Large Eddy Simulation of bluff body stabilized premixed and partially premixed combustion. Ionuţ Porumbel

Transcription

1 Large Eddy Simulation of bluff body stabilized premixed and partially premixed combustion A Thesis Presented to The Academic Faculty by Ionuţ Porumbel In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy School of Aerospace Engineering Georgia Institute of Technology December 2006

2 Large Eddy Simulation of bluff body stabilized premixed and partially premixed combustion Approved by: Prof. Suresh Menon, School of Aerospace Engineering Georgia Institute of Technology Committee Chair Prof. Tim Lieuwen, School of Aerospace Engineering Georgia Institute of Technology Prof. Jerry Seitzman, School of Aerospace Engineering Georgia Institute of Technology Dr. Saadat Syed, Pratt and Whitney Prof. Pui-Kuen Yeung, School of Aerospace Engineering Georgia Institute of Technology Date Approved : November 6, 2006

3 To my parents, Dumitru and Agatiea iii

4 ACKNOWLEDGEMENTS First, I would like to thank Prof. Suresh Menon for his advisement and support throughout the entire period I have spent at Georgia Tech, both material and professional, as well as for opening up for me the new and challenging world of computational combustion. I want to extend my thanks to my Thesis Committee members: Dr. Tim Lieuwen, Dr. Jerry Seitzman, Dr. Saadat Syed and Dr. Pui-Kuen Yeung for their time, patience and valuable suggestions. My gratitude also goes to the former and present members of the Computational Combustion Laboratory for their friendship and support in the difficult moments, for their help and collaboration in carrying on this work, and for their pertinent advice whenever I asked for it. To name just a few of them, with special contributions to this work, my special thanks go to Dr. Gilles Eggenspieler, Dr. Vaidyanathan Sankaran, Mr. Mehmet Kırtaş, Mr. Mathieu Masquelet, Mr. Rajat Kapoor, Mr. Hossam El-Asrag, Mr. Nayan Patel, Mr. Franklin Genin and Mr. Baris Ali Sen, as well as to the rest of the CCL group. I would also like to thank the Department of Defense High Performance Computing centers for providing the computational capabilities and to P ratt and W hitney for sponsoring my research throughout the most part of the six years I have spent at Georgia Tech, as well as for the internship opportunity it has offered me in For the help they have offered me during this period, I want to thank especially to Dr. Anuj Bharghava, Dr. Won-Wook Kim and Mr. Andrea Lentati. A great debt of gratitude I owe to my family and friends, here and at home, for their support and patience during these six years. I would never have been able to complete this degree without them. Finally, I want to thank the management and employees of S.C. Turbomecanica S.A., since without their understanding support and tolerance, this work would not have been possible. iv

5 TABLE OF CONTENTS DEDICATION ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE SUMMARY iii iv ix x xvi xxii I INTRODUCTION Overview Combustion and Turbulence Turbulent combustion Reynolds number invariance Scale separation hypothesis Premixed flame structure Regimes in premixed combustion Bluff Body Flame Stabilization Flame-holder blowout limits Bluff body stabilization mechanism Ignition delay and flame blow - off Turbulent flame extinction Partially premixed flames Modelling in Computational Fluid Dynamics Momentum equation closure Combustion modelling Motivations and Objectives Motivation Objectives Outline v

6 II MATHEMATICAL FORMULATION AND MODELLING Fluid Motion Governing Equations The Navier-Stokes equations LES filtered governing equations Subgrid closure of the LES equations Subgrid turbulent kinetic energy model Localized dynamic kinetic energy model (LDKM) Combustion Modelling The conventional closure: Eddy Break - Up model (EBU) The subgrid closure: Linear-Eddy Mixing model (LEM) III NUMERICAL IMPLEMENTATION Discretization of the Governing Equations The finite volume formulation Domain discretization The Runge - Kutta scheme The dual time stepping method The local time stepping convergence acceleration technique Boundary Conditions Characteristic boundary conditions Adiabatic and isothermal no-slip wall boundary conditions Periodic boundary conditions Linear Eddy Mixing Model Implementation The reaction diffusion equation The splicing algorithm The Direct Estimation of the Chemical Source Terms The Multiblock Grid Parallel Implementation Simulation Geometries and Computational Grids LDKM Model Coefficients IV ALGORITHM VALIDATION Results and Discussion of Non-Reactive Flow vi

7 4.1.1 Overview Spectral analysis Flow structures Time averaged results Results and Discussion of Reactive Flow Overview Spectral analysis Flow and flame structures Time averaged results V FLOW AND FLAME STRUCTURES IN PREMIXED AND PARTIALLY PREMIXED FLAMES Overview Results and Discussion of Constant Equivalence Ratio Reactive Flow Spectral analysis Flow and flame structures Sources of flame instability Intermittency Ignition delay effects Time averaged results Results and Discussion of Variable Equivalence Ratio Reactive Flow Spectral analysis Flow and flame structures Sources of flame instability Time averaged results VI CHEMICAL SOURCE TERMS ESTIMATION USING ARTIFICIAL NEURAL NETWORKS Introduction The ANN Training Numerical Results and Discussion Laminar 1-dimensional premixed methane flame Turbulent 3-dimensional premixed methane flame vii

8 6.3.3 Application of the ANN algorithm in complex geometries VII CONCLUSIONS AND RECOMMENDATIONS Conclusions Recommendations for Future Work BIBLIOGRAPHY 188 VITA 204 viii

9 LIST OF TABLES 3.1 Speedup factors for different methods of estimating the chemical source terms. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E5 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; DST E20 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; Computational grid dimensions. CF = Cold Flow; RF = Reactive Flow Reactive flow simulation cases Position of sampling points for the PDF of the axial velocity for Cases 2-4. Dimensions are in mm, measured from the centerline of the combustor and from the bluff body back wall Speedup factors for different methods of estimating the chemical source terms in a laminar, one-dimensional flame. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; ISAT = In Situ Adaptive Tabulation approach for determining the reaction rate; AN N = Reaction rates computed using the present ANN method ix

10 LIST OF FIGURES 1.1 Premixed flame structure Regimes in turbulent combustion [Peters, 2000] Recirculation zone behind a bluff body CFD methods and their applicability range. η is the Kolmogorov scale, l is the maximum size of the isotropic eddy, is the grid size and l is the inertial length scale Schematic representation of the triplet mapping Two-dimensional schematic of the LES and LEM domains Boundary conditions tests. Velocity wave evolution in time. Different colors represent different instants in time Boundary conditions tests. Pressure wave evolution in time. Different colors represent different instants in time Boundary conditions tests. Snapshots of the velocity field (vectors) and temperature (color and contours) Species field before and after the splicing of the cell (i,j) Comparative fuel reaction rate obtained by direct integration over the LES time step and by the direct estimation of the Arrhenius-type reaction rates at time steps equal to various fractions of the LES time step. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E5 = Direct estimation of the Arrhenius reaction rate at a time scale 5 times smaller than the LES time step; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; DST E20 = Direct estimation of the Arrhenius reaction rate at a time scale 20 times smaller than the LES time step; RMS error for the direct estimation of the Arrhenius reaction rate approach for different chemical time intervals Parallel algorithm efficiency in terms of speedup Schematic of the geometry. The solid symbol marks the position of the velocity probe used to capture the time signals in Figs. 3.11, 3.12, 5.4 and Spatial disposition of the computational domains Computational grid detail FFT of the axial velocity autocorrelation, E 11, for the cold flow, normalized by its maximum value x

11 3.12 FFT of the axial axial velocity autocorrelation, E 11, for the LEMLES reactive flow, normalized by its maximum value Instantaneous fields of the LDKM model coefficients predicted by the LEM- LES simulation Instantaneous spanwise vorticity field for the cold flow Instantaneous (left) and time-averaged (right) axial velocity fields Centerline variation of the normalized time-averaged axial velocity for the cold flow. The velocity is normalized by the inflow value, and the distance is normalized by the bluff body size, a Transverse profiles of the time-averaged normalized axial velocity for the cold flow at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the time-averaged normalized transverse velocity for the cold flow at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized rms of the axial velocity fluctuation intensity for the cold flow, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized rms of the transverse velocity fluctuation intensity for the cold flow, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized Reynolds stress for the cold flow, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The Reynolds stress is normalized by U0 2, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Instantaneous spanwise vorticity (solid color), instantaneous reaction rate (thick black line) and time-averaged temperature (color lines) for Cases a1 and 1b xi

12 4.10 Transverse profiles of instantaneous Karlovitz number in Case 1b at axial location m Instantaneous distribution of Karlovitz numbers and filter sizes on the flame surface in Case 1b. DNS = Direct Numerical Simulation, RFS = Resolved Flame Surface, CF = Corrugated Flamelets, TRZ = Thin Reaction Zone, BRZ = Broken Reaction Zone Instantaneous reaction rates. The color lines represents Case 1a and the black line Case 1b Three-dimensional view of the instantaneous (left) and time-averaged (right) reaction zone for Case 1b Three-dimensional view of the instantaneous spanwise vorticity field for Case 1b Centerline variation of the normalized time-averaged axial velocity for Cases 1a and 1b. The velocity is normalized by the inflow value, and the distance is normalized by the bluff body size, a Transverse profiles of the normalized time-averaged axial velocity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized time-averaged transverse velocity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized rms of the axial velocity fluctuation intensity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized rms of the transverse velocity fluctuation intensity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized Reynolds stress for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] Transverse profiles of the normalized time-averaged temperature for Cases 1a and 1b, at the normalized axial locations, from left to right: 3.75 a, 8.75 a and a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] xii

13 5.1 Equivalence ratio inflow profile for cases 3, 5 and Flame stability limits and the position with respect to them of Cases 2-4. The correlation data was reported by De Zubay [1950] and the stability parameter is defined in Eq Mean and variance of the instantaneous Karlovitz number for Cases 2-4. The solid circles indicate mean values and the lines indicate variance intervals. Notations are identical to Fig FFT of the axial velocity autocorrelation, E 11, for Cases 2-4, normalized by their respective maximum values. The three spectra are translated on the y-axis for clarity Ensemble averages of the FFT of the axial velocity autocorrelation, E 11, for Cases 2-4, normalized by their respective maximum values. The three spectra are translated on the y-axis for clarity Instantaneous spanwise vorticity (solid color), instantaneous reaction rate (thick black line) and time-averaged temperature (color lines) for Cases Instantaneous equivalence ratio fields for Cases Schematic of the effect of an antisymmetric (a) and symmetric (b) vortical pattern upon the flame surface Fluctuations of the position of the upper and lower flame fronts around the mean location for Cases Flame sheets fluctuation cross-correlation coefficients for Cases Instantaneous spanwise component of the baroclinic torque for Cases Unsteady dilatation for Cases Pressure fluctuation for Cases Instantaneous field of the unsteady dilatation contribution to the vorticity for Cases Instantaneous baroclinic torque and unsteady dilatation contribution to vorticity along the x axis in the shear layer region for Cases Time variation of the Rayleigh parameter for Cases 2-4. The parameter values are normalized by the maximum value Schematic displaying the placement of the probe points in used for the determination of the axial velocity PDF (Fig. 5.18) PDF of the axial velocity at 3 locations in the flame region for Cases Instantaneous variation of the reaction rate along the flame surface for Cases Instantaneous flame stretch for Cases xiii

14 5.21 Instantaneous variation of the flame stretch immediately downstream of the bluff body for Cases Centerline variation of the normalized time-averaged axial velocity for Cases Transverse profiles of the normalized time-averaged axial velocity for Cases 2-4, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a Transverse profiles of the normalized time-averaged transverse velocity for Cases 2-4, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a Transverse profiles of the normalized time-averaged temperature for Cases 2-4, at the normalized axial locations, from left to right: 3.75 a, 8.75 a and a Time-averaged vorticity along the flame surface for Cases FFT of the axial velocity autocorrelation, E 11, for Cases 5 and 6. For comparison, Case 3 is also included. The spectra are normalized by their respective maximum values and translated on the y-axis for clarity Ensemble averages of the FFT of the axial velocity autocorrelation, E 11, for Cases 5 and 6, normalized by their respective maximum values. For comparison, Case 3 is also included. The three spectra are translated on the y-axis for clarity Instantaneous spanwise vorticity (solid color), instantaneous reaction rate (thick black line) and time-averaged temperature (color lines) for Cases 5 and 6. For comparison, Case 3 is also included Instantaneous equivalence ratio fields for Cases 5 and 6. For comparison, Case 3 is also included Fluctuations of the position of the upper and lower flame fronts around the mean location for Cases 5 and 6. For comparison, Case 3 is also included Flame sheets fluctuation cross-correlation coefficients for Cases 5 and 6. For comparison, Case 3 is also included Time variation of the Rayleigh parameter in the combustor for Cases 5 and 6. For comparison, Case 3 is also included Instantaneous spanwise component of the baroclinic torque for Cases 5 and 6. For comparison, Case 3 is also included Pressure fluctuation for Cases 5 and 6. For comparison, Case 3 is also included Unsteady dilatation for Cases 5 and 6. For comparison, Case 3 is also included Instantaneous field of the unsteady dilatation contribution to the vorticity for Cases 5 and 6. For comparison, Case 3 is also included xiv

15 5.38 Instantaneous baroclinic torque and unsteady dilatation contribution to vorticity along the x axis in the shear layer region for Cases 5 and 6. For comparison, Case 3 is also included Centerline variation of the normalized time-averaged axial velocity for Cases 5 and 6. For comparison, Case 3 is also included Transverse profiles of the normalized time-averaged axial velocity for Cases 5 and 6. For comparison, Case 3 is also included. Axial locations are, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a Transverse profiles of the normalized time-averaged transverse velocity for Cases 5 and 6. For comparison, Case 3 is also included. Axial locations are, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a Transverse profiles of the normalized time-averaged temperature for Cases 5 and 6. For comparison, Case 3 is also included. Axial locations are, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a Time-averaged vorticity along the flame surface for Cases 5 and 6. For comparison, Case 3 is also included Schematic representation of the analogy between a biological neuron and an ANN processing element Temperature and species profiles used for ANN training Initial profile of reaction rates used for ANN training The architecture of the Artificial Neural Network. P denotes the processing element and L denotes the layer Reaction rates in a one-dimensional laminar flame. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; ISAT = In Situ Adaptive Tabulation approach for determining the reaction rate; AN N = Reaction rates computed using the present ANN method Reaction rates in a one-dimensional laminar flame for selected species using direct integration and ANN Instantaneous mass fraction contours for selected species in a three-dimensional, turbulent flame simulated using ANN Time averaged species mass fractions and temperature in a three-dimensional, turbulent flame simulated using ANN. For comparison, direct integration results are included Instantaneous mass fraction contours for selected species in a combustor simulation using ANN xv

16 NOMENCLATURE Roman Symbols A A k a C EBU C ɛ C λ C ν c c p c v D D sgs Da E E a E 11 e surface area Arrhenius pre-exponential constant bluff body size model coefficient for the Eddy Break-Up model model coefficient for subgrid dissipation scalar diffusivity model coefficient for turbulent viscosity speed of sound specific heat at constant pressure per unit mass specific heat at constant volume per unit mass mass diffusivity dissipation of turbulent kinetic energy Damkohler number total energy per unit mass activation energy axial velocity autocorrelation internal energy per unit mass F, F flux vectors f shedding frequency G, G flux vectors H total enthalpy per unit mass H, H flux vectors H i h enthalpy flux specific enthalpy per unit mass xvi

17 i, j, k computational grid indices J k Ka L l l f l G l m l δ Le M W N P p P r Jacobian of the coordinate system transformation turbulent kinetic energy Karlovitz number characteristic length integral length scale flame thickness Gibson length scale mixing length scale thickness of the flame inner layer Lewis number molecular weight total number of species production of turbulent kinetic energy pressure Prandtl number Q, Q state vector q i Re R u r S s St S L T t U heat flux Reynolds number universal gas constant spatial position rate of strain spatial coordinate along the LEM line Strouhal number laminar flame speed temperature time characteristic velocity u, v, w, u i Cartesian velocity vector components xvii

18 u v u sgs V V i,m rms velocity turbulent intensity subgrid scale turbulence intensity volume diffusion velocity of species m x, y, z Cartesian coordinate directions X Y Y i,m mole fraction mass fraction flux of species m Greek Symbols α α k γ δ ij h f h f ε φ Φ i,m η θ i,m κ λ thermal diffusivity Arrhenius temperature exponent ratio of specific heats grid size Kronecker delta specific heat enthalpy of formation per unit mass specific turbulent kinetic energy equivalence ratio convective mass flux of species m Kolmogorov length scale diffusive flux of species m thermal conductivity stirring frequency µ dynamic viscosity ν ν m kinematic viscosity stoichiometric coefficient xviii

19 ρ τ ij σ σ ij density viscous stress tensor wave reflection coefficient viscous work ξ, η, ζ spatial directions in computational space Υ sgs ω ω species-temperature correlation term turbulent frequency reaction rate per unit volume Subscripts cgs cm-gram-second measurement system i, j, k Cartesian tensor indices or species indices L LES m n laminar quantity at the LES level chemical index time step index t, T turbulent quantity u unburned quantity at infinity downstream Superscripts 0 reference quantity LEM sgs stir test quantity at the LEM level subgrid scale stirring test filter scale xix

20 Other Symbols partial derivative operator gradient operator divergence operator summation operator Π product operator difference operator characteristic wave amplitude M Mach number Favre spatial filter test filter space average, fluctuating quantity Abbreviations ANN CFD CFL CPU DNS DSTE DVODE EBU ISAT LDKM LEM LES LODI Artificial Neural Network Computational Fluid Dynamics Courant-Friedrichs-Lewy number Central Processing Unit Direct Numerical Simulation Direct Source Term Estimation Double precision Variable Coefficients Ordinary Differential Equations solver Eddy Break-Up model In Situ Adaptive Tabulation Localized Dynamic k-equation Model Linear-Eddy Mixing model Large-Eddy Simulation Local One-Dimensional Inviscid xx

21 MPI ODE PDE PDF PE RANS TKE rms Message Passing Interface Ordinary Differential Equation Partial Differential Equation Probability Distribution Function Processing Element Reynolds Averaged Navier - Stokes Turbulent Kinetic Energy Root Mean Square, variance xxi

22 SUMMARY Large Eddy Simulation (LES) of bluff body stabilized premixed and partially premixed combustion close to the flammability limit is carried out in this thesis. The main goal of the thesis is the study of the equivalence ratio effect on flame stability and dynamics in premixed and partially premixed flames. An LES numerical algorithm able to handle the entire range of combustion regimes and equivalence ratios is developed for this purpose. The algorithm has no ad-hoc adjustable model parameters and is able to respond automatically to variations in the inflow conditions, without user intervention. Algorithm validation is achieved by conducting LES of reactive and non-reactive flow. Comparison with experimental data shows good agreement for both mean and unsteady flow properties. In the reactive flow, two scalar closure models, Eddy Break-Up (EBULES) and Linear Eddy Mixing (LEMLES), are used and compared. Over important regions, the flame lies in the Broken Reaction Zone regime. Here, the EBU model assumptions fail. In LEMLES, the reaction-diffusion equation is not filtered, but resolved on a linear domain and the model maintains validity. The flame thickness predicted by LEMLES is smaller and the flame is faster to respond to turbulent fluctuations, resulting in a more significant wrinkling of the flame surface when compared to EBULES. As a result, LEMLES captures better the subtle effects of the flame-turbulence interaction, the flame structure shows higher complexity, and the far field spreading of the wake is closer to the experimental observations. Three premixed (φ = 0.6, 0.65, and 0.75) cases are simulated. As expected, for the leaner case (φ = 0.6) the flame temperature is lower, the heat release is reduced and vorticity is stronger. As a result, the flame in this case is found to be unstable. In the rich case (φ = 0.75), the flame temperature is higher, and the spreading rate of the wake is increased due to the higher amount of heat release. The ignition delay in the lean case (φ = 0.6) is larger when compared to the rich case (φ = 0.75), in correlation with the instantaneous xxii

23 flame stretch. Partially premixed combustion is simulated for cases where the transverse profile of the inflow equivalence ratio is variable. The simulations show that for mixtures leaner in the core the vortical pattern tends towards anti-symmetry and the heat release decreases, resulting also in instability of the flame. For mixtures richer in the core, the flame displays sinusoidal flapping that results in larger wake spreading. The numerical simulations presented in this study employed simple, one-step chemical mechanisms. More accurate predictions of flame stability will require the use of detailed chemistry, raising the computational cost of the simulation. To address this issue, a novel algorithm for training Artificial Neural Networks (ANN) for prediction of the chemical source terms has been implemented and tested. Compared to earlier methods, such as reaction rate tabulation, the main advantages of the ANN method are in CPU time and disk space and memory reduction. The results of the testing indicate reasonable algorithm accuracy although some regions of the flame exhibit relatively significant differences compared to direct integration. xxiii

24 CHAPTER I INTRODUCTION 1.1 Overview To date, the most important source of industrial power available to mankind is the chemical energy stored in hydrocarbon fossil fuels and their processed derivatives (e.g. oil, gasoline, diesel fuel). Therefore, the combustion process through which the fuel is burned and the chemical energy stored therein is released is of crucial importance to almost every engineering process, including aeronautical applications. The continuous drive for improvement that is innate to human nature, as is clearly illustrated by the history of science and technology, also drives the continuous improvement of all technological artifacts, and aircraft propulsion systems are no exception to this rule. Besides the continuing demand for stronger, lighter and more efficient engines, lower costs and better performances another driving force has become more and more important particularly for the aerospace industry: the concern for environmental safety and cleanliness. Over the last decade, the rules and regulations that were imposed on aircraft propulsion systems have become ever more drastic, and as this trend will most probably continue in the future, entire new areas of study become significant for the design of modern propulsion systems. In-depth analysis and investigation is required to further the understanding of both the chemical mechanism through which the fuel species react and are exothermally transformed into the product species and of the fluid mechanics of the flow through the combustor, as well as of the interaction between the combustion process and the fluid mechanics of the flow, including the turbulent processes. Generally, combustion can be classified according to several criteria. One of the most relevant classifications is made depending on the way the reactant species are mixed prior to entering the combustion chamber (i.e., premixed combustion, partially premixed and non-premixed combustion). The present work deals with premixed and partially premixed 1

25 combustion. In premixed combustion, all the reactants are homogeneously mixed before the combustion occurs, while in partially premixed combustion, the mixing of reactants is also achieved prior to combustion, but their ratio in the mixture may vary in time and space throughout the process. A second classification for premixed combustion is based on the velocity at which the combustion front (i.e. flame) propagates through the unburned mixture. When the flame propagates above the speed of sound, the phenomenon is termed detonation, while when the propagation velocity is subsonic, it is a def lagration phenomenon. The scope of this thesis is limited to deflagrations only. The proper functioning of a combustion engine depends strongly on having the flame anchored at the designed position in the combustion chamber. Several phenomena may occur within the device that may affect the flame stability: quenching, liftoff, blow - off and flashback, all of them having potentially catastrophic effects on the device. Hence, understanding and controlling the stability of the flame is of great importance for the design of propulsion systems. In practical applications, several methods for stabilizing the flame at the desired location exist, such as low-velocity by-pass ports, refractory burner tiles, swirl or jet induced recirculating flows, separated flows induced by rapid increases in flow area, and bluff-body flame holders [Turns, 1999]. This work will deal mainly with the bluff body stabilization mechanism. 1.2 Combustion and Turbulence The next sections will address some of the physical mechanisms that are encountered in a turbulent reactive flow, in order to give a better view of the phenomena that can be encountered in a bluff body stabilized flame, and to prepare the grounds for the interpretation of the simulation results that will be presented in later Chapters Turbulent combustion In most aerospace related applications, the Reynolds number characteristic of the fluid flow in the flame region is sufficiently high such that the combustion process occurs in a 2

26 turbulent flow field. The effects of the turbulence are generally advantageous for the efficiency of the combustion, since turbulence enhances the mixing of component chemical species and heat [Peters, 2000], but adverse effects upon combustion can also occur, if the turbulence level is sufficiently high to create flame extinction. In turn, combustion may enhance the turbulence through dilatation and buoyancy effects caused by the heat release. Thus, a thorough understanding of the combustion process occurring in a combustor, for instance, would require first understanding the interplay and interdependency between combustion and turbulence. However, the field of turbulent combustion is still an open research topic (according to Peters [2000] it is the most significant unresolved problem in classical physics ) and significant research efforts are currently underway towards this end Reynolds number invariance Even though the effect of turbulence on combustion is generally well understood, the reciprocal effect, the impact of combustion on turbulence is still a matter of some debate in the scientific community. First, the heat released by chemical reaction during the combustion process causes volumetric expansion and buoyancy in the surrounding flow. On the other hand, the increase in temperature causes an increase in the gas viscosity, and therefore, in the turbulent dissipation. Of even more fundamental concern is how the combustion process affects Kolmogorov s inertial range energy invariance hypothesis, which is at the core of every turbulence model developed so far. Fortunately, the empirical evidence gathered so far supports the invariance hypothesis for reactive flows as well [Peters, 2000]. For example, Damkohler [1947] showed that the ratio of the difference between the turbulent and the laminar flame speed and the turbulence intensity is invariant with the Reynolds number in the large Reynolds number limit. Sonju and Hustad [1984] have shown experimentally that the length of a non-buoyant turbulent jet diffusion flame is independent of the Reynolds number. Also, the NOx emission index of hydrogen - air diffusion flames was found to be Reynolds number independent [Peters and Donnerhack, 1981]. The validity of Kolmogorov s hypothesis for reactive flows is also supported by theoretical considerations, since it is postulated that combustion introduces no supplementary 3

27 viscous effects [Peters, 2000]. Therefore, it can be safely concluded, based on the evidence existing so far, that Kolmogorov s hypothesis also holds for reactive flows and the turbulent models developed for isothermal flows are valid for flows with heat release as well Scale separation hypothesis Another theoretical issue that arises when considering turbulent combustion problems is the interdependency of the various time and length scales involved. Besides the diversity of scales brought into play by the turbulence, combustion occurs at molecular levels and involves a multitude of elementary chemical reactions, each with its own characteristic length scale. If one is to consider simultaneously the entire range of scales involved, the problem becomes a lot more complex and also, when numerical simulation is considered, more computationally expensive. Therefore, a simplifying hypothesis was sought: the socalled hypothesis of scale separation [Peters, 2000]. The hypothesis assumes that in the inertial subrange the scales of the combustion process are separated from the scales characteristic for the turbulence. The idea behind it is that once the ignition point is reached and the chemical reaction moves on the upper branch of what the combustion literature calls the S shaped curve (for a definition see Peters [2000]) the chemical reactions are faster than any turbulent time scale and therefore the chemistry is independent of the inertial range turbulent mixing. However appealing through its simplicity, and popular for the variety of combustion models based on it, this hypothesis does not always hold true and accurate modelling of combustion processes occurring in a real application combustor requires more insight into the matter Premixed flame structure As stated earlier, the present work deals with premixed combustion. For a better understanding of the results to be presented and of the limitations inherent to various combustion models, several brief considerations on the structure of a premixed flame will be discussed next. As shown in Fig. 1.1, the structure of a high activation energy, laminar premixed flame 4

28 contains three major regions [Peters, 1997]: a preheat zone in the front of the flame, where the temperature starts to raise by heat diffusion from the flame front, an inner layer where the fuel is consumed and the radicals are formed and destroyed, and an oxidation layer where the final combustion products are formed. Figure 1.1: Premixed flame structure For the scale separation hypothesis to hold, the width of the inner layer, l δ, needs to be significantly smaller than the smallest eddy scale, the Kolmogorov scale, η. If this is not the case, turbulent eddies of sizes smaller than l δ will penetrate the inner layer and alter the combustion process Regimes in premixed combustion The above considerations imply that the validity of the scale separation hypothesis is not universal and that different situations occurring naturally in a combustion process may need different modelling. To define the validity limits of various proposed combustion models, 5

29 diagrams defining combustion regimes in terms of length and velocity ratios [Borghi, 1985, Peters, 1986] or Reynolds and Damkohler numbers [Williams, 1985b] were introduced in the combustion literature. In the following discussion, the combustion diagram of Peters [1999], shown in Fig. 1.2 will be used. Figure 1.2: Regimes in turbulent combustion [Peters, 2000] In Fig. 1.2, v is the turbulent intensity, S L is the laminar flame speed, l is the integral length scale of the turbulence, l F is the flame thickness, defined as l F = D/S L, where D is the average diffusion coefficient of the chemical species involved, η is the Kolmogorov length scale, l δ is the thickness of the inner layer, Re is the turbulent Reynolds number, defined, under the unity Schmidt number assumption, as Re = v l/s L l F, Da is the turbulent Damkohler number, defined as Da = S L l/v l F and Ka and Ka δ two Karlovitz numbers, defined as Ka = l F 2 /η 2 and Ka δ = l δ 2 /η 2,, respectively. To the left of the Re = 1 line, the combustion is laminar and flame propagation occurs 6

