled regime and at elevated pressures, is also well predicted by the current flame speed expression. The ability of the flame speed expression to predict values in close agreement with experimental data over a wide range of conditions indi- cates that the correct physics are incorporated into the mean scalar dissipation rate model. This model is a central modelling parameter in the strained laminar flamelet formulation described in Chapter 4. In the next chapter the strained flamelet formulation will be applied for turbulent premixed flame calculations to test its predictive capability. 90 6.5 Summary 0 1 2 3 4 5 6 7 0 1 2 3 0 1 2 3 0 1 2 3 4 (a) (b) (c) ST soL Expt. Eq. (6.3) ∗∗ ∗∗∗ ∗∗∗∗ ∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗ ∗ ∗∗∗ ∗∗ ∗∗∗∗∗∗ ∗ ∗∗ ∗ ............................................................................................................... ................................................................................................................. ....................................................................................... 0 1 2 3 4 5 6 7 0 1 2 3 0 1 2 3 0 1 2 3 4 (d) (e) (f) ST soL ∗∗ ∗ ∗∗∗∗∗∗∗∗∗∗ ∗ ∗ ∗∗∗∗∗ ∗∗∗∗ ∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ .................................................................................. ..................................................................................... ................................................................................ 0 1 2 3 4 5 6 0 1 2 0 1 2 0 1 2 3 (g) (h) (i) ST soL u′/soL u ′/soLu ′/soL ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗∗∗∗ ................................................................................... ..................................................................................................... ............................................................................... Figure 6.5: The predictions of turbulent flame speed expressions are compared to the experimental data of Smith & Gouldin (1978) for various equivalence ra- tios: (a)–(c) correspond to φ = 0.75; (d)–(f) correspond to φ = 0.85 and (g)–(i) correspond to φ = 1.0. The comparisons on the left, center and right columns correspond to Λ = 0.69mm, 0.95mm and 1.49mm respectively. 91 6.5 Summary 0 2 4 6 8 10 12 0 1 2 3 0 1 2 0 1 2 3 (a) (b) (c) ST soL Expt. Eq. (6.3) ¨ ¨ ¨¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨¨¨¨ ¨¨¨¨ ¨¨ ¨ ¨ ¨¨ ¨¨¨ ¨ ¨¨ ......................................................................................................................................................... ......................................................................................... ........................................................................................................... 0 2 4 6 8 10 12 0 1 2 3 0 1 2 3 0 2 4 6 8 (d) (e) (f) ST soL u′/soL u ′/soLu ′/soL ¨¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨¨ ¨ ¨ ¨¨ ¨¨ ¨¨¨ ¨¨¨ ¨¨ ............................................................................................. .............................................................................................................................................. ........................................................................................................ 0 2 4 6 8 10 12 0 1 2 3 0 1 2 3 4 (g) (h) ST soL u′/soLu ′/soL ¨¨¨ ¨¨¨ ¨¨ ¨ ¨ ¨¨ ¨¨¨¨ ¨¨ ¨¨ ............................................................................................................................................ .................................................................................................. Figure 6.6: The predictions of turbulent flame speed expressions are compared to the experimental data of Aldredge et al. (1998) for various equivalence ratios: (a) φ = 1.0; (b) 1.1; (c) 1.2; (d) 1.3; (e) 1.4; (f) 1.5; (g) 0.8; (h) 0.9. 92 6.5 Summary 0 1 2 3 4 0.0 0.5 1.0 1.5 (a) ST soL Expt. Eq. (6.3) MM M M M .................................................................................................................................. 0 2 4 6 8 10 12 0.0 1.0 2.0 3.0 4.0 5.0 (b) M MM M M M ............................................................................................................................... 0 2 4 6 8 10 12 14 0.0 1.0 2.0 3.0 (c) ST soL M M MM ............................................................................................................................................................. 0 5 10 15 20 25 30 0.0 3.0 6.0 9.0 12.0 (d) MM M M MM MM MM .......................................................................................................................... 0 5 10 15 20 25 30 35 0.0 3.0 6.0 9.0 12.0 (e) ST soL u′/soL M MM MM M ..................................................................................................................................... Figure 6.7: The variation of turbulent flame speed with u′/soL at different pres- sures: (a) 0.1 MPa; (b) 0.5 MPa; (c) 1 MPa; (d) 2 MPa; (e) 3 MPa. The experimental data of Kobayashi et al. (1996) are compared to the flame speed expression. 93 6.5 Summary 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 ST soL u′/soL Expt. Eq. (6.3) OO O O O O OO O O O O O O OO O O O O OO O O O O OO O O H H HH H HHH HH H HHH HH H HH H H H H H H H H HH H H H H Figure 6.8: Comparisons of turbulent flame speed prediction to the experimental data of Savarianandam & Lawn (2006). 0 10 20 30 40 50 60 70 0 10 20 30 40 50 ST soL u′/soL Expt.(Smooth) Expt.(Grid) Klimov Eq. (6.3) ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦◦◦ ◦ ◦ • • • • •• .... .... ... ..... ..... . ..... ..... . ...... ..... ...... ..... ...... ..... ...... ..... ....... .... ....... .... ....... .... ...... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ...... .... ...... ...... ...... ....... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . Figure 6.9: Comparisons of turbulent flame speed prediction to the experimental data of Il’yashenko & Talantov (1966). 94 6.5 Summary 0 3 6 9 12 15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (i) ST soL Expt. Eq. (6.18) Eq. (6.19) Eq. (6.3) ◦ ◦ ◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦ ◦◦◦◦◦ ◦◦ ◦ ◦◦ ◦ ◦◦◦ ◦◦ ◦◦ ◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦◦ ◦ ◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦ ...... ..... . ...... ...... .... .. ...... ...... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . .. .. ........ .............. ............. .......... ......... ........ ........ ......... ........ ........ ...... ........ ........ ........ ..... ....... ....... ..... ........ ....... ... ..................... ............. .......... ........ ........ ....... ...... ..... .. .... .... . .... .... .... .... .... .... .... ..... ..... ..... .... .... .... ..... ..... ..... ..... ..... .... ..... ..... ..... ..... ..... ..... ..... ..... .... .... ..... ..... ..... ..... ..... ..... ..... .. 0 5 10 15 20 25 0.0 1.0 2.0 3.0 4.0 5.0 6.0 (ii) ST soL u′/soL ◦ ◦ ◦ ◦ ◦◦ ◦◦◦ ◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ ◦ ◦ ◦ ◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ ◦ ◦◦ ◦ ◦ ◦◦◦ ◦ ◦◦ ◦ ◦◦◦ ...... ..... . ...... ...... . ..... .. .... .. .... .. .... . ..... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ..... .. ........ ........... ........... .......... .......... ......... ......... ......... ...... ....... ........ ........ ...... ....... ....... ..... ....... ....... ..... ........ ................ ............ .......... ........ ........ .... ....... ....... ..... ..... .... .... .... .... .... .... .... .... .... .... .... ..... .... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... Figure 6.10: Comparison of turbulent flame speed expressions to the experimental database of Abdel-Gayed et al. (1987) for two values of flame stretch parameter, K: (i) K = 0.053 and (ii) K = 0.15. 95 Chapter 7 Application of Strained Flamelet Formulation The strained flamelet formulation described in Chapter 4 is applied for turbulent premixed flame calculations in this chapter, to test its predictive ability. Cal- culations of statistically planar one–dimensional flames are considered first to compare the predictions of the strained flamelet mean reaction rate closure with other closures. Calculations of laboratory flames of different configurations are then performed and predictions of the flame brush structure and species concen- trations are compared with experimental data. An overview of the computational methodology adopted in the present calculations is described in the next section. 7.1 Computational methodology The calculations performed are in the Reynolds–Averaged–Navier–Stokes (RANS) paradigm with the standard k˜ − ε˜ turbulence closure (see Section 2.2.1). The governing equations solved are: • Favre averaged continuity equation; Eq. (2.7) • Favre averaged momentum equation; Eq. (2.8) • Favre averaged turbulent kinetic energy equation; Eq. (2.10) • Favre averaged kinetic energy dissipation rate equation; Eq. (2.11) 96 7.1 Computational methodology • Favre averaged progress variable equation; Eq. (2.15) • Progress variable variance equation; Eq. (3.1) These governing equations can be cast in a generalised form as ∂ρΨ ∂t + ∂ρu˜jΨ ∂xj = ∂ ∂xj ( Γ ∂Ψ ∂xj ) + SΨ (7.1) where Ψ = u˜i, k˜, ε˜, c˜, etc., and Ψ = 1 for mass conservation equation. The term involving Γ represents the diffusive flux typically including the turbulent and laminar diffusivities and SΨ represents the source terms. For example, when Ψ = ε˜, from Eq. (2.11), Γ = µ+ µt/Scε and SΨ = Cε1Pkε˜/k˜ − Cε2ρε˜2/k˜. The governing equations are discretised using the finite–volume formulation described by Patankar (1980). The pressure correction approach of Patankar (1980) is employed to obtain a solution for the flow field. This approach, applica- ble for low Mach number flows, allows a guessed pressure field to be corrected by the continuity equation, such that the resulting velocity field from the momentum equations satisfy the continuity equation. This iterative algorithm is described in detail by Patankar (1980) but it essentially involves the following steps: 1. Guess a mean pressure field p∗ 2. Using the guessed pressure field, solve the momentum equations to obtain a velocity field u˜∗i . 3. Solve the equation for pressure correction p′. This equation is obtained from the continuity equation in terms of the velocities u˜∗i . 4. Correct the velocities u˜∗i and pressure p∗ using p′. 5. Solve the equations for other Ψ. 6. Treating the corrected pressure as p∗, return to step 2 and repeat until convergence. In turbulent flame calculations, the flame influences the flow field via the mean density, ρ, which is calculated using the equation of state and this is incorporated at step 5 in the above sequence of calculation. 97 7.1 Computational methodology 7.1.1 Overview of modelling closures As noted earlier, the standard k˜ − ε˜ turbulence closure is employed and the Reynolds stresses are closed using Eq. (2.9). Two additional unclosed terms remain in Eqs. (2.10) and (2.11). The first of these - the pressure–dilatation term - is closed using the model of Zhang & Rutland (1995): p′ ∂u′′k ∂xk = 0.5 c˜ (τsoL) 2 ω˙ . (7.2) The second term, u′′, is closed as (Jones, 1994) u′′ = τ u˜′′c′′ 1 + τ c˜ . (7.3) The turbulent scalar flux is closed using the gradient transport hypothesis; Eq. (2.16). The turbulent flux term in the c˜′′2 equation - the third term on the l.h.s of Eq. (3.1) - is also closed using the gradient type closure as ρu˜′′i c′′ 2 = − µt Scc ( ∂c˜′′2 ∂xi ) . (7.4) The mean scalar dissipation rate term in Eq. (3.1) is closed using the algebraic model described in Chapter 5; Eq. (5.18). The values of the various model con- stants are given in Table 7.1. Table 7.1: Values of model constants. Constant Equation Value Cµ Eq. (2.9) 0.09 Sck Eq. (2.10) 1.0 Scε Eq. (2.11) 1.3 Cε1 Eq. (2.11) 1.44 Cε2 Eq. (2.11) 1.92 β′ Eq. (5.18) 6.7 C3 Eq. (5.18) 1.5 √ Ka/(1+ √ Ka) C4 Eq. (5.18) 1.1(1 + Ka)−0.4 K∗c /τ Eq. (5.18) 0.85 (hydrocarbon–air) 98 7.2 RANS of planar one–dimensional flames 7.2 RANS of planar one–dimensional flames A test case of statistically planar one–dimensional flames propagating into homo- geneous isotropic turbulence, is considered following many earlier studies (Anand & Pope, 1987; Bradley et al., 1994; Swaminathan & Bray, 2005). Physically, this problem corresponds to premixed flame propagating into a grid generated turbu- lence in a wind tunnel. This problem is inherently unsteady and thus one needs to consider spatio–temporal evolution of the flame brush. This problem offers a simple way to test reaction rate closures without clouding it with uncertainties in turbulence modelling since simple turbulence closure can be used. 7.2.1 Computational details The discretised governing equations and the computational algorithm described in Section 7.1 are implemented as a computer code using FORTRAN–77. This code was employed for one–dimensional flame calculations in the earlier study of Swaminathan & Bray (2005). The present formulation seems similar to the study of Swaminathan & Bray (2005), but it differs in the following respects, viz., (i) the turbulence in the reactant is not considered to be frozen, (ii) the variance transport equation, Eq. (3.1), is solved using the new scalar dissipation rate model in Eq. (5.18), whereas in Swaminathan & Bray (2005) the variance was taken to be c˜(1− c˜), and (iii) the reaction rate closures used here are entirely different. The mean reaction source terms, ω˙ and ω˙′′c′′, appearing respectively in Eqs. (2.15) and (3.1), are related via Eq. (3.2). Three closures for these terms are studied for the one–dimensional flame calculations: 1. Algebraic closure of Bray (1979); ω˙ closed via Eq. (2.30) and ω˙′′c′′ = (Cm− c˜)ω˙ with Cm = 0.7. The dissipation rate is obtained using Eq. (5.18). 2. Unstrained Flamelet model (UF) given by Eq. (4.1). According to this model ω˙c = ∫ 1 0 ζ ω˙o(ζ) p(ζ) dζ. (7.5) 99 7.2 RANS of planar one–dimensional flames 3. Strained Flamelet model (SF) using RtP flamelets given by Eq. (4.11). This model gives ω˙c = ∫ 1 0 ζ 〈ω˙|ζ〉 p(ζ) dζ. (7.6) The algebraic closure from the BML description of turbulent premixed flame is valid in the large Da limit and the turbulent flame front is treated to be a thin interface. The unstrained flamelet model allows one to include the flame front structure without the stretch effects. The strained flamelet model accounts for the stretch effects also. These effects can be clearly seen in the results to be presented in the next section. As noted in Section 4.3, the pdf p(ζ) is obtained from p˜(ζ) which is presumed to be a β–pdf. Table 7.2: Attributes of the one–dimensional flame cases. u′/soL Λ/δ Da Ka K=0.15 2.00 8.764 4.382 0.955 3.00 29.579 9.860 0.955 4.00 70.113 17.528 0.955 5.00 136.939 27.388 0.955 6.00 236.630 39.438 0.955 6.50 300.855 46.285 0.955 K=1.0 8.0 12.620 1.578 6.369 12.0 42.593 3.549 6.369 16.0 100.962 6.310 6.369 18.0 143.753 7.986 6.369 20.0 197.192 9.860 6.369 24.0 340.748 14.198 6.369 Turbulent combustion of stoichiometric methane–air mixture at 298 K and atmospheric pressure are considered. The thermo–chemical characteristics of this mixture are soL = 0.40 m/s, δoL = 0.41 mm and τ = 6.48. Flames corresponding to two values of the flame stretch factor, K: 0.15 and 1.0, are computed. It is evident from Eqs. (6.21) and (6.24) that K = 0.157 Ka. The attributes of the various cases computed are given in Table 7.2. The combustion conditions of these flames are also marked in the regime diagram in Fig. 7.1. It is evident from this figure that the flames with K = 0.15 are at the upper limit of the corrugated flamelets regime and the flames with K = 1.0 are in the thin reaction zones 100 7.2 RANS of planar one–dimensional flames 0.1 1 10 100 1000 0.1 1 10 100 1000 u′/soL Λ/δ ........................................................................................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . ................................................................................................................................................................................................................................. ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ....... ......... ......... ......... ......... ......... ......... ......... ¨ ¨ ¨ ¨ ¨¨ • • • •• • Re = 1 Ka = 100 Da = 1 Ka = 1 Wrinkled Corrugated Thin reaction zones Distributed reaction zones Figure 7.1: The regime diagram with the parameters of the one–dimensional computational flames for two values of flame stretch parameter K: 0.15 (¨) and 1.0 (•). regime. The length of the computational domain varies from 0.125m to 1.0m and the grid spacing varies from 0.13mm to 1.25mm and these values are dictated by the turbulent flame conditions. The turbulence characteristics in the unburnt mixture are specified by k˜ = 3u′2/2 and ε˜ = u′3/Λ for these flames. The grid spacing in the calculations is uniform and the number of cells is chosen to ensure that atleast 10 cells are inside min(Λ, δt). The turbulent flame brush thickness, δt, is defined as δt ≡ 1 (∂c˜/∂x)max . (7.7) Two and three dimensional look up tables of ω˙ are calculated a priori for the unstrained and strained flamelet models respectively. The tables have a resolution of 0.01 in the c˜ dimension and 0.02 in the g dimension. For the strained flamelet model the third dimension is ²˜c and the resolution is 5/s. Values of ω˙ from the table are obtained using linear interpolations for the values of c˜, g and ²˜c realised 101 7.2 RANS of planar one–dimensional flames in the RANS calculation. The unsteady calculations are initialised using an arbitrary c˜ profile going from 0 to 1 in the computational domain (see Fig. 7.2). The turbulence quantities, k˜ and ε˜, are set equal to the unburnt mixture values in the entire domain at the start of the calculations. Initial profiles of mean density and mean velocity are specified according to the c˜ profile. The mean velocity on the unburnt side of the flame brush remains zero at all times. The simulations are run until a steady propagation is observed and the time period for this depends on the flame and turbulence conditions specified at the start of the simulation. A value of 0.1 microsecond is used to time advance the equations in all cases. The time evolution is shown in Fig. 7.2 for the flame with K = 1 and u′/soL = 12 in the form of spatial variation of c˜ after every millisecond. The grid nodes are plotted for the last instant and it is evident that the flame brush is well resolved. After 0.0 0.2 0.4 0.6 0.8 1.0 0.15 0.17 0.19 0.21 0.23 0.25 c˜ x (m) ................................................................................ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ......................................... .. .. . . . . .. .. . . . . .. .. . . . . .. .. . . ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... .... .... ... .. . . .... .... ..... .......... . .. .. . . .. . . .. . . .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... .... .... .... ..... ... ... .... .... .... .... ..... . ... ........................ .... .... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... .... .... .... .... .... .... .... ... .... .... .... .... ..... ... ... .... .... .... .... ...... ......... . . ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... .... .... .... .... .... .... .... ... .... .... .... .... ..... .... ... ... .... .... .... .... ...... ......... ............... . .... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... .... .... .... .... .... .... ... ... .... .... ..... .... ..... ....... ............. ................ . ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ................. ............... ................ .............. .................. .......... ....... t t=0sec Figure 7.2: The progress of the solution in a typical one–dimensional flame cal- culation. Starting from an arbitrary initial profile (dashed line), the flame brush travels from right to left. some initial transients the flame propagation is steady and all the points on the c˜ profile move at the same speed. This is verified by tracking c˜ = 0.3, 0.5 and 0.7 with time. The rate of change of displacement of these tracked locations, Sd ≡ dx(c˜)/dt, is shown in Fig. 7.3 for two flames having u′/soL = 4, K = 0.15 102 7.2 RANS of planar one–dimensional flames −2 −1 0 1 2 3 4 0 0.001 0.002 0.003 0.004 (a) Sd ............................................................................................................. ........... .......... ....... ......... ......... ........ ........ ........ ........ ........ ......... ........... ............. ............... ................. ..................... ........................ ......................... ................................ .............................................. ..................................................................................................... .................................................................................................................................................................................................................................................................................................................... .............. ......... .......... ....... ....... ... ................................ .......................... ......................................................... ........... ........ ....... ..... ....... ...... ....... ....... ....... ....... ....... ....... ...... ..... .... ..... ........ ........ ................ .......... ................... ............ .......... ............ ........... ........... .......... ............ ........ ............ ........... ....................... ............. ............. .................................... ............. .............. ............... .............................................................................. .... .... ... .. . . ................ . . . ...... ............................ ................... .......... ................................................ ..... ....... ...... ..... .... .... .... ..... ... .. ... .. .... ..... .. ... .... ... ... ... .... .... ...... .... .... ....... ......... ....... ........ ....... ....... ....... ...... ........ ............ ....... ........ ...... .......... ........ .. ........ ............... .......... ........ ......... .......... ............. ...... .... ................ ........... . ......... .......... ........ ..... .. .......... ...................................................... ..... ... c˜ = 0.3 0 5 0.7 −2 0 2 4 6 8 0 0.002 0.004 0.006 (b) Sd t (sec) .................................................................................................. ..... ..... ... ..... ... ..... ... ..... ... ... ... ..... ..... ..... .... .... ..... ...... ........ .......... ............... ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ....... .... ..... ...................... ............ . ................................. .. .... .... .... .... .... ... .... .... .... .... .... ..... ... .... ..... .... ..... ......... .......... ........ ........ .................... ............ ...................................... .......... ..... ........................................................................................................................................................................................................................................................................................... . . . ... ..... . ....... ......... . . . . ..... ... ..... . ............... ... .... ... ... ... ... ... ... .. ... ... ... .... ... ... ... ... ... ... ... .... .... ...... . .. ..... ....... . ...... ........... ......... ....... ................ ....... . ...... .......... .................... . ....... ......... ......... .............. .............................................................................................................................c˜ = 0.3 0.5 0.7 Figure 7.3: History of displacement speed, Sd, in m/s for three c˜ values in two flames: (a) u′/soL = 4, K = 0.15 (b) u′/soL = 12, K = 1.0. The results are shown for the algebraic closure of the mean reaction rate. 103 7.2 RANS of planar one–dimensional flames and u′/soL = 12, K = 1.0. These two flames are close to the two extremes of the twelve flames marked in Fig. 7.1. The results shown in Figs. 7.2 and 7.3 are from algebraic model calculations and the similar behaviour is observed in all cases and with other models. The movement of iso–c˜ value is controlled by the diffusion and chemical reactions and thus the speed noted in Fig. 7.3 is the displacement speed, Sd. It is easy to show that this speed corresponds to the turbulent flame speed, ST , for the statistically planar flames considered here. Thus the stationary values such as those observed in Fig. 7.3 are taken to be ST . Equation (6.10), which is valid for this flame configuration, can be rewritten as ST = 1 ρu ∫ 1 0 ω˙(c˜) dc˜ , and hence the ST values will be dictated by the mean reaction rate closure. The ST values from calculations are compared to the experimental compilation Abdel- Gayed et al. (1987) and earlier computations of Bradley et al. (1994) in the next section. 7.2.2 Results and discussion Before comparing the turbulent flame speeds, let us study the structure of the flame brush predicted by the three reaction rate closures. In Fig. 7.4 the c˜ profiles of two flames with u′/soL = 4, K = 0.15 and u′/soL = 16, K = 1.0 are plotted as a function of a normalised spatial coordinate (x− x1)/δoL, where x1 corresponds to the location of c˜ = 0.5. The thickness of the profile given by the algebraic model is the largest and that given by the strained flamelet model is the smallest in Fig. 7.4. The normalised flame brush thickness, δt/δoL, shown in Fig. 7.5 confirms this observation for all cases. Alternative definitions of δt, such as that of Bray (1990): δt = ∫ ∞ −∞ c˜′′2dx , (7.8) or a definition based on mean reaction rate profile, yield the same trends as in Fig. 7.5. This figure shows that for a given K value, δt increases as u′/soL is increased. On the other hand for a given u′/soL, δt decreases as K is increased. The flame brush thickness, δt, is expected (Bradley et al., 1994; Peters, 2000) to 104 7.2 RANS of planar one–dimensional flames 0.0 0.2 0.4 0.6 0.8 1.0 −30 −10 10 30 50 (a) c˜ Algeb. UF SF ...... ...... ...... ..... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... .... .... .... .... .... .... ..... ...... ...... ...... . ..... ..... . ...... ... ... ...... . . . . . . . . ... ... ..... ... ... ..... ... ... .... .... . . ..... ... ... ..... ... ... ..... .. .. .... .. .. .... .. .. ...... . ... ........ ... ........ ... ........ ... .... .. ................................. .... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . .. .... .... .... .... ..... .... .... .... ..... ..... ..... ..... ..... ..... .... ..... ..... .... ..... ...... ......... ....................... ........................................... 0.0 0.2 0.4 0.6 0.8 1.0 −60 −20 20 60 100 (b) c˜ (x− x1)/δoL ...... ...... ...... .... .... ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . ..... . .... .... .... .... .... .... .... ...... ...... ...... ...... . ..... .... .. ...... . ..... ...... .. ........ ... .... .. .. ..... ... ... ..... ... ... ..... ... ... ..... ... ... ... ..... ... ... ..... ... ... ..... ... ... ..... ... ... .... .. .. .... .. . .. ........ ... ........ ... ........ ... ........ .. . .. ....................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... .. ... .... .... .... .... ... .... .... .... .... ..... ..... .... .... .... .... ..... ........ .......... ......................... ............... ... Figure 7.4: The spatial profiles of c˜ predicted by the various models plotted against the normalised distance for two cases: (a) u′/soL = 4, K = 0.15 (b) u′/soL = 16, K = 1.0. The location of c˜ = 0.5 in each case is denoted by x1. 105 7.2 RANS of planar one–dimensional flames 0 10 20 30 40 50 60 70 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 (a) δt δoL Algeb UF SF .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... . ◦ ◦ ◦ ◦ ◦ ◦ ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... M M M .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. • • • • • 0 30 60 90 120 150 0.0 5.0 10.0 15.0 20.0 25.0 (b) δt δoL u′/soL .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. ◦ ◦ ◦ ◦ ◦ .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... M M M ..... ..... ..... ..... ..... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . • • • • Figure 7.5: The normalised flame brush thickness, δt/δoL, predicted by various models at different u′/soL, for two values of flame stretch parameter: (a) K = 0.15 and (b) K = 1. 106 7.2 RANS of planar one–dimensional flames scale with the integral length scale, Λ. From the definition of the flame stretch factor it can be seen that (Λ/δ) ∼ (u′/soL)3/K2. Hence, for a fixed K value, Λ/δ increases with increase in u′/soL whereas, for a fixed u′/soL, Λ/δ decreases with increase in K. The present calculations hence seem to reproduce the scaling of δt with Λ. On the other hand, when Λ/δ is fixed, the calculations suggest that the flame brush thickness increases with u′/soL. This is evident by comparing the K = 0.15, u′/soL = 5 case with Λ/δ = 136.9 and the K = 1.0, u′/soL = 18 case with Λ/δ = 143.7. From Fig. 7.5 it can be seen that δt/δoL for the former case is 32 while for the latter it is 54. This trend is consistent with that observed in the DNS study of Veynante & Poinsot (1997). A higher turbulence intensity causes a greater wrinkling of the flame front resulting in a thicker flame brush (Veynante & Poinsot, 1997) which explains such a trend. Comparison of predicted flame brush thickness to its measured value for multidimensional laboratory flames will be discussed in Section 7.3.2.2. The density weighted pdf, p˜(ζ), from the strained flamelet calculations are shown in Fig. 7.6 for two cases at three locations inside the flame brush; c˜ = 0.1, 0.5 and 0.9. It is evident from this figure that there is a large contribution from the reactants compared to that from the products for the location near the unburnt side, c˜ = 0.1, and vice–versa for c˜ = 0.9. The contribution from the burning gases is non–negligible for c˜ = 0.5. This contribution can be quantified using ∫ 0.9 0.1 p˜(ζ) dζ, whose values are shown in the inset in Fig. 7.6b. A careful scrutiny of this figure shows that the burning mode contribution is increased at all locations inside the flame brush, which is more apparent for c˜ = 0.5, when the stretch rate K is increased. This is because of the flame front thickening at high stretch rates. The similarity of the pdfs in Fig. 7.6 to the experimental pdfs of Chen & Bilger (2002) is worth noting. The flame propagation speeds using the three ω˙ closures are compared with the experimental data of Abdel-Gayed et al. (1987) in Fig. 7.7. The predictions of the algebraic flame speed expression, Eq. (6.3), are also shown. The KPP analysis assumes a gradient transport for u˜′′c′′ and the variance is assumed to be given by the BML limit: c˜′′2 = c˜(1 − c˜). The present calculations with the algebraic reaction rate closure indeed yield c˜′′2 values close to the BML values 107 7.2 RANS of planar one–dimensional flames 0.1 1 10 0 0.2 0.4 0.6 0.8 1 p˜(ζ) ζ (a) c˜ = 0.1 + + + ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++ ++ +++ ++ + + + + c˜ = 0.5 × × ××××××××××××××××××××××××××××××××××××××××××××××××××× ×××××× ××× ×× ×× × × × c˜ = 0.9 ∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗ ∗∗∗∗ ∗∗∗ ∗ ∗ ∗ ∗ 0.1 1 10 0 0.2 0.4 0.6 0.8 1 p˜(ζ) ζ (b) + + + +++++++++++++++++++++++++++++++++++++++++++++++++ ++++ +++ +++ ++ + + + × × ×××××××××××××××××××××××××××××××××××××××××××××××××××× ××××× ××× ×× ×× × × ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗ ∗∗∗∗∗ ∗∗∗ ∗∗ ∗ ∗ 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 c˜ (a) ¤ ¤ ¤ ¤ (b) • • • • Figure 7.6: The shape of the density weighted pdf of progress variable, p˜(ζ), at three locations in the flame brush (c˜ = 0.1, 0.5 and 0.9) from the strained flamelet calculations for two cases : (a) u′/soL = 3, K = 0.15 and (b) u′/soL = 24, K = 1. The inset in (b) shows the values of ∫ 0.9 0.1 p˜(ζ; c˜) dζ for the two cases. 