Large Eddy Simulation of Turbulent Premixed and Partially Premixed Combustion. Eric Baudoin

Transcription

1 Large Eddy Simulation of Turbulent Premixed and Partially Premixed Combustion Eric Baudoin December, 2010

2 ii Thesis for the degree of Doctor of Philosophy in Engineering ISBN ISSN ISRN LUTMDN/TMHP-10/1074-SE Eric Baudoin, December 2010 Division of Fluid Mechanics Department of Energy Sciences Lund Institute of Technology Box 118 S LUND Sweden Printed by Media-Tryck, Lund, December 2010

3 iii Abstract In this thesis, a computational fluid dynamics (CFD) approach is used to study turbulent premixed and partially premixed combustion. The CFD approach is based on large eddy simulation (LES) in which the large-scale structures of the flow are resolved on a grid, leaving only the small-scale structures (subgrid scales) to be modeled. The combustion modelling is based on the flamelet concept in which the scale separation of the flow and the chemistry is assumed. This thesis work is made up of the following parts. First, focusing on the turbulent premixed combustion, the level-set G-equation flamelet model is applied to study high density ratio flames without explicit filter in order to capture the effect of thin reaction zone embedded in the LES subgrid. Conventional fractional step methods are shown to be numerically instable for high density ratio flames. A highly robust numerical method, which is known as the ghost fluid method (GFM), is implemented. The G-equation based flamelet model and the ghost fluid method are evaluated on a lean propane/air premixed flame stabilized by a bluff body. The methods are shown to be able to capture the density ratio effect on the flame dynamics, including the near flame holder wrinkling due to Kelvin Helmholtz (KH) instabilities, the downstream large scale wrinkling due to the lower frequency Bénard/von-Karman (BVK) instability at low density ratio conditions, and the suppression of BVK instability at high density ratio conditions. In LES, spatial filtering of the reaction zone leads to thickening of the reaction zone. It is shown that thickening of the reaction zone can lead to significant under-prediction of the turbulence intensity at high density ratio conditions. The effects of flame thickening are further studied for the cases of the flame/vortex interaction and hydrodynamic instability. Essentially, with thickening of the reaction zones, the development of flame wrinkling and hydrodynamic instability are suppressed. Second, focusing on the partially premixed combustion, a two-scalar flamelet approach for LES is developed. In this approach, the trailing edge of the flame is assumed to be controlled by diffusion of mass and heat and thereby it is modelled using a steady diffusion flamelet model. The stabilization of the flame is due to the propagation of the leading premixed flame front (triple flame) in turbulent flows. The leading premixed flame is modelled using the level-set G-equation. This model is applied to simulate partially premixed flames of various fuels in a conical burner to understand the structure and the dynamics of the turbulent partially premixed flames operating in the flamelet regime and in certain cases with thicker reaction zones and local flame extinction. It is found that in general partially premixed flames are more stable when the level of partial premixing of air to the fuel stream decreases. However, at high Reynolds number conditions, an optimal level of partial premixing is found where the flame is most stable. There are two possible flammable surfaces in the partially premixed flames in the conical burner, where the mixture is in stoichiometric condition. At low Reynolds number flows, the inner flame is observed experimentally whereas it is not possible to stabilize at high Reynolds number flows. Numerical results based on the twoscalar flamelet model correctly predicted the blowoff of the inner flame at high Reynolds number conditions. It is well known that LES results are sensitive to the inflow conditions. The sensitivity of LES results to inflow turbulence and the mean flow profiles is systematically investigated based on the conical burner. It is shown that in the proximity of the burner the onset of the flow instability is not only dependent on the shape of the mean profiles, but also on the anisotropy of the inflow turbulence and the integral length scale of the inflow turbulence. This calls for special care in validation of LES models and more detailed experimental data for the inflow conditions when preparing for the database for model development and validation.