30 at the laminar flame speed, S L. To the right of this line but below the v = S L line, the combustion is said to occur in the wrinkled f lamelet regime. Here, the turbulent intensity is low and laminar flame propagation is dominant, while the turbulence only wrinkles the flame surface. Above the Ka δ = 1 line lies the broken reaction zone regime, where the smallest eddies are smaller than the thickness of the inner layer. Thus, small eddies penetrate into the inner layer and the chemistry becomes strongly affected by the turbulence, the scale separation hypothesis being violated here. The region is of much interest especially for flame stabilization studies since in this regime the effects of turbulence (mainly the enhanced heat turbulent diffusion from the inner layer towards the preheat zone and radical, thermal and diffusive losses) can cause local flame extinction that may lead to flame quenching. The fact that in this region the scale separation hypothesis, which stands at the core of many combustion models, fails, deserves also careful consideration when the combustion model for the numerical simulation is selected. The region between these extremes is the f lamelet regimes zone, where two other regimes can be defined and where most of the modelling and experimental work took place so far. The region with Re > 1, Ka < 1 and v /S L > 1 represents the corrugated flame regime. Here, the flame width is smaller than the smallest eddy and the flame is entirely contained within Kolmogorov sized eddies and only perceives a laminar - like flow. If the Gibson scale is defined as l G = S 3 L /ε, where ε is the turbulent dissipation than, according to Peters [2000], eddies larger than the Gibson scale will wrap the flame around them, distorting it, while eddies smaller than l G will not be able to wrinkle the flame surface. Hence, the cutoff limit for the scalar spectrum function in this regime must be the Gibson scale [Peters, 1992]. Finally, for Re > 1, Ka > 1 but Ka δ < 1 the flame lies in the thin reaction zone regime. The smallest eddies can now penetrate into the flame, as η < l F, but not into the inner layer, as η > l δ. The infiltration of eddies into the preheat zone increases the scalar mixing. Peters [1991, 1999] introduced for this regime a mixing length scale, l m, defined as l m = εt 3 q, where t q is the quenching time given as the reciprocal of the strain rate required to quench a premixed flame. According to Peters [2000], eddies of sizes l m and 7

31 larger will transport preheated unburned mixture into the flame front, thus corrugating the flame front at scales equal to their size. In the last two regimes, the scale separation hypothesis holds and the flame can be regarded as a set of locally laminar stretched flames, called f lamelets, embedded in a turbulent flow [Williams, 1975a, 1985b]. 1.3 Bluff Body Flame Stabilization Flame-holder blowout limits Bluff body stabilized flames have been a research topic for quite a long time, due to their relevance to a wide range of aeronautical applications they have (e.g. turbojet afterburners, nozzle mixing burners, ramjets and SCRAM jets). The usual flow velocity in a ramjet or afterburner is several orders of magnitude higher than the highest flame speed for all practical fuels. Hence, some form of flame stabilization is needed to maintain combustion over the entire flight envelope. Over most of the flight regimes, the incoming temperature of the unburned mixture is much lower than the autoignition point, so a heat source must be provided for the ignition to occur. Also, the residence time of the unburned mixture in the high temperature region must be long enough to allow complete combustion in a compact physical device. The presence of a bluff body inside the high speed flow creates a recirculating wake structure, as shown in Fig. 1.3, that serves as a heat source for the continuous ignition of the incoming fresh mixture. Also, the turbulence in the shear layer of the bluff body generated wake enhances the mixing between the hot burned gases and the incoming fresh mixture and thus, increases the efficiency of the combustion process. Combustion also affects the geometry of the backflow region. The volumetric expansion caused by the chemical heat release causes an increase in the recirculation zone length and crossflow area, hence the residence time is further increased and the mass and heat transfer across the shear layer are reduced due to the heat release effects (reduced density and turbulent intensity). 8

32 Figure 1.3: Recirculation zone behind a bluff body Early experimental evidence indicates that several parameters can affect the geometry for recirculation zone and, therefore, the stability limits of the premixed combustion. Nicholson and Fields [1949] investigated the flow pattern in the wake of bluff bodies in both reactive (premixed) and non - reactive flows, showing that, for sufficiently large blockage ratios, vortices detach at the sharp corners of the bluff body in both cases and correlating the observed flame oscillations near ignition and rich blow-off to the duct dimensions. Winterfeld [1965, 1967] and Bovina [1959] have shown that for a cold flow the residence time depends on the bluff body shape. In a reacting flow, the shape effect on the recirculation flow time becomes negligible. The reason for this is that although the length of the recirculation region increases with the approaching velocity, the maximum diameter depends only on the shape of the bluff body and on the blockage ratio (defined as the ratio between the maximum crossflow area of the bluff body and the total crossflow area of the combustor). It can be shown that the volume to area ratio of a truncated ellipsoid approaches a constant value as its length over diameter ratio increases, hence the backflow region volume remains approximately constant. Further studies focused mostly on the flame stability limits, as well as on the stability mechanisms involved in the combustion process. Quantitative studies of the behavior of bluff flame stabilizers were reported by Wright [1958], Winterfeld [1965] and Solntsev [1963], 9

33 providing empirical correlations between the geometry of axisymmetric and two-dimensional bluff bodies and flow parameters associated with the recirculation region immediately downstream of the bluff body. The studies measured the influence of the inflow velocity, blockage ratio and flame holder shape on the geometry of the back-flow region, the residence time, the mean exchange velocity through the shear layer and the level of turbulence intensity at the recirculation region boundary. The conclusions reveal that two parameters are the most important in determining the stability, or lack thereof, of a bluff body stabilized premixed flame: the incoming flow velocity and the equivalence ratio of the fresh mixture, and they are generally used to define the stability limits of a flame holder. From a physical point of view, the controlling stability parameter is the ratio of the chemical time versus the residence time (i.e. the Damkohler number). The residence time is in turn related to the geometry of the recirculation zone and to the rate of mass exchange through the shear layer. An increase in the incoming velocity tends to reduce the residence time through increased momentum transfer from the main flow into the recirculation region [Winterfeld, 1965, 1967], thus reducing the Damkohler number. For equivalence ratios close to one, the residence time in the recirculation region approaches a maximum. For mixtures further away from the stoichiometric composition, the maximum temperatures are lower, hence the effect of heat release on the mass exchange rate reduction is less important. Also, the lower temperature implies lower reaction rates and hence, larger chemical times and larger Damkohler numbers. In conclusion, the incoming velocity for a stable flame is maximum for an equivalence ratio of (or close to) unity, while the limits of stability in terms of maximum and minimum equivalence ratio are wider as the approaching velocity of the main stream decreases. Besides the two major parameters controlling the flame stability discussed above, several other quantities can modify in some measure the flame holder stability. A large number of experimental studies were carried out during the middle of the last century to assess the correlations between the flame stability and the pressure [De Zubay, 1950, Scurlock, 1948], temperature [Haddock, 1951, Maestre and Barrere, 1954], turbulence [Petrein et al., 1955, Solokhin, 1963, Wright and Zukoski, 1960, Solntsev, 1963] combustion chamber exit 10

34 conditions [Maestre and Barrere, 1954, Petrein et al., 1955], bluff body geometry [Maestre, 1955] and fuel type [Winterfeld, 1967]. A detailed review of the work in this area in the decade is provided by Williams [1966]. The effects of those parameters on the stability limits can be inferred from considering their impact on the residence time (namely on the mass exchange rate) and on the recirculation zone geometry, as well as on the chemical time. For a detailed review of the impact of those parameters as well as for an empirical correlations database, see Ozawa [1971] Bluff body stabilization mechanism Fundamental insights into the stabilization mechanism in bluff body stabilized flames were gained through the study of Williams and Shipman [1953]. The changes in the chemical composition of the mixture in the immediate wake was found to correspond to a shift in the flame stability curves, hence the composition is considered as an important factor for flame stability. The separating region at the edge of the bluff body is observed to delimit the region of unburned mixture from the region of burned gases, and it is in this layer that the flame is initiated. Flame blowout occurs when unburned gases penetrates the flame front at the downstream end of the recirculation region and replace the hot products that should be brought back to sustain combustion. The flame is initiated and sustained in the free shear layer originating from the sharp corners of the bluff body. The flame stabilizes itself a short distance downstream of the bluff body due to a phenomenon called ignition delay that will be discussed later on. The shear layer separating from the buff body rolls up into large-scale coherent structures and significant flame-vortex interaction occurs in the wake of the body. The flame holding mechanism was studied by Zukoski and Marble [1955a, 1956] by analyzing the geometry of the back-flow region for various incoming flow velocities. The length of the recirculation region was found to increase as the inflow velocity was increased. Longwell et al. [1949] and, later, Westenberg et al. [1956] measured the range of equivalence ratios within which a premixed flame is stabilized by the presence of a bluff body. The effect of the inflow velocity, temperature, pressure, and shape and size of the bluff body 11

35 on this range was also analyzed, determining that the incoming velocity has a strong effect upon flame stability at a given equivalence ratio, and that the stability limits are extended by increasing the unburned mixture temperature. The mechanism of flame stabilization behind bluff bodies in premixed systems was also experimentally studied by Scurlock [1948], Williams et al. [1949] and Fabri et al. [1952]. Variations in the equivalence ratio, inflow velocity and pressure, flame-holder shape and size, fuel type and turbulence intensity were considered. The results of the studies have shown that turbulence increases the effective flame velocity, but the stability of the flame is decreased by increasing the turbulent intensity, that the shape of the bluff body has a small effect on the flame stability and that the blow-off velocity exhibits a direct dependence on the inflow pressure (blow-off velocity will increase when pressure increases). Experimental and theoretical studies on the blow-off velocity and the flame stabilization mechanism were reported by De Zubay [1950] and Zukoski and Marble [1955b, 1956]. In agreement with the earlier results, the ignition of the incoming fresh mixture was observed to occur in the shear layer by turbulent mixing with the recirculated hot combustion products and flame extinction occurs when the fresh mixture does not spend sufficient time in in the shear layer to allow ignition. Later studies supported the previous conclusion about the central role played by turbulent mixing. The flame spread in the region further downstream of the bluff body was investigated in detailed experiments by Wright and Zukoski [1960, 1962]. The influence of the laminar flame speed, density ratio, equivalence ratio and inflow temperature upon flame spreading are found to be insignificant, leading to the conclusion that the spreading rate of the flame is mainly controlled by the turbulence, in a manner similar to that of the spreading of a turbulent jet [Spalding, 1956]. Another interpretation of the flame stabilization and extinction in bluff body stabilized flames was proposed by Longwell et al. [1953]. Here, the wake of a bluff body is considered an homogeneous chemical reactor and extinction occurs when the time available for chemical reaction is less than the time required for sufficient heat release to raise the fresh mixture temperature to the ignition point. 12

36 Since the time spent by the fresh mixture in the shear layer and the residence time of the combustion products in the recirculation region are both proportional to the characteristic dimension of the flame holder [Lefebvre, 1999], both previous approaches towards explaining the bluff body flame stabilization mechanism yield similar conclusions when an extinction correlation is sought. The current general view, proposed by Ballal and Lefebvre [1979, 1981], is that blowout occurs in premixed systems when the rate of heat production in the combustion zone is not sufficient to raise the unburned gases temperature to the ignition point. In later years, a vast amount of more complex experimental studies on bluff body stabilized flames have been carried out and some, considered more relevant to the current study, are highlighted. Cheng et al. [1989], Cheng and Shepherd [1991], Cheng [1984] conducted a series of measurements of premixed bluff body stabilized flames, focusing mostly on the aspects of flame-turbulence interactions (e.g., velocity and transported scalar spectra, turbulence transport properties, and Reynolds stress). Another set of experimental studies were conducted by Sjunesson et al. [1991b,a, 1992] at VOLVO (Sweden) to develop a validation database for model validation and will be used extensively in this thesis. Various models for turbulent combustion were experimentally validated and assessed by Veynante et al. [1994, 1996] using bluff body flame stabilization techniques. Measurements of the flame surface density and burning rate were reported by Shepherd [1996], and several turbulent flame measurements in a bluff body stabilized premixed flame were also carried out by Fuji et al. [1978], Fuji and Eguchi [1981] and Nandula et al. [1996]. Furthermore, Hertzberg et al. [1991] studied the vortical pattern behind bluff bodies in reactive flow and the vortical interaction with the flame Ignition delay and flame blow - off Even for perfectly stable flames, experimental observations show that the flame does not stabilize itself immediately downstream of the bluff body trailing edge, but slightly downstream. This flame lift-off phenomenon is caused by the ignition delay. By analyzing the 13

37 mechanism governing this phenomenon, important insight into the flame blow - off mechanism can be gained. Several concurrent phenomena appear to contribute to this observed premixed flame liftoff. First, the quenching effect of the solid wall of the bluff body through heat and radical losses must be considered. However, its importance is expected to be diminished as the walls are hot and the fresh mixture comes into contact with the recirculated hot gases immediately after the bluff body trailing edge. Therefore, other mechanisms must be considered as well in order to explain the observed phenomenon. Another important factor is the fact that the cold unburned mixture requires a certain amount of time for its temperature to rise (by thermal diffusion) up to the ignition point and to initiate the combustion. If the convection velocity (controlled mainly by the approaching mainstream velocity) is too high, the flame liftoff distance can increase so much that the unburned gas exits the hot recirculation region before ignition temperature is reached, thus causing the flame to quench. A second model aimed at explaining the ignition delay and, eventually flame quenching, is based on the stretch rate to which the flame is subjected. In the case of bluff body stabilized flames, the strain rate reaches large values at the trailing edge of the bluff body. The extinction effect of high velocity gradients was first related to flame stretch by Karlovitz et al. [1953]. Later, Markstein [1959] considered flame curvature effect on stretch and its influence on the flame front stability. A rigorous mathematical definition of the stretch caused by the strain rate was introduced by Williams [1985a]: K = 1 da A dt (1.1) where A is the flame surface area, or it can be expressed in a non - dimensional form as a Karlovitz number: Ka = D 2 u SL 0 K (1.2) where D u is the diffusion coefficient of the unburned mixture and S 0 L is the un-stretched laminar flame speed. According to Law [1988], there can be three contributions to the stretch affecting a flame: the flow non-uniformity along the flame surface (aerodynamic straining), the flame 14

38 curvature (if the flame surface is not normal to the flow velocity) and the non-stationarity of the flame (flame motion), if the flame curvature is non - zero. The effects of these stretch components depend on where they occur with respect to the flame surface. Thus, if stretch occurs away from the flame, but close enough such that the flow non - uniformities affect the surface, (in the so called hydrodynamic zone) the stretch (termed hydrodynamic stretch) will displace and distort the flame surface so that the local propagation velocity balances the local normal flow velocity and modifies the volumetric burning rate of the flame [Law, 1988]. Positive stretch increases the burning rate and negative stretch decreases it. Closer to the flame in the preheat zone where the transport phenomena are dominating, the stretch (flame stretch) affects the normal mass flux entering the reaction zone and the residence time inside it. This stretch affects the rates of the chemical reactions occurring inside the reaction zone and is of critical importance for stretch induced flame extinction. The theoretical analysis of Law [1988], supported by experimental and numerical evidence [Sung et al., 1996], indicates that a flame without heat losses, with unity Lewis numbers for all species, and free to move unrestricted in response to changes in stretch will be unaffected by stretch and, therefore, cannot be quenched by it. For flame extinction to occur, other factors, such as incomplete reaction, heat loss, or preferential diffusion are required. In the case of a restricted flame, as is the case for a bluff body stabilized flame, the residence time decreases with increase in stretch, the flame becomes thinner and the combustion moves further away from completion until, eventually, flame extinction occurs. Heat loss (e.g., to a solid wall) also has a significant contribution to flame quenching, through the implied decrease in temperature and, therefore, the decrease in the chemical reaction rates. The coupling between the heat loss and the flame stretch is very strong [Libby and Williams, 1983], as aerodynamic stretch tends to displace the flame with respect to the heat sink (i.e., closer to the solid wall Law [1988]), while the curvature effect alters the heat flux through the flame surface [Law, 1988]. Finally, non - unity Lewis number flames (e.g. propane / air mixtures and, even more so, 15

39 hydrogen / air mixtures) are strongly influenced by stretch [Law, 1988, Sung et al., 1996]. Chung and Law [1988] have demonstrated by means of integral analysis that the local flame temperature can deviate from the adiabatic value if both stretch and preferential diffusion are simultaneously present in a flame, and their theoretical analysis is supported by substantial experimental evidence [Tsuji and Yamaoka, 1983, Sato, 1983, Law et al., 1986, Mizomoto et al., 1985]. Thus, for a negatively curved (convex with respect to the burned products) flame region (e.g. a stagnation flame), for Le < 1 the flame temperature will exceed the adiabatic value. For Le > 1 there exists a critical stretch value over which the flame temperature will decrease below the value required to sustain combustion, and the flame will extinguish. Conversely, for a positively curved (concave with respect to the burned products) flame region (e.g. a Bunsen flame), the flame temperature increases over the adiabatic value for Le > 1 and decreases under the extinction value for a large enough stretch rate if Le < Turbulent flame extinction From the standpoint of turbulent combustion, the laminar flamelets that form turbulent flames are highly convoluted and subjected to widely different aerodynamic stresses that change in time. Also, even for a perfectly premixed composition at the inflow, differential diffusion effects can change the local equivalence ratio in turbulent flames. Several experimental studies [Dinkelacker et al., 1998] and [Buschmann et al., 1994] and Direct Numerical Simulations (DNS) [Card et al., 1994] have shown that lean premixed flames can be quenched in a turbulent field even in the absence of heat losses. In turbulent premixed and partially premixed combustion, local flame quenching can occur, implying regions with low radical concentrations bordering the fresh mixture but maintained at a temperature close to the flame temperature by the stretch stabilization mechanism. On the other hand, depending on the variable flow conditions and flame geometry, this local extinction phenomenon may eventually develop subsequently into global flame extinction. 16

40 1.3.5 Partially premixed flames Partially premixed combustion is a relatively new research topic. In practical applications, the actual type of combustion may be neither premixed nor non-premixed combustion, but an intermediate type, where the fuel and oxidizer are mixed, but the mixture is not homogeneous, resulting in locally different equivalence ratio. Such a case can occur if fuel and the oxidizer enter the combustor separately, but partially mix by turbulence prior to reaching the flame front. Such cases of stratif ied combustion [Peters, 2000] can occur, for example, in aircraft gas turbines or in direct injection gasoline engines, where fuel droplets have time to vaporize and mix partially with air before igniting [Peters, 2000]. Partially premixed combustion is also present in the case of the lift-off and stabilization of turbulent jet diffusion flames [Peters, 2000]. In this case, a partially premixed field occurs at the base of flame and the flame propagation generates a structure called a triple flame. The point located at the leading edge of the flame is termed a triple point and propagates along a surface of equivalence ratio close to the stoichiometric value. A lean premixed flame exists on the lean side of this surface, and a rich premixed flame on its rich side. The diffusion flame stabilizes behind the triple point. The first studies dealing with the issue of partial premixing have studied the problem of turbulence - chemistry interaction at the lift-off height of the flame. The stabilization mechanism of a turbulent jet diffusion flame has been a matter of some debate in the combustion scientific community over the years. Wohl et al. [1949] proposed one stabilization mechanism, according to which the flame lifts when the mean velocity gradient exceeds a critical value and is stabilized at a height where the local flow velocity is balanced by the flame speed. A later study [Vanquickenborne and Van Tiggelen, 1966] showed that the fuel and air are fully premixed at the base of the flame and that stabilization is achieved at a location where the mean flow velocity in the vicinity of the stoichiometric mixture surface equals to the turbulent burning velocity of a stoichiometric premixed flame. Further proof of the fact that the stabilization mechanism is governed by the premixed flame propagation was reported by Eickhoff et al. [1984] in a study that provided measurements of time-averaged temperature, 17

41 species concentration and velocity at different locations upstream and downstream of the stabilization region. A different approach [Peters and Williams, 1983] suggests that flame stabilization is controlled by the extinction of the diffusion flamelets. They argued that the residence time at the flame base is insufficient to achieve a homogeneous mixture. Finally, Broadwell et al. [1984] proposed a stabilization mechanism controlled by the large scale turbulent structures. They transport hot reaction products to the edges of the jet where they ignite the fresh mixture. Another possible way of obtaining partial premixing comes from flow related instabilities or mechanical imperfections that can interact with the fuel feed-lines causing variations of the inflow equivalence ratio [Stone and Menon, 2003], and creating the conditions for a partially premixed flame. The unsteadiness thus introduced in the combustion process may, under certain conditions that will be discussed in detail later, develop into combustion instabilities that result in large pressure and velocity oscillations that are detrimental to the proper functioning of the engine and can lead even to global flame extinction or mechanical failure. Paschereit et al. [1999] experimentally studied partially premixed swirling flames and related the unstable combustion modes to flow instabilities in the recirculating region. A closed loop active control system was employed to suppress the pressure oscillations by decoupling the combustion process from the flow instability. Hermanson et al. [1997] reported experiments regarding the flame stability in the lean limit when partially premixed fuel injection is employed, showing an increase in the flame stability of bluff body flame holders mounted close to the combustor inlet. Experimental observations and numerical simulation of partially premixed bluff body stabilized flame were reported by Shen [2005], who studied the chemical reaction rates and the NOx emissions for a partially premixed, disk stabilized flame, providing a database for predicting NOx formation mechanism. More recently, Ferreira et al. [2005] studied pulsating partially premixed flames with the purpose of investigating the effects of combustion driven acoustic oscillations of the emission rates of the combustion products. 18

42 1.4 Modelling in Computational Fluid Dynamics With the recent advances in the computer technology, the numerical simulations of both reactive and non-reactive flow have become feasible and the bluff body stabilized flames were no exception to this rule. Reynolds Averaged Navier Stokes (RANS) simulations of reactive flows behind bluff bodies were reported by many researchers in the field (e.g. Bai and Fuchs [1994], Fureby and Moller [1995], Saghafian et al. [2003], Lien and Yee [2004], Frendi et al. [2004]) with varying degrees of success. However, important discrepancies in the wake configuration could not be avoided due to shortcomings stemming mostly from the RANS inability to handle the smaller turbulent scales that are key to the combustion process. The later development of the Large Eddy Simulation (LES) technique [Smagorinsky, 1963, Lilly, 1967, Deardorff, 1974] allowed a significant improvement of numerical simulation accuracy in general. In LES, the three dimensional large scale motion is resolved, hence the geometry dependent flow features are captured accurately, and only the small scales that exhibit local isotropy are modelled, as it will be discussed in detail in the subsequent Chapters. Due to its ability to resolve a larger range of turbulent scales, the method is more appropriate to simulate turbulent flows. In the case of a bluff body stabilized flame, the boundary layer that forms on the two sides of the bluff body separates at the bluff body edge and forms a pair of free shear layers surrounding a central recirculation region. The flow dynamics are largely controlled by the large scale fluctuations, caused by large eddies, often called coherent structures. Therefore, the mean fields alone, as resolved by RANS methods, are insufficient for accurate predictions. Also, previous studies [Menon, 2005] have shown that the full compressible LES equations (as employed in this study) allow the accurate capture of acoustic-vortex-entropy interactions that are expected to play a critical role in the flame stabilization mechanism behind a bluff body. Numerical simulations using this technique and dealing with bluff body flame holders were reported by Fureby and Lofstrom [1994], Fureby and Moller [1995], Fureby [1995a, 1996], Moller et al. [1996], Fureby [2000b,a], Ryden et al. [1993], Inage et al. [1998], Chakravarthy [2000], Giacomazzi et al. [2004]. 19

43 Stone and Menon [2002, 2003] and Duwig et al. [2005] conducted LES of partially premixed combustion in swirling flows using spatially and temporally variable inflow equivalence ratio. An analysis of the dynamic response of the combustor to these variations in equivalence ratio has shown that the heat release oscillations caused by the variation in the equivalence ratio can either enhance, or damp the pressure oscillations in the combustor. A novel approach for LES of partially premixed combustion has been proposed recently by Domingo et al. [2002] and Fiorina et al. [2005], based on the flame index approach that quantifies the occurrence in the flow of the premixed, or the non-premixed combustion regime. Dosing et al. [2005] proposed a combination of the conserved scalar approach and the G-equation model for partially premixed combustion and validated the method by simulating a bluff body stabilized flow. Recently, numerical studies of the dynamics of the premixed bluff body stabilized flame have been reported by Mehta and Soteriou [2003], Mehta et al. [2004], Erikson et al. [2006] using Lagrangian methods. Some DNS studies have also been reported over the last years [Vervisch et al., 2003], but they are only limited to small domains due to the important computational cost that makes DNS prohibitive for real-life combustor numerical simulations Momentum equation closure The fluid motion is governed by a set of equations generally known as the Navier Stokes equations. This system of partial differential equations is derived on physical basis from conservation laws (of mass, momentum, energy and chemical species) and, therefore, a mathematically exact solution would provide an exact description of the reactive fluid flow. However, with very few exceptions related to very simple cases, the Navier-Stokes equations have no known general analytical solution. To circumvent this problem, Computational Fluid Dynamics (CFD) proposes a numerical integration in time and space. Such a solution will not be mathematically exact, due to numerical errors and due to the truncation associated with the differencing process. However, over the years, especially since the the development of the high speed digital computer in the second half of the twentieth century, 20

44 a multitude of numerical methods and schemes have been developed such that today, the purely numerical error can be reduced to an insignificant value if sufficient computational resources are available [Tannehill et al., 1997]. The issue of availability of computational resources is, however, not trivial. For a numerical approach to exhibit temporal and spatial accuracy, the time and the length-scales at which the numerical integration is performed need to be at least of the same order of magnitude as the smallest time and length scales involved in the problem at hand. For laminar flows, where inertial forces are balanced by the viscous forces such that the flow is characterized by quasi-parallel streamlines, the smallest time- and length-scales are sufficiently large to be captured by reasonably dense grids. In a turbulent flow, however, inertial forces overcome the viscous forces and instabilities appearing in the flow field grow rapidly creating large, highly energetic, rotating structures characterized by the so-called integral length scale [Hinze, 1959], l. The large structures are subject to stretch and are broken down into smaller and smaller structures, up to the so called Kolmogorov limit, where the energy is dissipated by viscous action [Hinze, 1959]. If l is comparable to the largest length-scales in the flow and still in the range where reasonable grid resolutions can capture it, the Kolmogorov length-scale, η, is much smaller and numerical integration up to this scale becomes impractical from the resources standpoint. Such an approach, aimed at resolving all length-scales up to η is known in the literature as a DNS. The Reynolds number (defined as Re = L U / ν, where L and U are the flow characteristic length and velocity scale, and ν is the fluid viscosity) up to which DNS is achievable is generally about 6,000 [Pope, 2000] Historically, the first approach towards alleviating the resources issue was to solve the so called Reynolds Averaged N avier Stokes set of equations instead of the classical Navier - Stokes. In RANS, the Navier - Stokes equations are time-averaged and the numerical solution only resolves the time averaged flow field, thus eliminating the small fluctuating scales and allowing numerical solutions on coarse computational grids. However, the filtering process generates unclosed terms that require further study. Thus, the Reynolds stresses, u i u j, (where u i and u j represent different velocity vector components and represents the 21

45 time average) become unknowns in the RANS equations [Pope, 2000] and require modelling. To the date, a multitude of such turbulence models exists and a detailed review is given by Pope [2000]. Generally, turbulence models can be classified into several categories, as follows: T urbulent viscosity models This category is based on the turbulent viscosity hypothesis, introduced by Boussinesq in 1877, and according to which the deviatoric Reynolds stress is proportional to the rate of strain, u i u j = 2 u k u ( k 3 2 δ ui ij ν T + u ) j x j x i (1.3) where the proportionality constant ν T is called turbulent viscosity and requires modelling. Different approaches chosen to model the turbulent viscosity create different turbulent viscosity closure models (algebraic, turbulent kinetic energy models, k ε models, k ω models, the Spalart-Almaras model, etc.) However, the main flaw of turbulent viscosity models consists in the fact that Bousinesq s turbulent viscosity hypothesis fails for flows exhibiting rapid changes in the rate of strain [Bradshaw, 1973] and a need for better closure models became obvious. Reynolds stress models In this class of turbulence models, model transport equations for the Reynolds stresses are solved along with the RANS equations and along with a transport equation for some other quantity (usually the turbulent dissipation, ε, or the turbulence frequency, ω = ε / k, where k is the turbulent kinetic energy) that provides a turbulent length scale. The approach renders the turbulent viscosity hypothesis unnecessary, but its application creates supplementary unclosed terms that require modelling. Among Reynolds stress models, some of the better known are return to isotropy models [Rotta, 1951, Shih and Lumley, 1985, Sarkar and Speziale, 1990, Chung and Kim, 1995], pressure rate of strain models [Naot et al., 1970, Hanjalic and Launder, 1972, 22