108 7.2 RANS of planar one–dimensional flames and thus it is not surprising to see the flame speeds predicted by this closure close to the KPP values in Fig. 7.7. In general, the curves in Fig. 7.7a suggest that for a given stretch value ω˙SF < ω˙UF < ω˙Algeb with the inequality becoming larger as u′/soL increases. The difference in the mean reaction rate values and thus the turbulent flame speeds increases further in highly stretched flames when u′/soL is large as shown in Fig. 7.7b. This is because of greater influence of the fluid dynamic stretch effects in highly strained flames. Consequently, the algebraic closure overpredicts the mean reaction rate and the flame speed at high u′/soL as depicted in Fig. 7.7. A similar trend is seen for the unstrained flamelets (UF) closure. The prediction by this model seems acceptable as in Fig. 7.7a when the combustion is in the corrugated flamelets regime. However, for flames in the thin reaction zones regime there is a large over–prediction. The flame speeds predicted by the strained flamelets (SF) closure are closer to the experimental data as in Fig. 7.7, when the combustion is in the corrugated flamelets as well as in thin reaction zones regimes. Also, the departure of results for the algebraic, UF and SF closures from one another reinforces that the stretch effects become prominent at large u′/soL. The strained flamelets model using the scalar dissipation rate is shown to capture these variations nicely. Bradley et al. (1992) observe that the experimental data of ST are correlated better with the parameter KLe instead of K as in (Abdel-Gayed et al., 1987). The compiled data, when grouped based on KLe, shows far less scatter than when grouped on K (Bradley et al., 1992). The basis for this lies in the fact that the stretch rate effect on the burning velocity of a laminar flame has a dependence on the Lewis number, as noted in Section 4.2.11. This can be verified in the one–dimensional flame calculations by considering mixtures of different Lewis numbers. Accordingly, a few cases of propane–air mixture of equivalence ratio 0.8, with Le = 1.83, were considered. Three cases corresponding to KLe = 1.0 and u′/soL = 6, 8 and 12 were computed. The choice of lean propane–air mixture was also driven by the need to validate the strained flamelet closure for Le > 1 mixtures. As noted in Section 4.2.2, when Le > 1, the integration of the ω˙ vs Nc curve to obtain 〈ω˙|ζ〉 requires approximations (see Fig. 4.5b and associated 1This is also manifest in Eq. (4.2). The Markstein length, L, has a dependence on the Lewis number (Law, 2006). 109 7.2 RANS of planar one–dimensional flames 0 5 10 15 20 25 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 (a) ST soL Expt. KPP Algeb UF SF ◦ ◦ ◦◦◦◦◦◦◦ ◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦ ◦ ◦ ◦◦◦ ◦ ◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦ ◦ ◦◦◦ ◦ ◦◦◦ ◦ ◦◦ ◦ ◦◦◦ ◦ .............. .......... ........ ....... ....... ....... ...... ...... ...... ..... .... .... .... .... ..... .... .... .... ..... ..... ..... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... .... ..... .... ... ... ... ... ... ... .. .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... M M M ......... ......... ......... ......... ........ ........ ........ ........ ........ ........ ........ ........ ...... ..... .... • • • • • 0 10 20 30 40 50 0.0 5.0 10.0 15.0 20.0 25.0 30.0 (b) ST soL u′/soL ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦◦ ◦ ◦ ◦ .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . M M M .... .... .... .... .... .... .... .... .... ..... ..... . ..... . ..... . ..... . ..... . ..... . ..... ........ ........ ........ ........ ....... ..... ..... ..... ..... ..... .... .... .... .... .... .... ... • • • • ............ .......... ......... ........ ....... ....... ..... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... Figure 7.7: Comparisons of predicted flame speeds with the experimental data of Abdel-Gayed et al. (1987) for two values of flame stretch parameter: (a) K = 0.15 and (b) K = 1. 110 7.2 RANS of planar one–dimensional flames discussion). For the cases considered, the two approaches prescribed in Section 4.2.2, yield ST values that are nearly equal (within 5%). The ST values computed using approximation (ii) in Section 4.2.2 are plotted in Fig. 7.8, and these are compared with the values corresponding to KLe = 1 case of the stoichiometric methane–air mixture. Note that for the methane–air mixture, KLe ≈ K since Le ≈ 1. Figure 7.8 indeed suggests that for mixtures with different Lewis numbers, the flame speed is very well parametrised by KLe, as observed by Bradley et al. (1992). 0 10 20 30 40 50 0.0 5.0 10.0 15.0 20.0 25.0 30.0 ST soL u′/soL ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦◦ ◦ ◦ ◦ ........ ........ ........ ........ ....... ..... ..... ..... ..... ..... .... .... .... .... .... .... ... • • • • ........... ........... ........... ........... ....... ♦ ♦ ♦ Figure 7.8: Flame speeds predicted by the strained flamelet closure for two mix- tures: stoichiometric methane–air with K = 1.0 (•) and lean propane–air with KLe = 1.0 (♦). The experimental data of Abdel-Gayed et al. (1987) for K = 1.0 are also shown (◦). The predictions of the SF closure are compared with the results of Bradley et al. (1994) in Fig. 7.9. The mean reaction rate closure in Eq. (4.6) is used by Bradley et al. (1994) and the results were reported based on a parameter KLe. Their results for KLe = 0.14 and 1.0 are compared with the current results for KLe = 0.15 and 1.0 respectively in Fig. 7.9. As noted earlier, Eq. (4.6) includes the stretch effects via the burning rate factor, Pb. More importantly, second order turbulence closures are used by Bradley et al. (1994) by solving the transport equations for the Reynolds stresses and turbulent scalar flux as opposed 111 7.2 RANS of planar one–dimensional flames 0 5 10 15 20 25 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 (a) ST soL Bradley et al. (1994) ◦ ◦ ◦◦◦◦◦◦◦ ◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦ ◦ ◦ ◦◦◦ ◦ ◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦ ◦ ◦◦◦ ◦ ◦◦◦ ◦ ◦◦ ◦ ◦◦◦ ........ ........ ......... ......... ......... .......... .......... ......... ........ ........ ......... .......... .......... .......... ......... ......... ...... ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ......... ......... ......... ......... ........ ........ ........ ........ ........ ........ ........ ........ ...... ..... .... • • • • 0 10 20 30 40 50 0.0 5.0 10.0 15.0 20.0 25.0 30.0 (b) ST soL u′/soL ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ ◦ ◦◦ ◦◦◦ ◦◦ ◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦◦ ◦ ◦ ◦ .... .... .... .... .... ........ ........ ......... ... ∗ ∗ ∗ ∗ ∗∗ ........... ............ ............ ............ ......... ........ ........ ........ ........ ..... ..... ..... ..... ..... .... .... .... .... .... .... .... • • • • • • Figure 7.9: The comparisons of flame speeds predictions with the predictions of Bradley et al. (1994) for two values of flame stretch parameter: (a) KLe = 0.15 and (b) KLe = 1. to the simple closures used in the present calculations, and the dissipation rate of c˜′′2 in Eq. (3.1) is modelled by the classical linear relaxation model by Bradley et al. (1994). The calculations with the standard k˜–ε˜ closure in the work of Bradley et al. (1994) yielded higher flame speeds than with second order closures. Figure 7.9 suggests that the present calculations agree very well with the results of Bradley et al. (1994), which is surprising given the many differences in the implementation and details of turbulence and combustion models. Quantitatively, 112 7.3 RANS of laboratory flames the mean reaction rate across the flame brush depends on c˜′′2 , k˜ and ε˜ values inside the flame brush. This is evident from Eqs. (5.18) and (4.11) for the SF closure. For the closure of Bradley et al., this dependence arises via the pdf p(ζ) and the factor Pb. It is likely that the quantitative differences in the resulting ω˙ values are not large, and this could explain the agreement seen in Fig. 7.9. The comparisons in Figs. 7.7 and 7.9 clearly illustrate the benefit of accounting for local fluid–dynamic effects on the reaction rate for turbulent premixed flames. This becomes more important at high turbulence conditions because it can be expected that the local flamelet structure is disturbed by the straining induced by turbulence eddies to a larger extent from that of a freely propagating unstrained laminar flame. Flamelet models based on steady strained laminar flames seem promising in this respect. 7.3 RANS of laboratory flames The planar one–dimensional flame calculations clearly showed the benefits of strained flamelets and the use of scalar dissipation rate, by capturing the re- duction in burning rate and the flame speed as stretch rate increases. This be- haviour is captured well for both the corrugated flamelets and thin reaction zones regimes of turbulent premixed combustion. In this section, calculations of labora- tory flames are discussed with the aim of examining the predictive ability of this strained flamelets approach across the combustion regimes by using a single set of values for model parameters. Two different laboratory flames: a rod stabilised V–flame of Robin et al. (2008) and pilot stabilised Bunsen flames of Chen et al. (1996) are considered. The V–flame conditions correspond to the corrugated flamelets regime and the Bunsen flame conditions correspond to the thin reaction zones regimes in the combustion regime diagram shown in Fig. 7.10. While the applicability of flamelet models in the former regimes is intuitive, their applica- bility to the latter requires a suitable description of fluid–dynamics influence on the laminar flame structure. This was appropriately addressed in the strained flamelets model and this will be tested using the Bunsen flames calculations. 113 7.3 RANS of laboratory flames 0.1 1 10 100 1000 0.1 1 10 100 1000 u′/soL Λ/δ ........................................................................................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . ................................................................................................................................................................................................................................. ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........ ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ....... ......... ......... ......... ......... ......... ......... ......... • ¨ ¨¨F1F2 F3Re = 1 Ka = 100 Da = 1 Ka = 1 Wrinkled Corrugated Thin reaction zones Distributed reaction zones Figure 7.10: The regime diagram with conditions of the V–flame (•) of Robin et al. (2008) and the Bunsen flames (¨) of Chen et al. (1996). 7.3.1 Computational details The finite volume form of the discretised transport equations are implemented in Rolls-Royce proprietary CFD code PRECISE (Anand et al., 1999). Appropriate sub–routines were added to this code for the implementation of the strained flamelets reaction rate model. 7.3.1.1 V–flame calculations The V–flame experiments of Robin et al. (2008) were designed mainly to study stratified flames but measurements for premixed flames were also reported. A schematic of the flame and the coordinate directions are shown in Fig. 7.11. The z direction is the streamwise direction and x is the transverse direction. The origin of the coordinate system is located at the centre of the stabilising rod. The mean quantities are assumed not to vary much in the y direction and hence the flames can be treated as two–dimensional with symmetry about the x = 0 plane. 114 7.3 RANS of laboratory flames 11 0 m m 70 mm 80 mm x z CH4­air (ø = 0.6) Figure 7.11: The schematic of the computational domain for the V–flame calcu- lation. The ‘O’–grid in the region close to the rod is also shown. A rectangular computational domain in the region -10 mm ≤ z ≤ 100 mm and 0 ≤ x ≤ 70mm is considered as shown in Fig. 7.11. Note that in the x direction the computational domain is wider than the half burner width (40mm) to allow for the entrainment of ambient air. The computational domain is discretised into a structured grid of nearly 30000 cells with the smallest cell size of 0.05mm × 0.1mm nearest to the rod and the cell size gradually increases away from the rod as shown in Fig. 7.11. The grid generation is performed using the ‘O’–grid utility of the commercial package ANSYS–ICEMCFD. The flame calculated in the present study is that designated as “HE06” by Robin et al. (2008) and it is shown in the regime diagram in Fig. 7.10. It corre- sponds to a lean methane–air mixture of equivalence ratio φ = 0.6 with unstrained laminar flame speed of soL = 0.13 m/s and the Zeldovich thickness of δ = 0.163 115 7.3 RANS of laboratory flames mm. Turbulence at the burner exit was generated by different grids and for the flame HE06 the mean exit velocity is U = 3.14 m/s, the normalised turbulence RMS velocity is u′/soL = 2.14 and the turbulence integral length scale is 6.1 mm. In the calculations, uniform conditions at the inlet boundary (z = -10 mm, 0 ≤ x ≤ 40 mm) are specified based on the values reported by Robin et al. (2008) and they are, k˜ = 0.15 m2/s2 and ε˜ = 10 m2/s3. The entrainment is facilitated by setting a small inlet velocity of 0.3 m/s in the region x > 40 mm. The fluid density can be influenced by combustion as well as mixing with ambient air. To account for this mixing, the transport equation for mean mixture fraction, ξ˜ is also solved and it’s inlet values are specified to be ξ˜ = 1 for 0 ≤ x ≤ 40mm and ξ˜ = 0 for x > 40mm. Note that ξ˜ = 1 here corresponds to a reacting mixture of equivalence ratio 0.6. It was assumed that the flame brush never interacts with the entrainment air which was verified from the ξ˜ field calculations. Such an assumption simplifies the calculation of the mean density via the formula ρ = [ (1− ξ˜) ρair + ξ˜ ρreac ]−1 , (7.9) where the density of air is ρair and the density of the reacting mixture, ρreac , is related only to c˜ via Eq. (2.27). The unburnt density of the reacting mixture is ρu. Hence the combustion affects the flow field only via ρ which is related to ξ˜ and c˜ using the above relations. The flame is anchored numerically by patching a small region of cells (5 × 5) just adjacent to the rod with burnt conditions (c˜ = 1). This is reasonable since the flame is anchored by a heated rod in the experiment. The grid independence of the solution is also verified. 7.3.1.2 Bunsen flames calculations The pilot stabilised Bunsen flames of Chen et al. (1996) have been computed in earlier studies using different modelling approaches such as the Flame Surface Density approach (Prasad & Gore, 1999), the level set G–equation approach both in RANS (Herrmann, 2006; Schneider et al., 2005) and LES (De & Acharya, 2008; Pitsch & de Lageneste, 2002) contexts and the transported PDF approach (Lindstedt & Vaos, 2006). The flames correspond to stoichiometric methane air mixtures and the same mixture is used for the pilot flames (Chen et al., 1996). 116 7.3 RANS of laboratory flames Here, all three flames designated as F1, F2 and F3 by Chen et al. (1996) are calculated. The Reynolds number based on jet exit diameter and bulk mean velocity is 52000, 40000 and 24000 for flames F1, F2 and F3 respectively. These flame conditions, shown in Fig. 7.10, lie in the thin reaction zones regimes and are challenging to compute with flamelet models in RANS as noted earlier. The centreline value of the turbulence RMS velocity, u′, and the lateral integral length scale of 2.4 mm measured (Chen et al., 1996) near the nozzle exit are used to calculate the flame conditions in Fig. 7.10. 