4 iv List of Papers This thesis is based on the following papers, which will be referenced by Roman numerals in the text. The papers are referenced and appended to the thesis in the order listed here. I. E. Baudoin, R. Yu, K.J. Nogenmyr, X.S. Bai, C. Fureby. Comparison of LES models applied to bluff body stabilized flame. Submitted to AIAA journal. II. E. Baudoin, R. Yu, X.S. Bai. Numerical simulation of premixed flame/turbulence interaction at high density ratio conditions. To be submitted for journal publication. III. B. Li, E. Baudoin, R. Yu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Aldén, M.S. Mansour. Experimental and numerical study of a conical turbulent partially premixed flame. Proc. Combust. Inst. 32: , IV. E. Baudoin, X.S. Bai, B. Yan, C. Liu, A. Lantz, S.M. Hosseini, B. Li, A. Elbz, M. Sami, Z.S. Li, R. Collin, G. Chen, L. Fuchs, M. Aldén, M.S. Mansour. Effect of partial premixing on stabilization and local extinction of turbulent methane/air flames. Submitted to Combustion and Flame, V. B. Yan, B. Li, E. Baudoin, C. Liu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Aldén, G. Chen, M.S. Mansour. Structures and stabilization of low calorific value gas turbulent partially premixed flames in a conical burner. Experimental Thermal and Fluid Science 34: , VI. E. Baudoin, C. Duwig, X.S. Bai. Large eddy simulation of turbulent jet flows: a sensitivity study of inflow boundary conditions. Submitted to Int. Journal of Heat and Fluid Flow, Other related work by the author E. Baudoin, R. Yu, X.S. Bai, J. Hrdlicka, S.I. Möller, M. Alden. Large eddy simulation and measurements of methane/air combustion in a pulsating combustor. Proceedings of ECCOMAS Computational Combustion Conference, Delft, The Netherlands, E. Baudoin, R. Yu, K.J. Nogenmyr, X.S. Bai, C. Fureby. Comparison of LES models applied to bluff body stabilized flame. Proceedings of 47th AIAA Aerospace Sciences Meeting, AIAA , Orlando, Florida, 2009.

5 v Contents 1 INTRODUCTION Context and motivation Purpose and objectives Scope and outline of the thesis BASIC FEATURES OF REACTING FLOWS Chemical kinetics Chemical reaction mechanism The rate of chemical reaction Chemical time scales Conservation equations for reacting flows Transport equations for momentum Conservation of mass and transport equations for species Transport equation for the energy Laminar premixed flames Laminar premixed flame structure Laminar premixed flame properties Introducing the level-set G-equation to study flame propagation Steady premixed flamelet library generation Premixed flame stabilization Laminar non-premixed flames Laminar non- premixed flame structure Laminar non-premixed flame properties Flamelet description of the laminar non-premixed flame Non-premixed flame stabilization Laminar partially premixed flames Partially premixed flame definition Partially premixed flame structure and stabilization BASIC FEATURES OF TURBULENT REACTING FLOWS Turbulence Phenomenological description Statistical description of turbulence Scales in turbulence Large eddy simulation modelling approach Filtered transport equations for reacting flows in the LES framework Transport equations for the momentum Conservation of mass and transport equations for the species Modelling of the subgrid scale stress tensor Modelling of the unresolved scalar transport: species fluxes Problem of reaction rate closures Flame and turbulence interaction: combustion regime classifications Regimes in turbulent premixed combustion Regimes in turbulent non-premixed combustion Partially premixed combustion: a combination of regimes MODELLING OF TURBULENT PREMIXED COMBUSTION USING LES APPROACH Overview on the combustion models Description of turbulence mixing Flame front topology Probability density function Finite rate chemistry Level set G-equation model G-equation in the LES framework Propagation of the filtered flame front Subgrid scale turbulence velocity Limitations of the method...33