46 Launder et al., 1975, Shih and Lumley, 1985, Fu et al., 1987, Jones and Musonge, 1988, Speziale et al., 1991], algebraic stress models [Rodi, 1972, Girimaji, 1997] and nonlinear turbulent viscosity models [Girimaji, 1996, Taulbee, 1992, Gatski and Speziale, 1993, Yoshizawa, 1984, Speziale, 1987, Rubinstein and Barton, 1990, Craft and Launder, 1996] P robability Density F unction (PDF) models In PDF methods, the mean velocity and the Reynolds stress are considered the first and the second moments, respectively, of the Eulerian PDF of velocity, f( V ; x, t) [Pope, 2000], and a model transport equation is solved for such a PDF. In this approach, all convective transport appears in closed form, and the model equation for the PDF can be obtained in closed form using the generalized Langevin model [Pope, 2000]. Stochastic Lagrangian models (such as the Langevin equation) are usually employed to model the velocity PDF. Since the velocity PDF alone contains no information on the time-scale of the turbulence, supplementary information is required. This information can be obtained either by employing a dissipation model, or by considering instead of the velocity PDF, the joint PDF of velocity and turbulence frequency [Pope, 2000]. A third approach, which is actually a compromise between DNS and RANS is provided by LES. The main disadvantage of the classical RANS approach resides in the scarcity of information provided by the simulation. By only resolving the averaged flow characteristics, RANS methods fail to capture information about flow unsteadiness and vortex dynamics that are crucial to the interaction between the flow and the chemistry in a reactive flow. Since DNS is currently out of reach even for the most advanced computers in use, the only approach that is both reliable in terms of accuracy and feasible in terms of computational cost remains the LES. LES resolves both the large, geometry dependent turbulent scales (as RANS does as well) and a fraction of the smaller energy containing scales within the inertial range, up to a level dictated by the resolution of the numerical grid, and only the remaining scales are modelled. If the grid resolution is appropriately chosen, the unresolved 23

47 scales, by Kolmogorov s hypothesis [Kolmogorov, 1941] are isotropic and, therefore, more amenable to modelling. This approach not only provides a lot more information but since the energy contained in the unresolved scales is much less than in the RANS approach, is also less sensitive to modelling hypotheses. A more detailed presentation of the method will follow in the next Chapter. A schematic of the 3 CFD methods and their applicability range in terms of length scale is presented in Fig Figure 1.4: CFD methods and their applicability range. η is the Kolmogorov scale, l is the maximum size of the isotropic eddy, is the grid size and l is the inertial length scale Combustion modelling In both RANS and LES, unclosed quantities in the reacting flow equations need to be addressed by using combustion models. This section is aimed at reviewing some of the most successful such models The Eddy Break - Up model (EBU) The model was developed by Spalding [1971] and is based on the assumption that the chemistry is much faster than the mixing and, thus the rate controlling phenomena is the turbulent mixing. Since this model will be used in this study, more details will follow in the next Chapter. 24

48 The Eddy Dissipation Model (EDM) This model is a variant of the EBU model developed for non-premixed combustion by Magnussen and Hjertager [1977]. The mean chemical source term is computed in a similar manner to EBU, except for the fact that the reaction rate is limited by a deficient species (i.e., fuel, oxidizer or combustion products) [Veynante and Vervisch, 2002]. The model has been employed in turbulent combustion simulations with some degree of success [Byggstoyl and Magnussen, 1988, Fureby, 1995a, Sloan and Sturgess, 1994] The Probability Density Function (PDF) transport equation models The method is based on the derivation of a joint PDF transport equation for the velocity and the reactive scalars as well as, in some approaches, their gradients or other quantities considered of interest. To the date, several models in this category have been proposed: Pope [1990] proposed a joint PDF of velocity, viscous dissipation and reactive species, Dopazo [1994] used a joint pdf of velocity, velocity gradient, reactive scalars and their gradients. The method has the advantage that is valid for both premixed and non - premixed combustion but its accuracy is strongly dependent on the models employed for the remaining unclosed terms. The method have been used for both RANS [Pope, 1985, Anand and Pope, 1987, Vervisch et al., 1994, Sakak and Schreiber, 1996, Jones and Prasetyo, 1996] and in LES [Gao and O Brien, 1993, Frankel et al., 1993, Cook and Riley, 1998]. The main disadvantage of such methods is the high computational cost in a LES formulation [Chakravarthy, 2000]. Another shortcoming of the PDF approach is the fact that the molecular diffusion is not in closed form and requires the employment of a mixing model The Monotone Integrated LES (MILES) The model was developed by Grinstein and Kailasanath [1995] and solves the unfiltered Navier-Stokes equations for a global chemical reaction mechanism. The method uses no subgrid closure models but employs the inherent numerical scheme dissipation to account for the energy transferred to the subgrid scales. Several numerical schemes have been used, such as Flux Corrected Transport [Boris et al., 1992, Fureby et al., 2001], Jameson, TVD - 25

49 MUSCL, ENO [Tenaud et al., 2000] Laminar flamelets models In certain combustion regimes, namely when the scale separation hypothesis is satisfied, a turbulent flame can be considered as a set of stretched laminar diffusion flames. The view was first proposed by Williams [1975a], and Liew et al. [1981] were the first to use laminar flame profiles to compute turbulent flame mean and variance. The methods in this class are concerned with computing the location of the inner layer that defines the flame surface through an iso - surface of some non - reacting scalar (usually denoted G for premixed combustion and Z in non - premixed combustion) or a progress variable. Once the distribution of this scalar is known, the reactive scalar profiles normal to the surface are determined from laminar flamelet computations. The flamelet equations as functions of G were derived for premixed combustion by Peters [1993]. The biggest advantage of these methods is there is no chemical source terms to be considered (since G and Z are non - reactive) and the only modelling issue remaining concerns the unclosed terms pertaining to the G (or Z) field resulted from the LES or RANS filtering. More detailed reviews of the large variety of models in this class can be found in the review articles of Peters [1986] and Bray and Peters [1994] The Linear Eddy Mixing (LEM) model Originally developed by Kerstein [1988, 1989, 1990, 1991a,b, 1992] and extended to a subgrid combustion model for LES by Menon et al. [1993], LEM is a stochastic approach aimed at simulating the turbulent mixing, molecular diffusion and the chemical reaction in a one - dimensional domain embedded in the LES cells of the computational domain (LEMLES). The basic assumption of the LEM model is that an accurate representation of the turbulent combustion requires a model able to discriminate and resolve the various physical processes involved in the process, such as large scale advection, small scale mixing, molecular diffusion and chemical reaction. Since reaction kinetics and scalar mixing are used directly, without filtering, LEMLES can account for the scalar anisotropy in the small-scale [Menon et al., 2004]. Additionally, 26

50 for premixed systems, it is possible to actually predict the turbulent burning rate in the subgrid rather than just a priori specifying it, as is done in many flame speed models [Yakhot, 1988, Pocheau, 1994]. When the local turbulence level is high, the premixed flame structure can change from corrugated flamelet to thin-reaction-zone or to broken-reaction-zone regime without requiring any model changes [Menon, 2003]. This ability is particularly important when attempting to simulate flame stability near the lean-blow-out limit [Eggenspieler and Menon, 2005]. LEM is the only known combustion model that does not use the scale separation hypothesis and is, therefore, valid even in regimes where the hypothesis fails. Also, the model is highly compatible with the LES technique [Menon et al., 1993] and very flexible in terms of the chemical reaction mechanism used to describe the chemical reactions. In the implementation used here, it is important to note that the LEMLES has no ad hoc adjustable parameters for either momentum or scalar transport equations. The k sgs model plays a critical role in the LEM closure by providing estimates for the sub-grid turbulence level needed for both small-scale stirring and large-scale transport [Chakravarthy and Menon, 2000b]. Nevertheless, the approach has some limitations. Most importantly, LEMLES is relatively much more expensive than conventional LES models, such as EBULES. However, it is highly scalable, so the overall computation time can be decreased by increasing the number of processors. Laminar molecular diffusion across LES cells is not included but this limitation is significant only in laminar regions, whereas LEMLES is designed for high Reynolds number turbulent flow applications. Also, the viscous work is neglected in the sub-grid temperature equation, but it is explicitly included in the LES energy equation, Eq. 2.14c, which is used to ensure total energy conservation. Finally, the flame curvature effect is not explicitly present in the sub-grid. If the flame is highly wrinkled in the sub-grid, multiple flames can be present in the 1D line, and the distance between the flames can be approximated as twice the local radius of curvature [Menon and Kerstein, 1992]. However, this situation will only occur if the LES grid is very coarse in regions of very high turbulence, where the LES of momentum transport is expected to fail much before any sub-grid 27

51 flame related effects become prominent. For the cases simulated here, the LES resolution is chosen to reasonably resolve the turbulence in the flame holding region. As a result, most (if not all) of the flame curvature effects are resolved at the LES level. The LEM model will be used extensively in this work, therefore further details on the model will follow in the next Chapters. 1.5 Motivations and Objectives Motivation The reactive flow behind bluff bodies is a very complex one, associated with high pressure gradients, separation and recirculation. The recirculation region behind a bluff body provides an intense turbulent mixing of mass, momentum and energy, with beneficial effects for the stability of the flame. The modern tendency towards continuous reduction of combustion generated NOx, driven by increasingly more restrictive regulations controlling burning emissions, is shifting combustor design towards fuel lean operating conditions, where lower flame temperatures are able to significantly decrease NOx production. However, combustors operating close to the lean extinction limit are more sensitive to variations in the pressure, velocity or species composition fields, and an accurate assessment of the effect of such fluctuations upon the stability of the combustion process becomes essential. Partially premixed flames are used in numerous practical applications. Also, local extinction and re-ignition in non-premixed combustion may involve partially premixed regions with a very important effect upon the dynamics of the combustion. In practical premixed combustion systems fuel is injected upstream of the flame holder and is mixed with the incoming air using various techniques (usually swirlers). Turbulent fluctuations of the pressure and velocity occurring in the premixing device of the combustor may interact with the fuel and / or air feed lines and create variations in the respective mass - flow rate, and thus locally alter the equivalence ratios entering the combustion chamber and creating spatial gradients of the equivalence ratio. Also, imperfect mixing, due to geometrical errors or other unforseen factors can as well alter the uniformity and the local value of the 28

52 equivalence ratio at the inlet. Such fluctuations would be convected into the flame region and will create instabilities in the heat release rate, local temperature and flame speed, and flame location. Lord Rayleigh, [Rayleigh, 1878] showed that the combustion process becomes unstable when the pressure (p ) and heat release fluctuations (q ) are in phase (i.e., when p q > 0). In this case, acoustic energy is added to the system and drives up, in a positive feedback loop, the amplitude of the instability, endangering the overall flame stability as well as the mechanical integrity of the system. The phenomenon was observed experimentally [Yang and Culick, 1986] and studied both theoretically [Lieuwen and Zinn, 1998, Lieuwen et al., 2001] and numerically [Stone and Menon, 2001, 2003]. Therefore, understanding the combustion process under partially premixed conditions is of significant research interest. Even though bluff body stabilized flames were extensively studied both experimentally and numerically, simulations of partially premixed combustion near the flammability limits are still scarce. As the equivalence ratio decreases below the extinction limit for a given inflow velocity, local or global extinction and re-ignition may occur. Local flame extinction causes incomplete combustion and, therefore, high pollutant emissions and inefficient fuel consumption. Also, as the heat release pattern is altered by periodic extinction and re - ignition, changes in the dynamics of the combustor may occur, causing from high levels of noise to even mechanical failure of the combustor. In non-premixed systems, local flame quenching can be modelled by comparing the scalar dissipation rate to a quenching dissipation rate [Peters, 1984]. Several attempts at developing quenching criteria for premixed flames have also been reported. An extinction model was developed by Poinsot et al. [1991] (named the Intermittent Turbulence Net Flame Stretch) and successfully applied for pollutant prediction but has the disadvantages that it only considers positively stretched flames, which is usually not true for the entire turbulent flame. It also assumes heat losses downstream of the flame front and is, therefore, not applicable for quasi - adiabatic combustors. Tajiri and Menon [2001] have used as a quenching criteria the comparison of the LES strain rate to a known quenching value of the stretch determined for stagnation point flame but, again, this approach does not take into 29

53 account opposite sign curvatures. Pitsch and Duchamp De Lageneste [2002] improved the flamelet model described earlier to account for heat losses to the walls and for mixing of burned gasses with the co-flowing air in the post-flame region. The scalar dissipation rate can be determined from the progress variable variance gradient and can be compared to a quenching scalar dissipation, similarly to the approach used for diffusion flame quenching. Turbulent flame quenching in partially premixed systems is, however, a relatively new research area. In order to address the challenges raised by partially premixed turbulent flames close to extinction, this work aims at simulating premixed and partially premixed bluff body stabilized flames in regions close to the extinction limit and studying flame interaction with the turbulent flow in the combustor, the stabilization mechanism and the flame response to turbulence. The main goal of the thesis is to study partially premixed flames close to the lean limit and how the local variations in the equivalence ratio affect the flame dynamics and stability. For this, a numerical algorithm able to handle the entire range of combustion regimes and equivalence ratios was developed. It is important to note that the numerical algorithm requires no adjustable model parameters. Thus, when the inflow conditions are changed, the algorithm adapts automatically to changes in the flow conditions, requiring no user intervention. The development of such a physics-controlled, non-empirical numerical algorithm for both premixed and partially premixed combustion is considered as the most important achievement of this thesis and as a step towards making possible accurate predictions of realistic combustion problems. To accomplish this goal, the achievement of several objectives, described in the following, was required Objectives Objective 1: LES of non - reactive flow behind a bluff body. The first step towards the the achievement of the previously mentioned goal of this thesis is the development of a computationally efficient, high fidelity, time accurate LES numerical algorithm. The algorithm is based on an explicit five - stages Runge - 30

54 Kutta numerical scheme with local time stepping and uses a multi - domain, parallel, implementation. The non-reactive flow validation is achieved in a first stage by the numerical simulation of a non - reactive flow behind a bluff body and comparison with experimental data. Objective 2: LES of bluff body stabilized flame in the fast chemistry assumption. The numerical simulation is performed in the assumption that the chemical reactions are much faster than the turbulent mixing, and the combustion model employed is the Eddy Break - Up Model (EBU). A five species - one step chemical mechanism provides the turbulent mean reaction rate. Due to the inherent limitations of the model, the EBULES approach is not be expected to accurately capture the combustion process details, but its usefulness becomes obvious in terms of computing time, rendering the EBULES numerical solution as a useful first appraisal of the reactive flow field and a good starting point for simulations using more complex models. Validation of the numerical algorithm is done at this stage through comparison with reactive flow experimental data. Objective 3: LEMLES of bluff body stabilized flame. The next step is to replace the EBU model by the Linear Eddy Mixing as the combustion model. The same five species - one step chemical mechanism provides the chemical source terms in the reactive species equation. The effect of the combustion model (LEM vs. EBU) is analyzed in terms of accuracy of prediction versus measured data and computational cost. Also, comparisons of flame structure are carried on, in order to assess flame dynamics close to the flammability limits and to prove the LEMLES ability to handle combustion regimes out of range of the EBULES. The final validation of the numerical algorithm is completed at this stage by comparing the simulation results to previous experimental and numerical data. Objective 4: LEMLES of premixed combustion under different inflow conditions. 31

55 Numerical simulations of premixed flames at equivalence ratios that represent the estimated limits of variation for the incoming fuel / air mixture are first conducted and the impact of the equivalence ratio on flame stability and dynamics is evaluated. The impact of the change in inflow equivalence ratio on the predicted flame stability is appraised. The ability of the numerical algorithm to respond to this changes is demonstrated to prepare the ground for partially premixed simulations. Objective 5: LEMLES of partially premixed combustion. Next, partially premixed flames with equivalence ratios varying spatially within the same range are simulated and studied. The flame structure and its dynamic response is analyzed. Potential sources of combustion instabilities are considered by evaluating their impact upon flame stability. The numerical algorithm demonstrates its capacity to adapt not only to different constant inflow conditions, but also to transient states in this range and to predict, without changes in the model, both stable and unstable flames. Objective 6: Development and implementation of an LEMLES algorithm using Artificial Neural Networks. Even more insight into the subtle details of the bluff body flame stabilization mechanism detailed, multi-step chemical kinetics will be eventually needed. Even though the numerical algorithm is able to capture flame instability, the predicted flame stability limits may not be accurate. Radical species play a central role in flame quenching and re-ignition, and detailed chemistry is required for reliable quantitative predictions of flame stability. The main challenge raised is the high computational cost. One method to reduce this cost is the tabulation of the chemical source terms. In the In-Situ Adaptive Tabulation (ISAT ) approach [Pope, 1997], this tabulation is performed on - line by storing the direct integration results once computed and retrieving them when the same initial conditions occur in the simulation. Preliminary studies [Eggenspieler, 2005] indicate a reduction in CPU time by a factor of 30 for a 16 steps methane mechanism, but issues related to the CP U memory and algorithm 32

56 accuracy in the limit of low fuel fractions were found to diminish the attractiveness of the method [Eggenspieler, 2005]. An Artificial Neural Network (ANN) algorithm applicable for LEMLES is developed, implemented and tested. However, further testing and validation will be required in the future. The approach is based on the preliminary steps taken in this direction by Kapoor et al. [2001], Kapoor and Menon [2002], and the basic idea is to model the chemically reacting system by predicting the temporal evolution of its reactive scalars using a previously, off-line trained ANN. The main advantages of the ANN method are in CPU time and disk space and memory reduction, but the correct choice of the correct ANN structure and parameters becomes critical for achieving the desired proficiency and accuracy of the ANN predictions. 1.6 Outline The second chapter of this thesis presents the mathematical formulation of the LES technique and the combustion models. The third chapter presents the numerical implementation used in this work as well as the geometries and operating conditions for the problem studied. The fourth chapter focuses on the non-reactive and reactive validation study performed in a geometrical setup reproducing the so-called V OLV O experiment [Sjunesson et al., 1991b,a]. The reactive flow simulation is conducted using two different scalar closure models that are compared against each other and with the experimental data. In Chapter V, the effect of the inflow equivalence ratio upon the flow and flame structure in premixed and partially premixed systems is studied. First, the flame structures and holding mechanisms are compared for three constant equivalence ratio premixed flames, in a geometrical setup similar to the V OLV O experiment. Next, the dynamic behavior of flames with spatially variable equivalence ratio is presented and analyzed, in comparison with a baseline, constant equivalence ratio case. Chapter VI revisits the issue of numerical estimation of the chemical source terms and proposes a new and numerically efficient method based on ANN. Final conclusion and future research objectives are presented in Chapter VII. 33

57 CHAPTER II MATHEMATICAL FORMULATION AND MODELLING 2.1 Fluid Motion Governing Equations The equations governing the motion of an unsteady, compressible, reacting, multiple-species fluid are the Navier-Stokes equations The Navier-Stokes equations The fully compressible Navier-Stokes equations describing the conservation of mass, momentum, total energy and conservation of N chemical species are: ρ t + ρu i x i = 0 ρu i t ρe t ρy m t [ ] + x j ρu i u j + pδ ij τ ji = 0 ] + x i [(ρe + p )u i + q i u j τ ij = 0 [ ( )] + x i ρy m u i + V i,m = ẇ m,where m = 1,N (2.1) In the above equations, u i is the i-th velocity component, ρ is the mass density, p is the pressure, Y m is the species mass fraction of the m-th species, V im is the diffusion velocity of the m-th species in the i-th direction, E = e (u ku k ) is the total energy per unit mass, and τ ij is the viscous stress tensor, defined as: ( ui τ ij = µ + u ) j 2 x j x i 3 µ u k δ ij (2.2) x k where δ ij is the Kronecker function (δ ij = 0 if i j and δ ij = 1 if i=j). Also, e is the internal energy per unit mass computed as: N e = Y m h m p ρ m=1 where h m is the species enthalpy per unit mass given by: (2.3) 34

58 In the above, h 0 f,m h m (T ) = h 0 f,m + T T 0 c P,m (T )dt (2.4) is the enthalpy of formation per unit mass of the m-th species at the reference temperature T 0, c P,m is the specific heat at constant pressure for the m-th species. Returning to Eq. 2.1, ω m is the mass reaction rate per unit volume of the m-th species: ω m = MW m L k=1 ( ) ( ν mk ν mk A k T α k e ( E a,k/r ut ) Π N Xm p ) ν n=1, m = 1, N (2.5) R u T where L is the number of chemical reactions of the considered mechanism and N is the number of species, MW m is the mass fraction of the m-th species, ν mk and ν mk are the stoichiometric coefficients of the m-th species and for the k-th chemical reaction on the product and reactant side, respectively. A k, α k and E a,k are the Arrhenius rate pre-exponential coefficient, temperature exponent and activation energy for the k-th chemical reaction, respectively, T is the temperature and R u is the universal gas constant. X m is the molar fraction of the m-th species. The heat flux vector in Eq. 2.1 contains the thermal conduction (I), enthalpy diffusion (i.e. diffusion of heat due to species diffusion) (II), the Dufour heat flux and the radiation heat flux. Dufour heat flux and radiation heat flux are neglected, therefore: q i = κ T x }{{} i (I) + ρ N h m Y m V im m=1 } {{ } (II) (2.6) where κ= c P µ/p r is the mixture averaged thermal conductivity. c P = N m=1 Y mc P,m is the mixture averaged specific heat at constant pressure and P r is the mixture Prandtl number. The pressure p is directly derived from the equation of state for perfect gas: R u p = ρrt = ρ T = ρr u T MW mix N m=1 Fick s Law is used to determine the species diffusion velocity: Y m MW m (2.7) V im = D m Y m Y m xi (2.8) 35

59 where D m is the m-th species molecular diffusion coefficient. Gradients of temperature and pressure can also produce species diffusion (Soret and Dufour effects, respectively) but these two contributions are neglected hereafter. The viscosity is determined using Sutherland s law: T µ µ 0 = (2.9) T S +T T 0 T 0 3/2 where µ 0 is the reference viscosity at T 0 and T S = K. Finally, total mass conservation is ensured by enforcing:: N Y m = 1 (2.10) m=1 N V im = 0, i = 1, 2, 3 (2.11) m= LES filtered governing equations The earliest application of the LES methodology was performed by Smagorinsky [1963] and important further developments of the method were introduced by Germano et al. [1990], Moin et al. [1991], Erlebacher et al. [1992] and Menon [1992]. The separation between the large and the small scales is determined by the grid size ( ). Therefore, the Navier-Stokes equations have to be filtered with respect to the grid size in order to obtain the LES governing equations. A Favre spatial top-hat filter (appropriate for finite-volume schemes) is employed to derive the LES equations. Thus, any variable (f) is decomposed into a resolved quantity ( f) and a unresolved quantity (f ) such that f = f + f. More details regarding the LES filtering and the different techniques are given by Ghosal [1993] and Pope [2000]. The Favre filtering, represented in the following by the symbol f is defined, for any flow variable f, by: f = ρf ρ (2.12) where the over-bar represents spatial filtering and is defined as: 36

60 f( x, t) = f( x, t)g f ( x, x )dx (2.13) D where D is the entire computational domain, x is the position vector and G f is the top-hat filter kernel defined as: G f ( x, x ) = 1 if x x < 2 0 otherwise Thus, by applying the above described filtering process to Eq. 2.1, the LES filtered Navier-Stokes equations can be written as [Erlebacher et al., 1992]: ρ t + ρũ i x i = 0 ρũ i t ρẽ t ρỹm t ] = 0 [( ) + x i ρẽ + p ũ i + q i ũ j τ ji + H sgs i [ + x j ρũ i ũ j + pδ ij τ ij + τ sgs ij [ + x i ρỹmũ i + ρd Ỹm m x i + Φ sgs i,m + Θsgs i,m ] + σ sgs i = 0 ] = ρ ẇ m, where m = 1,N (2.14) The filtered LES equations contain terms representing the effects of the unresolved scales on the resolved motion, resulting from the filtering process and denoted henceforth by the superscript sgs. Thus, if k sgs is the subgrid turbulent kinetic energy, defined as: k sgs = 1 ( ) u k u k ũ k ũ k 2 (2.15) the total energy per unit mass can be written as: Ẽ = ẽ (ũk ũ k ) + k sgs (2.16) where, for calorically perfect gases: ẽ = N m=1 ( ) c v,m Ỹ m T + Ỹ m h f,m (2.17) In the above, h f,m constant pressure. = h 0 f,m c p,mt 0, and c v,m is the specific heat os species m, at Also, the filtered viscous stress tensor can be written in terms of filtered velocities and temperature as: 37

61 Similarly, the filtered heat flux vector becomes: ( ũi τ ij = µ + ũ ) j 2 x j x i 3 µ ũ k δij (2.18) x k q i = κ T x i + ρ N m=1 h m Ỹ m Ṽ im + N m=1 q sgs im (2.19) where the diffusion velocities are computed using the resolved gradient of the species mass fraction (Ỹk) and q sgs ik represents the heat transfer via turbulent convection of species. Finally, the equation of state can be written in a filtered form as: p = ρr u N m=1 Ỹ k T MW m + Υ sgs (2.20) Summarizing, the LES filtered governing equations contain several unclosed terms that need to be modelled: the subgrid shear stress tensor, τ sgs ij, the subgrid enthalpy flux, H sgs i, the subgrid viscous work, σ sgs i, the subgrid convective mass flux, Φ sgs j,m, the subgrid diffusive mass flux, Θ sgs jm, the subgrid heat flux, qsgs i,m and the subgrid temperature - species correlation term, Υ sgs τ sgs ij ] = ρ [ũ i u j ũ i ũ j H sgs i = ρ[ẽui Ẽũ i σ sgs i = u j τ ji ũ j τ ji [ ] Φ sgs jm = ρ Ym u j Ỹmũ j [ jm = ρ Ym V j,m ỸmṼj,m Θ sgs q sgs jm = [ h m D m Y m x j ] ] + [pu i pũ i ] h m Dm Ỹm ) Υ sgs = N m=1 (Ỹk T Ỹk T x j ] (2.21) Subgrid closure of the LES equations The closure of subgrid terms is a major area of research and many approaches have been proposed in the past. The initial LES formulation [Smagorinsky, 1963] used an algebraic eddy viscosity model for the LES closure equations under the assumption that subgrid kinetic energy production and dissipation balance each other. In general, since the small scales 38

62 primarily provide dissipation for the energy that cascades from the large scales through the inertial range, an eddy viscosity type subgrid model appears appropriate for modelling. However, the underlying assumption that subgrid kinetic energy production and dissipation balance each other is somewhat questionable. This requirement is only satisfied in the dissipation range and, in most cases, especially for high Reynolds number cases, the LES grid resolution is insufficient. For a computationally affordable, coarser grid, the kinetic energy contained in the subgrid scales is significant and the subgrid field requires a relaxation time before responding to the resolved scales changes, so the above hypothesis is not accurate. Therefore, an alternate, non - equilibrium, choice of the velocity scale is needed. A non - equilibrium model based on the subgrid kinetic energy transport equation was developed by Schumann [1975]. This study will follow the implementation given by Kim and Menon [1999], Kim et al. [1999] and Menon et al. [1996]. Basically, the model assumes isotropic turbulence at the subgrid scales and tracks the subgrid kinetic energy (k sgs ) using a transport equation that will be described in the next section. The subgrid length scale is given by the filter size,, the velocity scale, V sgs is determined from k sgs and the eddy viscosity is modelled as: The subgrid stress tensor τ sgs ij gradient diffusion model at the grid cutoff scale: ν t = C ν k sgs (2.22) is then closed using the subgrid eddy viscosity and a τ sgs ij = 2ρν t ( Sij 1 3 S kk δ ij ) ρksgs δ ij (2.23) where S ij = 1/2( ũ i / x j + ũ j / x i ) is the resolved strain rate. It is important to note that even though the τ sgs ij model employs a gradient diffusion assumption, the large scale counter-gradient effects are accounted for, since the large scales are resolved. The subgrid enthalpy flux H sgs i eddy viscosity ν t : is also modelled using a gradient assumption and the H sgs i = ρ ν t P r t H x i (2.24) 39

63 where H = h + ũ i ũ i /2 + k sgs is the total enthalpy, P r t is a turbulent Prandtl number. In this case, the turbulent Prandtl number is assumed unity, but it is important to note that P r t can actually be dynamically computed based on ν t and the thermal diffusivity. h is the specific mixture enthalpy, evaluated as: h = N m=1 hm Ỹ m (2.25) It was also proved [Fureby, 1995b, Veynante et al., 1996] that Υ sgs can be neglected for low heat release cases but may become important otherwise. However, due to significant modelling difficulties it will henceforth be neglected. Also, the subgrid work, σ sgs i neglected here. Finally, closures for the filtered reaction rate ẇ m for the subgrid viscous work σ sgs i, for the sub-grid convective (Φ sgs i,m ) and diffusive (θsgs i,m ) species fluxes, and the sub-grid heat flux, q sgs i,m will be described in the next section Subgrid turbulent kinetic energy model A transport equation is formally derived for k sgs and solved along with the rest of the LES equations. Since k sgs evolves locally and temporally in the flow, the equilibrium assumption is relaxed. Hence, the grid resolution needs to resolve scales up to the inertial range instead is of the dissipation range. The model has been successfully employed for high Reynolds number flows in the past [Menon et al., 1996, Kim and Menon, 1999, Kim et al., 1999, Kim and Menon, 2000, Eggenspieler and Menon, 2003, 2005]. The k sgs transport equation is: ( ρk sgs t p p ) u i ũ i x i x }{{ i } III + ( ρũ i k sgs) = ( ) ρkui ρ kũ i x } i x {{}} i {{} I II τ ) ij ũ i + (ũj τ sgs ) ij τ sgs ũ j ij x i x i x i ( τ ij ui } {{ } IV } {{ } V x }{{ i } V I (2.26) where k=1/2ũ k ũ k and the subgrid kinetic energy is defined by Eq It is important to note that although k sgs is resolved at the LES level, it represents a subgrid quantity. The 40