70  D 45 D r x Pilot (68 mm) D = 12 mm Figure 7.12: The schematic of the computational domain for the Bunsen flames calculations. The diameter of the nozzle jet is D. A two–dimensional axisymmetric computational domain of 70D in the axial direction, x, and 45D in the radial direction, r, was considered where D = 12 117 7.3 RANS of laboratory flames mm is the diameter of the nozzle jet. The diameter of the pilot is 68 mm. The domain was discretised into a structured grid of non–uniform cell sizes with the smallest cells of 0.5mm×1.0mm close to the nozzle inlet plane. The schematic of the computational domain is shown in Fig. 7.12. The value of the standard k˜–ε˜ model constant Cε1 is changed from 1.44 to 1.52 to account for the round–jet anomaly (Pope, 1978). The mean axial velocity profiles measured at the nozzle exit (Chen et al., 1996) are used to set the correpsonding velocity boundary conditions. The measured turbulent RMS velocity profiles and the longitudinal length scale are used to set the boundary conditions for k˜ and ε˜. A uniform velocity for the pilot stream was set based on the volume flow rate through the pilot. A small velocity of 0.2 m/s was specified for the coflow air to allow for ambient air entrainment. The experimental conditions present a difficulty for modelling with adiabatic flamelets since the temperatures of the pilot stream are sub–adiabatic due to losses to the burner wall and the presence of a cooling water circuit. To overcome this difficulty the following methodology is adopted. Transport equations for the mean and variance of the normalised fuel mass fraction, cf = 1 − Yf/Yfu are solved instead of the progress variable based on the temperature. The equations of c˜f and c˜′′ 2 f are identical to that of c˜ and c˜′′ 2 since the Lewis number is close to unity for the reactant mixtures. Adiabatic flamelets are still used but to account for sub–adiabatic temperatures at the pilot boundary, the transport of the Favre averaged total enthalpy, h˜, is considered. The transport equation of h˜ for low speed flows is (Libby & Williams, 1980) ∂ρh˜ ∂t + ∂ρu˜ih˜ ∂xi = ∂ ∂xi [ µt Pr ∂h˜ ∂xi ] . (7.10) In the above equation, the effects of viscous dissipation, body forces and radiation have been assumed to be negligible. The total enthalpy is defined as the sum of sensible and chemical enthalpies of the mixture, h˜ = Cpmix(T˜ − T o) + ∆hofmix (7.11) where Cpmix = ∑ α Y˜α Cp,α , (7.12) 118 7.3 RANS of laboratory flames and, ∆hofmix = ∑ α Y˜α ∆h o f,α . (7.13) The reference temperature is T o (298 K). Here too it is assumed that the en- trained ambient air mixes only with the fully burnt mixture since the pilot is large compared to the main jet. Similar to the V–flame calculations, the effect of entrainment is accounted via the mean mixture fraction, ξ˜, whose value is 1 in the main jet and the pilot stream and 0 in the ambient air stream. The values of Cpmix and ∆hofmix are then approximated as: Cpmix ≈ ξ˜ Cpreac + (1− ξ˜) Cpair , ∆hofmix ≈ ξ˜ ∆hofreac + (1− ξ˜) ∆hofair , (7.14) where Cpreac and ∆hofreac are for the reacting component of the mixture and Cpair and ∆hofair are for the air. Since the air stream is assumed to be comprised of only O2 and N2 at 298 K, ∆hofair = 0. The values of Cpreac and ∆h o freac are obtained from laminar flame calculations and tabulated along with the mean reaction source terms as functions of c˜f , c˜′′ 2 f and ²˜cf . Thus the temperature is calculated from the local values of c˜f , c˜′′ 2 f , ²˜cf , ξ˜ and h˜ via Eqs. (7.11) and (7.14). The mean density is then obtained from the ideal gas relation. Such a formulation allows for specification of sub–adiabatic temperature bound- ary conditions in the pilot region through h˜. However, the measurements of Chen et al. (1996) indicate a radial variation in the pilot temperature values. At a radial location of 10 mm (r/D ∼ 0.83) the measurements indicate that the tempera- ture is less than 1500 K while at 20 mm it is about 2000 K. Lindstedt & Vaos (2006) considered two values for the pilot temperature: 1785 K and 2005 K in their calculations while Herrmann (2006) used a value of 1936 K. Measurements of the radial variation of the pilot stream temperature, ideal from the boundary condition point of view, have not been reported by Chen et al. (1996). In the present calculations we choose an ad–hoc value of 1950 K as the uniform pilot temperature in the reacting flow calculations. The approach described above in Eq. (7.14) is an approximate way to account for the influence of the ambient air whilst using a purely premixed model for 119 7.3 RANS of laboratory flames turbulent combustion. The calculations of Herrmann (2006) support such an approximation for the lower Reynolds number flame F3. For higher Reynolds numbers the entrainment is larger and there could be some interaction of the ambient air with the flame front, especially at locations far from the nozzle. A more accurate way to account for this mixing is to consider the influence of stratification on the flame front characteristics and reacting mixture composition. However, our focus in the present study is on a purely premixed model and hence we proceed with the above approximation. Accordingly the tabulation for ω˙, ω˙c, Cpreac and ∆hofreac is based only on laminar flame calculations of stoichiometric methane–air mixture. 7.3.2 Results and discussion 7.3.2.1 V–flame The cross stream profiles of longitudinal and transverse velocities at three z lo- cations from the V–flame calculations are compared with experimental data in Fig. 7.13. The velocity predictions of the unstrained and the strained flamelet formulations are very close to each other. The quantitative agreement with the experimental velocity profiles is good over most of the x range except a small disagreement (about 5%) for the longitudinal velocity in the vicinity of x = 0. A similar disagreement of centreline velocities were reported by Robin et al. (2008) as well as the V–flame DNS of Bell et al. (2005). This was attributed to the recirculation zone behind the rod. However the centreline velocity in Fig. 7.13 is qualitatively similar to the DNS result (Bell et al., 2005) whereas the velocity profile calculated by Robin et al. (2008) (not shown here) has a local minimum at x = 0. This could be because the stabilising rod has a semi–circular cross section in Robin et al. (2008) whereas it is circular here. The reason for choosing a semi–circular cross section in Robin et al. (2008) is unclear. Nonetheless the velocity profiles are well predicted in the present calculations and in particular the transverse velocities induced due to the streamline divergence through the flame brush are close to the experimental values. The computed Reynolds mean progress variable, c, is compared with the ex- perimental data for two z locations in Fig. 7.14. The experimental values obtained 120 7.3 RANS of laboratory flames 3.0 3.2 3.4 3.6 3.8 3.0 3.2 3.4 3.6 3.8 3.0 3.2 3.4 3.6 3.8 4.0 -20 -10 0 10 20 x(mm) ........ ......... ........ ....... ...... ....... .................... .... ........................................................................ ........ ......... ........ ....... ....... ......... ......... ........ ..... ......................................................................... ......... ........ ....... ....... ........ ........ ....... ...... .... .... .... ................................................................................ .. ....... ....... .... ....... ...... ..... .... . ..... ..... .. .... . .... . .... . ..... ..... ..... ..... ..... ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ z = 20mm 30mm 40mm -1.0 -0.5 0.0 0.5 -1.0 -0.5 0.0 0.5 -1.0 -0.5 0.0 0.5 1.0 -20 -10 0 10 20 x(mm) ............................................. .... ..... ..... ...... ............. ..... ..... ..... ............................................. ......................................... .... ..... ..... ..... ................... ..... ..... ..... ..... ..................................... ................................... .... ..... .... .... ...... ..................... ..... .... .... .... ........ ............................ ....... ....... ....... ....... .. ...... ....... ....... ....... ... ....... ....... ....... ....... ....... ....... ....... .... ....... ....... ....... . .. ... . .. .. .. .. ... ....... ....... ....... ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦ z = 20mm 30mm 40mm Figure 7.13: The calculated longitudinal (left) and transverse velocities (right) in m/s using the unstrained flamelets (dashed lines) and strained flamelets (solid lines) models are compared with the experimental data (◦) for the V–flame (Robin et al., 2008) at three downstream locations. from binarised tomographic images are shown in the figure. The Reynolds mean which is reported in the experiments, is obtained from the Favre mean using c = c˜+ τ c˜′′2 1 + τ c˜ . (7.15) This equation results from the presumed β shape for p˜(ζ) and it is derived in the Appendix A. Figure 7.14 suggests that the flame brush width is also well predicted by both the unstrained and the strained flamelet formulations although the former predicts slightly wider flame brush than the latter. The planar one–dimensional flame calculations in Section 7.2.2 showed similar trend in flame brush thickness predicted by the two models. The small difference in the unstrained and strained flamelets predictions is not surprising since the combustion conditions correspond 121 7.3 RANS of laboratory flames 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -40 -20 0 20 40 c c x(mm) ............................................. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ....................................................................................................................................... ................................... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ..................................................................................................................................... ... .... ... .... ... .... ... .... ... .... .. ....... ....... ....... ....... ....... ... .... . .... ... .... ... .... ... .... ... ..... .. ..... .. .. .... ....... ....... ....... ....... ....... ....... ... ······ ···· ···· ···· ···· ···· ································· ······ ···· ···· ···· ···· ···· ····· ··································· 42mm 70mm Figure 7.14: The calculated mean progress variable using the unstrained flamelets (dashed lines) and strained flamelets (solid lines) models are compared with the experimental data (dots) of the V–flame (Robin et al., 2008) at two streamwise locations. to the corrugated flamelets regime. The planar flame calculations have shown that for low turbulence intensities, the mean reaction rate values predicted by both the closures are not very different. The contours of ω˙/ρ shown in Fig. 7.15 provide further evidence for this. The contours in the left half of Fig. 7.15 (x < 0) correspond to the strained flamelets and those in the right half (x > 0) correspond to the unstrained flamelets models. The values of ω˙/ρ are nearly the same in the flame brush with the two models. The predictions of the reaction rate by these two models are likely to be significantly different for high turbulence intensities as observed in the planar flame calculations. 7.3.2.2 Bunsen flames Calculations of non–reacting flow were performed to verify the turbulence mod- elling and the boundary conditions discussed in Section 7.3.1.2. Results for the F2 case are shown in Fig. 7.16, where the predictions of normalised mean axial velocity and mean turbulent kinetic energy, k˜/ko are compared to the experimen- tal data. The bulk mean velocity at nozzle exit is Uo and the turbulent kinetic 122 7.3 RANS of laboratory flames Figure 7.15: The contours of ω˙/ρ (s−1) from the V–flame calculations. The contours in x < 0 region are for the strained flamelets while those in x > 0 region are for the unstrained flamelets. 123 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.5 U Uo r/D .......................................................................................................................................................................... .................................................................................................................................................. ........................................................................................................................................ ................................................................................................................................................. ......................................................................................................................................... ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 15 k ko r/D ....... .... ..... .... .... .... .... .... .... .... .... ..... .............................................................................................................................................. .... ..... ..... ..... ..... ..... ..... ..... .... .......................................................................................................................... ....... .... .... ...... ................................................................................................................... .............. ................................................................................................................................... ......................................................................................................................................... ◦◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 Figure 7.16: The normalised mean axial velocity and normalised mean turbulent kinetic energy from non–reacting flow calculations (solid lines) of case F2 are compared with the experimental data (◦) of Chen et al. (1996). 124 7.3 RANS of laboratory flames energy at the centreline of the nozzle exit is ko and their values for flame F2 are 50 m/s and 10.8 m2/s2 respectively (Chen et al., 1996). The radial spread of the jet can be clearly seen in the velocity profiles which is in very good agreement with the experimental data. The centreline value seems to be slightly underpre- dicted at x/D = 6.5 and 8.5. The generation of turbulence in the shear layer is evident from the turbulent kinetic energy profiles showing a peak at roughly r/D = 0.5. Both the location and the magnitude of this peak are very well predicted upto x/D = 8.5. At x/D = 10.5, the predicted values are slightly lower than the experiment at all radial locations while the qualitative trend is well captured. Experimental data for F1 and F3 cases was not reported in (Chen et al., 1996). However, assuming the jet obeys self–similarity, the experimental data of U/Uo and k/ko for flame F2 are compared with the predictions for flames F1 and F3 in Figs. 7.17 and 7.18 respectively. The values of Uo are 65 m/s and 30 m/s and those of ko are 12.7 m2/s2 and 3.82 m2/s2 for F1 and F3 respectively (Chen et al., 1996). Figures 7.17 and 7.18 indeed show the self–similar behaviour of the jet and the predictions for these flames are also in good agreement with the experimental data. The overall good prediction of the non–reacting flow provides validation for the turbulence modelling. The results of reacting flow calculations are presented next. Unless specified otherwise, the results in the following discussion are for the strained flamelets model. The normalised mean velocity, U˜/Uo, and turbulent kinetic energy, k˜/ko, computed for the three flames are compared to the experimental values in Figs. 7.19 and 7.20 respectively. While the mean velocities are well predicted at all axial locations, the values of the turbulent kinetic energy are overpredicted for loca- tions close to the nozzle. At x/D = 2.5 this overprediction is about 40 % for the flame F1 while it is nearly a factor of 3 for the F3 flame. The agreement improves further downstream and for the flames F1 and F2, the agreement is very good at x/D = 8.5 and 10.5. Furthermore, even when the magnitudes are overpredicted, the radial location of the peak due to shear generated turbulence is well captured for all flames and at all locations. The values of k˜ predicted by Herrmann (2006) using a standard k˜–ε˜ model with round jet correction, are lower than the values shown in Fig. 7.20. The present values, however, are closer to the predictions of Lindstedt & Vaos (2006) using Reynolds stress closure. 125 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.5 U Uo r/D ............................................................................................................................................................................. .................................................................................................................................................... ........................................................................................................................................ ................................................................................................................................................. ......................................................................................................................................... ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 15 k ko r/D ...... .... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ...... ................................................................................................................................................................ .... ..... ..... ..... ..... .... .... .... .... .... .... ..... .... ..... .................................................................................................................................................. ...... .... ..... ..... .... ....... ............................................................................................................................ ............ ......... ..................................................................................................................... ....................... ........................................................................................................................... ◦◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 Figure 7.17: The normalised mean axial velocity and normalised mean turbulent kinetic energy from non–reacting flow calculations (solid lines) of case F1 are compared with the experimental data (◦) of Chen et al. (1996). 