6 vi 5 MODELLING OF TURBULENT PARTIALLY PREMIXED COMBUSTION USING LES APPROACH The reaction progress variable approach based models Reaction progress variable for partially premixed flames Simplification of the progress variable transport equation Modelling partially premixed flames within flammability limit A two-scalar variable flamelet model NUMERICAL METHODS Grid system Discretization of the scalar and momentum equations Fifth-order Weighted Essentially Non Oscillatory (WENO) scheme Discretization of the G-equation Spatial derivatives Time integration Re-initialization step Boundary conditions Wall boundary condition Turbulent inlet condition Outflow condition Solution algorithm Handling high density gradient: ghost fluid method (GFM) The GFMIT approach The GFMFT approach Parallel computation RESULTS AND SUMMARY OF PUBLICATIONS Modelling of turbulent premixed flame at high density ratios Problem set-up Summary of the corresponding papers Development and validation of LES model for turbulent partially premixed flames Development of the optical burner Modelling of the mixing process in the mixing chamber Summary of the corresponding papers Influence of inflow boundary in LES of low Re jets Contributions by the candidate to the papers in this thesis CONCLUDING REMARKS AND FUTURE PERSPECTIVES...57

7 vii Nomenclature Latin characters a speed of sound, [m s 1 ] q energy flux, [kg s 3 ] A flame surface area, [m 2 ] Q heat source term, [kg m 1 s 3 ] A p pre-exponential factor, [m 3 kmol 1 s 1 ] R u universal gas constant: 8.314, [J K 1 mol 1 ] c progress variable, [-] s mass stoichiometric ratio, [-] C p specific heat capacity, [m 2 s 2 K 1 ] S characteristic rate of strain, [s 1 ] C s Smagorinsky constant, [-] S local displacement speed, [m s 1 ] d D mass diffusion coefficient, [m 2 s 1 ] S ij rate of strain tensor, [s 1 ] D t turbulent diffusion coefficient, [m 2 s 1 ] S laminar flame speed, [m s 1 ] L 0 E expected value (first moment), [-] S unstretched laminar flame speed, [m s 1 ] L E a activation energy, [J kmol 1 ] ST, Δ subgrid scale turbulent flame speed, [m s 1 ] f autocorrelation function, [-] t time, [s] F cumulative distribution function, [-] t c chemical time scale, [s] g filter function, [-] t 0 turbulence integral time scale, [s] G premixed flame coordinate, [m] t η Kolmogorov time scale, [s] h specific enthalpy, [m 2 s 2 ] t Δ subgrid turbulence time scale, [s] H high pass filter, [-] t total stress tensor, [kg m -1 s -2 ] 0 h specific enthalpy of formation, [m 2 s 2 ] T temperature, [K] f k reaction rate coefficient, [m 3 kmol 1 s 1 ] u, v, w velocity components, [m s 1 ] ' k subgrid scale kinetic energy, [m 2 s 2 ' ] u velocity fluctuation, [m s 1 ] K flame stretch, [s 1 ] u characteristic velocity scale, [m s 1 ] * l subgrid length scale, [m] Δ u η Kolmogorov velocity scale, [m s 1 ] ' l characteristic length scale, [m] * u Δ turbulence subgrid scale velocity, [m s 1 ] l turbulence integral length scale, [m] V diffusion velocity, [m s 1 ] 0 L a Markstein length, [m] x, y, z space, [m] n surface normal vector, [-] Y mass fraction, [-] p dynamic pressure, [kg m 1 s 2 ] Z mixture fraction, [-] P probability, [-] probability density function, [-]