64 symbol will be henceforth omitted for simplicity. In Eq. 2.26, term (I) represents the resolved convection, terms (II) and (III) the subgrid turbulent convection, term (IV) is the sum of the subgrid stress work (IV 1 ) and the turbulent kinetic energy dissipation (D sgs ), term (V) represents the subgrid transport of turbulent kinetic energy and (VI) the turbulent kinetic energy production (P sgs ). Terms (II) to (VI) require closure. The three subgrid turbulent convection terms (II), (III) and the subgrid stress work (IV 1 ) are modelled together as: II + III + IV 1 x i [ ρ (ν + ν t) σ k k sgs x i ] (2.27) Here, and σ k is a constant assumed to be unity Chakravarthy and Menon [2001b]. The production of subgrid turbulent kinetic energy (P sgs ) can be expressed in terms of the subgrid stress tensor: P sgs = τ sgs ũ i ij (2.28) x j Finally, the dissipation of subgrid turbulent kinetic energy (D sgs ) is modelled as: D sgs = C ɛ ρ (k sgs ) 3 (2.29) In the previous equations there are two model coefficients C ɛ and C ν that must be either prescribed as constants, or obtained dynamically as a part of the solution [Kim and Menon, 1995, Kim et al., 1998] Localized dynamic kinetic energy model (LDKM) In the present study, C ɛ and C ν are dynamically computed as a part of the solution and this technique is described in the following section. The approach was developed by Kim and Menon [1995] for incompressible flows and was extended by Nelson and Menon [1998] for compressible flows. This model was successfully used in many non-reacting [Menon et al., 1996, Patel et al., 2003] and reacting [Kim et al., 1999, Kim and Menon, 2000, Eggenspieler and Menon, 2005, Menon and Patel, 2005] studies. 41

65 Although the LDKM is not the only method available for the dynamic evaluation of C ɛ and C ν (see, for instance the model developed by Germano et al. [1991]), its advantage consists in the fact that no spatial averages are required, and makes the model easy to implement in complex geometries. The LDKM model assumes that the resolved and the unresolved small scales behave in a similar manner and, thus, the model coefficients can be computed using similarity relationships. First, a test filter, denoted, operating in a region close to the cutoff scale, at the small, but still resolved, scales is defined. Usually, the size of the test-filter is twice the LES resolution: =2. Since the turbulent quantities are known at the testfilter level, the LES model coefficients can be determined by comparing quantities resolved at. Experimental observations in high Reynolds number turbulent jets [Liu et al., 1994] have shown that the subgrid stress τ sgs ij the test filter level (Eq. 2.30) are self-similar. at the grid filter level and the Leonard s stress at L ij = ρũ i ũ j ρũ i ρũj ρ (2.30) However, a simple scale-similar model of the form τ sgs ij = C L L ij, where C L is a model constant [Liu et al., 1994] does not yield the proper turbulent dissipation and a better approach needs to be found. Consider the subgrid stress tensor τ sgs ij, given by At the test-filter level, Eq can be re-written as: τ sgs,test ij = ρũ i ũ j ρũ i ρũj ρ ( ρ ũk ũ k 2C ν ρ 2 ρ ρũ k 4 ρ ρ + 1 ( 3 ) 1 ( 2 Sij 1 ) S kk δ ij 3 ρ ũ k ũ k ρũ kρũ k ρ ) δ ij (2.31) where Sij is defined as: S ij = 1 ( ) ρũj + 2[ ( )] ρũi x i ρ x j ρ (2.32) Also, at the test filter level, the sub-grid kinetic energy is: 42

66 k test = 1 2 ( ρũk 2 ρ ρũ k 2 ρ 2 ) (2.33) In LDKM, it is assumed that the self-similarity between the subgrid stress and Leonard s stress also holds when both quantities are evaluated at the test filter level: Hence, Leonard s stress can be written as: τ sgs ij = ĈLL ij (2.34) Assuming ĈL L ij = τ sgs ij Ĉ L = 2 ρ C ν k test ( Sij 1 ) S kk δ ij Ĉ L ρk test δ ij (2.35) 3 Ĉ L = 1 and using Eq. 2.35, C ν can be determined using a least-square method Lilly [1992] as: C ν = D ijm ij 2D ij D ij (2.36) where M ij and D ij are the exact and modelled test-filter subgrid stress tensor, respectively: M ij = ρ ũ j ũ i ρũ j ρũi 1 ( ρ ρ ũ k ũ k ρũ ) k ρũk δ ij 3 ρ D ij = ρ ( ρ ũk ũ k ρũ k ρũ ) 1 ( k 2 Sij 1 ) S kk δ ij 2 ρ 2 ρ 2 ρ 3 (2.37) The value of C ν must be constrained such that τ sgs ij is positive and finite [Nelson, 1997]. To satisfy this constraints, several conditions must be enforced [Vreman et al., 1994]: τ sgs 11 0 τ sgs 22 0 τ sgs 33 0 τ sgs 12 2 τ sgs 11 τ sgs 22 τ sgs 13 2 τ sgs 11 τ sgs 33 τ sgs 23 2 τ sgs 22 τ sgs 33 det(τ sgs ij ) 0 (2.38) 43

67 C ɛ is determined in a similar manner, using an expression for the rate of k sgs dissipation ɛ sgs [Kim and Menon, 1999, Kim et al., 1999]. Thus: C ɛ = (µ + µ t ) ( ũ ) j ρ k test 3/2 T ij Tij ũj x i x i (2.39) In the above, µ is the molecular viscosity, µ t = ν t ρ is the eddy viscosity at the test level and the tensor T ij is defined at the test filter level as: T ij = ũ i + ũ j 2 x j x i 3 ũ k δ ij (2.40) x k LDKM is locally stable in both in space and time without smoothing. Past studies [Fureby et al., 2004, James et al., 2006], and recent commercial code evaluation by Kirpekar and Bogy [2005] has demonstrated the reliability and accuracy of the LDKM closure. The model was found to be also effective in the near wall region in turbulent wall bounded flows [Fureby et al., 2004, Patel and Menon, 2006] recovering the exponential wall damping [Van Driest, 1956]. 2.2 Combustion Modelling Several terms in Eqs and 2.19 still remain open at this point: the subgrid convective (Φ sgs i,m ) and diffusive (θsgs i,m ) species fluxes, the subgrid heat flux, qsgs i,m and the filtered reaction rate, ẇ m. In this study, two approaches towards closing these terms are considered: a conventional closure at the LES resolved scale and a closure directly at the subgrid scales The conventional closure: Eddy Break - Up model (EBU) The conventional closure for the species equations employs for the sub-grid convective scalar flux an eddy diffusivity closure, D T = ν T /Sc T, where Sc T is a turbulent Schmidt number that was set to unity in this study. Note that, since ν t is obtained dynamically, D t is also dynamically obtained in this closure. Thus: Φ sgs i,m = ρν t Sc t Ỹm x i (2.41) 44

68 Since no conventional closure models exist [Poinsot and Veynante, 2001] for the sub-grid heat flux, q sgs i,m, the sub-grid diffusive species flux, θsgs i,m and the sub-grid temperature-species correlation, Υ sgs, these terms have been neglected at present. Although many models for the filtered reaction rate have been proposed, especially for premixed combustion [Pope, 1990, Sethian, 1996], this study uses an approach based on a sub-grid EBU approach developed earlier by Fureby and Moller [1995]. The basic idea is that for combustion to occur, two processes need to take place simultaneously: chemical reaction and scalar mixing, and the rate controlling phenomena will be the slower of the two. The turbulent mixing rate is given by Peters [2000]: ε ω p = ρc EBU k Y F (2.42) where ε is the specific turbulent dissipation, k is the turbulent kinetic energy, Y F is the minimum reactant mean mass fraction and C EBU is a model constant, set to 2 in this study [Magnussen and Hjertager, 1977]. The approach is cost effective and easy to implement since the sub-grid turbulent mixing time scale can be directly estimated using k sgs. Since the effects of the super-grid turbulence on the scalar fields are resolved, k in Eq can be replaced by the known sub-grid kinetic energy, k sgs, and ε by ε sgs = D sgs /. Hence, the mixing rate equation becomes: k sgs ω p = ρc EBU C ɛ Y F (2.43) The chemical reaction rate is given for the problem at hand, that uses a 5-species, 1-step, reduced chemical mechanism for propane / air combustion: with a reaction rate given by Westbrook and Dryer [1981]: C 3 H 8 + 5O 2 3CO 2 + 4H 2 O (2.44) ω c = ν mw m ρ ( A k exp E ) a [C 3 H 8 ] c 1 [O 2 ] c 2 (2.45) R u,cgs T where ν m is the stoichiometric coefficient, A k is a pre-exponential factor equal to 8.6 x 10 11, E a is the activation energy, equal to 3.0 x 10 4 calories / g, R u,cgs is the universal 45

69 gas constant expressed in calories / gram Kelvin, [X] represents the molar concentration of species X, in moles / cm 3, and c 1 and c 2 are two coefficients, set to 0.1, respectively 1.65 for this case [Westbrook and Dryer, 1981]. Finally, the slowest of these two rates is used to represent the filtered reaction rate, ω m = min( ω p, ω c ). Although this approach (called EBULES hereafter) can provide reasonable results [Fureby and Moller, 1995], it also has well known limitations. First, it must be noted that small-scale scalar mixing, molecular diffusion and chemical kinetics all occur at the small scales and are not resolved in LES. Furthermore, the scalar fields at the sub-grid level are, unlike the turbulent scales, strongly anisotropic, thus rendering the use of the eddy diffusivity closure questionable. Also, in highly turbulent regions (e.g. in regions of high shear) an over-estimation of the reaction rate is likely to occur. In regions of low turbulence, or for very high grid resolutions approaching DNS, the sub-grid turbulent kinetic energy k sgs decreases towards zero. If the molecular diffusion effect is neglected, on grounds that it is much smaller than the turbulent mixing, the mixing rate given by Eq will tend to zero, and so will the filtered reaction rate. It could be argued that using the sub-grid kinetic energy in Eq to determine the turbulent mixing rate is still acceptable, since the super-grid turbulence effects on the scalar fields are resolved, but this reasoning does not consider the fact that the sub-grid turbulent mixing may decrease up to the point that neglecting molecular diffusion is no longer justifiable, rendering the reaction rate inaccurate. Most importantly, the EBU model is based on the scale separation hypothesis and it cannot be expected to provide accurate results where the hypothesis is invalidated The subgrid closure: Linear-Eddy Mixing model (LEM) The Linear-Eddy Mixing model governing equations A more comprehensive closure of the scalar mixing and combustion is based on the LEM model proposed by Kerstein [1989] and developed into a sub-grid model by Menon et al. [1993]. This approach, called LEMLES hereafter, has been developed in the past few years to offer a closure directly at the sub-grid scales for all combustion processes. It has been successfully applied to scalar mixing [Menon and Calhoon, 1996, Chakravarthy 46

70 and Menon, 2001b, Sankaran and Menon, 2005a], premixed combustion [Chakravarthy and Menon, 2000b, Sankaran and Menon, 2005b, Eggenspieler and Menon, 2005], non-premixed combustion [Menon and Calhoon, 1996, Calhoon et al., 1995], pollutant emission [Eggenspieler and Menon, 2005] and spray combustion [Menon, 2004, Menon and Patel, 2005], with little or no change to the basic structure of the model. LEM is a stochastic approach aimed at simulating, rather than modelling the effects of turbulence on the chemistry, and it is not limited by the scale separation hypothesis [Peters, 2000]. The parameters controlling the LEM turbulent mixing model require only the validity of the Reynolds number independence of free shear flows in the limit of large Reynolds numbers, which is a safe assumption for any flow of engineering interest [Peters, 2000]. Due to this extended validity range, the LEM model can be expected to perform well in any combustion regime, and to be able to accurately handle flames near to, or even outside, the flammability limits. In LEMLES, the scalar equations are not filtered, and instead the large scale advection, turbulent mixing by eddies smaller than the grid size, molecular diffusion and chemical reaction are resolved at their appropriate length and time scales inside each LES cell. While the LES filtered conservation equations for mass, momentum and energy are numerically integrated on the LES grid, the evolution of the species fields is tracked using a two-scale, two-time numerical approach. For any scalar an exact and unfiltered Eulerian transport equation can be written as: ρ ψ t = ρu ψ i ( ψ ) ρd ψ + ω ψ (2.46) x i x i x i where the first right hand side term represents the total convection, the second is the molecular diffusion (D ψ is the species dependent diffusion coefficient) and the last term is the unfiltered chemical reaction source term. The velocity vector can be decomposed into: u i = ũ i }{{} I ( + ) face u i } {{ } II ( + u i ) LEM } {{ } III (2.47) where term (I) represents the LES resolved velocity, term (II) is the sub-grid velocity at the interface between LES cells determined using the known subgrid kinetic energy, and term 47

71 (III) is the small scale velocity fluctuation inside the LEM domain, unresolved at the LES level. By using Eq and regrouping the terms in Eq. 2.46, equations characterizing the large (Eq. 2.48), respectively small scale processes (Eq. 2.49) can be written: t+ LES ψ n+1 = ψ + 1 t ρ ρ ψ ψ n ψ n ( + ρũ i + ρ t LES x i [ ( ρ u i u i ) LEM ψ n x i ) face ψ n = 0 (2.48) x i ψ (ρd n ) ] ψ ω ψ dt (2.49) x i x i In the above, t LES is the LES time step, ψ n and ψ n+1 are consecutive time values of the scalar ψ evolution, ψ is an intermediate solution, after the large scale convection is completed. In Eq. 2.49, the first term under the integral represents the sub-grid stirring, the second is the sub-grid molecular diffusion and the last accounts for the reaction kinetics The small scale processes Molecular diffusion and chemical reaction contribution to the small scale transport are resolved on a one-dimensional grid inside each LES cell at a resolution much finer than the LES resolution, and approaching the Kolmogorov scale. The 1-D computational domain is aligned in the direction of the flame normal inside each LES cell, ensuring an accurate representation of flame normal scalar gradients [Kerstein, 1989]. On this domain (denoted the LEM domain hereafter) molecular diffusion (term A below), chemical reactions (term B), diffusion of heat via species molecular diffusion (term C), heat diffusion (term D) and chemical reaction heat release (term E) are resolved, according to the equations below: LEM Y LEM m ρ t ρ LEM c P T LEM t + F stir m + F stir T = + ( ρ LEM Y LEM ) m D m } s {{ s } A N = ẇ m W m }{{} B )( T LEM ) s ( Y LEM m ρc p,m D m s s k=1 }{{} C N h m ẇ m W m m=1 } {{ } E T LEM (κ s (2.50) ) } {{ } D (2.51) Here, the superscript LEM indicates values at the sub-grid LEM level, and s is the spatial coordinate along the LEM domain. The chemical source terms are computed using 48

72 the chemical mechanism described by Eqs and Fm stir and FT stir represent respectively the effect of the sub-grid turbulence on the species m mass fraction field and on the temperature field. The current implementation assumes a calorically perfect gas, and the sub-grid pressure, p LEM is assumed constant over the LEM domain, and equal to the supergrid value, p, which is a valid assumption in the absence of strong pressure gradients [Sankaran et al., 2003]. Hence, the sub-grid density is computed from the equation of state at the sub-grid level : p LEM = ρ LEM T LEM N k=1 Y LEM k Radiation effects are neglected. The small-scale turbulent stirring (F stir m R u W k (2.52) and FT stir ) is implemented explicitly on the same grid using stochastic re-arrangement events that mimic the action of an eddy upon the scalar field using a method known as triplet mapping and designed to recover the 3D inertial range scaling laws [Smith and Menon, 1996a]. Kerstein [1989] demonstrated that triplet mapping is able to accurately capture the increase in the scalar gradient while maintaining the mean scalar field value. Since the effects of the rearrangement are applied to the one-dimensional LEM line, the subgrid turbulence is inherently assumed isotropic. Mathematically, the triplet mapping can be expressed as a function that, when applied to the initial scalar field Ψ 0 (x, t), transforms it into a new scalar field, Ψ(x, t): Ψ 0 (x, t) = Ψ 0 (3x 2x 0, t) x 0 x x 0 + l/3 Ψ 0 ( 3x 2x 0 + 2l, t) x 0 + l/3 x x 0 + 2l/3 Ψ 0 (3x 2x 0 2l, t) x 0 + 2l/3 x x 0 + l Ψ 0 (x, t) otherwise (2.53) where the mapping interval is [x 0 ; x 0 + l]. The location of this stirring event is chosen from a uniform distribution. The frequency at which stirring events occur is given by Kerstein [1989]: λ = 54 5 νre C λ 3 [( /η) 5/3 1] [1 (η/ ) 4/3 ] (2.54) 49

73 1 1 Subgrid eddy LEM field LEM field LEM domain LEM domain (a) Before rearrangement (b) After rearrangement Figure 2.1: Schematic representation of the triplet mapping where C λ stands for the scalar turbulent diffusivity, set to [Chakravarthy and Menon, 2000a]. The eddy size, l, ranges from the Kolmogorov scale, η, to the grid size, with a distribution given by Kerstein [1989]: f(l) = (5/3)l 8/3 η 5/3 5/3 (2.55) where the Kolmogorov scale is determined as η = N η Re 4/3 and N η is an empirical constant that reduces the effective range of scales between the integral length scale and η but without altering the turbulent diffusivity [Smith and Menon, 1996a]. The value used for this study is 5 [Smith and Menon, 1996a]. A schematic of this rearrangement is shown in Fig A schematic of this rearrangement is shown in Fig The large scale processes Equation 2.48 is modelled using a Lagrangian transport of the scalar field across the LES cells that ensures exact mass conservation and called splicing [Sankaran et al., 2003]. Thus, once the LES computations are completed at a given time step, LEM domain cells (and/or cell fractions) are exchanged between the LES cells in a manner that accounts for the mass fluxes across the LES cell faces. Thus, LEM cells are transferred between the LES volumes 50

74 1 (dρv/dy) j+1/2 0.5 (dρu/dx 1 ) i-1/2 (dρu/dx) i+1/2 (dρv/dy) j-1/ Figure 2.2: Two-dimensional schematic of the LES and LEM domains accounting for the mass fluxes through the LES cell faces. The order in which the cell transfer on each of the spatial directions is performed is dictated by the magnitude of the mass flux in the respective direction at the resolved level [Chakravarthy and Menon, 2001a, Sankaran, 2002]. Next, the number of LEM cells containing the mass flux to be transported to the adjacent cell is determined. If a fractional number is obtained, the LEM cell is split so that exact mass conservation is achieved. A schematic representation of the process is shown in Fig The sufficiently small LES time step ensures that scalars are transported from one LES cell only to an adjacent LES cell, thus drastically reducing the complexity of the problem. The specifics of the implementation will be described in the next Chapter The thermal expansion and the re-gridding procedure Since the pressure in the LEM domain is assumed constant and there is no pressure gradient term in Eqs or 2.51, the volumetric expansion of the LEM cell needs to be modelled 51

75 separately, after each diffusion step, to account for the increase in volume through thermally generated pressure waves. This is done by changing the LEM cell volume according to the equation [Sankaran, 2002]: V n+1 i = ρn i ρ n+1 i where n and n + 1 are two consecutive diffusion steps. (2.56) Both the thermal expansion and the splicing procedure cause the LEM linear grids to be neither uniform, nor have the same number of cells in different LES cells. However, the triplet mapping procedure and the discretization method used to integrate the reaction diffusion equation require an uniform grid and a variable number of LEM cells in the computational domain will increase unnecessarily the complexity of the parallel numerical algorithm. Hence, a re-gridding procedure aimed at producing uniform grids with equal numbers of cells is applied [Sankaran, 2002]. Two things must be noted on this issue. First, the linear interpolation used for the re-gridding procedure is known to produce spurious diffusion. Second, the re-gridding is applied after each LES time step but the thermal expansion effects are determined after each subgrid integration step, as the subgrid temperature evolves through time integration The large - scale / small scale - coupling The coupling between the large scales resolved in the LES formulation and the small scales modelled by LEM is achieved in two ways. First, the LEM model implementation uses the supergrid pressure, subgrid kinetic energy (modelled based on supergrid quantities) and supergrid convection velocities. On the other hand, the LEM model provides to the supergrid level Favre averaged (over the LES cell) values of species mass fractions. Based on those mass fractions, the supergrid temperature is computed. The sub-grid scalar fields in each LES cell are ensemble averaged to obtain the LES-resolved scalar field, Ỹm, which is used in the LES energy equation and equation of state. 52

76 CHAPTER III NUMERICAL IMPLEMENTATION 3.1 Discretization of the Governing Equations The finite volume formulation The LES filtered Navier - Stokes differential equations (Eq. 2.14) together with the subgrid kinetic energy transport equation (Eq. 2.26) are solved using a finite volume 5 stages modified Runge - Kutta scheme. For this, the computational domain is divided into small volumes using a Cartesian grid and the conservation equations (Eq and Eq. 2.26) are applied to these control in an integral form: QdV + (FdS x + GdS y + HdS z ) = ΦdV (3.1) t V S V where V is the control volume delimited by surface S. Here, Q is the state vector: ρ ρũ Q = ρṽ ρ w (3.2) ρẽ ρk sgs ρỹm F, G, H are the fluxes on the three spatial directions, x, y and z, composed of an inviscid part, a viscous part and a subgrid contribution: 53

77 F = F i + F v + F s G = G i + G v + G s H = H i + H v + H s (3.3) In the above, the inviscid fluxes are defined as: ρũ ρũũ + p F i = ρũṽ ρũ w (3.4) ρũ(ẽ + p) ρũk sgs ρũã ρṽ ρũṽ G i = ρṽṽ + p ρṽ w (3.5) ρṽ(ẽ + p) ρṽk sgs ρṽã ρ w ρũ w H i = ρṽ w ρ w w + p (3.6) ρ w(ẽ + p) ρ wk sgs ρ wã The viscous fluxes are, with the notations in the previous Chapter, are: 54

78 0 τ xx F v = τ xy τ xz (3.7) ũτ xx + ṽτ xy + wτ xz q x 0 ρd m Ỹm x 0 τ yx G v = τ yy τ yz (3.8) ũτ yx + ṽτ yy + wτ yz q y 0 ρd m Ỹm y 0 τ zx H v = τ zy τ zz (3.9) ũτ zx + ṽτ zy + wτ zz q z 0 ρd m Ỹm z The subgrid contributions to the fluxes, with the notations in the previous Chapter, are defined as: 55

79 0 τ sgs xx F s = τ sgs xy τ sgs xz (3.10) H sgs x K sgs x Y sgs x,m 0 τ sgs yx G s = τ sgs yy τ sgs yz (3.11) H sgs y K sgs y Y sgs y,m 0 τ sgs zx H s = τ sgs zy τ sgs zz (3.12) H sgs z K sgs z Y sgs z,m Finally, the source terms in Equation (3.1) are: 56

80 0 0 Φ = 0 0 (3.13) 0 P sgs D sgs ω A Domain discretization The numerical simulations presented herein where carried out on structured, three-dimensional, Cartesian grids. In this approach, the computational cell is a rectangular hexahedron. Each grid element represents a control volume, V, delimited by 6 surfaces da i, i=1,6. The physical space (x, y, z) is transformed into a computational space (ξ, η, ζ) of uniform unity length by Vinokur [1989]: x = ξ x ξ + η x η + ζ x ζ y = ξ y ξ + η y η + ζ y ζ z = ξ z ξ + η z η + ζ z ζ (3.14) where ξ x, η x, ζ x, ξ y, η y, ζ y, ξ z, η z, ζ z are the grid metrics. Applying (3.14) to the Navier - Stokes equations (3.1) yields: where: V ( F t Q dv + V ξ + G η + H ) dv = ΦdV (3.15) ζ V Q = 1 J ( Q ) F = 1 J ξ x F + ξ y G + ξ z H ( ) G = 1 J η x F + η y G + η z H ( ) H = 1 J ζ x F + ζ y G + ζ z H (3.16) 57

81 where J is the Jacobian of the coordinate transformation: J = x y z ξ η ζ + x ζ y z ξ 1 η + x y z η ζ ξ x y z ξ ζ η x η y z ξ ζ x y ζ η z ξ (3.17) The Runge - Kutta scheme Equations (3.15) are solved using a cell centered, second order discretization on a multiblock grid, using a numerical algorithm based on a 5 stage modified Runge - Kutta scheme with artificial dissipation, following the work of Jameson et al. [1981], Alonso and Jameson [1994], and Yao et al. [2001], that allows for a higher numerical stability at larger CFL numbers, thus reducing the solution time. Usually, Runge-Kutta schemes are employed for solving ordinary differential equations (ODE), but they can also be applied to partial differential equations (PDE) [Lomax et al., 1970] by converting them into so called pseudo ODEs. This achieved by separating out the partial derivative with respect to time and placing the remaining of the equation into a term that depends upon the dependent variable [Tannehill et al., 1997]: where α is any dependent variable. α t = R(α) (3.18) Thus, once the time differencing on the left hand side of Eq is completed, the partial differential contained in the right hand side term can also be spatially differenced and any ODE integration scheme, including Runge - Kutta, is now applicable. Generally, the modified m stages Runge - Kutta scheme can be formulated as: Q (0) = Q n Q (k) = Q (0) α k tr (Q (k 1) ) (3.19) Q n+1 = Q (m) where n and n + 1 are two consecutive time steps, k is the current Runge - Kutta stage, Q is the state vector, α k are the Runge - Kutta scheme coefficients, for the case of a 5 stage scheme defined, in turn, as: 1 4, 1 6, 3 8, 1 2, 1 and R (Q) is the residual term that includes the contribution from Euler, viscous and subgrid fluxes and the source terms. To eliminate 58

82 spurious fluctuations of the state vector, second and fourth order artificial dissipation terms, based on pressure switches, D(Q k ), are added to the residual term [Jameson et al., 1981, Martinelli and Jameson, 1988]: where R e (Q k ) = R(Q k ) + D(Q k ) (3.20) D(Q k ) = β k D(Q k ) + (1 β k )D(Q k 1 ) (3.21) with coefficients β k set to 1.0, 0.0, 0.56, 0.0, 0.44 for maximum numerical stability The dual time stepping method The advance of the numerical solution in time can be achieved numerically in two generic ways: implicit or explicit. Explicit schemes use information at a given time t to compute the new value of the different variables at time t + t. On the other hand, implicit schemes use the information at time t + t to compute the value of the different variables at time t + t. The explicit method has the advantage of simplicity in implementation but it has significant numerical stability constraints, usually requiring times steps smaller than the physically achievable ones. Reversely, implicit methods are less constrained by numerical stability but are far more difficult to implement, especially in parallel algorithms. Jameson [1991] proposed a method combining the advantages of the afore mentioned approaches, known as the dual time stepping approach. The governing equations, are discretized implicitly with a second order backwards equation: Q τ + 3 Q n+1 4Q n + Q n 1 + R( 2 t Q n+1 ) = 0 (3.22) Jameson s approach was to consider the second and third terms in the above equation as the total residual of the state vector on a fictitious pseudo time τ: Q τ = R (Q) (3.23) 59

83 where R (Q) = 3 Q n+1 4 Q n + Q n 1 2 t + R( Q n+1 ) (3.24) Thus, by solving Equation (3.23) with a standard explicit method and using the maximum locally available time step until convergence to the steady state is iteratively achieved, the solution of Equation (3.22) can also be advanced in time with a much larger time step. Specifically, it has been observed that in regions where momentum is close to zero the convergence of Equation (3.23) is significantly delayed compared to the rest of the computational domain. In order to avoid an important increase in computation time with only marginal improvements in the accuracy of the solution, the maximum number of iterations performed in solving Equation (3.23) is limited to 250 after which convergence is declared. Obviously, this introduces some inaccuracy in the solution, but it impacts strongly on reducing the computation time. To eliminate spurious fluctuations of the state vector, second and fourth order artificial dissipation terms, based on pressure, are added to the residual term [Jameson et al., 1981]. Previous applications of the Runge - Kutta, dual time stepping algorithm were, to the knowledge of the author, used for external flows. Here, an implementation of the method for internal flows is presented and validated with good results. A new set of inflow and outflow boundaries, specific to internal flows, had also to be implemented and will be discussed later The local time stepping convergence acceleration technique In order to accelerate the pseudo - time convergence an acceleration technique known as local time stepping has been implemented and used for EBULES. The technique uses the locally maximum available time step to evolve the solution in pseudo-time. Thus, at each point in the flow, the maximum available pseudo-time step can be determined as: 1 τ = CF L λ ξ + λ η + λ ζ + 2(ν + ν T )J 2 (Sξ 2 + S2 η + Sζ 2) (3.25) where CF L is the Courant-Friedrichs-Lewy number, λ ξ, λ η, λ ζ are the spectral radii in the ξ, η and ζ directions, defined (with c being the local speed of sound) as: 60