126 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.5 U Uo r/D .......................................................................................................................................................................... ................................................................................................................................................. ........................................................................................................................................ ................................................................................................................................................. ......................................................................................................................................... ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 15 k ko r/D ...... .... .... ..... .... .... .... .... .... .... .... ..... ...... ......................................................................................................................................... .... ..... ..... ..... ..... ..... ..... ..... ..... ......................................................................................................................... ....... .... .... ........ ............................................................................................................... ............... .................................................................................................................................. ......................................................................................................................................... ◦◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 Figure 7.18: The normalised mean axial velocity and normalised mean turbulent kinetic energy from non–reacting flow calculations (solid lines) of case F3 are compared with the experimental data (◦) of Chen et al. (1996). 127 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.5 U˜ Uo r/D ..................................................................................................................................................................... ........................................................................................................................................... ......................................................................................................................................... ......................................................................................................................................... ........................................................................................................................................... ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.5 r/D ............................................................................................................................................................... ............................................................................................................................................... .......................................................................................................................................... .......................................................................................................................................... .......................................................................................................................................... ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.5 r/D ..................................................................................................................................................... ........................................................................................................................................... .......................................................................................................................................... ......................................................................................................................................... ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 Figure 7.19: The normalised mean axial velocity from reacting flow calculations of flames F1 (left), F2 (center) and F3 (right) are compared with the experimental data of Chen et al. (1996). 128 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 15 k˜ ko r/D ...... .... .... ..... ..... ..... ..... ..... ..... .... .... ..... ..... .... .............................................................................................................................................. ....... .... .... .... .... .... ..... ..... ..... ..... ..... ..... .... ..... ............................................................................................................................. ........... ...... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... .... .................................................................................................................. ............. ....... .... .... .... ..... ..... ..... ..... ..... ..... ..... .... .... ........................................................................................................ ............. ....... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... .... ...... ...................................................................................... ◦◦◦◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦ ◦ ◦◦◦ ◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦◦◦◦ ◦ ◦◦◦◦◦◦◦◦ ◦ ◦ ◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 12 r/D ...... .... .... ..... ..... ..... ..... .... .... .... .... .................................................................................................................................... ......... ..... ..... ...... .... .... .... .... .... .... ...... .............................................................................................................. ............ ........ ....... .... .... .... .... ..... ..... ..... .... .... .......................................................................................................... ............... ........ .... .... ..... ..... ..... ..... ..... ..... .... .... ...... ........................................................................................ ................ ....... .... .... .... ..... ..... ..... ..... ..... ..... ..... .... ................................................................................ ◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦◦ ◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10 r/D ...... .... ..... ..... ..... ..... ..... ..... .... .... .... .... ....... .................................................................................................................................. ........ .... .... ...... ....... .... .... .... ..... .... .... .................................................................................................................. ............. ......... ........ ...... .... .... ..... ..... ..... ..... ..... .... ........................................................................................................ .................. ......... ....... .... .... .... ..... ..... ..... ..... ..... ..... ..... .... ............................................................................................ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦ ◦◦◦ ◦◦◦ ◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 Figure 7.20: The normalised mean turbulent kinetic energy from reacting flow calculations using the strained flamelets model of flames F1 (left), F2 (center) and F3 (right) are compared with the experimental data of Chen et al. (1996). 129 7.3 RANS of laboratory flames Figure 7.21: The contour plot of temperature (K) from the Bunsen flame calcu- lations for case F3. The plot for the entire computational domain (top) and the region close to the nozzle exit (bottom) are shown. 130 7.3 RANS of laboratory flames In Fig. 7.21 the contour plots of temperature from the reacting flow calcula- tions of flame F3 are shown. The plot above in Fig. 7.21 shows the temperature variation in the entire domain while the region close to the nozzle exit is shown in the plot below. This figure clearly shows that while the temperature at the pilot exit (0.5 < r/D < 2.83) is sub–adiabatic (1950 K), the temperature in- creases downstream as more heat is released from the chemical reactions. The measurements of Chen et al. (1996) show the same behaviour. This feature of the pilot–stabilised flames is very well captured in the calculations via the transport equation for the mean enthalpy, h˜. The interaction of burnt products with the ambient air and the resultant drop in temperature is also well captured in the simulation. The calculated Reynolds mean progress variable, c, and mean CH4 mass frac- tion are compared with the experimental data for the three flames in Figs. 7.22 and 7.23 respectively. The agreement for CH4 mass fraction is very good for all the cases indicating that the transport of c˜f is well predicted. The quantity c from the calculations is obtained from Eq. (7.15) with an assumption c˜′′2 ≈ c˜′′2f . Such an assumption seems, at first, reasonable since the diffusivities of c and cf are nearly equal for methane–air mixtures. The radial decrease in calculated temperature due to mixing with the entrained cold air is clearly seen in Fig. 7.22. The prediction of c is reasonable and the rise in centreline temperatures is well predicted. However, there are some notable discrepancies between the calculated and measured values. For locations close to the burner exit, the calculated peak temperature is higher and the temperature rises quickly as in Fig. 7.22. The most plausible reason for this is the overprediction of c˜ and c˜′′2 , both of which are attributable to the pilot temperature boundary condition used. As noted earlier, the measured temperature at r/D = 0.83 is about 1500 K which is considerably lower than the value of 1950 K specified in the calculations. Consequently, at a given radial location in the flame brush, this leads to a higher mean temperature in the calculation compared to the experiment. Furthermore, the assumption c˜′′2 ≈ c˜′′2f results in an overprediction of c˜′′2 , because of sub-adiabatic pilot stream temperatures used to specify the boundary conditions for the enthalpy equation given in Eq. (7.10). These factors contribute to an overprediction of c through Eq. (7.15). A similar overprediction of mean temperature is seen in earlier studies 131 7.3 RANS of laboratory flames (Herrmann, 2006; Lindstedt & Vaos, 2006; Pitsch & de Lageneste, 2002; Schnei- der et al., 2005) as well. Lindstedt & Vaos (2006) found that a pilot temperature of 2005 K was too high and hence performed calculations with a much lower value of 1785 K. However, this boundary condition mainly affects predictions close to the nozzle and its influence diminishes further downstream. For x/D ≥ 6.5 the agreement is good for all three flames similar to previous study (Lindstedt & Vaos, 2006). The overprediction of c˜′′2 due to the pilot temperature boundary condition is evident in the comparisons of root mean square fluctuations, √ c′2 , with experi- mental data shown in Fig. 7.24. The c′2 value is obtained using c′2 = c˜2 + c˜′′2 1 + τ c˜ { 1 + τ c˜2(1− c˜) + c˜′′2(2− c˜) c˜′′2 + c˜(1− c˜) } − c2 (7.16) and the computed value of c˜′′2f which is taken equal to c˜′′ 2 . Equation (7.16) is derived in Appendix A. Similar to c, values of c′2 are overpredicted at x/D = 2.5 and 4.5, but the prediction improves further downstream due to the reasons described above. However, the location of the peak in c′2 profiles is well predicted for all flames at all locations. While Fig. 7.22 suggests some discrepancies in the prediction of mean tem- perature it will be interesting to compare the gradient of c with the experiments. Chen et al. (1996) define a flame brush thickness as δt ≡ (∂c/∂r)−1max, and its calculated values are compared to the experimental data in Fig. 7.25. For flames F1 and F2 the experimental values of δt do not vary much while the calculated values show a slight increase with x/D. For flame F3, Chen et al. (1996) suggest that δt variation in the experiment follows the variation of peak c′2 value, which increases with x/D. The calculations show a similar correspondence between δt and the peak c′2 values. In general, the calculated flame brush thicknesses seem reasonably close to the experimental values for all flames. The predictions of c and mean CH4 mass fraction from the calculations using the unstrained flamelets closure are compared with the experimental data for the flame F2 in Fig. 7.26. The CH4 comparisons indicate that the calculated flame is “hungrier” and consumes reactants much faster compared to the experiment. At x/D = 8.5 nearly all the fuel is consumed in the calculations, but a significant 132 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0 c r/D ................. .... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ................................................................................................................................. ........... .... ..... .... .... .... .... .... ..... .... .... .... .... .... .... ..... .... ...... ......................................................................................................... ........ .... .... .... .... .... ..... ..... ..... ..... ..... ..... .... ..... ..... ...... ................................................................................................ ........ .... .... .... ..... ..... ..... ..... ..... .... ..... .... .... .... ........ ............................................................................................. ....... ..... ..... ..... ..... ..... ..... ..... ..... ..... .... ..... ......... ............................................................................................... ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0 r/D ................. ..... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... .... ............................................................................................ ............ ..... ..... ..... .... .... .... ..... .... .... .... .... .... .... ..... ..... ..... ..... .... ................................................................................................................... ......... .... ..... ..... .... ..... .... .... .... .... ..... ..... .... .... ..... ..... .... ..... .................................................................................................... ........ .... .... ..... .... .... ..... .... .... .... .... ..... ..... ..... ..... .... ...... ............................................................................................. ....... ..... .... ..... ..... .... ..... ..... ..... ..... ..... ..... ..... .... ....... ................................................................................................. ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦ ◦◦◦ ◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0 r/D ................. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ........................................................................ ............ .... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... .... .... ................................................................................................... .......... .... .... ..... ..... ..... .... .... .... ..... .... .... .... .... .... ..... .... .... .... ..... ..... .... .... ............................................................................................................. ......... .... .... .... ..... .... .... .... .... .... .... .... .... ..... ..... .... .... .... .... ..... .... .... ........................................................................................ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦◦ ◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦ ◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 Figure 7.22: The mean progress variable, c, using the strained flamelets model is compared with the experimental data of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). 