8 viii Greek characters α thermal diffusion coefficient, [m 2 s 1 ] ν kinematic viscosity, [m 2 s 1 ] Γ efficiency function, [-] ΞΔ subgrid scale wrinkling factor, [-] δ ij Kronecker symbol, [-] ρ density, [kg m -3 ] 0 δ unstretched laminar flame thickness, [m] Σ L Δ subgrid scale flame surface density, [m -1 ] δ R [m] inner layer thickness, τ ij viscous tensor, [kg m -1 s -2 ] Δ LES filter width, [m] R τ ij subgrid scale stress tensor, [m 2 s 2 ] ε TKE dissipation rate, [m 2 s -3 ] Φ equivalence ratio, [-] η Kolmogorov length scale, [m] χ scalar dissipation rate, [s 1 ] θ reduced temperature, [-] ω reaction rate, [kg m 3 s 1 ] κ flame curvature, [m -1 ] λ thermal conductivity, [kg m s 3 K 1 ] λ t Taylor scale, [m] μ dynamic viscosity, [kg/m s] μ n nth central moment, [-] Non-dimensional numbers t Da = t Ka = 0 c t c t η ( ) 2 Damkhöler number Karlovitz number δ Ka δ = R 2 Kalovitz number (inner layer) η D Le = α Lewis number Re u ν * * = l Reynolds number ' u 0 Re t = l ν turbulent Reynolds number Re Δ ' uδδ = ν subgrid scale Reynolds number ν Sc = D Schmidt number Abbreviations BVK Bénard Von-Karman PaSR partially stirred reactor CDF cumulative distribution function PDF probability density function CFD computational fluid dynamics PLIF planar laser induced fluorescence DNS direct numerical simulation RANS Reynolds averaged Navier-Stokes EBU eddy break up RMS root mean square EDC eddy dissipation concept SGS subgrid scale FID flame index SSM scale similarity model FT finite thickness TKE turbulent kinetic energy GFM ghost fluid method TFM thickened flame model IT infinitely thin TVD total variation diminishing KH Kelvin Helmholtz UHC unburned hydrocarbon LCV low calorific value VR validation rig LES large eddy simulation WENO weighted essentially non oscillatory LHV Lower heating value

9 1 1Chapter 1 Introduction 1.1 Context and motivation Combustion and its control are essential to our existence on this planet today as we know it. Our strong dependency on fossil fuels (coal, oil and natural gas) is obvious as we are surrounded in our every day life by its benefits: heating, transportation and indirectly electricity production. From the industrial revolution, it has been a driving force for development of humanity. Nevertheless it does not come without costs. First, fossil fuels which were yesterday considered as cheap and abundant are today associated to a reality of no sustainability. Fossil fuels are being consumed at a greater rate of what they naturally form. Although a depletion of this source of energy is to be considered in the long term (complex and difficult to evaluate), modern and improving extraction techniques try to keep up with the growing global demand. Nevertheless fossil fuel price is increasing and will remain high. Therefore a better usage and efficiency of the combustion devices are required. Second, the downside issue associated with combustion is the environmental pollution. The major pollutants produced by combustion are unburned and partially burned hydrocarbons, nitrogen oxides (NO and NO 2 ), carbon monoxide, sulfur oxides (SO 2 and SO 3 ), and particulate matter in various forms. Depending on the considered devices, in most of the cases, they are subjected to increased legislated controls. Primary pollution concerns relate to specific health hazards, smog, acid rains and ozone depletion [1]. Those problems can be avoided by carefully designing the combustion devices or use of an external system. Third, on the industrialization time scale, carbon dioxide (CO 2 ) has been proven to be the most important greenhouse gas that threatens to substantially change the climate on this planet. Again refining current technology is a necessity. Finally as combustion devices evolve towards more complexity in both design and operating conditions, stability issues can be encountered due to the complex dynamics of the flow and combustion interactions. These unwanted instabilities lead to both noise pollution and in some extreme cases mechanical failures. In order to overcome those issues, a constantly improved understanding of combustion is needed. This is not a simple task. Indeed, combustion could be technically defined as a sequence of exothermic chemical reactions between a fuel and an oxidizer accompanied by the production of heat and conversion of chemical species. This definition, although rather simple, underlines that combustion involves highly complex phenomena. They are produced by the interplay of several elementary phenomena, each associated with a different aspect of the physical world: thermodynamics, heat and mass transfer, radiation, fluid mechanics and chemistry all play a part [2]. Besides the experimental approach, which can be limited by the access in the considered devices, a numerical approach of combustion problems is possible, based on Computational Fluid Dynamics (CFD). The most fundamental consideration of the CFD is how to treat continuous fluid in a discretized manner on a computer. The basis of almost all CFD problems is the Navier-Stokes equations. When combustion is considered additional equations are required to describe the process. Since the 1940s, analytical solutions were