84 ( λ ξ = U + c ξ x ( λ η = V + c η x ( λ ζ = W + c ζ x ) 2 + ( ξ y ) 2 + ( η y ) 2 + ( ζ y ) 2 ( ) 2 + ξ z ) 2 + ( η z ) 2 + ( ζ z ) 2 ) 2 (3.26) Also: S ξ = ( x ξ ) 2 + ( y ξ ) 2 ( ) 2 + z ξ S η = ( x η S ζ = ( x ζ ) 2 + ( y η ) 2 + ( y ζ ) 2 ( ) 2 + z η (3.27) ) 2 ( ) 2 + z ζ In terms of actual algorithm efficiency, for the simulations presented here, the dual time - stepping method, including the acceleration techniques previously mentioned provided a speedup factor of 14.6 when compared to the classical Runge - Kutta algorithm. However, a possible deficiency of the dual stepping scheme is that when a time averaged solution is desired for data comparison with experimental results, the time step achieved by the dual time stepping method is too large compared to the resolved turbulence time scales to allow for an accurate time averaging. Thus, a final computation without the dual time stepping is required for accurate averaged results. It is also important to note that the current implementation does not support the use of the local time stepping acceleration technique for LEMLES simulations, due to complications related to determining the exact Lagrangian mass fluxes between cells at different pseudo-times, required for the LEMLES splicing algorithm. Hence, LEMLES simulations had to be carried on using the classical Runge - Kutta method. 61

85 3.2 Boundary Conditions The boundary conditions used here are a combination of Dirichlet and Neumann boundary conditions and can be classified in three categories: adiabatic or isothermal viscous wall boundary conditions, periodic boundaries, and inflow-outflow boundary conditions, the latter being treated according to the method developed by Baum et al. [1994] for reactive, compressible, three-dimensional flows Characteristic boundary conditions Assuming that the characteristic boundary conditions are applied for a boundary located in the x 2 x 3 plane and following Poinsot and Lele [1992], Eq can be written as: ρ t + d 1 + ρũ 2 x 2 = 0 ρũ 1 t + ũ 1 d 1 + ρd 3 + ρũ 1ũ 2 x 2 + ρũ 1ũ 3 x 3 ρũ 2 t + ũ 2 d 1 + ρd 4 + ρũ 2ũ 2 x 2 + ρũ 2ũ 3 x 3 ρũ 3 t + ũ 3 d 1 + ρd 5 + ρũ 3ũ 2 ρẽ t + ρũ 1 d 3 + ρũ 2 d 4 + ρũ 3 d ( + d 2 γ 1 + x 2 + N m=1 ρd 6+m ρẽ )ũ + p 2 + [ ẽ m = τ 1j x j = τ 2j x j x 2 + ρũ 3ũ 3 x 3 = τ 3j x ( j ) (ũ1 ũ 1 ) x 1 + (ũ 2ũ 2 ) x 2 + (ũ 3ũ 3 ) x 3 ( [( ] x 3 ρẽ )ũ + p 3 + ẽ a2 γ(γ 1) )d 1 R u T MW m(γ 1) ] = q j x j + (ũ iτ ij ) x j ρk sgs t + k sgs d 1 + ρd 6 + (ρũ 2k sgs ) x 2 + (ρũ 3k sgs ) x 3 = ( ) k x j ρν sgs t x j + P sgs D sgs (ρũ 2Ỹm) x 2 ρỹm t + Ỹmd 1 + ρd 6+m + ( ) x j ρd Ỹm m x j + ρ ẇ m, m = 1, N + (ρũ 3Ỹm) x 3 = (3.28) where d i are the various partial derivatives with respect to the x 1 th direction: 62

86 d = d 1 d 2 d 3 d 4 d 5 d 6 d 6+m = ρũ 1 x 1 ρc 2 ũ 1 x 1 + ũ 1 p x 1 ũ ũ 1 1 x p ρ x 1 ũ 1 ũ 2 x 1 ũ 1 ũ 3 x 1 ũ 1 k sgs x 1 ũ 1 Ỹm x 1 = 1 (L 5 + L 1 )] [L c ) (L 5 + L ρc (L 5 L 1 ) L 3 L 4 L 6 L 6+m (3.29) where L i s is the characteristic waves amplitude associated with the eigenvalue λ i : λ = λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 6+m where c is the speed of sound (c= γ R u T ) and: = ũ 1 c ũ 1 ũ 1 ũ 1 ũ 1 + c ũ 1 ũ 1 (3.30) λ 1 is the velocity of the positive sound wave; λ 2 is the velocity of the entropy wave (convection velocity); λ 3 is the velocity of ũ 2 advection in the x 1 -direction; λ 4 is the velocity of ũ 3 advection in the x 1 -direction; λ 5 is the velocity of the negative sound wave; λ 6 is the velocity of k sgs advection in the x 1 -direction; λ 6+m is the velocity of Ỹm advection in the x 1 -direction, where m goes from 1 to the number of chemical species considered in the problem.. 63

87 The amplitudes of the characteristic waves, L i are: L 1 L 2 L 3 L 4 L 5 L 6 L 6+m = ( ) λ p 1 x 1 ρc ũ 1 x 1 ( ) λ 2 c 2 p x 1 p x 1 λ 3 ũ 2 x 1 λ 4 ũ 3 x 1 ( ) λ p 5 x 1 + ρc ũ 1 x 1 λ 6 k sgs x 1 λ 6+m Ỹm x 1 (3.31) From the analysis of the above equations, the Local One-Dimensional Inviscid (LODI) relations can be determined [Poinsot et al., 1991] and values of the wave amplitude variations in the viscous, three-dimensional Navier-Stokes equations can be inferred: ρ t + 1 c 2 [L ( L5 + L 1 ) ] = 0 p t + 1 2( L5 + L 1 ) = 0 ũ 1 t + 1 2ρc( L5 L 1 ) = 0 ũ 2 t + L 3 = 0 ũ 3 t + L 4 = 0 (3.32) Based on the LODI equations 3.32 various types of characteristic boundary conditions can be derived [Poinsot et al., 1991]. For the purposes of this work, only two such boundary condition types were needed and will be described in the following Subsonic inflow boundary conditions For the inflow, the gas density is computed based on the flow information, while all other values are specified. The inflow boundary condition employed in this study required the a-priori specification of the velocity components, u 1, u 2 and u 3 as well as of the inflow temperature T and pressure, p and of the chemical composition of the incoming fluid, Y m. For a subsonic, three dimensional, N species reacting flow, 5 + N characteristic waves, L 2, L 3, L 4, L 5, L 6 and L 6+N enter the domain and L 1 leaves the domain at λ 1 =u 1 c. 64

88 Since u 1, u 2, u 3, T and Y m are known at the inflow, the second, third and fourth relation from Eq become irrelevant and can be discarded. The characteristic boundary condition determines the temporal change in the incoming fluid density, ρ as follows: Using the known quantities mentioned earlier, the wave amplitudes can be determined: ( p L 1 = (u 1 c) ρc ũ ) 1 x 1 x 1 (3.33) L 5 = L ρc ũ 1 t (3.34) where L 2 = 1 2 ( γ 1 )( L5 + L 1 ) + c 2 ρ t ρ t = ρ T T t (3.35) (3.36) Finally, d 1 can be now determined as: d 1 = 1 c 2 [ L ( L1 + L 5 ) ] (3.37) and: dρ = d 1 t (3.38) Partially reflecting subsonic outflow At the outflow, second order accurate, partially reflecting conditions are enforced by allowing a pressure wave coming from downstream to enter the computational domain. For the conditions of this study, the pressure is imposed downstream of the outflow, as p. Using this, the amplitude variation of in ingoing wave, L 1, can be determined. Thus, if the current outflow pressure p is different from p, reflected waves will enter the domain to bring the pressure back to p. Therefore: L 1 = β ( p p ) (3.39) where: β = σ ( 1 M 2) c L (3.40) 65

89 and M is the maximum Mach number in the flow, L is a characteristic length scale chosen in this study to be equal to the length of the computational domain and σ is the a reflection coefficient chosen for this study to be 0.15 Sankaran [2002] Boundary condition testing The accuracy of the inflow and outflow boundary conditions was tested in several ways. First, in a single species implementation (non-reactive flow), a travelling acoustic wave was sent upstream such that it hits the inflow, it reflects and than it travels towards the outflow exiting the domain. For testing purposes, the coefficient controlling the amplitude of the reflected wave at the outflow was set to zero (non-reflecting outflow). The test was also repeated in a multiple species implementation, both with non - reflecting and partially reflecting boundary conditions. The pressure (Fig. 3.2) and velocity (Fig. 3.1) profiles along the domain centerline for the multiple species, non - reflective outflow case are shown, indicating accurate reflection of the acoustic wave by the boundary, in both wave amplitude and frequency. A first order accurate implementation was also tested, but the results show strong numerical oscillations at both the inflow and the outflow. Finally, a region of high temperature was convected through the both the first and second order accurate outflow boundary, a temperature and velocity profile obtained for the second order test being shown in Fig. 3.3: Adiabatic and isothermal no-slip wall boundary conditions Assuming again that the wall boundary condition is imposed on a surface located in the x 2 -x 3 plane, for a finite volume scheme, the adiabatic no-slip wall boundary conditions are: 66

90 Velocity (m.s -1 ) 20 Velocity (m.s -1 ) Distance (m) Distance (m) Velocity (m.s -1 ) Distance (m) Distance (m) Figure 3.1: Boundary conditions tests. Velocity wave evolution in time. Different colors represent different instants in time p i+1 = p i ũ 1,i+1 = ũ 1,i ũ 2,i+1 = ũ 2,i ũ 3,i+1 = ũ 3,i (3.41) T i+1 = T i k sgs i+1 = ksgs i Ỹ m,i+1 = where it has been assumed that i is the last cell in the computational domain, and i + 1 is the boundary condition Ỹm,i For an isothermal wall, Eqs still apply, except for the temperature boundary condition that becomes, in this case: 67

91 Pressure (Pa) Pressure (Pa) Distance (m) Distance (m) Distance (m) Distance (m) Figure 3.2: Boundary conditions tests. Pressure wave evolution in time. Different colors represent different instants in time T i+1 = T wall (3.42) where T wall is the specified temperature of the isothermal wall Periodic boundary conditions Under the same assumption, and if the computational domain contains cells from 1 to N, with cells 0 and N +1 as boundary conditions, periodic boundary conditions can be written as: 68

92 Figure 3.3: Boundary conditions tests. Snapshots of the velocity field (vectors) and temperature (color and contours) ρ N+1 = ρ 1 ũ 1,N+1 = ũ 1,1 ũ 2,N+1 = ũ 2,1 ũ 3,N+1 = ũ 3,1 (3.43) T N+1 = T 1 k sgs N+1 = ksgs 1 Ỹ m,n+1 = Ỹm,1 respectively ρ 0 = ρ N ũ 1,0 = ũ 1,N ũ 2,0 = ũ 2,N ũ 3,0 = ũ 3,N (3.44) T 0 = T N k sgs 0 = k sgs N Ỹ m,0 = Ỹm,N 69

93 3.3 Linear Eddy Mixing Model Implementation The reaction diffusion equation The first requirement for the numerical implementation of LEM is the numerical integration of the reaction diffusion Eq or Previous studies [Sankaran, 2002] have shown that implicit schemes are very expensive in terms of memory requirements, so an explicit method is used in this study. The method uses an operator splitting technique developed by Smith and Menon [1996a] and Calhoon et al. [1995] and is based upon a sequential application of individual operators describing different physical phenomena, each at the appropriate time scale. In the reaction - diffusion Eq four distinct phenomena are identifiable, therefore four separate time scales: A. Molecular diffusion. The time scale is associated to species and temperature transport by diffusion and is the largest timescale involved [Sankaran, 2002]. s 2 t diffusion = κ max(d k ) (3.45) where s is the LEM grid size, D k is the diffusion coefficient of species k and κ is a model constant, set here to 0.25 for reasons of numerical stability [Sankaran, 2002]. B. Chemistry. This time scale, t chem, is associated to the chemical reaction rates and it is usually the smallest time scale [Sankaran, 2002]. Its value is determined by the stiffness of the chemical reaction rates equation system. The implementation in the current study uses a chemical time step 10 times smaller than the molecular diffusion time scale, t diffusion. A previous implementation of the LEM algorithm used an adaptive time step, direct integration subroutine called DVODE [Brown et al., 1989] to integrate the reaction rates determined at the chemistry time scale over the diffusion time. However, the approach was very expensive in terms of computational time and it has been showed [Eggenspieler, 2005] that a direct 70

94 estimation of the Arrhenius-type reaction rate Eq at a time step one order of magnitude below t diffusion drastically reduces the computational expense while maintaining reasonable accuracy. The effect of replacing the DVODE direct integration by the direct estimation technique for the case of the Propane mechanism employed here (Eq. 2.44) will be addressed in the next section. C. Thermal expansion. The time scale is associated with the volumetric expansion induced by the increase in temperature through chemical heat release. In the current implementation it is assumed that, in the fast chemistry limit, the heat release is controlled by the molecular mixing [Sankaran, 2002], therefore: t expansion = t diffusion (3.46) D. Turbulent stirring. The time scale is associated to the turbulent convection by small (subgrid) eddies and is defined by: τ stir (x) = 1 λ where λ is the turbulent stirring frequency, given by Eq (3.47) To explicitly solve all these processes appropriately, an operator splitting method [Smith and Menon, 1996b, Calhoon and Menon, 1996] is used. This technique allows for decoupled time resolution of the chemical, diffusion and turbulent processes. Thus, at each LES time step the diffusion and stirring time scales are determined. The chemical species source terms are determined and integrated over the diffusion time step. With this term known, the reaction - diffusion equation, less the turbulent stirring contribution is integrated at the diffusion time step. After each integration time step, thermal expansion is implemented as described in Chapter 2. At a frequency given by 2.54, the integration process in interrupted by the triplet map rearrangement of the scalar fields simulating the turbulent eddy. From a numerical standpoint, the LEM domain resolution is a function of the LES resolution and the turbulence intensity, to be discussed later. The length of the LEM 71

95 domain, L LEM is defined as: L LEM = V 1 3 LES N LEM (3.48) where N LEM is the number of LEM cells per LES cell and V LES is the volume of the LES cell. For the triplet mapping algorithm to work, the resolution on the LEM domain has to be uniform and the number of LEM grid points has to at least 6 and a multiple of 3. The value used for this study is 12, for reasons that will be discussed in later sections The splicing algorithm After the subgrid algorithm is completed at each LES time step and for each LES cell, a Lagrangian advection of the scalar LEM fields is performed using the LES resolved velocities. The splicing species transport between adjacent LES domains is achieved by the algorithm that splices the species field successively in the three spatial directions: Splicing is done once in every spatial direction. Splicing is performed using an upwind scheme. The absolute value and the sign of the term ρũ i x i determines the order in which splicing is performed. The largest negative flux will be the first one to exit the LES cell, while the largest positive flux will be the last one to enter the LES cell. As an example, in 2-D, it can be assumed that : where: ρũ x (i 1 < ρũ 2 ) x (i+ 1 1 δm 1 δt LES V LES = F i 0 1 δm 2 δt LES V LES = F i 1 1 δt LES δm 3 1 δt LES δm 4 < 0 < ρṽ 2 ) y = ρũ x = ρũ (j 1 x V LES = F j 0 = ρũ y V LES = F j 1 = ρũ y (i 1 2 ) (i+ 1 2 ) (j 1 2 ) < ρṽ 2 ) y (j+ 1 2 ) (3.49) (j+ 1 2 ) (3.50) 72

96 1 1 j+1 δm 4 j j δm 1 δm j δm 3 j-1 i i-1 i (a) Before splicing j-1 i i-1 i (b) After splicing Figure 3.4: Species field before and after the splicing of the cell (i,j) The effect of the splicing process in the 2-D representation considered previously is presented in Fig Further details on the splicing numerical procedure are given by Sankaran [2002]. Finally the re-gridding procedure presented in Chapter 2 is carried out to ensure uniform LEM grid resolution and constant number of LEM cells in all LES cells. 3.4 The Direct Estimation of the Chemical Source Terms The technique was proposed by Eggenspieler [2005] and is aimed at reducing the computational cost of the chemical source term evaluation. As mentioned in the previous section, the method, called Direct Source Term Estimation (DSTE) hereafter, replaces the computationally expensive time-integration of the Arrhenius-type reaction rates by a direct estimation of the terms. The evaluation of the source terms is made at a given, constant temperature, pressure and chemical composition, assuming that changes in these parameters over the estimation time interval can be neglected. The effect of replacing the direct integration by DSTE for various estimation times is shown in Fig As indicated in Fig. 3.5, the current study recovers for the Propane - air chemistry the conclusion obtained by Eggenspieler [2005] for Methane, that a chemical time step of 10 % of the diffusion time step is sufficiently accurate. If the chemical time step is reduced to 5 % of the diffusion time step the improvement in the reaction rate accuracy is only marginal, as indicated by Fig. 3.6, while the computational expense increases by 73

97 Reaction rate [1/s] -2e+06-1e+06 0 DVODE DSTE1 DSTE5 DSTE10 DSTE Temperature [K] Figure 3.5: Comparative fuel reaction rate obtained by direct integration over the LES time step and by the direct estimation of the Arrhenius-type reaction rates at time steps equal to various fractions of the LES time step. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E5 = Direct estimation of the Arrhenius reaction rate at a time scale 5 times smaller than the LES time step; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; DST E20 = Direct estimation of the Arrhenius reaction rate at a time scale 20 times smaller than the LES time step; 25 %. The computational cost for the 5 discussed source term integration approaches are presented in Table 3.1, normalized by the DVODE computational time. 3.5 The Multiblock Grid Parallel Implementation Generally, the geometry describing combustors used in real applications may become quite complicated. Various methods that avoid body fitting grids have been described in the literature (e.g. unstructured grids, or immersed boundaries [Moin, 2002]), but they are either extremely expensive in terms of computational time, or lack in accuracy due to the 74

98 Rms error Number of chemical time steps per diffusion time step Figure 3.6: RMS error for the direct estimation of the Arrhenius reaction rate approach for different chemical time intervals interpolations involved in the process, especially in a geometry such as the one studied herein, where the vortices shed at the sharp trailing edges control the entire flow dynamics downstream. Hence, a body fitting grid was selected for the numerical simulations herein. To allow for this while avoiding unnecessary clustering of grid points in flow areas that do not require fine resolutions, a parallel multiblock approach was implemented in the algorithm. The computational cost associated with LES computations can be quite large, and even more so in the case of LEMLES. The advent of modern computers provided a solution for this problem by allowing the computational load to be divided among a number of processors, thus decreasing the computational time. In the present implementation, this division is done by dividing the computational grid is into separates sub-elements. This raises the problem of allowing the separate processors to communicate with each other, such information as the global LES time step ( t LES ), or the state vector values and 75

99 Table 3.1: Speedup factors for different methods of estimating the chemical source terms. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E5 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; DST E20 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; Method Speedup DVODE DSTE DSTE DSTE DSTE variables at the boundary of each sub-element of the grid, made possible by the use of the Message-Passing Interface (M P I) protocol. Combining the multiblock approach with the MPI parallel implementation, the computational domain is divided into geometrically significant blocks and processors are assigned to each block in such a way that it ensures a balanced computational load per processor. In order to avoid interpolation at the block boundaries, known to decrease numerical accuracy, the number and distribution of grid points at the block interfaces is identical in the two adjacent blocks. Once the blocks are set up, the algorithm identifies for each processor its neighbors irrespective of the domain they may be part of and performs point - to point, MPI controlled communication among them. Another advantage of the implementation is that it allows to easily implement different boundary conditions along the same spatial direction (e.g. an adiabatic wall that becomes, at a given location, an isothermal wall) by assigning the two different regions to two separate blocks. The only limitation of the implementation is in its requirement that the number of processors in the two directions along the interface plane is the same in the two adjacent blocks. This may create some difficulties in balancing the work load per processor, but a careful initial design of the block distribution will avoid this problem. In terms of computational efficiency, the cold flow simulations required 36.5 single CPU microseconds, per time step, per grid point. Compared to the McCormack predictor - corrector scheme, that has been timed at 12 microseconds, the Runge - Kutta implementation 76

100 is about 4 times slower per iteration due, mainly, to the 5 intermediate steps required by the algorithm, compared to the 2 in the McCormack scheme, and to the supplementary computation of Jameson artificial dissipation terms. However, it needs to be noted that since the Runge - Kutta scheme allows larger CFL numbers, the two algorithms yielded, on the same grid and with the same initial conditions a computed physical time of 5.5 ms, in the Runge - Kutta formulation and 0.8 ms in the McCormack formulation. Thus, one second of actual physical time would be computed in 6.6 s per grid point by the Runge - Kutta scheme and in 15.0 s per grid point by the McCormack algorithm, which makes the new algorithm 2.27 times faster in terms of actual simulated physical time, even without the dual time stepping acceleration technique. The advantage of the new Runge - Kutta scheme comes, however, from the much larger time steps the dual time stepping acceleration technique allows. For comparison, if dual time stepping is used, the cold flow simulation requires 47 single CPU microseconds, per time step (including pseudo-time iterations), per grid point, for a time step of 1 ms and 1 second of simulated physical, meaning an increase in computational speed by a factor of 33.3 compared to the McCormack algorithm. Typically, EBULES simulations required about 1.44 times, and LEMLES about 8 times more computational time than the cold flow simulations. All the timing computations discussed above were based on simulations performed on a Compaq SC40 machine with MHz Alpha EV68 processors. The solver performance was also evaluated in terms of parallel implementation efficiency. Thus, repeated simulations were carried on and timed on a 32 x 32 x 32 computational grid (with 12 LEM cells for each LES cell) and the results are shown in Fig The ideal speedup is considered achieved when doubling the number of processors halves the computational time. Previous results [Stone and Menon, 2001] have shown that, above a critical number, the marginal speedup obtained by adding more processors decreases due to message passing between processors that becomes more significant as the total number of processors increases. The results obtained for this study confirm the observation, the critical number for this algorithm being around 50 CPU. However, the results were obtained through a 1-domain simulation and in a multiple domain simulation the restriction imposed 77

101 on the processor distribution among domains discussed previously may alter this number. Time per step [s] Ideal Actual Number of processors Figure 3.7: Parallel algorithm efficiency in terms of speedup 3.6 Simulation Geometries and Computational Grids The validation numerical simulations, both non-reactive and reactive, and presented in Chapter 4 where performed in a geometry that reproduces the V olvo experiment [Sjunesson et al., 1991b] and consists of a rectangular duct of size 1.0 m x 0.24 m x 0.12 m with a triangular prism that extends between the two lateral walls of the combustor, as shown in Fig The side of the bluff body triangular base, a, measures 0.04 m. The computational domain was divided into 10 blocks, and each block was discretized by a body fitting grid with dimensions given in Table 3.2 and positioned according to Fig For the reactive flow the spanwise resolution was increased to resolve the three-dimensional flame structure. 78

102 Figure 3.8: Schematic of the geometry. The solid symbol marks the position of the velocity probe used to capture the time signals in Figs. 3.11, 3.12, 5.4 and 5.27 Table 3.2: Computational grid dimensions. CF = Cold Flow; RF = Reactive Flow Block I J K (CF/RF) Block I J K (CF/RF) / / / / / / / / / /90 Figure 3.9: Spatial disposition of the computational domains The computational grid is stretched both axially and transversally and provides the maximum resolution in the two separated shear layers, immediately downstream of the bluff body, resolved by about 20 grid points. The grid stretching is maintained under 5 79

103 percent for reasons of numerical stability. A detail of the computational grid is given in Fig. (3.10). Figure 3.10: Computational grid detail In order to assess the appropriateness of the computational grid a Fast Fourier Transform was applied to the time signal of the axial velocity autocorrelation, E 1 1. The resulting energy spectra are presented in Figs and 3.12 and show that the energy decay scales with the inertial range scaling [Pope, 2000], k 5/3, where, k represents the wave-number. The recovery of the 5/3 slope proves that the current grid is reasonable for LES. For the LEMLES, 12 LEM cells are used in each LES cell. Using the predicted k sgs and 80

104 Hz Normalized amplitude E 11-5/3 law Frequency [Hz] Figure 3.11: FFT of the axial velocity autocorrelation, E 11, for the cold flow, normalized by its maximum value the local, the maximum local sub-grid Re is 130 and η = 25 x 10 6 m. Thus, scales down to about 3 η are resolved in the sub-grid. Due to heat release, the local Re in most of the grid will be lower than this value and hence, the sub-grid resolution is considered acceptable. 3.7 LDKM Model Coefficients As mentioned earlier, the present numerical simulations employ a turbulence model based on the transport equation of the subgrid turbulent kinetic energy (TKE) [Schumann, 1975]. The model assumes isotropic turbulence at the subgrid scales and solves a subgrid TKE transport equation to track the temporal and spatial variation of the subgrid TKE. Thus, the turbulent viscosity can be determined locally, using a length scale given by the local filter size, and a velocity scale given by the local subgrid TKE (Eq. 2.22). Also, the turbulent dissipation can be determined locally, based on the same length and velocity scales (Eq. 81

105 Hz Normalized amplitude 0.01 E 11-5/3 law Frequency [Hz] Figure 3.12: FFT of the axial axial velocity autocorrelation, E 11, for the LEMLES reactive flow, normalized by its maximum value 2.29). Both these equations require each a model constant, respectively C ν and C ε. In the current approach, these constants are determined dynamically as a part of the solution [Kim and Menon, 1995, Kim et al., 1998], as shown previously. The instantaneous values of the two LDKM model coefficients (C ν and C ε ) obtained from the LEMLES simulation are presented in Fig It is obvious that the values of both coefficients vary significantly throughout the flow, within ranges from 0.42 to 1.0 for the turbulent viscosity coefficient, and from 0.0 to 2.0 for the dissipation coefficient. A strong correlation between the distribution of the coefficient values and shear layer and vorticity pattern is observed, due to the fact that the turbulent kinetic energy, both resolved and subgrid and including the test level, is peaking in this region. Negative values of the C ν, indicating energy backscatter are also present in the field. However the fraction of the total grid points where negative model coefficients appear, known to cause numerical instabilities [Kim and Menon, 1999] is reduced, allowing a numerically stable simulation. In terms of 82

106 (a) Turbulent viscosity coefficient (b) Dissipation coefficient Figure 3.13: Instantaneous fields of the LDKM model coefficients predicted by the LEM- LES simulation mean values, the turbulent viscosity coefficient, C ν recovers in the shear layer a mean value of 0.067, value determined previously by Chakravarthy and Menon [2000b] using spectral closure theoretical formulations in the inertial and dissipation range [Kraichnan, 1976]. This indicates that the shear layer resolution is sufficiently large so that the LES filter size falls in the isotropic region of turbulence, supporting the previous observations. In the low turbulence regions upstream of the bluff body, the C ν mean value is much reduced (0.021), due to the low intensity of the subgrid turbulent kinetic energy (TKE). The dissipation coefficient, C ε has a mean value of in the shear layer, which is significantly smaller than the value reported by Chakravarthy and Menon [2000b] (0.916). The reason for this may be the high shear layer resolution, which is significantly decreasing the amount of subgrid TKE forcing the turbulent dissipation to decrease to smaller values as well, via the decreasing of the dynamically computed C ε. In the low turbulence regions, the dissipation coefficient decreases in a similar manner to C ν, to a mean value of

107 CHAPTER IV ALGORITHM VALIDATION 4.1 Results and Discussion of Non-Reactive Flow Overview As shown in the previous chapter, the geometry of the simulations herein reproduces the V olvo experiment [Sjunesson et al., 1991b]. The goal of this chapter is to accurately simulate a non-reactive flow behind a bluff body and, in doing so, to validate the LES algorithm to be used for numerical studies in the following chapters. For comparison, the LES numerical results obtained recently by Giacomazzi et al. [2004] are also included, where available. The inflow velocity is 17.3 m/s with a 2 percent turbulence intensity under standard atmospheric conditions. The reference Reynolds number based on inflow velocity and bluff body height is 45, 500. The inflow consists of air. The simulations are carried out for three flow-through times before the flow statistics are collected, and the time averaged data presented herein are collected over a period equal to five flow-through times. Typically, cold flow simulations require about 75 single-processor hours for a single flow-through time on a IBM P655 Power 4+ cluster Spectral analysis A Fast Fourier Transform was applied to the kinetic energy contained in the axial mode and the resulting energy spectrum is shown in Fig The spectrum presents a peak at 102 Hz, which represents the alternate vortex shedding frequency and compares well with the experimental value of 105 Hz reported by Sjunesson et al. [1992]. The Strouhal number, defined as: St = fa U 0 (4.1) is found to be equal to 0.24, in good agreement with earlier numerical [Giacomazzi et al., 84

108 2004] and experimental [Sjunesson et al., 1992] studies. In the above, f is the shedding frequency, a is size of the bluff body, and U 0 is the inflow velocity Flow structures As seen in Fig. 4.1, asymmetric vortex shedding occurs from the two shear layers formed at the trailing edge of the prism in a von Karman vortex street pattern. The observed vortices develop as a result of the roll-up of the two vortex sheets formed at the trailing edges of the bluff body. Further downstream, the vortices break down into smaller eddies that eventually dissipate, due to vortex stretching and viscosity effects. Figure 4.1: Instantaneous spanwise vorticity field for the cold flow The instantaneous and the averaged velocity fields are presented in Fig. (4.2). The flow is accelerated due to the obstruction caused by the bluff body and further downstream undergoes sudden expansion at the trailing edge. The two sharp trailing edges will cause the flow to separate even at very low Reynolds numbers and create a recirculating wake downstream. In the far field, away from the bluff body, both the recirculation zone and the accelerated regions surrounding it disappear and the flow velocity tends toward the uniform inflow velocity as viscous momentum transfer diminishes the velocity gradients. 85