133 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10 Y CH4 r/D .................................................................................................................................................. ............................................................................................................................................. .................................................................................................................................... ................................................................................................................................... ................................................................................................................................ ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10 r/D ................................................................................................................................................... ............................................................................................................................................. ........................................................................................................................................ ...................................................................................................................................... .................................................................................................................................... ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10 r/D ................................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................ ........................................................................................................................................... ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 Figure 7.23: The mean CH4 mass fractions from strained flamelets calculations are compared with the experimental data of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). 134 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 √ c′2 r/D ...... ..... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..................................................................................................................... ...... ..... ..... .... .... .... .... .... .... .... .... .... .... .... .......................................................................................................................... ..... .... .... .... ..... .... .... ..... ..... .... ................................................................................................................... ...... .... .... .... ..... .... ...... .................................................................................................................. ....... .... .... .... ..... .................................................................................................................. ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 r/D ....... .... ..... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ........................................................................................................................................... ...... ..... .... .... .... .... .... .... .... .... .... .... ..... ..... ..................................................................................................................... ...... ..... ..... .... .... .... .... ..... ..... ..... ...... ................................................................................................................. ...... ..... ..... .... .... ..... .... ..... ................................................................................................................... ...... ..... ..... ..... ..... .... ....... ............................................................................................................ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦◦◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 r/D ......... ..... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... .................................................................................................................................. ...... ..... ..... .... .... .... .... .... .... .... .... .... ..... .... ......................................................................................................................... ....... .... .... ..... .... .... ..... .... .... ..... ..... .... ........................................................................................................................ ...... ..... ..... .... .... .... .... ..... .... ..... ..... ....................................................................................................................... ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 Figure 7.24: The root mean square progress variable fluctuations, √ c′2 , from strained flamelets calculations are compared with the experimental data of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). 135 7.3 RANS of laboratory flames 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 δt x/D x/D x/D F1 F2 F3 .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ...... ........ ........ ........ .... .... .... .... .... .... ..... ..... ..... ..... ........... ........... .......... .......... .......... ....... ....... ....... ....... ..................... ............................. .. .. .. .. .. .. .. .. .... ...... ...... ....... ........ ........ ..................... ......... ........ ... .... ................................ ... ... ... ... ... ... . . ... .... .... .... ..• • • • • • • • • • • • • • • Figure 7.25: The turbulent flame brush thicknesses, δt (mm), from the strained flamelets calculations are compared with the experimental data (•) of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). amount of fuel is present even at x/D = 10.5 in the experiment. The resultant temperature rise is also quicker compared to the experiment. The centreline value of the temperature at x/D = 8.5 is close to the fully burnt value in the calcula- tions while it is close to the unburnt value in the experiment. This indicates that the mean fuel consumption rate is higher in the unstrained flamelets calculations resulting in a shorter and narrower flame brush compared to the experiment. The calculations of flames F1 and F3 (not shown) show similar results. Instantaneous experimental images in Chen et al. (1996) show that the flame front is strongly stretched by the turbulent eddies which are likely to attenuate the reaction rate. While the strained and unstrained flamelet models yield comparable mean reac- tion rates for low turbulence intensities (see Fig. 7.15), for very high turbulence intensities the unstrained flamelets overpredict the mean reaction rate as one would expect. This is consistent with the planar flame results reported earlier. The major species predictions are compared to the experimental data in Fig. 7.27. As with Eq. (7.14), the influence of mixing with entrained air on the species mass fractions is accounted by Y i ≈ ξ˜ Y Ri + (1 − ξ˜)Y Ai where Y Ri is for the reacting component of the mixture and Y Ai is for air. Note that Y Ai is zero for all species except O2 and N2. The agreement is quite good for flame F3 136 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.0 c r/D ...... .... . ..... .. ..... .. ..... .. ..... .. ..... .. ..... .. .... . . ...... ...... ..... ...... ..... ..... ..... ..... ..... ..... ...... .... .. ..... .. ..... .. .... . .... . .... . . ...... ..... ..... ....... ..... ....... ....... ....... ..... ..... ...... ... ....... ....... ....... ....... ....... ....... ..... ..... ..... ..... ..... ..... ..... ..... ..... ....... ....... ... .... ....... ....... ..... ..... ..... ..... ..... ..... ..... ..... ..... ....... ....... ....... ....... ....... ..... ..... ..... ..... ..... ..... ..... ...... ... ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦◦◦ ◦◦ ◦◦◦ ◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10 Y CH4 r/D ....... ..... ..... ....... ..... ..... ...... ....... ....... ....... ....... ....... ....... ...... ....... ..... ..... ..... ....... ....... ....... ....... ....... ....... ....... ....... ..... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ..... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ..... ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 Figure 7.26: The mean progress variable, c, (left) and mean CH4 mass fraction (right) from the unstrained flamelet calculations are compared with the experi- mental data of Chen et al. (1996) for flame F2. 137 7.3 RANS of laboratory flames for all three major species: O2, CO2 and H2O. For flames F1 and F2, H2O is well predicted, but there is an overprediction of O2 and underprediction of CO2 for x/D ≥ 6.5 which is probably due to the approximate treatment of mixing with entrained air. Nonetheless, the approximation seems reasonable since the qualitative trends are very well predicted. For instance, the rise in O2 and drop in H2O mass fractions in the radial direction due to mixing with entrainment air are well captured in the calculations. The O2 mass fraction for flame F1 at x/D = 2.5 is overpredicted for r/D ≥ 0.8, which is similar to the predictions of Herrmann (2006) and Lindstedt & Vaos (2006), the reasons for which are not clear. The overall level of agreement for the major species predictions is good. The predictions of the minor species: CO, H2 and OH, are compared with the experimental data in Fig. 7.28. While H2 and OH are well predicted for all three flames, CO is considerably overpredicted. For flame F3 the CO overprediction is by nearly a factor of 2.5 while for flames F1 and F2 it is by a factor of 2. The CO predictions of Herrmann (2006) are similar to those shown in Fig. 7.28. The CO predictions of Lindstedt & Vaos (2006) are also similar to those shown in Fig. 7.28 for flame F1. However, their predictions are closer to experimental data for flame F3 and a typical comparison is shown in Fig. 7.29. Figure 7.29 also shows typical sensitivity of minor species predictions to the RtP flamelets boundary conditions. The results using the equilibrium mixture composition for the product stream is shown in Fig.7.29a, whereas the Fig.7.29b shows the results when the product stream contains only CO2 and H2O. As one can observe in this figure, minor species predictions are sensitive to the flamelets boundary conditions, but the major species do not show such a sensitivity. For the results reported in this study, the equilibrium composition is used as product stream for the flamelets. This sensitivity may be reduced if one uses a second progress variable to represent the chemical species with slow time scales, since the progress variable used here represents the overall progress of the chemical reactions. The carbon monoxide is clearly identified to have a slow chemical time scale by Peters & Williams (1987). The choice for the second progress variable is unclear and will be explored in future. Since an overall progress variable is used to represent the combustion, its influence on CO prediction is verified using an 138 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 Y i r/D .................................................................. ...... ... ..... ... ... ..... .. ... ..... ... ... ..... ... ..... ... ....................... ................ ............................................................... ... .... .. .... .. .... ... .... ... ..... ... ........ ................... ............................................................ .. .... ... .... ... .... ... ..... ... ....... ....... ....... ...... ...................................................... ... ..... .. ..... ... ..... ..... ...... ........ ........ .... .................................................... ...... ...... .... ....... ...... ...... ........ ........ ....... ... ..................... .... ..... .... .... .... ..... ... .... ............................................................................................... ................ ...... .... ..... ..... .... ...... ............................................................................................ ............. ....... ..... .... .... ...... ....... ..................................................................................... ............ ....... ...... ...... ...... ....... ............. ....................................................................... ........... ....... ...... ...... ....... ........ ................................................................................... ... .. ... ... ... ... .. ... .. .................... .. ...... ... .. .. .. ... ..... ..... ..................... ......... ... ... ... ... ...... ........ ................................ ........ ... ... ... .... ...... ....... .................................... .. ........ ... .... ...... ............. ............................................... ◦◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦ ◦ ◦◦ ◦ ◦ ◦ ◦◦◦◦◦ ◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦ ◦◦ ••• • • •• •••••••• •• • • • •• •••••••• •• • • ••••••••••• • •• •• •••••••••• • •• ••• ••••••••• MMM M M M MMMMMMMMM MM M M MM MMMMMMMMM MM M MM MMMMMMMMMM MM MM MMMMMMMMMMM MM MM MMMMMMMMMMM x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 r/D ........................................................................................ ... .. .... .. ..... .. .... ... .... ... .... ... .... ... ... ...................................................................... .. .... ... ... ..... ... ..... ... ... .... .. .... ... .... .... .................. .................................................... ........... .... .. .... ... .... ... .... ... .... ... ..... ..... ....... ...... ............................................................... ..... .. .... ... .... ... ..... ... ...... ...... ...... ........ ............................................................ ...... .... ..... ... ..... ..... ...... ........ ....... ...... . ..................... .... ..... ..... ..... ..... .... ... .... .... ...... ....................................................................................... ................. ..... .... ..... ..... ..... ..... .... ....... .................................................................................... .............. ....... .... .... .... .... ..... ....... ............. ....................................................................... ............. ...... ..... .... .... ..... ...... ...... ........................................................................... ........... ...... ..... ..... ..... ...... ....... .................................................................................. ... .. ... ... ... ... ... .. ... ............................ .............................. ..... ... .. .. .. .. ... ... ..... ....... ................. ......... ... ... ... ... ... ... ..... ......................................... ......... ... ... ... ... ...... ..... ........................................... ........ ... ... ... .... .... .... .. ................................................ ◦◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦ ◦◦ ◦ ◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦◦ ◦◦◦◦◦◦◦◦◦ ◦◦ ••• • • •• •••••••• •• • • •• ••••••••• •• • • ••• •••••••• •• • •• •••••••••• •• ••• •••••••••• MM M M M MMM MMMMMMM MM MM MM MMMMMMMMM MM M M MM MMMMMMMMM MM M MMM MMMMMMMMM MM MMM MMMMMMMMMM x/D = 2.5 x/D = 4 5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 r/D ...................................................................................................................................... ...... . ................................................................................................... .... .... ... ..... ... ... ..... ... .... ... .... ... . ...................................................................................... ..... ... .... .. .... .. .... ... .... ... .... ... .... ... ... ................................................................................ .... ..... .. .... ... .... ... .... ... .... ... ..... ... ..................... .... ..... .... .... .... .... .... ..... .... .... ...... ........ ............................................................... .................. ..... .... .... ..... ..... ..... ... .... ..... ..... ........ .......................................................................... ................ ...... ..... .... .... .... .... .... .... ..... ........... ............................................................................. .............. ..... ..... .... .... .... .... .... ..... ....... .................................................................................. ... .. ... ... ... ... ... .. ... ........ .............................................. ...... ... .. .. .. .. .. .. ... .. ................................... ................. ......... ... ... .. .. .. ... .. ... . . ................................................ . ......... ... .. .. .. ... .. .... .. ........................ ................................. ◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ •••• • • • ••••• ••• •• • •• •••••••••• •• • •• ••• •••••• • •• •• ••• •••••••• MM M M MM MMM MMMMMM MM M MM MMMMMMMMMM MM MMM MMMM MMMMMM MM MMM MMMMMMMMMM x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 Figure 7.27: The major species mass fractions: O2 (◦, ), CO2 (•, ) and H2O (M, ), from strained flamelets calculations are compared with the experimental data (symbols) of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). 139 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 Y i r/D ............ .... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ............................................................................................................... ........ .. ..... ... ... .... .. ... .... ... ... .... ... ... .... ... ... ..... ... ..................................................................................................... ...... ... ..... ... ... ..... ... .... .... ... .... ... ... ..... ... ... ........................................................................................... ..... ... ..... ... ... ..... ... ... ..... ... ..... ... ... .... .................................................................................. .... ... ... ..... ... ..... ... ... ..... ... ... ..................................................................................... . ..... .... .... ..... ..... ..... ................................................................ ... ... . ...... .... .... ..... .... ...... ............................................................................ . . . ........ ..... ..... ..... ....... .................................................................................................... .......... ....... ....... ....................................................................................................... ........... ............................................................................................................. ......... .. ... ... ... ... ... ... ... ... ... ... ........ . ............................... . . ............ ... ... ... ... ... .. ... ... ... ... ....................................... . .......... ... .. ... ... ... ... ... .. ... .................... .................................... ......... ... .. .. .. .. ... .. .. ... ..... .................. ................................... ......... ... .. .. .. .. .. .. ... .... ..................... ........................................ ◦◦ ◦ ◦ ◦ ◦◦ ◦◦◦◦◦◦◦◦ ◦◦ ◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦ ◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ •• • • • • ••• • • •••• • • ••••••••••••• ••• •••••• ••••• • • ••••••••••••• • •••• ••••••••••• MMM M M M MMMMMMMMM MM M M M MM MMM M M M MM M M M M MMMMMMMM M MM M M M M MMM MMMMM M MM M M M MM MMMMMM M MMM x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 r/D ............. ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ... .................................................................................................. ........ .. ..... ... ... .... .. ... .... ... ... .... ... ... .... ... ... ..... ... ... .... .. ...................................................................................................... ....... .. ..... ... ... ..... .. ... .... .. ... .... ... ... .... ... ... ..... ... ............................................................................................ ..... ... ..... ... ... .... ... ... .... ... ... .... ... ... ..... ... ... ..................................................................................... .... ... ... ..... ... .... .... ... ..... ... ... ..... ... ...... ................................................................................. . . ... ..... .... ..... ..... ..... ................................................................................................... ..... .... ..... ..... .... ...... ............................................................................... ... .. .. ... ..... .... .... ...... ...................................................................................................... .......... ...... ...... .......... ................................................................................................ ........... ........ .......................................................................................................... .......... .. ... ... ... .. ... ... ... ... ... ... ... .. .... . ............................................................ ............. ... ... ... ... ... .. ... ... ... ... .. ... ............................................. ............ ... ... ... ... ... ... ... ... ... ... ... .............. ....................................... ............ .. .. ... ... ... ... .. ... ... .. ... ....................................................... ......... ... ... ... ... ... ... ... ... ... ... .... ............... ..................................... ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ • • • • •• • •• •••••• • • •••• ••••••••• •• •••• ••••••••• ••• •••••••••••• •••••••••••• ••• MM M M M M MMMMMMMMM M M M M M MMMMMM M M MM MM M M M MMMMMM MMMM MM M MM MMMMMM M MMM M M M MM MMMMMMMM MM x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 r/D ............. ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ............................................................................................. ......... ..... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ... ... ................................................................................. ........ .. ..... ... ... .... .. ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ... .................................................................................. ...... .. ..... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ... ..... ............................................................................. . . .... ..... ..... ..... ..... ..... .............................................................................................. .......... . .... ..... .... .... ...... ............................................................................................... ......... .... .... .... ....... .............................................................................................. ......... ...... ..... ....... .................................................................................................. .......... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... .. ....................................................... ............... .. ... ... ... ... ... .. ... ... ... ... ... ... .. ... ... .. .. ... .......................................................... .............. ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... .......................................................... ............ ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ..... .............................................. ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ •• • • • ••• •• ••••• • • ••••••••••••• • • •••••••• ••••• • • • ••• ••• •••••• MM M M M M MMMMMMMM M M M M M MM MMMMMM M M M MM M MM MMMMMMMMM M MM M MM MMMMMMMMM M x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 Figure 7.28: The minor species mass fractions: 10×CO (◦, ) , 100×H2 (•, ) and 75×OH (M, ), from strained flamelets calculations are compared with the experimental data (symbols) of Chen et al. (1996) for flames F1 (left), F2 (center) and F3 (right). 140 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 Y i r/D ............ .... .... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... .... ........................................................................................................................................................................................ .............. ..... .... ..... ..... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ..... ..... ..... .... .............................................................................................................................................................................. .......... ..... ... ... ..... ... ... .... .. ... .... ... ... .... ... ... .... ... .. .... ... ... .... .. ... ..... .. . ....................................................................... ................................... .......... ..... ... ... ..... ... .. ... .... .. ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... .. ... ............................................................................................................... ......... . .. ..... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .......................................................................................... ......... . .. ..... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ................................... ..................................................... ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦ (b) (a) Figure 7.29: The predictions of 10 × YCO using strained flamelets model ( ) are compared with predictions of Herrmann (2006) ( ) and Lindstedt & Vaos (2006) ( ) for flame F3 at x/D = 4.5. The sensitivity of strained flamelets pre- dictions to RtP flamelet boundary conditions are shown: (a) equilibrium mixture composition for product stream, (b) product stream comprising of only CO2 and H2O. 141 7.3 RANS of laboratory flames 0.0 0.4 0.8 1.2 1.6 2.0 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 0.25 Y i r/D ............. ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ... .................................................................................................. ........ .. ..... ... ... .... .. ... .... ... ... .... ... ... .... ... ... ..... ... ... .... .. ...................................................................................................... ....... .. ..... ... ... ..... .. ... .... .. ... .... ... ... .... ... ... ..... ... ............................................................................................ ..... ... ..... ... ... .... ... ... .... ... ... .... ... ... ..... ... ... ..................................................................................... .... ... ... ..... ... .... .... ... ..... ... ... ..... ... ...... ................................................................................. . .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ................................................................................................ .... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ................................................................. . . . ...... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... .............................................................................. ..... .. ... ... ... ... ... ... ... ... ... ... ... ................................................................................. .. ... ... ... ... ... ... ... .... ............................................................................. ◦◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ x/D = 2.5 x/D = 4.5 x/D = 6.5 x/D = 8.5 x/D = 10.5 Figure 7.30: The predictions of 10×CO from strained flamelet calculations using two definitions of progress variable: cf ( ) and cp ( ) are compared with the experimental data of Chen et al. (1996) for flame F2. 142 7.4 Summary alternative definition, given by Fiorina et al. (2003) cp = YCO + YCO2 YCO,eq + YCO2,eq . (7.17) It is necessary to include YCO2 in the above definition to ensure that cp varies monotonically in the flamelets. The CO values calculated using the cp progress variable are compared with those using cf for flame F2 in Fig. 7.30. It appears that the improvement in CO predictions is marginal. 7.4 Summary RANS calculations of turbulent premixed flames are performed to assess the strained flamelets model described in Chapter 4. The mean scalar dissipation rate, which is a modelling parameter, is closed via the algebraic model derived in Chapter 5. Calculations of statistically planar one–dimensional flames are performed to test the mean reaction rate closure and laboratory flame calculations are used to test the overall predictive ability of the model. A single set of model parameters is used in all the calculations. Calculations of planar one–dimensional flames are performed to compare the predictions of the strained flamelets closure with those of algebraic closure of Bray (1979) and the unstrained flamelets closure. Two sets of flames, one in the corrugated flamelets regime with the value of flame stretch factor K = 0.15 and the other in the thin reaction zones regime with K = 1.0, are considered for stoichiometric methane–air mixtures. The results show that the flame brush thickness, δt, increases with u′/soL for a given K value, as one would expect, for all three reaction rate closures. The algebraic closure viewing the flame front as a thin interface predicts the broadest flame brush while the closure proposed here predicts the narrowest flame brush. The turbulent flame speed, ST , predicted by the three closures is compared to the experimental data of Abdel-Gayed et al. (1987). The algebraic closure overpredicts the mean reaction rate and hence the turbulent flame speed for all flames considered here. The unstrained flamelets closure predicts turbulent flame speed reasonably well for flames in the corrugated flamelets regime but overpredicts for flames in the thin reaction zones regime. This is because the fluid dynamic straining reduces the burning rate. This effect 143 7.4 Summary is well captured by the strained flamelets closure and the flame speeds predicted by this closure are close to the experimental data for flames both in the corrugated flamelets and the thin reaction zones regimes. Calculations for lean propane–air mixture with a Lewis number greater than unity confirm the hypothesis of Bradley et al. (1992) that the flame speeds correlate better with the parameter KLe than with K. The flame speeds predicted by this strained flamelets closure are close to the values of Bradley et al. (1994) who used stretch rate to parametrise the flamelets and employed second order turbulence closure by solving Reynolds stresses and turbulent scalar flux transport equations. The agreement in the flame speed values of Bradley et al. (1994) and that of the present study, despite a vast modelling difference, is likely due to insignificant quantitative differences in the predicted mean reaction rate values for the range of flow and flame conditions considered here. However, it is clear that the stretch effects are to be included in the modelling and the scalar dissipation rate based approach seems attractive for this. Laboratory flames of two different configurations: rod stabilised V–flame of lean methane–air (Robin et al., 2008) and pilot stabilised Bunsen flames of sto- ichiometric methane–air mixtures (Chen et al., 1996) are also calculated. The conditions of the V–flame correspond to the corrugated flamelets regime and the Bunsen flame conditions correspond to the thin reaction zones regime. The results of V–flame calculations using the strained flamelets model are in good agreement with the experimental data. The longitudinal and transverse velocities and the Reynolds mean progress variable, c, are well predicted. Results of an unstrained flamelets model are close to the results of the strained flamelets model. This is be- cause the mean reaction rate values predicted by the two models are nearly equal for the V–flame, which is in the corrugated flamelets regime. This is consistent with the planar flame results. Calculations of Bunsen flame with the unstrained flamelet model predicted a flame brush which is much shorter and narrower compared to the experiment, indicating an overprediction of the mean reaction rate. The flame front in these flames is strongly stretched by the turbulent eddies which are likely to attenuate 144 7.4 Summary the reaction rate. This is not accounted by the unstrained flamelets model result- ing in an overprediction of mean reaction rate. On the other hand, the strained flamelets model predictions of the flame brush dimensions are in good agreement with the experimental values. From flamelet modelling point of view, this un- derscores the importance of appropriately accounting for the local fluid–dynamic effects on the flamelet characteristics, and the scalar dissipation rate is a good parameter to represent this. The mean velocity and turbulent kinetic energy of the Bunsen flame are rea- sonably well predicted by the strained flamelet calculations. The mean and RMS of the progress variable are also well predicted although a slight overprediction is observed at locations close to the nozzle which is attributed to the higher pilot temperatures used in the calculations as in earlier studies. However, the flame brush thickness, which is related to the gradient of the mean progress variable, is well predicted. The predicted major and minor species mass fractions are also in reasonable agreement with the experimental data. It is assumed in the present calculations that the combustion is in a purely premixed environment and the mixing of entrained air is only with the combustion products. This assumption yields good results for the lower Reynolds number flame F3 while for flames F1 and F2, some discrepancies in the major species predictions are observed due to such an approximate treatment. However, CO mass fraction is consistently over- predicted which is likely due to the misrepresentation of the slow CO oxidation in the flamelet by an overall progress variable. Calculations with a progress variable based on a combination of CO and CO2 yielded little improvement over calcula- tions with a fuel based progress variable. The predictions of minor species such as CO are also sensitive to the RtP flamelets boundary conditions. In the present calculations, the product stream in RtP flamelets has an equilibrium composition. However, considering RtP flamelets whose product stream comprises of only CO2 and H2O yields considerably different predictions for the minor species while the major species predictions are not affected. The current flamelet modelling needs to be improved to capture the statistics of the species with slow chemical time scales such as CO and the minor species. 