10 2 available for most of dynamics problems in simplified or idealized situations. The usage and capability of the CFD approach rose with the improvement of the computational resources. Recent combustion devices, e.g. IC engines, gas turbines, etc., involves complex flows (turbulence), complex mixture (lean or partially premixed), complex geometries. As described above they are subjected to new environmental legislations. CFD approach can play a great role to match the increasing demand from the industry for insight in their evolving combustion devices and new concepts. This thesis deals with CFD modelling of premixed and partially premixed combustion processes. These processes are currently adopted in modern combustion engines as they offer possibilities of having low pollutant emissions and high fuel economy. However, there is a lack of fundamental understanding of the physical and chemical processes, and most importantly a lack of modelling tool for analysis and simulation of these processes. In order to develop and validate modelling tools for premixed and partially premixed combustion processes, lean premixed bluff-body stabilized flames and partially premixed flames stabilized in a conical burner are considered in this thesis as they present almost the same physics as those encountered in the complex devices. Such configurations often allow combining both numerical and experimental approaches and can therefore help to enable a better understanding of the combustion process. 1.2 Purpose and objectives The purpose of this thesis work is therefore to improve the understanding of premixed and partially premixed flames, and to develop and validate models for premixed and partially premixed flames. The numerical simulations are carried out within the large eddy simulation framework (in low Mach number in-house CFD code [3]), which is a high fidelity approach demanding high temporal and spatial resolution. The approach demands sophisticated modelling of the sub-filter scale effects, such as transport of mass, momentum and turbulence chemical reaction interaction. Within the model development and validation in the LES framework, the objectives are: to improve existing level-set G-equation flamelet model to support high density ratio without explicit filtering of the density, to develop a two scalar flamelet approach for partially premixed combustion in a conical burner. With experimental support, the LES models are used to investigate the effect of density ratio on the dynamics of a bluff body stabilized flame, investigate partially premixed combustion in a conical burner. 1.3 Scope and outline of the thesis This thesis work focuses on gaseous phase combustion, where light hydrocarbon gases are considered (CH 4, C 3 H 8 and low calorific gas). Despite the small size of those molecules, the chemical reactions involves in the combustion processes are complex as presented in Chapter 2. In order to analyse the flame structure and to develop models (by means of libraries), after introducing the governing equations for reacting flows, laminar premixed and nonpremixed flames are considered in Chapter 2. It is relevant to study these two idealized modes as in partially premixed flames, one of the main focuses of this thesis, both features can be encountered. In Chapter 3, turbulence is introduced and its effects on combustion are discussed with focus on the LES framework. In this chapter, combustion regimes are identified especially the

11 flamelet regimes in which stable and efficient burning are obtained in many combustion devices [4]. Furthermore the flamelet based models used in this work are applicable in those regimes i.e. when the chemical time scale is smaller than the smallest time scale in the flow e.g. at high Damköhler number, Da. Due to the scale separation, the turbulent flames can be viewed as an ensemble of laminar flames. Those approaches offer a good compromise between computational cost, ease of implementation and incorporation of rather complex chemistry [5]. Chapter 4 and 5 describe the combustion models used in this work for respectively the turbulent premixed (level-set G-equation) and partially premixed (two scalar flamelet) combustion. The numerical methods are explained in Chapter 6. The improvement on the existing level-set G-equation in order to handle high density ratio are also emphasised. To validate the level-set G-equation flamelet model for premixed combustion, the validation rig (VR-1) case from Volvo is considered, which consists of a V-shaped flame stabilized by a bluff body in a rectangular duct. This rig has been used in many model validation tests previously [6, 7]. To develop and validate combustion models for partially premixed combustion, methane/air flames in conical burner have been studied experimentally and numerically. Selected results are presented in Chapter 7 while the main results are discussed in the appended papers. Finally Chapter 8 contains a summary of the contributions as well as suggestions for future work. 3