109 (a) Instantaneous (b) Time-averaged Figure 4.2: Instantaneous (left) and time-averaged (right) axial velocity fields Time averaged results Comparison with the existing experimental data provides a better appraisal of the accuracy of the simulation. Figure 4.3 presents the normalized time-averaged axial velocity profile along the combustor centerline behind the bluff body. The numerical result matches closely the experimental data. Immediately downstream of the bluff body, the velocity is negative and reaches a negative maximum of about 64% of the inflow velocity at around 0.75 a. The length of the recirculation region is about 1.25 a. After the end of the reverse flow zone, the mean axial velocity increases upstream to gradually approach the free stream value, as the velocity deficit induced by the bluff body disappears. Figure 4.4 shows several transverse profiles of the normalized time-averaged axial velocities at several locations downstream of the bluff body. The axial velocity profiles capture accurately the recirculation region, in both shape and intensity, as well as the free stream values both in the near and the far field. The velocity deficit behind the bluff body, namely the difference between the axial velocity directly behind the obstacle and that of the stream flowing around it decreases with the increasing axial location, due to the effect of the viscous forces that ensure momentum transfer along the velocity gradient. The free stream velocity, accelerated in the convergent section created by the prismatic obstacle immediately downstream of the bluff body also decreases in such a manner that sufficiently far downstream the inflow velocity tends to be recovered in the entire domain, reflecting the overall energy conservation. 86

110 Normalized mean axial velocity Experimental Numerical Normalized x coordinate Figure 4.3: Centerline variation of the normalized time-averaged axial velocity for the cold flow. The velocity is normalized by the inflow value, and the distance is normalized by the bluff body size, a The transverse velocity profiles, shown in Fig. 4.5 reflect the vortical pattern discussed previously (Fig. 4.1). As the data in Fig. 4.5 is time averaged, the von Karman vortex street is not captured by the transverse velocity plot and only indicates a pair of counter - rotating vortices at the trailing edge of the bluff body. In terms of axial variation, as the vortices are shed at the backwall of the bluff body, their intensity increases over a distance of about 1.5 a, leading to an increase in the transverse velocity up to about 50% of the axial inflow value. After this peak, the transverse velocity decreases as the perturbation induced by the bluff body tends to disappear. The overall agreement with both the experimental data of Sjunesson et al. [1991a] and the earlier numerical result of Giacomazzi et al. [2004] is generally good. Some discrepancies can be observed at the first axial location (x/a = 0.375), and the reason for this is not quite clear. The numerical results indicate that the transverse velocity peaks a distance from the centerline corresponding to the bluff body edge, while the experimental data captures the peaks in locations closer to the centerline. Since the 87

111 shear layer created by the bluff body is most likely located at the bluff body edge, it is likely that the discrepancies are the result of some minor measurement errors. The variation along the transverse axis of the normalized axial and the transverse velocity fluctuation rms intensity and of the Reynolds stress, u v, are presented, respectively, in Figs and 4.8 at the same five axial locations. The agreement with the measured experimental data and with the previous numerical simulation remains good for both velocity components. The accuracy of the simulation is better for the axial stations closer to the bluff body and becomes slightly worse further downstream, where the grid resolution also decreases due to grid stretching. Both axial and transverse fluctuations peak in the shear layers, at about 1.0 a for the axial fluctuations and 1.5 a for the transverse component, with a maximum value of about 65% of the inflow velocity, for the axial rms, and about 95% for the transverse rms. The velocity fluctuations have the tendency to become isotropic further downstream, as the influence of the large scale turbulent structures created by the obstruction becomes less important, supporting the previous conclusion that the computational resolution is sufficient for capturing some of the isotropic, inertial range scales. The Reynolds stress peaks in the two shear layers that delimit the recirculation region, reaching a maximum of about 20% of the squared inflow velocity at 1.5 a, and decreases downstream, as turbulence decays and turbulent structures become less coherent. 4.2 Results and Discussion of Reactive Flow Overview As discussed in Chapter 2 two approaches towards closing these terms are considered in this study: a conventional closure at the LES resolved scale (EBULES) and a closure directly at the sub-grid scales (LEMLES). The simulation results for those two closure models will be presented in the following and compared against each other, as well as against experimental and numerical data. A total of seven reactive flow simulations have been carried out for the purpose of this thesis and the various conditions for each of those are described in Table 4.1. In all reactive 88

112 cases, the inflow consists of a propane - air mixture. Table 4.1: Reactive flow simulation cases Case Model Φ Span (mm) 1a EBU b LEM LEM LEM LEM LEM variable 30 6 LEM variable 30 The simulations are carried out under the same inflow conditions as the non-reactive flow, except for the chemical composition of the inflow gases. Cases 1a and 1b are conducted mainly for validation purposes and their results will be discussed in the remainder of this chapter. The simulations are carried out for five flow-through times before the flow statistics are collected, and the time averaged data presented herein are collected over a period equal to five flow-through times. Typically, reactive flow simulations require around 108 and 740 single-processor hours on a IBM P655 Power 4+ cluster for EBULES (Case 1a) and LEMLES (Case 1b), respectively Spectral analysis The Fast Fourier Transform of the axial velocity autocorrelation E 11, presented in Fig indicates that the dominant frequency is at 139 Hz, which correlates well with the dominant frequency reported by earlier experimental [Fureby, 2000b] and numerical [Giacomazzi et al., 2004] studies. As a result, the Strouhal number (Eq. 4.1) based on this frequency is As in the cold flow, an important region of the energy spectrum recovers the k 5/3 scaling suggests adequacy of the LES resolution Flow and flame structures The instantaneous spanwise vorticity and reaction rate, together with the time-averaged temperature fields are shown in Fig. 4.9 for EBULES (Case 1a) and LEMLES (Case 1b), 89

113 respectively. The instantaneous vorticity field in the reactive Cases 1a and 1b is found to be quite different from the non-reactive result shown in Fig. 4.1 in several key points. McMurtry et al. [1985] have shown that combustion occurs rapidly at the vortex core, causing the density to drop via thermal expansion. Since the angular momentum is conserved, an increased in the vortex area results in a decrease in its intensity. The strong baroclinic torque resulting from non-aligned pressure and density gradients, strongly weakens the Kelvin-Helmholtz instability responsible for the von Karman street [Chakravarthy and Menon, 1999]. The numerical results show that the vortex shedding that occurs at the corners of the bluff body in the non-reactive case is now suppressed, and a pair of stationary vortices forms at the two edges, phenomenon also observed by previous researchers [Fureby and Lofstrom, 1994]. Further downstream, flow instabilities in these stationary vortices lead to a symmetrical vortical pattern, correlating with previous observations by Fureby and Moller [1995], and consistent with previous data [Fureby and Lofstrom, 1994, Veynante et al., 1996]. In the initial phase of vortex development the vorticity is relatively low and the turbulent mixing and burning are reduced. After this, the vortices undergo pairing and during this process the flame is wrinkled and its surface area increases rapidly, enhancing both turbulent mixing and combustion. The flame, represented in Fig. 4.9 by the reaction rate, tends to wrap around the vortical structures [Poinsot et al., 1991]. The flame predicted by LEMLES (Case 1b) is significantly more wrinkled than the EBULES (Case 1a) flame and the accompanying spreading of the wake is larger further downstream due to this effect in Case 1b but is absent in the (Case 1a). The near field instantaneous temperature profile follows the flame structure and also follows the vorticity profile. Hence, as will be shown later, the EBULES (Case 1a) time-averaged temperature profile presents a reduced level of spreading when compared to LEMLES (Case 1b). Understanding the reasons behind the differences in the results of the two simulations and assessing which of the two discussed combustion models is a more accurate representation of the physical reality, requires some analysis of the simulated flame structure and of the underlying assumptions behind the two closures. 90

114 From a modelling standpoint, the fundamental assumption of the Eddy Break-Up model is the validity of the scale separation hypothesis, introduced in Chapter 1. For the scale separation hypothesis to hold, the width of the inner layer of the flame reaction zone needs to be significantly smaller than the smallest eddy scale, the Kolmogorov scale. For turbulent flames with turbulent intensities u significantly larger than the laminar flame speed S L, (in the present case u /S L in the flame region ranges between 25 and 70), two combustion regimes can be defined: the Thin Reaction Zone (TRZ) and the Broken Reaction Zone (BRZ) regimes. In TRZ, the validity of the scale separation hypothesis is maintained and the influence of the small, unresolved eddies on the preheat zone is accounted for by the EBULES model, as it becomes mixing controlled in the regions of high turbulence. However, for larger Karlovitz numbers, the combustion regime moves into the BRZ regime. In that case, the EBU assumptions are violated, and the model is expected to fail. The instantaneous Karlovitz number, computed as [Pitsch and Duchamp De Lageneste, 2002]: Ka = ( u S L ) 3 lf (4.2) exceeds the 100 limit over significant regions of the flame, especially in the two shear layers just downstream of the bluff body, as shown in Fig Therefore, in those regions, the EBU combustion model becomes unreliable. On the other hand, LEM maintains its validity over the entire range of combustion regimes, as shown in Chapter 1. Fig presents its instantaneous profile along the y axis at m from the bluff body, as obtained from the LEMLES simulation. An instantaneous distribution of LES filter sizes and Karlovitz numbers on the surface of the flame simulated in this work is presented in Fig. 4.11, superimposed on the regime diagram proposed by Pitsch [2006]. In the figure, η is the Kolmogorov length scale, δ is the thickness of the flame reaction zone, l G is the Gibson length scale, l m is the thickness of the broadened flame [Pitsch, 2006] and Re and Da are the Reynolds, respectively Damkohler numbers defined in terms of the filter size. The points are scattered over a significant range of Karlovitz numbers and filter size 91

115 to flame thickness ratios. The scatter in the filter size comes from the variable size of the grid cell size, which is used as the LES filter by the current LES implementation. The filter size varies from of the flame thickness to over 3 l f. The lower range of grid sizes ensured 3 5 LES cells in the flame region, a number considered appropriate to accurately resolve the flame. Also, since as mentioned before, 12 LEM cells are contained within each LES cell, about 4 6 LEM cells can accurately resolve the flame reaction zone. However, this resolution is only achieved in the region just downstream of the bluff body, and it deteriorates significantly downstream, where grid stretching occurs. For this reason, a significant portion of the grid (five bluff body lengths) is maintained uniform in order to accurately capture the relevant aspects of the reactive flow. In terms of the Karlovitz number, the scatter in the data points is even more significant. Thus, it can be noted that although the majority of the flame resides in the thin reaction zone regime, a significant portion of the flame extends deep into the broken reaction zone. As noted earlier, in this regime a majority of the combustion models are invalidated, and the only known model that maintains validity is the Linear Eddy Model. Thus, it was chosen for most of the numerical simulations presented herein. The extension of the flame into the Broken Reaction Zone regime causes the kinetics to become strongly affected by the turbulence. As heat diffusion from the inner layer towards the preheat zone is significantly enhanced, the reaction zone thickness is overpredicted by EBULES (Case 1a) and, as a thicker reaction zone is less susceptible to turbulent fluctuations [Chakravarthy and Menon, 2000a], EBULES predicts a much smoother flame than LEMLES (Case 1b) and than experimentally observed by Sjunesson et al. [1992], as shown in Fig It is obvious that even though the predicted flame thicknesses are relatively equal to each other in some regions, there are significant portions of the flame where LEMLES (Case 1b) reaction zone is significantly thinner and those regions also correspond to regions where the flame is strongly convoluted. A more clear view at the flame shape and its interaction with the vorticity field in Case 1b is given by Figs and 4.14 which present a 3-dimensional views of the instantaneous 92

116 and time - averaged reaction rate, and, respectively, of the vorticity field Time averaged results Figure 4.15 presents the normalized axial velocity profile along the combustor centerline, behind the bluff body. Compared to the cold flow, the volumetric expansion caused by the chemical heat release causes an increase in the recirculation zone length and crossflow area. The length of the recirculation region is about 3.75 a, and the maximum absolute value of the negative velocity reaches about 0.75 of the inflow velocity at about twice the size of the bluff body, in agreement with the experimental data. The far-field free stream velocity is about three times larger than the inflow velocity, due to the addition of chemical energy through combustion. Both combustion models yield equally accurate results, although the LEMLES (Case 1b) appears to show slightly better agreement, which is due to a better temperature prediction in the far field, as it will be shown later. In the transverse direction, the accuracy of both EBULES and LEMLES is also acceptable, as seen in Fig As mentioned before, the negative velocity region is found to be wider than in the non-reactive flow. In the near field, LEMLES (Case 1b) predicts a more accurate axial velocity in the free stream, correlating well with the improved temperature predictions, to be shown later. Also, far downstream, the centerline velocity becomes overpredicted by EBULES (Case 1a), indicating that the predicted acceleration rate is slightly off. In the case of the transverse component of the time-averaged velocity, the agreement with the experimental data of Sjunesson et al. [1991b] is generally good as shown in Fig The large scale vortical structures created by the presence of the bluff body decrease in intensity with the distance from the obstruction, as the wake momentum deficit diminishes and the flow tends to recover its initial axial direction. Generally, the magnitude of the transverse component is lower than in the non-reactive case, as an effect of the reduced vorticity magnitudes observed in Fig The maximum transverse velocity is about 30% of the inflow velocity, and is achieved closer to the bluff body, at about 0.4 a, due to the 93

117 increased viscous effects resulting from the higher temperature. A notable feature of the flow field present in Cases 1a and 1b, is the sudden decrease of the transverse velocity at the flame front, not captured in the non-reactive data. This behavior correlates with the pair of stationary, counter-rotating vortices that form at the sharp edge of the bluff body. The effect disappears further downstream, where the intensity and the coherence of the vortices weaken. The axial, the transverse velocity fluctuations and the Reynolds stress u v along the transverse axis for Cases 1a and 1b are presented, respectively in Figs. 4.18, 4.19 and 4.20, together with previous experimental [Sjunesson et al., 1991b] and numerical [Giacomazzi et al., 2004] data. For the axial component, the EBU model tends to underpredict the velocity fluctuations at the centerline, while LEMLES predicts values in significantly better agreement with the experimental data due to the more accurate modelling of the flame - turbulence interaction. In the shear layer, the velocity fluctuations are generally overpredicted by both models, although the overprediction decreases downstream, especially for Case 1b. However, the overprediction of the EBULES (Case 1a) is about twice as large in the near field when compared to LEMLES (Case 1b) for both axial and transverse fluctuations. The Reynolds stress peaks in the two shear layers created at the bluff body sharp corners, at a value of about 10% of the squared inflow velocity, in agreement with the experimental observations. Again, the LEMLES provides a more accurate result. Overall, these results appear to suggest only modest improvements when using LEMLES. However, the improvements do appear in regions where mixing between the products and the reactants is occurring and where the flame structure exists. This subtle fact becomes clearer when the time-averaged temperature profiles are compared in Fig (Here, the temperature is normalized by the inflow value, and the distance is normalized by the bluff body size, a.) It can be seen that the EBULES (Case 1a) under-predicts the mean temperature in the centerline region. As a consequence of the overestimated flame thickness discussed earlier, the EBULES flame is slower to respond to the turbulent fluctuations, and the 94

118 intermittency effect is not captured accurately. As a consequence, the turbulent flame brush is not captured accurately and Case 1a shows a smaller spreading rate in the transverse direction that becomes more prominent further downstream. The LEMLES (Case 1b), on the other hand, predicts the temperature field more accurately in the flame zone, which affects the velocity field as well. Also, EBULES (Case 1a) tends to underpredict the centerline values by as much as 10%. It can be noted that the centerline region is also the region of low turbulent kinetic energy and EBULES will predict here a reduced turbulent mixing rate. However, the experimental data show that the temperature maintains its high value over a large portion of the domain, so even with a reduced mixing rate the premixed reactants entrained in this region should burn at a high rate. The LEM model, on the other hand, avoids estimating the controlling rate and simulates the involved processes, thus allowing for a more accurate prediction of the temperature. The more accurate prediction of heat release results in more accurate spreading rates, which is reflected in the velocity field, as discussed earlier. These results demonstrate the subtle and global effects of using a more comprehensive combustion and mixing model as in LEMLES. 95

119 Experimental Numerical (P) Numerical (G) Normalized y distance Normalized axial velocity Figure 4.4: Transverse profiles of the time-averaged normalized axial velocity for the cold flow at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 96

120 Experimental Numerical (P) Normalized y distance Normalized transverse velocity Figure 4.5: Transverse profiles of the time-averaged normalized transverse velocity for the cold flow at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 97

121 Normalized y distance Experimental Numerical (P) Numerical (G) Normalized axial velocity rms Figure 4.6: Transverse profiles of the normalized rms of the axial velocity fluctuation intensity for the cold flow, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 98

122 Normalized y distance Experimental Numerical (P) Normalized transverse velocity rms Figure 4.7: Transverse profiles of the normalized rms of the transverse velocity fluctuation intensity for the cold flow, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The velocity is normalized by the inflow value, U 0, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 99

123 Normalized y distance Experimental Numerical (P) Normalized Reynolds stress Figure 4.8: Transverse profiles of the normalized Reynolds stress for the cold flow, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The Reynolds stress is normalized by U0 2, and the distance is normalized by the bluff body size, a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 100

124 (a) EBULES (b) LEMLES Figure 4.9: Instantaneous spanwise vorticity (solid color), instantaneous reaction rate (thick black line) and time-averaged temperature (color lines) for Cases a1 and 1b Karlovitz number BRZ TRZ Y coordinate [m] Figure 4.10: Transverse profiles of instantaneous Karlovitz number in Case 1b at axial location m 101

125 Figure 4.11: Instantaneous distribution of Karlovitz numbers and filter sizes on the flame surface in Case 1b. DNS = Direct Numerical Simulation, RFS = Resolved Flame Surface, CF = Corrugated Flamelets, TRZ = Thin Reaction Zone, BRZ = Broken Reaction Zone Figure 4.12: Instantaneous reaction rates. The color lines represents Case 1a and the black line Case 1b 102

126 (a) Instantaneous (b) Time-averaged Figure 4.13: Three-dimensional view of the instantaneous (left) and time-averaged (right) reaction zone for Case 1b Figure 4.14: Three-dimensional view of the instantaneous spanwise vorticity field for Case 1b 103

127 Normalized mean axial velocity Experimental EBULES LEMLES Normalized x coordinate Figure 4.15: Centerline variation of the normalized time-averaged axial velocity for Cases 1a and 1b. The velocity is normalized by the inflow value, and the distance is normalized by the bluff body size, a 104

128 Normalized y distance Experimental EBULES (P) LEMLES (P) Numerical (G) Normalized axial velocity Figure 4.16: Transverse profiles of the normalized time-averaged axial velocity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 105

129 Normalized y distance Experimental EBULES (P) LEMLES (P) Normalized transverse velocity Figure 4.17: Transverse profiles of the normalized time-averaged transverse velocity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 106

130 Normalized y distance Experimental EBULES (P) LEMLES (P) Numerical (G) Normalized axial velocity rms Figure 4.18: Transverse profiles of the normalized rms of the axial velocity fluctuation intensity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 107

131 Normalized y distance Experimental EBULES (P) LEMLES (P) Normalized transverse velocity rms Figure 4.19: Transverse profiles of the normalized rms of the transverse velocity fluctuation intensity for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 108

132 Normalized y distance Experimental EBULES (P) LEMLES (P) Normalized Reynolds stress Figure 4.20: Transverse profiles of the normalized Reynolds stress for Cases 1a and 1b, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 109

133 1.5 Experimental EBULES (P) LEMLES (P) Numerical (G) Normalized y coordinate Normalized mean temperature Figure 4.21: Transverse profiles of the normalized time-averaged temperature for Cases 1a and 1b, at the normalized axial locations, from left to right: 3.75 a, 8.75 a and a. The symbol P indicates results obtained in the present study, and the symbol G indicates results published by Giacomazzi et al. [2004] 110

134 CHAPTER V FLOW AND FLAME STRUCTURES IN PREMIXED AND PARTIALLY PREMIXED FLAMES 5.1 Overview The goal of this chapter is to revisit the reactive flow problem to assess the effect of changing the inflow equivalence ratio φ, on the reactive flow, and on the flame structure and stability. First, perfectly premixed cases with equivalence ratios of φ = 0.6, φ = 0.65 and φ = 0.75, as shown in Table 4.1, were simulated using the LEMLES approach, and the results are presented and analyzed in terms of the effect of the equivalence ratio upon the general flow parameters (time averaged and rms velocities, recirculation region geometry, vortical pattern, mean temperature, etc.) as well as upon the flame dynamics and stabilization mechanism. The second part presents the effect of spatial variation in the inflow equivalence ratio. For this, two simulations using the inflow equivalence ratio profiles described in Fig. 5.1 were carried out, and their results are presented in contrast with the constant equivalence ratio Case 3. To enable proper comparison, the overall equivalence ratio was maintained at Figure 5.2 presents Cases 2, 3 and 4 with respect to the blow-off flame stability limits obtained through correlation of experimental data for propane, premixed, bluff body stabilized flames by De Zubay [1950]. The stability parameter S introduced in Fig. 5.2 is defined as: S = C g C d U 0 a p ( 0 T0 ) 1.5 p T (5.1) where C g is a geometry related constant, equal, for triangular bluff bodies to 1.73, C d is a dimensional constant, equal here to 2.41, U 0 is the inflow velocity, a is the bluff body size, p 0 is a reference pressure equal to 1 atmosphere, p is the inflow pressure, T 0 is a reference 111

135 Normalized y distance Case 3 Case 5 Case Inflow equivalence ratio Figure 5.1: Equivalence ratio inflow profile for cases 3, 5 and 6 temperature equal to K and T is the stagnation temperature at the inflow. All the constant and reference values in Eq. 5.1 are set following Ozawa [1971]. Figure 5.2 also presents a sixth case, at a constant equivalence ratio of φ = In this case, the flame was blown off and the results of this simulations are not included here. As Fig. 5.2 shows, the blow-off predicted by the numerical simulation occurs at an equivalence ratio higher than that experimentally measured (in the range , compared to about ) for the stability parameter simulated here, of about So, even if the simulations prove the ability of LEMLES to handle the transition from a stable flame to blow-off with no other modifications than the change of the inflow equivalence ratio, the predictions may not be quantitatively accurate. The reason for this likely resides in the simplicity of the chemical mechanism employed herein, that only accounts for the major species. The radical species, that play a central role in flame extinction and re-ignition [Turns, 1999] are unaccounted for and may be responsible for the overprediction of the lean 112

136 Equivalence Ratio φ = 0.75 φ = 0.65 φ = 0.60 φ = 0.55 Correlation data Stability Correlation Parameter, S Figure 5.2: Flame stability limits and the position with respect to them of Cases 2-4. The correlation data was reported by De Zubay [1950] and the stability parameter is defined in Eq. 5.1 stability limit observed here. The combustion regime of the simulated flames is also affected by the changes in the equivalence ratio. Thus, Fig. 5.3 presents the spatially averaged mean and variance of the instantaneous Karlovitz number for Cases 2, 3 and 4 overlapped on the combustion diagram shown in Fig The effect of the change in inflow equivalence ratio on the Karlovitz number is twofold. The decrease in equivalence ratio and, therefore, in the amount of heat release reduces the laminar flame speed and thicken the flame. From Eq. 4.2, it can be noted that both these tendencies result in an increase in the Karlovitz number, and the effect can be observed in Fig. 5.3, where the leanest flame (φ = 0.60) reaches the highest Karlovitz numbers. Thus, leaner flames extend deeper into the Broken Reaction Zone. According to the observations in Chapter 4, LEMLES is the only valid combustion model for the Broken Reaction Zone, and since flame quenching is expected to occur as the flame becomes leaner and leaner, 113

137 Figure 5.3: Mean and variance of the instantaneous Karlovitz number for Cases 2-4. The solid circles indicate mean values and the lines indicate variance intervals. Notations are identical to Fig LEMLES is the only combustion model that can be expected to handle properly the flame extinction problem. The geometry and the inflow conditions of the simulations generally reproduces the V olvo experiment [Sjunesson et al., 1991b] used in Chapter 4, except for the equivalence ratio and the spanwise dimension, that was reduced, for these cases, to 0.03 m in order to decrease the computational cost. Accordingly, the number of grid cells in the k direction was reduced to 32. The lateral walls were removed and replaced by periodic boundary conditions, while the rest of the walls were considered adiabatic. The simulations are carried out using LEMLES, for 5 flow-through times before the flow statistics are collected, and the time averaged data presented herein are collected over a period equal to three flow-through times. 114

138 5.2 Results and Discussion of Constant Equivalence Ratio Reactive Flow Spectral analysis A Fast Fourier Transform is applied to the time signal of the LEMLES predicted axial velocity autocorrelation, E 11 (Fig. 5.4). The spectral analysis shows little change in the dominant frequencies. The peak values for the three cases being, respectively, 134 Hz, 139 Hz and 140 Hz. Although there is a slight increase in the dominant frequency with the increasing equivalence ratio, the differences are very small (under the time resolution of the Fourier transform) and all very close to Case 1b value. This indicates that the domimamt frequency is not significantly affected by the inflow equivalence ratio value when this change is within the small range simulated here. Normalized amplitude Case 2 Case 3 Case Frequency [Hz] Figure 5.4: FFT of the axial velocity autocorrelation, E 11, for Cases 2-4, normalized by their respective maximum values. The three spectra are translated on the y-axis for clarity. In order to reduce the uncertainty contained in the spectra in Fig. 5.4, the time series 115

139 data was ensemble averaged. The raw data was divided into a number of ensembles, the spectral density was obtained for each ensemble and subsequently averaged. The results for Cases 2-4 are presented in Fig. 5.5 when using 2 (Fig. 5.5a) and 4 (Fig. 5.5b) ensembles for averaging. Normalized amplitude Case 2 Case 3 Case 4 Normalized amplitude Case 2 Case 3 Case Frequency [Hz] (a) 2 averaged ensembles Frequency [Hz] (b) 4 averaged ensembles Figure 5.5: Ensemble averages of the FFT of the axial velocity autocorrelation, E 11, for Cases 2-4, normalized by their respective maximum values. The three spectra are translated on the y-axis for clarity As expected, the uncertainty is reduced and the fluctuations in the spectra in Fig. 5.5 are somewhat reduced in comparison to those in Fig. 5.4, the effect being more evident stronger for the spectra in Fig. 5.5b, where 4 ensembles were averaged. The peak frequencies occur at the same frequencies for Fig. 5.5a, confirming the previous results, even if the intensity of the peaks is decreased by the averaging process. However, in Fig. 5.5b the spectral maxima in the low frequency region disappear. The reason for this is the insufficient temporal resolution used for data collection. Thus, the raw data was collected over a time interval of 15 ms. If 4 ensembles are created to be used for averaging, the time interval reduces to.25 ms, which translates into a minimum frequency of about 260 Hz, which is above the expected peak frequencies. As a result, the spectra in Fig. 5.5b fail to capture the dominant frequencies in all three cases Flow and flame structures The instantaneous spanwise vorticity fields in Cases 2-4 are shown in Fig The vortical pattern remains symmetric for all three cases, but its intensity changes 116

140 (a) Case 2 (b) Case 3 (c) Case 4 Figure 5.6: Instantaneous spanwise vorticity (solid color), instantaneous reaction rate (thick black line) and time-averaged temperature (color lines) for Cases 2-4 with the equivalence ratio, as the stronger vortices are seen in the lower equivalence ratio Case 2 (φ = 0.6), and the weaker vortices appear in the high equivalence ratio Case 4 (φ = 0.75). The equivalence ratio fields for the three cases are presented in Fig The decrease in vortical intensity between Cases 3 and 4 is less that between Cases 2 and 3, showing that the effect becomes stronger closer to the lean extinction limit. Two causes contribute to this effect. First, the combustion that takes place at the vortex core releases a smaller amount of heat in Case 2 than in Case 4. The vortex thermal expansion 117

141 (a) Case 5 (b) Case 6 (c) Case 3 Figure 5.7: Instantaneous equivalence ratio fields for Cases 2-4 is, thus, reduced. Since the angular velocity is conserved, a larger vortex will be weaker, as is the situation in Case 4. A second reason for the observed higher vorticity in Case 2 is the turbulent dissipation that is higher for higher temperature. Thus, the vortical structures will dissipate faster in Case 4, where temperatures are higher. In each case, the flame is convoluted and wrapped around the vortices, but the flame shape differs considerably from case to case. In Case 2, as the large vortices move downstream, the flame is broken and combustion occurs in pockets of un-reacted mixture interspersed with regions of unburned 118