145 Chapter 8 Conclusions and Future work In this thesis laminar flamelet modelling of turbulent premixed flames was stud- ied in the RANS context. A strained flamelet model is proposed and the scalar dissipation rate of the progress variable is used to parametrise the flamelets which are represented by Reactant–to–Product opposed flow laminar flames. Shapes are presumed for the pdfs of the progress variable and the conditional scalar dissi- pation rate. Using physical arguments, the conditional mean scalar dissipation rate is closed by relating it to the unconditional mean dissipation rate, which is a modelling parameter. An algebraic model for the unconditional mean scalar dissipation rate, valid for large Reynolds and Damköhler number flames, was de- rived based on recent findings of scalar dynamics in turbulent premixed flames which are different from the passive scalar scenario. The mean scalar dissipation rate model lead to an algebraic expression for the turbulent flame speed which, in this study, is defined as the propagation speed of the flame brush leading edge. The predictions of the mean scalar dissipation rate model were in reasonable agreement with DNS data of turbulent premixed flames in different combustion regimes. The predictions of the turbulent flame speed expression were also in good agreement with experimental data from a wide range of flame configura- tions and conditions, with the same set of model constants, providing an indirect validation for the scalar dissipation rate model. RANS calculations of turbulent premixed flames provide evidence supporting the central hypothesis of the model formulation in this work; flamelet models that appropriately account for the turbulence fluid dynamics influence on the flame 146 8.1 Recommendations for future work front characteristics have a wider regime of applicability than those that do not. Calculations of statistically planar one–dimensional flames show that for condi- tions in the corrugated flamelets regime, the unstrained flamelets model predicts turbulent flame speeds reasonably close to experimental data, whereas for con- ditions in the thin reaction zones regime the flame speed is overpredicted. This is because the reaction rate attenuation due to fluid dynamic stretch, likely to be considerable in the latter regime, is not captured by the unstrained flamelets model resulting in an overprediction of the mean reaction rate. That the strained flamelets model using a scalar dissipation rate parametrisation is able to capture this is encouraging. Calculations of laboratory flames reflect the same character- istics of the models. For a flame in the corrugated flamelets regime, the V–flame, the strained and unstrained flamelets models predict flame brush structures in close agreement with experimental data but for the Bunsen flames which are in the thin reaction zones regime the unstrained flamelets model predicts a smaller flame brush as a result of the mean reaction rate overprediction. The flame brush structure predicted by the strained flamelets model is in much better agreement for these flames. The strained flamelets model is also able to give good predic- tions of the major species mass fractions although minor species predictions show room for improvement. The minor species predictions appear sensitive to prod- uct stream composition in RtP flamelets while major species predictions are not. Furthermore, for species such as CO which is known to have slow chemistry in the post flame region, a single progress variable is probably insufficient to capture its thermochemistry accurately. A formulation involving two progress variables, one representing the slow and the other representing the fast time scales, might be more successful in this regard and this could be explored in future studies. 8.1 Recommendations for future work There are many topics worthy of further investigations based on the contributions of the present work. Some of these are discussed below. 147 8.1 Recommendations for future work 8.1.1 Lewis number effects The nature of the flamelet response to straining entailed considerations of the mixture Lewis number. This was appropriately addressed in the strained flamelet model, as evidenced from the one–dimensional flame calculations for the lean propane–air mixture in Chapter 7. However, non–unity Lewis number effects were not explicitly accounted for in the modelling of the mean scalar dissipation rate. Accordingly, data corresponding to mixtures with Lewis numbers close to unity were considered for the validation of this model in Chapters 5 and 6. The Lewis number dependence of the mean scalar dissipation rate needs to be inves- tigated further. A recent study by Chakraborty et al. (2009) suggests that the turbulence–scalar interaction term, a leading order term in the transport of mean scalar dissipation rate, shows a Lewis number dependence. Such investigations can form the basis for incorporating an explicit Le dependence in the dissipation rate model. 8.1.2 Sensitivity to turbulence closure The influence of the underlying turbulence model on the predictions using the strained flamelet model need to be investigated. In the present study all calcula- tions employed the standard k˜–ε turbulence closure. The calculations of Bradley et al. (1994) using the stretch rate based flamelet model show that predictions with standard k˜–ε closure can be quite different from those with second order turbulence closure. The burning rate factor, Pb, a central quantity in the flamelet model of Bradley et al. (1994), is directly influenced by the turbulence quantities k˜ and ε˜. In the scalar dissipation based model studied in the present work, the influence of the turbulence field on the mean thermochemical quantities is pre- dominantly via the mean scalar dissipation rate. Hence it might be worthwhile to investigate the extent to which second order turbulence closures affects the predictions of the strained flamelets model. 148 8.1 Recommendations for future work 8.1.3 Unsteady strain, non–adiabaticity and curvature The strained flamelets model employs steady strained adiabatic RtP laminar flames as the canonical flamelet configuration. In reality however, the turbulent flame front experiences unsteady strain rate. Hawkes & Chen (2006) study the response of a laminar flamelet to unsteady strain by imposing inlet velocity os- cillations at a particular frequency, and find that point–wise quantities in the flamelet are likely display quasi–steady behaviour. While this result is by no means general, it is encouraging and lends some validity to the use of steady strained flamelets. However, one suspects that beyond a certain frequency the quasi–steady behaviour might break down and unsteady response might have to be rigorously incorporated in the flamelet formulation. Such a formulation might be critical for instance in the prediction of thermo–acoustic oscillations of premixed flames. Another important factor to consider is curvature. Traditionally turbulent eddies are thought to influence the flame front via strain rate and curvature, together imposing a total stretch rate. It can be argued that the present formu- lation prescribes the scalar dissipation rate description as an alternative to the strain rate + curvature description. Still, the flamelet structure in highly curved portions of a turbulent flame might not correspond to that given by the scalar dissipation parametrisation of a purely strained laminar flame. In other words, the choice of the RtP flamelet configuration, while accounting for the strain rate directly, does not explicitly account for curvature effects. Whether it is necessary at all to treat these two influences separately in the context of scalar dissipation rate parametrisation is worth examining using DNS data. If it is, then one could address this by considering a canonical curved laminar flamelet in addition to the purely strained flamelet. Also, in many practical scenarios effects of non–adiabaticity could be impor- tant. For instance, such effects can occur for flames in confined geometries with significant flame wall interactions. The asymptotic analysis of Libby et al. (1983) considers non–adiabatic effects in RtP flames in great detail. For flamelet model formulation, following van Oijen & de Goey (2000), one could consider calculat- ing RtP flamelets for different product stream enthalpies and generate a look–up 149 8.1 Recommendations for future work table with enthalpy as the additional dimension. 8.1.4 Stratified combustion A similar approach can be adopted to extend the strained flamelets closure for application to stratified combustion. Working with the progress variable defini- tion of Bray et al. (2005) that accounts for local stoichiometry, together with a mixture fraction, the strained flamelet model can be employed in calculations of stratified flames. This would then involve calculating RtP flamelets of various equivalence ratios and generating the look–up table with the mixture fraction moments as the additional dimensions. 8.1.5 Statistically non–planar flames An interesting result obtained in the thesis from studying the applicability of KPP analysis to multi–dimensional flames is the influence of mean curvature on the propagation speed of flame brush leading edge. The analysis in Chapter 6 indicates that a positive mean curvature enhances the leading edge propagation speed (see Eq. (6.17)). While it is difficult to study statistically steady curved flames in experiments, the KPP result can still be compared with calculations of statistically non–planar one–dimensional flames. Such calculations would also facilitate investigating the effects of mean curvature in great detail. 8.1.6 Modelling for CMC As noted in Chapter 2, implementation of CMC for premixed combustion involves two main modelling challenges: closure for the conditional mean dissipation rate and conditional mean reaction rate of the progress variable. The closure for the former based on physical arguments, presented in Chapter 4, can be evaluated for applicability with CMC. This closure can be described simply as estimating the conditional mean scalar dissipation rate from the unconditional mean quantity based on flamelet assumption. As a first step, this closure can be validated with DNS data by taking the unconditional mean dissipation rate value from the DNS. 150 8.1 Recommendations for future work If such a validation is promising, the closure together with the algebraic mean scalar dissipation rate model presented in Chapter 5 can be assessed by making comparisons with DNS data. Alternatively, obtaining a conditional mean scalar dissipation rate closure from the analysis of the pdf transport equation, Eq. (4.21), is also worth exploring. 8.1.7 Modelling for LES For turbulent flames with complex flow features LES is becoming increasingly preferred. However, as noted in the Introduction, the chemical reactions occur at scales that are unresolved in LES and hence RANS type combustion models are relevant for LES. Indeed, many combustion models employed in LES are based on extension of RANS modelling ideas (Poinsot & Veynante, 2001). Typically, for LES of premixed flames, the balance equation for filtered progress variable is solved. In flame surface density based approaches the filtered reaction rate, combined with the diffusive flux term (terms on the r.h.s of Eq. (2.13)) are closed via the displacement speed sd (see Eq. (6.8)) as (Boger et al., 1998; Fureby, 2005) ρ̂sd|∇c| ≈ ρusoLΞ|∇ĉ| , (8.1) where a ̂ denotes filtered quantities. The flame wrinkling factor, Ξ, is closed as in RANS models. Another approach is to provide a closure of the filtered reaction rate, ̂˙ω, using an FPI tabulation (Domingo et al., 2005) ̂˙ω = ∫ 1 0 ω˙o(ζ) p̂(ζ) dζ , (8.2) where p̂(ζ) is the filtered pdf whose shape is presumed. The limitations of such a closure based only on unstrained flamelets are clearly documented in this thesis. The attenuation of reaction rate due to turbulence straining is likely to affect ̂˙ω just as it affects ω˙. Therefore it will be worthwhile to extend the scalar dissipation based strained flamelet approach to provide a closure for the filtered reaction rate,̂˙ω. 151 Appendix A Relations between Reynolds and Favre averaged quantities The Reynolds average quantities can be obtained from Favre averaged quantities if the relation between p(ζ) and p˜(ζ) and the shape of the latter is known. We know that p(ζ) = 1 + τζ 1 + τ c˜ p˜(ζ), and hence c = ∫ 1 0 ζ p(ζ) dζ = 1 1 + τ c˜ ∫ 1 0 ζ(1 + τζ) p˜(ζ)dζ. (A.1) The pdf p˜(ζ) is presumed to be a β–pdf i.e. p˜(ζ) = Cζa−1(1− ζ)b−1, where C = 1 β(a, b) , a = c˜ ( 1 g − 1 ) , b = (1− c˜) ( 1 g − 1 ) and the variance parameter is g ≡ c˜′′2/c˜(1 − c˜). The beta function is written in terms of the gamma functions (Davis, 1970) as β(a, b) = ∫ 1 0 ζa−1(1 − ζ)b−1dζ = Γ(a)Γ(b)/Γ(a+ b). Substituting for p˜(ζ) in Eq. (A.1) and using the above defini- tions yields c(1 + τ c˜) = Γ(a+ b) Γ(a)Γ(b) { Γ(a+ 1)Γ(b) Γ(a+ b+ 1) + τ Γ(a+ 2)Γ(b) Γ(a+ b+ 2) } . (A.2) 152 Using the property of the gamma function Γ(x + 1) = xΓ(x) (Davis, 1970) and simplifying the above equation yields c(1 + τ c˜) = a a+ b { 1 + τ a+ 1 a+ b+ 1 } . (A.3) After some algebra one gets c = c˜+ τ c˜′′2 (1 + τ c˜) . (A.4) It is interesting to note that the BML relationship (Bray, 1980), c = c˜(1+τ)/(1+ τ c˜), is recovered when c˜′′2 = c˜(1− c˜). The variance can be written as c′2 = ∫ 1 0 ζ2p(ζ)dζ − c2 . (A.5) Repeating the above steps yields (c′2 + c2)(1 + τ c˜) = a(a+ 1) (a+ b)(a+ b+ 1) { 1 + τ a+ 2 a+ b+ 2 } (A.6) with the final result as c′2 = c˜2 + c˜′′2 1 + τ c˜ { 1 + τ c˜2(1− c˜) + c˜′′2(2− c˜) c˜′′2 + c˜(1− c˜) } − c2 . (A.7) It is easy to see that for the case of constant density (τ = 0), the above relations recover the equalities c = c˜ and c′2 = c˜′′2 . 153 Appendix B List of Publications • Kolla, H. & Swaminathan, N. (2010). Strained flamelets for turbulent pre- mixed flames, I: formulation and planar flame results. Combustion and Flame, In press. • Kolla, H. & Swaminathan, N. (2010). Strained flamelets for turbulent pre- mixed flames, II: laboratory flame results. Combustion and Flame, Under review. • Kolla, H., Rogerson, J.W. & Swaminathan, N. (2010). Validation of a tur- bulent flame speed model across combustion regimes. Combustion Science and Technology, In press. • Kolla, H., Rogerson, J.W., Chakraborty, N. & Swaminathan, N. (2009). 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