12 4

13 5 2Chapter 2 Basic features of reacting flows This thesis work focuses on flames. A common definition states that a flame is produced when combustion occurs in a limited zone, smaller than the total volume of the system, and from which the flame then propagates [2]. Such flames can be divided into two idealized regimes: premixed flames and non-premixed flames. However in many cases the flames are a mixed of both premixed and non-premixed flames, which may be classified as partially premixed and stratified premixed flames. Fig. 1 shows simple examples of freely propagating premixed flames and counter-flowing non-premixed flames. streamline flame front oxidizer stream a. fuel and oxidizer stream S L unburned region burned region x<0 x>0 x=0 (t=0) x products stream b. flame front staganation plane x streamline fuel stream Figure 1. a. Sketch of a planar flame propagating in a straight pipe b. Sketch of counter-flow diffusion flame. In premixed flames, the fuel and the oxidizer are mixed prior to the reaction. In a tube (Fig. 1a), the flame front (yellow region) propagates from the burned region to the unburned region as a deflagration wave. The hot products are found behind the front. In diffusion flames, the fuel and the oxidizer are mixed by diffusing in opposite directions towards the flame front (yellow region). In a counter-flow configuration (Fig. 1b), fuel is supplied on one side (lower stream) of the flame front while air is supplied in the opposite direction (upper stream) on the other side of the flame front. Products are present on both sides of the front. In Sec. 2.1, it will be shown that the combustion process is controlled by small scale phenomena (mixing at molecular level) that are identical in both regimes. The problem of complex reactions, involving multiple species in multiple reactions, will be highlighted. Additionally, the related multi-component transport processes will be described in Sec Hence, it will be shown in Sec. 2.3 and 2.4 that distinct assumptions (or properties) can arise by considering either premixed flames or non-premixed flames in laminar flows. As mentioned previously, in most of practical applications such ideal flames do not exist but flames are rather in a mixed regime called partially premixed combustion which is discussed in Sec Chemical kinetics Combustion is a globally exothermic process and therefore releases energy when it occurs. If one is only interested in the thermodynamic aspects of the combustion (adiabatic temperature

14 6 reached, etc.), there is no real need to understand the actual chemical mechanisms involved. However without the knowledge of the chemical kinetics, it is not possible to determine [2]: whether or not combustion can actually occur under the given conditions whether it can reach complete chemical equilibrium in the time available Chemical reaction mechanism The balance reaction for the global reaction (combustion) of methane with air can be written as followed, CH 4 + 2( O N 2 ) CO2 + 2H 2O N (2.1) 2 In its current formulation only major species of the initial system prior to the reaction (left hand side reactants) and of the final system (right hand side products) are considered. In reality, the combustion process can not occur in a single step but involves a complex kinetic mechanism with numerous of elementary reactions. As an example a reduced mechanism for the above global reaction involving 12 reactions is given below: CH4 + M CH3 + H + M (2.2) O2 + H OH + O (2.3) CH4 + H CH3 + H (2.4) 2 CH4 + O CH3 + OH (2.5) CH4 + OH CH3 + H2O (2.6) CH3 + OH CH2 + H2O (2.7) CH3 + O CH2O+ H (2.8) CH2 + O CHO+ H (2.9) CH2 + OH CH + H2O (2.10) CHO + H CO + H 2 (2.11) CHO + OH CO + H 2O (2.12) CO+ OH CO2 + H (2.13) Reactions ((2.2) - (2.6)) describe how the fuel and oxygen react initially. CH 4 dissociates first to form radical H; radical H reacts with O 2 to form more radicals, O, OH in reaction (2.3). Radicals H, O, OH are very active and they participate to most of the later reactions until the formation of final products CO 2 and H 2 O. A reaction that creates an excess of radicals is called chain branching reaction [8] e.g. in reaction (2.3). On the other hand if radicals are consumed the reaction is called chain terminating reaction. As long as the chain branching reactions overcome the chain terminating reactions, the combustion is sustained. The temperature at which this event occurs first is referred as the cross-over temperature. Note that the above reduced mechanism is incomplete. A typical detailed mechanism for methane is the GRI-MECH 3.0 [9] involving 325 elementary reactions The rate of chemical reaction Each elementary reaction corresponds to a reaction rate i.e. the rate at which the reactants will collide with sufficient kinetic energy (activation energy) to change their structure and form new products. Considering the bimolecular reaction in which the reactants A and B form the products C and D: A + B C+ D (2.14)