142 mixture. The Case 3 flame is similar to Case 1b, but with a reduced spreading angle due to the increased axial velocity that convects the flame to more acute angles with respect to the burner. The flame in Case 4 is also continuous, and the lateral extent of the region of burned gases is the largest of all Cases 2-4, due to the higher combustion rate and laminar flame speed, corresponding to the higher equivalence ratio. The lower intensity vortices allow bulkier regions of burned products to form downstream of the bluff body, regions that travel downstream along the flame, creating the different time-average temperature pattern that can be noted in Fig The effect of decreasing vorticity with the increase of the temperature in a bluff body stabilized flame was also observed in a recent study by Erikson et al. [2006]. In order to quantitatively estimate the degree of symmetry (or antisymmetry) of the vortical patterns in the three cases, the fluctuations of the flame fronts were considered. First, the instantaneous location of the flame front was extracted from the LES data for both the upper, L u i (x, t), and the lower, Ll i (x, t), flame sheets, at various time instants, along the axial direction, by determining the maximum reaction rate over each transverse grid line. The time-averaged flame locations, (L u a(x) for the upper flame sheet and L l a(x) for the lower one) were next computed and the flame location fluctuation around the mean values were, then, determined as L u (x, t) = L u i (x, t) Lu a(x), respectively L l (x, t) = L l i (x, t) Ll a(x). Since the flame front is wrapped around the vortical structures, the cross-correlation between the fluctuations in the positions of the upper and the lower flame sheets is expected to provide a measure of the correlation between the vortices rolling up at the two edges of the bluff body. Thus, as shown in Fig. 5.8 a value of the cross-correlation coefficient of +1 will indicate a perfectly antisymmetric vortical pattern, while a value of 1 indicates a symmetric one. The flame front location fluctuations for Cases 2-4 are presented in Fig. 5.9: The cross-correlation coefficient, defined as: C(x) = { t1 t1 t=t0 L u (x, t)l l (x, t) ] 2 }{ t=t0 [L u t1 ] 2 } (5.2) (x, t) t=t0 [L l (x, t) 119

143 (a) Symmetric pattern (b) Antisymmetric pattern Figure 5.8: Schematic of the effect of an antisymmetric (a) and symmetric (b) vortical pattern upon the flame surface 1 1 Normalized flame front fluctuation Upper branch Lower branch Normalized flame front fluctuation Upper branch Lower branch Normalized x coordinate Normalized x coordinate (a) Case 2 (b) Case 3 1 Normalized flame front fluctuation Upper branch Lower branch Normalized x coordinate (c) Case 4 Figure 5.9: Fluctuations of the position of the upper and lower flame fronts around the mean location for Cases 2-4 for Cases 2-4 is presented in Fig. 5.10: In the near field, where the combustion process occurs, the cross-correlation coefficient is negative for all three cases, except for a small region located around 2 a. It is important to note that the flame surface is broken in the same region, and it can be concluded that the vortical pattern tends to return to antisymmetry whenever the reaction rate tends to 120

144 Cross - correlation coefficient Case 2 Case 3 Case Normalized x coordinate Figure 5.10: Flame sheets fluctuation cross-correlation coefficients for Cases 2-4 0, emphasizing the critical role of the heat release process in suppressing the antisymmetric vortical pattern. Also interesting is the fact that the tendency towards symmetry increases as the heat release effect is stronger, the negative values of the cross correlation coefficients being closer to 1 in Case 4 than in Case 2. Far downstream, the cross-correlation coefficient tends to 0, as the two flame fronts tend to become uncorrelated due to the dissipation of the driving vortices Sources of flame instability Combustion instability are defined [Zinn and Lieuwen, 2005] as large amplitude oscillations of one or more natural acoustic modes of the combustor. These natural acoustic modes can be excited by the combustion process and may lead to severe vibrations of the combustor, oscillations in thrust, or flame blow-off or flash back. Early studies [Chu and Kovasznay, 1958] have identified three mechanisms that are responsible for wave generation in a combustor: acoustic fluctuations, vortical action and unsteady heat release. These mechanisms 121

145 interact with each other in a complex, nonlinear and unsteady fashion [Menon, 2005] and can excite the natural modes of the combustor leading to combustion instabilities. For each of the mechanisms mentioned previously, various sources and sinks can be identified by analyzing the governing equations [Menon, 2005]. Thus, vorticity may be generated by shear flows, as is the case in the bluff body stabilization problem studied here, compressibility can create pressure oscillations triggered by the variations in density, and combustion related oscillations are responsible for entropy waves. The baroclinic torque ( ρ p/ρ) can contribute to the enhancement or suppression of turbulent fluctuations that are responsible for the small scale turbulent motion of the flame [Menon, 2005]. An instantaneous visualization of the z-component of the baroclinic torque for Cases 2-4 is presented in Fig An analysis performed by Menon [2005] on the acoustic equations has shown that vorticity is enhanced by positive values of the baroclinic torque and suppressed by negative values. In all the cases studied, both positive and negative regions can be observed alternating closely on the flame surface. The baroclinic torque pattern follows closely the flame pattern, but no significant difference (other than the flame shape) can be observed between the studied cases. The unsteady dilatation ( v, where v is the fluctuating velocity vector, defined as difference between the instantaneous and the time-averaged velocity, presented in Fig. 5.12) presents a similar pattern, closely following the flame. The waves that are formed at the flame surface are propagating in the transverse direction, towards the two solid walls where they reflect and return to the domain, interfering with the incoming ones. The wave-length of these waves can be observed to vary for the three cases, Case 2 showing the lowest wavelength (highest frequency) and Case 4, the highest. Also noteworthy is the intense and numerous dilatation waves that form in the downstream region for Case 2. The spatial region also coincides with a local extinction region, where the breaking up of the flame surface occurs, so it can be concluded that the acoustic mode, excited by the unsteady dilatation takes part, along with the vorticity, in the flame front instability noted in Fig. 5.6a. The significant impact of the dilation upon 122

146 (a) Case 2 (b) Case 3 (c) Case 4 Figure 5.11: Instantaneous spanwise component of the baroclinic torque for Cases 2-4 the dynamic response of the flame was also highlighted by Mehta and Soteriou [2003], in a recent numerical study using a Vortex Element method. Previous studies [Menon, 2005] have shown that the unsteady dilatation and the fluctuating pressure field are closely related, a crest in the fluctuating pressure corresponding to a trough in the dilatation. Fig confirms the presence of the pressure fluctuations in the same region where the dilatation waves are occurring. The intensity of the pressure fluctuation is higher in Case 2 (φ = 0.6) and lower in Case 4 (φ = 0.75), correlating with the previous observation concerning the local extinction observed for the Case 2 flame. The reason for this is the lower amount of heat released 123

147 (a) Case 2 (b) Case 3 (c) Case 4 Figure 5.12: Unsteady dilatation for Cases 2-4 through combustion in Case 2, by which the turbulent fluctuations are allowed to maintain higher values through a reduced dissipation. To estimate the relative contribution of the various vorticity sources and sinks, consider the unfiltered vorticity transport Eq. 5.3 [Menon, 2005]: DΩ Dt = (Ω ) V Ω( V ρ p ) + ρ 2 (5.3) where Ω = V is the instantaneous vorticity, and V is the instantaneous velocity field. 124

148 (a) Case 2 (b) Case 3 (c) Case 4 Figure 5.13: Pressure fluctuation for Cases 2-4 The first term on the right hand side is the vortex stretching. If the strain element produced by the velocity gradient acts to stretch the material linear element aligned with Ω, then the vorticity magnitude increases. The second term is the thermal expansion term, which is non-zero only in compressible and / or reacting flow [Menon, 2005]. The term can be further decomposed into a mean dilatation contribution, Ω( V ), and a contribution from the unsteady dilatation, Ω( v ). The instantaneous field of the unsteady dilatation contribution to the vorticity is shown, for the three cases, in Fig

149 (a) Case 2 (b) Case 3 (c) Case 4 Figure 5.14: Instantaneous field of the unsteady dilatation contribution to the vorticity for Cases 2-4 Finally, the last term on the right hand side of Eq. 5.3 represents the baroclinic torque, presented previously in Fig A comparison between the instantaneous values of the baroclinic torque and unsteady dilatation contribution in the axial direction is shown in Fig Generally, the baroclinic torque contributions are larger than the unsteady dilatation contributions. Immediately downstream of the bluff body, over a region of about of 126

150 3e+08 Baroclinic torque Unsteady dilatation contribution 3e+08 Baroclinic torque Unsteady dilatation contribution Magnitude [1/s sq.] 0 Magnitude [1/s sq.] 0-3e Normalized x distance -3e Normalized x distance (a) Case 2 (b) Case 3 3e+08 Magnitude [1/s sq.] 0 Baroclinic torque Unsteady dilatation contribution -3e Normalized x distance 4 5 (c) Case 4 Figure 5.15: Instantaneous baroclinic torque and unsteady dilatation contribution to vorticity along the x axis in the shear layer region for Cases 2-4 the bluff body size, the baroclinic torque acts to decrease the vorticity. Over the same range, the strong unsteady dilatation acts in an opposite manner, enhancing the vorticity, but with a lower intensity. Further downstream, regions of both positive (vorticity enhancing) and negative (vorticity suppressing) values alternate for both terms. Overall, for both terms considered here, the magnitudes are increasing with the increase in inflow equivalence ratio, following the increasing amount of heat release. The overall stability of the flame can be estimated by evaluating the Rayleigh criterion. The criterion states that combustion oscillations will grow whenever the pressure oscillation and the unsteady heat release are in phase, and the heat release can add energy to the pressure oscillation. If this addition of energy surpasses the losses due to viscous dissipation and waves exiting the combustor, the oscillation will be amplified, leading to flame front instability (local extinction) and, eventually to combustion instabilities. If the Rayleigh 127

151 parameter is defined as: Normalized amplitude R(t) = V p q dv (5.4) the resulting time variation of the parameter can be found in Fig Case 2 Case 3 Case Time [s] Figure 5.16: Time variation of the Rayleigh parameter for Cases 2-4. The parameter values are normalized by the maximum value The integral in Eq. 5.4 is taken over the entire volume of the combustor, while p and q are the unsteady pressure, respectively heat release. The data was normalized by the maximum value of the previously defined Rayleigh parameter over the three cases, R max. The figure indicates both time intervals when the Rayleigh parameter is negative and the oscillations are attenuated, and periods when the value of the parameter is positive, and energy is added to the oscillation. Overall, the amount of time when the Rayleigh parameter is positive appears to decrease with the increase in the inflow equivalence ratio, supporting the previous observation that Case 2 is unstable, while Cases 3 and 4 are stable. To better estimate this, the three Rayleigh parameters where integrated over time, yielding a value 128

152 of for Case 2, of 0.05 for Case 3 and of 0.16 for Case 4, confirming the previous observations Intermittency Another unsteady phenomenon related to combustion is the external intermittency. Due to the unsteadiness present in the flow, the flame does not hold the same position in time and oscillates around a mean location. Hence, fixed points in the flame region change from the unburned to the burned region and back, as the flame front responds to turbulent fluctuations. As an effect, the velocities in regions close to the flame front will alternate between the value in the unburned region and the larger value in the post-flame region due to acceleration caused by the flame. The predicted axial velocity was stored at three spatial locations for each of the Cases 2-4, as shown in Table 5.1. The significance of the three regions (A, B and C) is described in Fig Table 5.1: Position of sampling points for the PDF of the axial velocity for Cases 2-4. Dimensions are in mm, measured from the centerline of the combustor and from the bluff body back wall Case Region A Region B Region C Direction x y x y x y Figure 5.18 shows the Probability Distribution Functions (PDF) of the axial velocity in the three regions, A, B and C, defined as per Fig The PDFs show a single-peaked distribution for Regions A and C that are sufficiently remote from the flame surface, on either side, to remain unaffected by the intermittency phenomenon. Region B, on the other hand, falls within the intermittency region and the axial velocity PDFs show a doublepeaked distribution corresponding to the unburned, respectively burned region. Although the PDFs in Fig do not seem to be affected significantly by the variation in equivalence 129

153 Figure 5.17: Schematic displaying the placement of the probe points in used for the determination of the axial velocity PDF (Fig. 5.18) ratio of Cases 2-4, a closer examination of Table 5.1 indicates that, in order to capture the intermittency, the three points defining the unburned, intermittent, and burned regions had to be shifted in the transverse direction further away from the centerline as the equivalence ratio was increased. The reason for this is that the angle that the mean position of flame forms with the combustor centerline increases with the increasing equivalence ratio, as an effect of higher combustion rates and laminar flame speeds Ignition delay effects Behind a bluff body, even for perfectly stable burners, experimental observations have shown that the flame stabilizes itself a short distance downstream of the bluff body back wall, due to the phenomenon known as the ignition delay [Peters, 2000]. The numerical simulation presented herein is able to capture this phenomenon, as shown in Fig. 5.19, presenting the variation of the instantaneous reaction rate along the flame surface. Thus, the distance from the bluff body to the flame is of 2.5 mm for Case 2, of 1.5 mm for Case 3, and decreases to 1.0 mm for Case 4. In a real device, the reasons behind this effect are multiple. On one hand, the presence of a solid wall implies heat losses from the surrounding flow towards it and, also, triggers and favors radical recombination thus inhibiting the 130

154 Region A Region B Region C 0.08 Region A Region B Region C Probability Probability Axial Velocity [m/s] Axial Velocity [m/s] (a) Case 2 (b) Case Region A Region B Region C Probability Axial Velocity [m/s] (c) Case 4 Figure 5.18: PDF of the axial velocity at 3 locations in the flame region for Cases 2-4 chemical reactions. However, neither of these mechanisms is included in the numerical simulation, so the results in Fig must be attributed to a different phenomenon. A suitable explanation is provided by Law [1988] and is based on the quenching effect of the high velocity gradients occurring at the trailing edge of the bluff body. This effect is related to the concept of flame stretch, and it is defined as the fractional rate of change of an area element, A, on the flame surface, as per Eq. 1.1 [Williams, 1975b]. The analysis performed by Law [1988] indicates that the stretch affecting a flame can be due aerodynamic straining, flame curvature, if the flame normal is not aligned with the flow, and, for curved flames, the flame non-stationarity. If the stretch occurs in the hydrodynamic zone (just close enough for the flow non-uniformities to affect the flame surface), the flame surface will be distorted and displaced and the volumetric burning rate will be altered. The effect of the flame deformation is to ensure that the local propagation velocity is balanced by the flow velocity component normal to the flame surface. Also, in the preheat zone the 131

155 Reaction rate [1/s] 0 2e+06 4e+06 6e+06 Case 2 Case 3 Case Axial distance [m] Figure 5.19: Instantaneous variation of the reaction rate along the flame surface for Cases 2-4 stretch can affect the normal mass flux entering the reaction zone and the residence time, thus affecting the reaction rates of the chemical reactions. The value of the stretch on any point on the flame surface is given by Eq If V f is the velocity of the flame surface, according to Law and Sung [2000], the stretch can also be expressed as: K = t V f + ( V f n)( n) (5.5) where t is the tangential gradient operator over the flame surface, and n is the flame surface normal: n = T T (5.6) 132

156 After further manipulation and assuming the flame velocity component tangent to the flame surface equals to the flow velocity component in the same direction [Law and Sung, 2000], Eq. 5.5 becomes: K = v t + ( V f n)( n) (5.7) where v t is the flow velocity component tangential to the flame surface. Thus obtained, the instantaneous strain for the 3 simulated cases is shown in Fig. 5.20, while its streamwise variation along the flame immediately downstream of the bluff body at the same instant in time is presented in Fig (a) Case 2 (b) Case 3 (c) Case 4 Figure 5.20: Instantaneous flame stretch for Cases 2-4 In Fig. 5.20, the flame stretch pattern is, as expected, following closely the vorticity 133

157 15000 Case 2 Case 3 Case 4 Flame stretch [Hz] Normalized x coordinate Figure 5.21: Instantaneous variation of the flame stretch immediately downstream of the bluff body for Cases 2-4 pattern. Besides the very high stretch region occurring at the sharp edges of the bluff body, regions of high stretch can be found at various locations in all 3 cases. When the flame stretch exceeds a critical value, local flame extinction may occur in these regions. A closer look at Fig indicates that, for all cases, the reaction rate becomes negligible for regions where the instantaneous flame stretch exceeds a value of a 2, 000 Hz that appears to be strong enough to produce flame quenching. In a recent study, Most et al. [2002] has found that a similar value for the critical strain rate. Thus, a stretch under 1, 800 Hz allows stable flames, while higher values lead to flame extinction. Dinkelacker et al. [1998] has also shown that lean turbulent flames can be quenched in a turbulent field even in the absence of heat losses, supporting the current observations. As mentioned earlier, simulations of even leaner flames have been conducted, but are not included here. For an inflow equivalence ratio of 0.55 the flame is blown off. Based on the previous conclusion, this can be attributed to a further increase in the flame stretch 134

158 causing the increase of the ignition delay distance and, eventually, flame blow-off Time averaged results The axial velocity profiles along the centerline for Cases 2-4 are shown in Fig The length of the recirculation region increases with the increase of the equivalence ratio, from a value of 2.75 a for Case 2, to 2.98 a for Case 3 and, finally, to 3.33 a for Case 4, as an effect of the increased thermal expansion of the backflow bubble, due, in turn to the larger heat release obtained for the higher equivalence ratio. The far field velocity is also affected by the inflow equivalence ratio because of the increased chemical energy released in the flow by the higher equivalence ratio combustible mixture. Thus, the Case 2 flow is accelerated to a value of 3.8 U 0, the Case 3 flow accelerates to 4.4 U 0, and the Case 4 flow reaches 5.0 U 0. 6 Normalized mean axial velocity Case 2 Case 3 Case Normalized x coordinate Figure 5.22: Centerline variation of the normalized time-averaged axial velocity for Cases 2-4 Figures 5.23 and 5.24 present the transverse plots of time-averaged axial and respectively transverse velocity for Cases 2-4. The tendency of the axial velocity component to increase 135

159 with the equivalence ratio noted previously can also be observed here. The free stream velocities are lower in Case 2 (3.70 U 0 ) than in Case 3 (4.21 U 0 ) and in Case 4 (4.78 U 0 ), and the recirculation bubble is also narrower in the leanest case (0.775 a for Case 2, a for Case 3 and a for Case 4). When compared to Case 1, the axial velocity is higher even for the lower equivalence ratio Case 2, especially in the free stream region. In Case 1, the axial velocity in the free stream is found to be by 36% lower than for Case 3. The reason for this is not entirely clear. Since the only difference between Case 3 and Case 1b is the absence of the lateral walls that have been replaced by periodic boundary conditions in the latter case, the change in the free stream axial velocity may be attributed to the lack of lateral confinement this introduces. The boundary layers that form on the spanwise walls in Case 1b are missing for Cases 2-4, impacting on the flow pattern. Also, three-dimensional heat release effects may not be captured accurately in a reduced span domain and may also contribute to the observed effect. The different equivalence ratios introduce, also, significant differences in the time averaged temperature profile, presented for Cases 2-4 in Fig As expected, the average temperature increases with the equivalence ratio. Thus, the maximum centerline temperature for Case 2 is 1558 K, for Case 3 is 1628 K and for Case 4 is 1804 K. Also, the spreading of the hot wake region is larger for the high equivalence ratio Case 4. The reason behind this is the higher heat release of the higher equivalence Case 4 ratio, and its effect is dual, both by increasing the temperature gradient between the fresh mixture and the burned gases, and by increasing the thermal expansion of the flame region. Comparing again Case 1b to Case 3, the spreading of the temperature profile is found to be larger in the confined case than in Case 3. Due to the larger axial velocities, the Case 3 flame stabilizes at a more acute angle, and the temperature profile spreading is delayed, occurring at a lesser rate. The time-averaged vorticity profile along the flame sheet for the three studied cases is presented in Fig The average vorticity reaches its maximum immediately downstream of the bluff body, 136

160 Normalized y distance Case 2 Case 3 Case Normalized axial velocity Figure 5.23: Transverse profiles of the normalized time-averaged axial velocity for Cases 2-4, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a 137

161 Normalized y distance Case 2 Case 3 Case Normalized transverse velocity Figure 5.24: Transverse profiles of the normalized time-averaged transverse velocity for Cases 2-4, at the normalized axial locations, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a 138

162 1.5 Case 2 Case 3 Case 4 Normalized y coordinate Normalized mean temperature Figure 5.25: Transverse profiles of the normalized time-averaged temperature for Cases 2-4, at the normalized axial locations, from left to right: 3.75 a, 8.75 a and a and the value of this maximum is higher for lower equivalence ratios, confirming the trend observed for the instantaneous vorticity discussed previously. Further downstream, the vorticity decreases continuously, eventually reaching regions of opposite sign far downstream, an effect also observed experimentally by Nair [2006], that explains the decrease in the timeaveraged voticity through the effect of flame generated baroclinic torque that weakens the wall generated vorticity. As the equivalence ratio decreases, the magnitude of the temperature jump across the flame diminishes, so the time-averaged effect of the baroclinic torque decreases, such that the low equivalence ratio flame experiences a stronger time-averaged vorticity field near the bluff body that also persist for a longer distance downstream. 139

163 Case 2 Case 3 Case 4 Vorticity [Hz] Normalized x coordinate Figure 5.26: Time-averaged vorticity along the flame surface for Cases Results and Discussion of Variable Equivalence Ratio Reactive Flow Spectral analysis Figure 5.27 presents the Fourier transforms of the axial velocity autocorrelation at a fixed point in space in the shear layer downstream of the bluff body, placed at the same location as for the previous frequency studies. The spectra are normalized by their respective maxima and shifted along the y axis for clarity. The peak frequencies in the pressure spectra are around 139 Hz, 106 Hz and 111 Hz, respectively, for the three cases (3, 5 and 6). For all spectra, after the peak is reached the energy content of higher frequencies decreases following the well known inertial range law: E = Ck 5/3. where C is a proportionality constant. As noted earlier, the reduced span Case 3 agrees well with Case 1a, Case 5 is very close to the non-reacting vortex shedding frequency Giacomazzi et al. [2004], Fureby [2000b], while Case 6 has an intermediate value, between the previously mentioned extremes but 140

164 Figure 5.27: FFT of the axial velocity autocorrelation, E 11, for Cases 5 and 6. For comparison, Case 3 is also included. The spectra are normalized by their respective maximum values and translated on the y-axis for clarity closer to the non-reactive frequency as well. As before, further analysis was performed by ensemble averaging the density spectra for the three cases. The results for Cases 5 and 6 are presented in Fig when using 2 (Fig. 5.28a) and 4 (Fig. 5.28b) ensembles for averaging. Case 3 is also included for comparison. As for the constant equivalence ratio cases, the fluctuations in the spectra diminish slightly as the number of ensembles used for the averaging increases, the peaks occur at the same frequencies for the 2 ensemble averages spectra (Fig. 5.28a) and disappear if 4 ensembles are used for averaging (Fig. 5.28b). The reason for this is the the same as for the constant equivalence ratio spectra, namely the insufficient temporal resolution used for data collection. 141

165 Normalized amplitude Case 3 Case 5 Case 6 Normalized amplitude Case 3 Case 5 Case Frequency [Hz] (a) 2 averaged ensembles Frequency [Hz] (b) 4 averaged ensembles Figure 5.28: Ensemble averages of the FFT of the axial velocity autocorrelation, E 11, for Cases 5 and 6, normalized by their respective maximum values. For comparison, Case 3 is also included. The three spectra are translated on the y-axis for clarity Flow and flame structures More insight into the effect of changing the inflow equivalence ratio on flame stability is obtained by viewing the instantaneous spanwise vorticity overlaid by the reaction rate (Fig. 5.29). There are significant differences in the size and roll-up features for the three cases. Experimental observations Sjunesson et al. [1992] indicate that the heat release process is cyclic in nature. In the initial phase of vortex development, the vorticity is relatively low and the turbulent mixing and burning are reduced. During the vortex roll - up phase, as vorticity increases, the flame is entrained into the vortices and wrinkled, its surface area increases rapidly, enhancing both turbulent mixing and combustion. The augmented rate of chemical heat release, in turn, decreases vorticity and the cycle repeats itself. This periodic local heat release can have effects upon the combustor if the oscillations are in phase with the pressure fluctuations. The near wake central region presents low vorticity levels due to the combined effects of combustion related phenomena [Fureby, 2000a]. Further downstream. the large scale vortices break down into small, incoherent vorticity. For Case 5, the fluid in the shear layer region will be leaner than in the baseline Case 3 (φ = 0.60) (see Fig. 5.30b) and, once entrained into the hot region its combustion will release a lower amount of heat, and, therefore, an antisymmetric vortex shedding as in non-reacting flow is seen. Since in Case 2 vortical pattern is symmetric, this cannot be only 142

166 (a) Case 5 (b) Case 6 (c) Case 3 Figure 5.29: Instantaneous spanwise vorticity (solid color), instantaneous reaction rate (thick black line) and time-averaged temperature (color lines) for Cases 5 and 6. For comparison, Case 3 is also included an effect of the leaner mixture in the shear layer, but rather a consequence of the spatial variability in the equivalence ratio. The flame wraps around these large structures but as the large vortices move downstream, the flame structure is broken up creating regions of burned fluid interspersed with regions of unburned mixture. Combustion in this case is unstable and further proof for this observation will be presented in the following. In contrast, as Fig. 5.30c indicates, the shear layer consists of a richer mixture (φ = 0.75) 143

167 (a) Case 5 (b) Case 6 (c) Case 3 Figure 5.30: Instantaneous equivalence ratio fields for Cases 5 and 6. For comparison, Case 3 is also included for Case 6, and the combustion of the fluid entrained by vorticity into the hot, central region will release more heat than in Case 3. This causes the large scale vortical structures to be significantly reduced in intensity and the flame front is continuous. The vortices roll-up simultaneously from the two shear layers but the symmetry is broken further downstream by the non-uniformities existing in the flow field. In the far field, vortices of alternating sign cause the flame-vortex region to exhibit a sinusoidal transverse flapping model, which on the average shows a spreading rate larger than in the Case 3. The higher amount of 144

168 heat release also causes the large scale vortical structures to collapse faster into incoherent vorticity further downstream. To quantitatively estimate the degree of symmetry (or antisymmetry) of the vortical patterns in the variable equivalence ratio cases, the same procedure applied previously is carried on. The flame front location fluctuations for Cases 5 and 6 are presented in Fig. 5.31: Normalized flame front fluctuation Upper branch Lower branch Normalized flame front fluctuation Upper branch Lower branch Normalized x coordinate Normalized x coordinate (a) Case 5 (b) Case 6 1 Normalized flame front fluctuation Upper branch Lower branch Normalized x coordinate (c) Case 3 Figure 5.31: Fluctuations of the position of the upper and lower flame fronts around the mean location for Cases 5 and 6. For comparison, Case 3 is also included With the definition given by Eq. 5.2, the cross-correlation coefficient between the upper and lower flame sheets position fluctuations for Cases 5 and 6 are presented in Fig. 5.32: The correlation coefficient is negative very close to the bluff body indicating an antisymmetric vortical pattern in this region, but it becomes positive shortly downstream, in agreement with the vortical pattern shown in Fig For Case 5, this is an effect of the discontinuous combustion occurring for the low equivalence ratio case, as the amount of 145

169 Cross - correlation coefficient Case 3 Case 5 Case Normalized x coordinate Figure 5.32: Flame sheets fluctuation cross-correlation coefficients for Cases 5 and 6. For comparison, Case 3 is also included heat release is now insufficient to completely destroy the antisymmetry characteristic to the non-reactive case. For Case 6, the reason for the antisymmetric pattern that entrains the observed flame flapping mode is uncertain and can only be attributed to the effects of the changing amount of heat release determined by the variable equivalence ratio. Regions of low equivalence ratio are entrained into the combustion zone decreasing the amount of heat release through combustion and, consequently, the baroclinic torque associated with the the pressure and density gradients, as it will be discussed later. Thus, the effect of the baroclinic torque is diminished, and the Kelvin - Helmholtz instability remains strong enough to cause the antisymmetric vortex pattern. As for the constant equivalence ratio cases, far downstream, the cross-correlation coefficient tends to zero, as the vortices break-up into incoherent vorticity. 146

170 5.3.3 Sources of flame instability As mentioned earlier, the flame instabilities will grow if the pressure oscillations (p ) are in phase with the heat release fluctuation (q ). The stability of the combustion process in these three cases can be determined by computing the Rayleigh parameter. Normalized amplitude Case 3 Case 5 Case Time [s] Figure 5.33: Time variation of the Rayleigh parameter in the combustor for Cases 5 and 6. For comparison, Case 3 is also included Figure 5.33 presents a portion of the time variation of the Rayleigh parameter normalized by its maximum value over the three cases. As the figure indicates, for all three cases, there are time intervals where R(t) is positive, but it is apparent in Fig that these intervals are more significant for Case 5 than for the other two cases. A global Rayleigh parameter can be determined as: R = t0 + t t 0 t R(t)dt (5.8) where t is a relevant time interval (chosen here to be the flow-through time) and can be regarded as a measure of the global stability. Under these assumptions, the following values 147

171 were obtained: R = 0.053, and for Cases 3, 5, and 6, respectively. Thus, it appears that Case 5 is susceptible to instability, and also that the stability of Case 6 is reduced in comparison to Case 3. Usually, the unsteady flow field can be decomposed into three canonical types of disturbances [Lieuwen and Zinn, 2005]: vortical, acoustic and entropy. As for Cases 2-4, further insight into the flame dynamics characteristic to the three simulated flames can be gained by analyzing these potential sources of disturbances. Thus, Fig presents the z-component of the baroclinic torque, for Cases 5 and 6 in comparison with Case 3. As expected, the instantaneous snapshots show non-zero regions of the baroclinic torque appearing in close correlation with the flame front, in all cases. Both regions of vorticity enhancement (appearing in black in Fig. 5.34) and regions of vorticity suppression (in white) are visible in the flow. In Case 5, a region of very strong baroclinic torque magnitude (positive and negative) can be observed in Fig. 5.34b, in the region where the flame surface is breaking up. Large positive values appear on the flame front in this region, magnifying the local vorticity followed closely by large negative values in the adjacent regions, where vortical damping through the effect of the baroclinic torque is occurring. Overall, since Case 5 exhibits significant regions of strong baroclinic torque values coinciding with the flame break-up region, is reasonable to infer that the two phenomena are related, and the the baroclinic torque, along with the vorticity it influences, takes part in the observed flame front discontinuity. For Case 6, the same close correlation between the baroclinic torque pattern and the flame front is present. In the downstream region where the flame approaches the upper wall, a wave-like structure of the baroclinic torque field can be observed. The waves that impact the wall are reflected back with a change in phase that is also captured in Fig. 5.34c. Another potential source of flow disturbance, this time driving the acoustic mode, is the pressure fluctuation, presented for Cases 3, 5 and 6 in Fig Overall, the highest level of pressure fluctuation is achieved in Case 5, in correlation with previous observation about the flame instability tendency in this case, but the higher fluctuation intensity is more likely to be an effect of the observed flame front instability 148

172 (a) Case 5 (b) Case 6 (c) Case 3 Figure 5.34: Instantaneous spanwise component of the baroclinic torque for Cases 5 and 6. For comparison, Case 3 is also included than a driving cause. In all 3 cases, pressure waves are formed in the flame region and are propagating in the transverse direction in the flow, interacting with the solid walls. In close correlation with the unsteady pressure field is the unsteady dilatation [Menon, 2005] (Fig. 5.36). Here, the differences between the 3 studied cases are very significant. In both variable equivalence ratio Cases 5 and 6, the unsteady dilatation reaches higher values than in the constant φ Case 3. Dilatation waves are generated at the flame fronts and travel outwards 149

173 (a) Case 5 (b) Case 6 (c) Case 3 Figure 5.35: Pressure fluctuation for Cases 5 and 6. For comparison, Case 3 is also included interacting with the solid walls and with each other. The higher intensity of the unsteady dilatation in Cases 5 and 6 may be explained by the spatially and temporally variable amount of heat release at the flame front, as it encounters regions of different equivalence ratio of the unburned mixture, as shown in Fig and the impact of the flow disturbance generated by the strong unsteady dilatation may be held responsible for the reduced flame stability exhibited by Cases 5 and 6, as shown by the Rayleigh criterion analysis discussed earlier. 150

174 (a) Case 5 (b) Case 6 (c) Case 3 Figure 5.36: Unsteady dilatation for Cases 5 and 6. For comparison, Case 3 is also included The instantaneous field of the unsteady dilatation contribution to the vorticity (Eq. 5.3) is shown for the two variable equivalence ratio cases in Fig. 5.37, and its variation along the axial direction is presented in Fig along with the variation of the baroclinic torque, for comparison. No significant differences can be found between the variable equivalence equivalence ratio cases and the reference, constant equivalence ratio case. The baroclinic torque contribution remains generally more significant than that of the unsteady dilatation. In the near field, the 151

175 (a) Case 5 (b) Case 6 (c) Case 3 Figure 5.37: Instantaneous field of the unsteady dilatation contribution to the vorticity for Cases 5 and 6. For comparison, Case 3 is also included baroclinic torque is negative, decreasing the magnitude of the vorticity, while the unsteady dilatation term acts in an opposite manner, enhancing it. In Case 6, after an initial region of about 1.5 times the bluff body length where positive and negative values of the baroclinic term alternate, a region of low values for both contributions, extended over about one bluff body length, follows. This low value of the instantaneous baroclinic torque could provide a reason for the antisymmetric vorticity pattern observed in Fig. 5.29, as discussed earlier. 152

176 3e+08 Baroclinic torque Unsteady dilatation contribution 3e+08 Baroclinic torque Unsteady dilatation contribution Magnitude [1/s sq.] 0 Magnitude [1/s sq.] 0-3e Normalized x distance -3e Normalized x distance (a) Case 5 (b) Case 6 3e+08 Baroclinic torque Unsteady dilatation contribution Magnitude [1/s sq.] 0-3e Normalized x distance 4 5 (c) Case 3 Figure 5.38: Instantaneous baroclinic torque and unsteady dilatation contribution to vorticity along the x axis in the shear layer region for Cases 5 and 6. For comparison, Case 3 is also included Time averaged results Figure 5.39 compares the centerline variation of the time-averaged axial velocity for Cases 5 and 6, and for comparison, for Case 3. The variable φ Cases 5 and 6 both show longer (by 21% and respectively by 5%) recirculation regions in comparison to Case 3. In terms of the peak negative axial velocities, Case 5 presents a lower value, of only 68% of the peak negative value of the constant equivalence ratio Case 3, while Case 6 has a peak value lager by 38% than the Case 3 value. The axial velocity increases in the far wake at a rate that has the highest value for Case 6 and the lowest value for Case 5, but the far field velocity is nearly identical in all cases, as an effect of the identical overall equivalence ratio in the flow and, thus, of the identical total amount of chemical energy added to the system. In the transverse direction, the time averaged velocity profiles of both the axial and 153

177 6 Normalized mean axial velocity Case 3 Case 5 Case Normalized x coordinate Figure 5.39: Centerline variation of the normalized time-averaged axial velocity for Cases 5 and 6. For comparison, Case 3 is also included transverse velocity components are shown in Figs and respectively As a general trend, the free stream axial velocities are lower in Case 5 and higher in Case 6, when compared to the reference Case 3, but, as for Cases 2-4, they are all exceeding the Case 1 free stream velocities for reasons discussed earlier. The most significant difference is observed at the second axial station (0.95 a), where the Case 5 free stream value is 10% lower than the reference value, and the Case 6 value is higher by 14%. The difference between the free stream values in the three cases tends to diminish further downstream. The differences between the three considered cases are less significant for the time-averaged transverse velocities. The situation is reversed for the transverse component, where at the first axial station Case 5 presents higher transverse velocity values, that can be related to a more intense vortical activity. Further downstream the differences between the 3 cases become less and less important. The mean temperature profiles at various axial locations are shown in Fig for 154

178 Normalized y distance Case 3 Case 5 Case Normalized axial distance Figure 5.40: Transverse profiles of the normalized time-averaged axial velocity for Cases 5 and 6. For comparison, Case 3 is also included. Axial locations are, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a 155

179 Normalized y distance Case 3 Case 5 Case Normalized transverse velocity Figure 5.41: Transverse profiles of the normalized time-averaged transverse velocity for Cases 5 and 6. For comparison, Case 3 is also included. Axial locations are, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a 156

180 the three cases. For the axial locations shown here, the maximum temperature in Case 6 is always larger than the flame temperature of the reference, constant φ, Case 3, by as much as 5% at the centerline in the second axial station because the combustible mixture is richer in the region directly behind the bluff body, so the combustion process is more intense here. The far field spreading of this case is also larger. In contrast, Case 5 shows a lower maximum temperature, the maximum deviation from the Case 3 flame temperature being of 10%, due to the unstable combustion that allows time intervals when the burning occurs to alternate with periods when the flame is extinct at a fixed position in space. As an effect, Case 5 also presents a rather slow wake spreading. Wake spreading is the fastest in Case 6 and, since the global equivalence ratio is the same for Cases 3, 5 and 6, the higher temperatures predicted downstream in the near wall region are, probably, and effect of the flame flapping observed for this case, phenomenon that enhances the turbulent flame brush. The time-averaged vorticity profile along the flame sheet for Cases 5 and 6 is presented in Fig In a similar manner as for the previous constant equivalence ratio case, the average vorticity reaches a maximum immediately downstream of the bluff body, and constantly decreasing towards opposite sign values further downstream, due to the effect of the flame generated baroclinic torque. In the reference, constant equivalence ratio Case 3 the vorticity is stronger near the bluff body, but its rate of decrease is larger than in the two variable equivalence ratio Cases 5 and 6, for the same global equivalence ratio, Sice previous experimental observations Nair [2006] associated the decrease in the mean vorticity with the baroclinic torque, this also implies that the baroclinic torque magnitude is smaller in Cases 5 and 6 than in Case 3 in the region immediately downstream of the bluff body, fact that could represent an explanation for the antisymmetric, flame flapping pattern observed in Fig

181 1.5 Case 3 Case 5 Case 6 Normalized y coordinate Normalized mean temperature Figure 5.42: Transverse profiles of the normalized time-averaged temperature for Cases 5 and 6. For comparison, Case 3 is also included. Axial locations are, from left to right: a, 0.95 a, 1.53 a, 3.75 a and 9.4 a 158

182 10000 Case 3 Case 5 Case 6 Vorticity [Hz] Normalized x coordinate Figure 5.43: Time-averaged vorticity along the flame surface for Cases 5 and 6. For comparison, Case 3 is also included 159

183 CHAPTER VI CHEMICAL SOURCE TERMS ESTIMATION USING ARTIFICIAL NEURAL NETWORKS 6.1 Introduction In general terms, an Artificial Neural Network (ANN) is a system modelled after the structure of the human brain. Its basic component, called processing element, (PE), and analogous to the biological neuron, is a non - linear element that receives a number of inputs (representing the dendrites of its cellular counterpart), as shown in Fig Each input connection has a corresponding weight that modifies the contribution of each input signal to the total PE input. The result of this weighted summation is then processed through the application of a transfer function, and the information is passed along either to another PE or to the ANN output. Figure 6.1: Schematic representation of the analogy between a biological neuron and an ANN processing element The processing elements that form a neural network are organized into groups called 160

184 layers and are linked to some or all of the neighboring neurons with various degrees of connectivity representing the strength of the connection. The different layers of the ANN are serially interconnected, the first layer accepting the input from the external environment and the last one is producing the ANN output. By adjusting the values of these strength coefficients through a process called network training, or learning the network is able to generate an output that is consistent with the expected, known, result. Once the training phase is completed, the ANN can be further applied for similar problems with satisfactory accuracy to predict the results for unknown problems, meaning a new set of input parameters, in a manner similar to the functioning of the human brain and termed recalling. The ANN capabilities of modelling highly complex non - linear functions have already been established in various applications in domains other than the CFD, such as noise reduction, image processing, non - linear controls or forecasting, where ANN techniques were developed formerly. In CFD, and more specifically in combustion, ANN applications still constitute a novel field, with very few applications so far. In combustion CFD applications, previous past work was mainly carried on addressing the ability of ANN to accurately map the composition space [Christo et al., 1995, Blasco et al., 1998, 1999]. More recently, Christo et al. [1996] used an ANN approach combined with a modelled velocity-scalar joint PDF to numerically simulate a H 2 CO 2 premixed flame, and Chen et al. [2000] used a combination of In-Situ Adaptive Tabulation and ANN to study a partially stirred reactor. Kapoor et al. [2001] and Kapoor and Menon [2002] employed ANN to estimate the chemical source terms in LEMLES of turbulent flames. Flemming et al. [2003] used ANN for chemistry representation by replacing the flamelet chemistry tables with a set of off-line trained neural networks. In different applications, but while still remaining in the field of turbulent combustion, Choi and Chen [2005] successfully employed ANNs to model the ignition delay in homogeneously charged compression ignition engines, and Sarghini et al. [2003] used neural networks to create an LES subgrid model for the near wall region. In this work, ANN is used to model the chemically reacting system in predicting the temporal evolution of its reactive scalars. The proven ability of ANN to model highly non - 161

185 linear problems makes it a suitable choice for modelling the time variations in temperature and species concentrations that are expected to occur during a chemical reaction. The main advantage of the ANN method resides in the reduction of both CPU time and disk storage space, but the correct choice of ANN structure and parameters becomes critical for achieving the desired proficiency and accuracy of ANN predictions. The method used for the algorithm presented here is based on the perceptron concept proposed by Rosenblatt [1962], and employs the back propagation learning technique. The perceptron was initially developed as a pattern classification system aimed at optical pattern recognition, capable of limited learning and generalization and possessing a great deal of robustness and plasticity. The back-propagation technique [Parker, 1985] assumes that any error appearing in the output has to be assigned to all processing elements and connections, and the responsibility for the error is affixed by propagating it backwards through the existing connections to the previous PE layer, until reaching the input layer, using a transfer function and with the goal of minimizing a global function describing the error between the desired output and the network output. For the purpose of this work, two types of transfer functions have been used: the sigmoid function, defined by Eq. 6.1 and the hyperbolic tangent function, given by Eq. 6.2, but any differentiable function may serve the error back-propagation purpose [NeuralWare, 2001]. f(z) = e z (6.1) f(z) = ez e z e z + e z (6.2) For the case discussed here, that of a chemistry simulation problem, previous studies [Kapoor et al., 2001, Kapoor and Menon, 2002] have shown that the accuracy of the results predicted by the artificial network is strongly affected by several requirements that need to be met: The initial training data set must cover the entire species concentration / temperature space that will be explored during the CFD computation. 162

186 A suitable neural net training algorithm is needed for developing the ANN. A validation data set is needed to determine the post-training network accuracy. This set of points must not have been used for the initial training in order to correctly assess the ANN validity. The best input / output range from the standpoint of ANN accuracy is in the interval [-1;1]. For this reasons, all the data sets must be transformed first through a standardization procedure into a standard, zero mean and unity variance set and next through a normalization procedure into an [-1,1] interval. Finally, a logarithmic transformation (similar to a histogram redistribution) is applied on the data set in order to smooth out the sudden changes in the reactive scalar values and in their rates around a certain temperature value, as it is usual for a combustion mechanism. The ANN approach in solving a reactive flow problem described here consists in modelling the temporal evolution of the reactive scalars and temperature in the computational domain for a premixed flame of a given equivalence ratio and initial temperature and pressure. The aim is to predict the chemical reaction rate within a given time step, and for a given input species composition and temperature. The time step for the calculation of the chemical evolution is kept constant in the ANN training, but is allowed to vary around the training value in the CFD main solver, under the assumption that the variation in species concentrations and temperature is linear for small changes in the time interval. Once trained, the ANN will provide the main CFD solver with correlation coefficients able to model the source terms in the species conservation equations. Those correlation coefficients will be computed off - line, stored in the form of weights and biases and read by the main solver at its initialization, thus allowing a significant reduction in computational costs. For each species in the chemical mechanism, several ANNs are trained. The composition domain is divided into sub-domains defined in terms of the temperature field. Previous results [Kapoor et al., 2001] have shown that the training of multiple neural nets 163

187 separately for each species increases the ANN results accuracy. The choice of the number of layers and number of neurons in each layer is an open question, and has to be optimized iteratively. 6.2 The ANN Training The standard back-propagation training algorithm requires that an input set and a desired output set be presented to the network. In the case of the current ANN, the input consists of chemical composition mass fractions and temperature, while the desired output is the corresponding chemical reaction component for one chemical species. According to the previous section, an accurate ANN implementation has to span the entire realizable domain of input states, meaning that the input set has to include chemical compositions describing states starting with the fresh, unburned mixture and up to completely burned gases. In order to achieve this, the CHEMKIN P remix package [Kee et al., 1996] developed at the Sandia National Laboratories was employed to determine realistic species concentrations and temperature throughout a premixed flame of specified equivalence ratios. The species - temperature profiles thus obtained are interpolated to provide a number of data points sufficiently large for an accurate ANN training and testing. At each point, the chemical reaction rates are integrated over a previously specified time interval using a previously selected chemical mechanism. Once the reaction rates for each species are known for each data point, the overall domain of realizable states is divided into subsets defined in terms of limiting temperatures in such a way that in each such subset (or bin) every reaction rate profile maintains as much as possible a unique elementary shape (such as a line, a parabola, circle, etc.). Next, a logarithmic transformation is applied to both the input information and to the desired output in order to increase network performance, and a final linear transformation is applied to bring the results in the [-1;1] interval. Finally, the network architecture is determined. The typical back-propagation network consists of an input layer, several, intermediate, hidden layers, and an output layer. Previous work [NeuralWare, 2001] indicates that 3 hidden layers are sufficient for solving a problem 164

188 with any degree of complexity, so this choice will be made for the architecture of the present problem. The number of PEs in each layer is determined as follows: In the input layer, the number of PE must be equal to the number of reactive species (excepting Nitrogen), plus one (temperature). In the hidden layers, the number of PEs will be decided by trying to improve the ANN accuracy, as it will be discussed later. In the output layer, there should be 1 PE, corresponding to one reaction rate. As a recommendation based on a multitude of trials performed by the author, the total number of PE s should be about the same as the number of PE s in the input layers, distributed decreasingly between the hidden layers. So far there exists no established methodology or guiding principle for determining the optimum network structure and the ANN optimization still remains a trial and error procedure for each problem at hand. Once the architecture of the ANN s was selected, for each species reaction rate and for each temperature bin, an ANN will be trained and tested, using the same number of layers and processing elements. The ANN implementation described here uses the NeuralW orks P rofessional II P lus software for network training and initial testing, as briefly described in the following: From the InstaN et menu, chose Backpropagation Set the chosen number of layers Set the number of processing elements in each layer Select the desired type of transfer function Select the Connect prior box to ensure full inter-connectivity of the processing elements 165

189 Select the MinMax T able box to insure network input and output lies in the [-1;1] interval Select the Bipolar Inputs box to allow the network to accept both positive and negative inputs Select the learning rule to be used by the network Set all other network parameters. Similarly to the selection of the network architecture, their choice is made by trial and error, in such a way as to improve network accuracy, i.e. to minimize the network prediction rms error with respect to the desired output Using half the input data set, perform actual network training by using the Run command Test the trained network using the remaining half the input data set, by using the T est command At the testing phase, monitor the network rms error. While the desired accuracy is not met, adjust the network parameters set at the previous steps When the desired accuracy is achieved save the network and store the network parameters for future use in the CFD solver 6.3 Numerical Results and Discussion Laminar 1-dimensional premixed methane flame The first test of the proposed ANN algorithm consisted in reproducing a one-dimensional premixed methane flame, at a stoichiometric equivalence ratio and inflow at standard atmospheric conditions. As discussed previously, the P remix CHEM KIN package was employed to generate an initial profile of species and temperature throughout the flame, as shown in Fig Next, the profile is interpolated and expanded to a set of 200,000 data points (see Fig. 6.2) and for each point the reaction rate of each species is integrated over a typical time 166

190 CHEMKIN data points Interpolated data points T 0.2 H2 Mole fraction / Temperature [K] H OH HO2 CH e O2 H2O H2O2 CH CO CO CH2O C2H C2H4 C2H6 Distance [cm] Figure 6.2: Temperature and species profiles used for ANN training 167

191 step of 2 x For the purpose of this work, the 16 species, 12 steps, methane reaction mechanism [Sung et al., 1998] detailed below was used: O 2 + 2CO 2CO 2 H + O 2 + CO OH + CO 2 H 2 + O 2 + CO H + OH + CO 2 HO 2 + CO OH + CO 2 O 2 + H 2 O 2 + CO OH + HO 2 + CO 2 O C 2 H 2 H + CO 2 O 2 + CH 3 + CO + C 2 H 4 CH 4 + CO 2 + CH 2 O + 0.5C 2 H 2 O 2 + 2CH 3 H 2 + CH 4 + CO 2 O 2 + 2CH 3 + CO CH 4 + CO 2 + CH 2 O O 2 + CH 3 + CO H + CO 2 + CH 2 O O 2 + CO + C 2 H 6 CH 4 + CO 2 + CH 2 O H + OH H 2 O (6.3) The individual reaction rates of each reaction in the above mechanism can be found in the previously quoted reference and the resulting reaction rates are shown in Fig The temperature domain was divided into 14 bins and for each bin and each reactive species (Nitrogen was excluded) a neural network was trained, leading to a total of 210 ANNs. The network architecture, presented in Fig. 6.4 consisted in 3 hidden layers containing, respectively, 8, 4 and 2 processing elements. The transfer function used for network training was the hyperbolic tangent (Eq. 6.2) and the maximum allowable rms error at network testing was 1%. After the training of the neural network was completed as described in the previous section, the resulting ANN were implemented into the Runge-Kutta algorithm described in the previous Chapters and the simulation of a one-dimensional laminar flame was carried on. The simulations were performed on a uniform, one-dimensional grid consisting of 1000 points over a length of 0.02 m. Several techniques for the estimation of the chemical source terms were employed and the results are compared in Fig

192 0 50 Reaction rate [1/s] H OH HO2 CH3 CO CH2O H2 O2 H2O H2O2 CH4 CO2 C2H2 0 C2H C2H Temperature [K] Figure 6.3: Initial profile of reaction rates used for ANN training 169

193 Figure 6.4: The architecture of the Artificial Neural Network. P denotes the processing element and L denotes the layer The first approach was to directly integrate the reaction rates determined at the chemistry time scale over the LES time step using a Variable coefficients Ordinary Differential Equation solver (DVODE) [Brown et al., 1989]. The method has maximum accuracy and it will be used henceforth as a baseline in determining the accuracy of various techniques presented in Fig. 6.5, but it becomes extremely expensive especially for complex chemical mechanisms. The second technique tested in this Chapter was initially proposed by Eggenspieler [2005] and consists in a direct estimation of the Arrhenius-type reaction rate equations under the assumption that the LES time step is small enough so that the changes in the temperature and in the mixture composition within a time step are sufficiently small to not affect significantly the reaction rate. Obviously, since the chemical time scales are much smaller than the usual LES time-step, the method (denoted DST E1 hereafter) cannot be expected to be of acceptable accuracy, and Fig. 6.5 proves this, the reaction rates predicted by DST E1 being significantly different from those computed by DV ODE. However, Eggenspieler [2005] has shown that if the chemical source term evaluation time scale is chosen to be one order of magnitude lower than the LES time step the results become quite accurate 170

194 and the reduction in computational time when compared to DV ODE is very important. In the present work, the LES time step was divided into 10 equal size periods and the reaction rates where estimated successfully. As seen in Fig. 6.5, the method (denoted DST E10 hereafter) provided results very close to the direct integration, with the exception of some of the radicals. This indicates that the approach is acceptable but in the case of complex mechanisms further testing is required, since the time scales for radical formation and destruction can be significantly smaller than those implied in the major species chemical kinetics, thus requiring even smaller divisions of the LES time step. Next, the In Situ Adaptive Tabulation approach proposed by Pope [1997] is tested for the same flame. The main idea behind the ISAT approach is to store the reaction rates determined by direct integration and to retrieve them later in the computation if the point in the mixture composition - temperature where the reaction rates are needed lies within a certain maximum distance from one of the previously stored points. Thus, using an efficient searching algorithm, the retrieval of the previously computed reaction rates is much faster than the direct integration leading to speed-up factors of the order of 30 [Sankaran and Menon, 2000]. As direct integration is always employed if no suitable previously computed state is found, the accuracy of the method is fully controlled by the previously mentioned tolerance. For the purpose of this study, the initial data set used for ANN training was also used to build up an ISAT table, in order to create appropriate conditions for method comparison. As seen in Fig. 6.5, ISAT achieves results practically indistinguishable for DV ODE, the accuracy of the method being extremely good. Finally, the simulation was repeated once more, using the ANN approach described previously. For a better appraisal, ANN results for the reaction rates CH 4, CO 2, CO and OH are presented in Fig. 6.6 in comparison with DV ODE results. The overall accuracy is acceptable, but for some species there are regions in the flame where the ANN prediction differs from the direct integration data by as much as 15%. A closer look to those maximum error regions shows that they are characterized by mixture compositions lying outside the range corresponding to the given temperature bin. This indicates the need for improving the binning algorithm, maybe considering species information 171

195 as well as temperature when distributing the realizable domain to the various ANNs. Since the errors are large only at the bin limits, it might be worth trying removing the binning altogether and training one single network for the entire temperature range. In terms of computational efficiency, the comparative simulation time obtained for the 5 methods discussed previously are presented in Table 6.1, over one flow through time and normalized by the ANN time in order to facilitate comparison. Table 6.1: Speedup factors for different methods of estimating the chemical source terms in a laminar, one-dimensional flame. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; ISAT = In Situ Adaptive Tabulation approach for determining the reaction rate; AN N = Reaction rates computed using the present ANN method Method Speedup DVODE DSTE DSTE ISAT 3.52 ANN 1.00 The ANN method ranks second in terms of computational efficiency, being surpassed only by DST E1 which, however, was found to be unacceptable in terms of accuracy. So far, the ANN algorithm developed here looks promising, even though a more accurate prediction of the reaction rates over certain regions of the realizable space remain Turbulent 3-dimensional premixed methane flame The next step for testing the ANN algorithm developed in this study was the numerical simulation of a three-dimensional, turbulent, flame. The simulation was conducted in a cubic domain, of size m employing a uniform Cartesian grid of 64 cells in each direction. Inflow and outflow boundary conditions of the type described in Chapter 3 where imposed in the axial direction and periodic boundary conditions on the remaining lateral walls. The simulation was conducted over 3 flow-through times. 172

196 Figure 6.7 presents instantaneous snapshots of mass fraction for several major and minor species. The turbulence existing in the flow is captured by numerical simulation and impacts upon the flame shape. The flame is not a planar sheet, as in a laminar case, but instead is wrinkled by the small vortices present in the flow as it adapts itself to the turbulent fields. A more clear view of the effect of the ANN algorithm, allowing a better appraisal of its accuracy can be gained from analyzing Fig. 6.8, presenting the time-averaged values of temperature and species mass fractions across the flame, in comparison with results obtained using the direct integration of the chemical source terms. First thing to be noted is the very good agreement of the temperature profile, the two lines being almost identical. Also, very good agreement between ANN and DV ODE is achieved for all major species (CH 4, O 2, H 2 O and CO 2 ). Some of the radical species (H 2, OH, CH 3, CO, C 2 H 2, C 2 H 4, C 2 H 6 ) are also reasonably well modelled by the ANN, due to a good prediction of the reaction rate, this being the reason for the good match of the temperature profile. However, there are several exceptions (most notably H, but also HO 2, CH 2 O and H 2 O 2 ) where the ANN prediction is significantly different from the direct integration result. It is important to note that the chemical species that exhibit the largest errors are also among those with very short chemical time scales. Hence, most likely, the ANN relative failure for those species may be related to an inappropriate time step used for the direct integration of the reaction rates for the training data set. Nevertheless, if a smaller time step would have been used, it would have been too small for the major species reaction rates, and might have led to worse overall results. It is important to note that the current implementation of the ANN algorithm assumes a constant reaction rate over the entire LES time step, so errors are introduced both by an inappropriately large, and by an inappropriately small time step. Overall, considering the very significant decrease in the computational time ANN provides over the direct integration of the reaction rate equations, and even over the ISAT approach, combined with the small amount of memory and disk space required for ANN implementation, the ANN performance can be considered promising. 173

197 6.3.3 Application of the ANN algorithm in complex geometries Finally, the ANN algorithm was applied to the numerical simulation of a full combustor, the device used for the simulations presented in the previous Chapters and reproducing the V OLV O experiment Sjunesson et al. [1991b]. Figure 6.9 presents instantaneous contours of species mass fractions for several major and minor species. Since no 0.65 equivalence ratio propane - air ANNs where available, the simulation used the existing stoichiometric methane - air artificial networks, so the results cannot be expected to be accurate in any way. The reason for their inclusion herein is illustrative only, in order to demonstrate the ability of the presented method to handle complex geometries using large chemical mechanisms. 174

198 Reaction rate [1/s] DVODE DSTE 1 DSTE 10 ISAT ANN H OH HO2 CH3 CO CH2O H2 O2 H2O H2O2 CH4 CO2 C2H2 0 C2H C2H Temperature [K] Figure 6.5: Reaction rates in a one-dimensional laminar flame. DV ODE = Direct integration of the reaction rate equation using an ordinary differential solver; DST E1 = Direct estimation of the Arrhenius reaction rate at the LES time scale; DST E10 = Direct estimation of the Arrhenius reaction rate at a time scale 10 times smaller than the LES time step; ISAT = In Situ Adaptive Tabulation approach for determining the reaction rate; AN N = Reaction rates computed using the present ANN method 175

199 Reaction rate [1/s] Direct integration ANN Temperature [K] Reaction rate [1/s] Direct integration ANN Temperature [K] Direct integration ANN (a) CH 4 (b) CO Direct integration ANN Reaction rate [1/s] Reaction rate [1/s] Temperature [K] (c) CO Temperature [K] (d) OH Figure 6.6: Reaction rates in a one-dimensional laminar flame for selected species using direct integration and ANN 176

200 (a) CH 4 (b) CO 2 (c) CO (d) OH Figure 6.7: Instantaneous mass fraction contours for selected species in a threedimensional, turbulent flame simulated using ANN 177

201 DVODE ANN 5e T 0.3 H2 Mass fraction / Temperature [K] e H OH HO2 CH e O2 H2O H2O2 CH CO CH2O CO2 C2H C2H4 C2H6 Distance [m] Figure 6.8: Time averaged species mass fractions and temperature in a three-dimensional, turbulent flame simulated using ANN. For comparison, direct integration results are included 178

202 (a) CH 4 (b) CO 2 (c) CO (d) OH Figure 6.9: Instantaneous mass fraction contours for selected species in a combustor simulation using ANN 179