15 7 Experimentally it has been observed that the rate of this reaction is proportional to the product of the concentrations of the reactants each raised to a power equal to their stoichiometric coefficient (here 1) in the reaction [2] such as: da [ ] (2.15) = k( T)[ A][ B] dt where [] refer to species concentration and k(t) is the rate coefficient that depends on temperature. Typically the rate constant is given by an empirical Arrhenius expression: n E (2.16) a kt ( ) = AT p exp RuT where A p, E a and R u are respectively the pre-exponential factor, the activation energy and the molar gas constant. A, E and n are the three constants determined experimentally. p a Chemical time scales For the modelling of combustion, insight can be gained from the knowledge of the chemical time scales τ c especially in comparison to other time scales i.e. time scales of flow (Sec. 3.3). Recalling reaction of (2.14) and its rate k(t) in Eq. (2.16), the chemical time scales can be defined as the time required for the concentration of A to fall from its initial value to a value equal to 1/e times the initial value [1], i.e.: ln ( e + ( 1 e) )([ A] [ B] ) (2.17) tc = B A k( T) ([ ] [ ] ) When global or reduced reaction mechanisms are considered similar time scales can be evaluated. 2.2 Conservation equations for reacting flows In a combustion system of multiple constituents, some species will be consumed and some will be generated in chemical reactions, as discussed previously. The combustion process (involving also transport) is governed by three basic conservation laws the conservation of mass, momentum and energy. Transport equations for the species mass fraction, momentum and energy can be derived from these basic conservation laws [10], as discussed below Transport equations for momentum The momentum equation is the same in reacting and non-reacting flows: p τ ij ρuj + ρuu i j = + t x x x i j i (2.18) where u j is the velocity component in the j-direction, p is the pressure, τ ij is the viscous tensor (the body forces are neglected in this work). This tensor is defined as, 2 u k u u (2.19) i j τ = + + ij μ δij μ 3 x k x j xi where kinematic viscosity is ν = μ ρ.

16 Conservation of mass and transport equations for species The continuity equation (mass conservation) is written as: ρ ρu (2.20) i + = 0 t xi The transport equations for species are written as: ρyk (2.21) + ( ρ( ui + Vk i ) Yk ) =, ωk for k=1,,n t xi where Yk is the mass fraction of species k, V k, i is the i-component of the diffusion velocity Vk of species k and ω k is the reaction rate of species k given by Eq. (2.15). By definition: N N (2.22) Y k V k, i = 0 and ω k = 0 k = 1 k = 1 Using Fick s law, the diffusion velocity V, can be modelled as: k i Y (2.23) k Vk, iyk = Dk xi where D k is the diffusion coefficient of species k into the mixture. Finally Eq.(2.21) becomes, ρyk ρuiyk Y (2.24) k + = ρd k + ωk t xi x i x for k=1,,n i Transport equation for the energy Since a single gas phase is considered in this work and with the assumptions that the body forces and the Dufour effect can be neglected, the transport equation for the energy is: ρh ρuh i Dp qi ui + = + τ ij + Q (2.25) t x Dt x x i i j where h is the enthalpy of the mixture, Dp Dt is the pressure material derivative, q i is the energy flux and Q is the heat source term due to radiative flux. The enthalpy of the mixture h is, N T (2.26) 0 h = Yh with h h C dt k k k = 1 = + k f, k p, k To 0 where h f, k is the enthalpy of formation of species k at a reference temperature T 0 and C pk, is the heat capacity of species k. Similarly the heat capacity C p of the mixture is given by, N (2.27) C p = Cpk, Yk k = 1 The energy flux q i is, N h 1 Y (2.28) k qi = ρα + ρα 1 hk x j k = 1 Lek x j where α is the thermal diffusion coefficient ( α λ ρc p ), λ is the thermal conductivity. The Lewis number of species k, Le k, is defined as: D ρc k pd (2.29) k Lek = α λ

17 9 It represents the ratio of mass diffusion to thermal diffusion. Similarly the ratio of the viscous forces to the thermal diffusion leads to the Schmidt number Sc k : ν (2.30) Sck Dk Equality of those numbers is a useful (and essential) assumption when studying laminar premixed and non-premixed flames as shown below. 2.3 Laminar premixed flames Laminar premixed flame structure As shown on Fig. 1a, premixed flames are characterized by a flame front (yellow region) separating unburned and burned gases. The thickness of this front is referred as the unstretched laminar flame thickness, δ. The red line in Fig. 1a, represents the thin reaction 0 L zone (relative to the flame front) in which most of the chemical reactions occur. In fact the flame front can be divided into three sub-regions: the preheat zone, the inner layer (reaction zone) and the oxidation layer as described in Fig. 2. In the preheat zone, the temperature is below the cross-over temperature (1000K), therefore radical recombination reactions are dominant over radical branching reactions. Hence none or few chemical reactions occur in this region. The deviation of temperature and species such as combustion product CO 2 are due to diffusion of heat and mass from the reaction zone to the preheat zone. It is in the reaction zone that the fuel (here C 3 H 8 ) and the oxidizer (O 2 ) are rapidly consumed and the temperature increases rapidly (high heat release). It is important to point out that most of the chemical reactions occur in this thin inner layer, whose characteristic thickness δ R is typically estimated asδ 0 R = δl 10. Thus, a peak of concentration of intermediate species such as CO and other radicals (CH 2 O, etc.) is observed as shown on Fig. 3. CO and other intermediate species are oxidized soon after in the oxidation layer. In this region, the product species are approaching steady state. Figure 2. Composition and temperature for a C 3 H 8 -air laminar flame at 1 atm, φ = 0. 6 and with unburned mixture preheated at 600K.

18 10 Finally, in the post flame zone, the temperature and the combustion products such as CO 2 exhibits a rather flat profile (Fig. 2) indicating an almost zero net heat release and most of the chemical reactions have reached equilibrium. However, due to high temperature, a few reactions occur such as NO formation and OH radical oxidation as shown on Fig. 3. A control over the post flame zone temperature (or residence time of the gas) is necessary in order to prevent formation of harmful pollutant such as NO x or the incomplete combustion of CO or unburned hydrocarbons (UHC). Figure 3. Composition of radicals, NO and density for a C 3 H 8 -air laminar flame at 1 atm, φ = 0. 6 and with unburned mixture preheated at 600K Laminar premixed flame properties Solving Eqs. (2.18), (2.20), (2.24) and (2.25) for a reacting fluid is computationally expensive as it depends strongly on the size of chemical kinetic mechanisms (Sec ) used to describe the chemical reactions but it can give valuable insight. On the other hand, in the context of a one dimensional freely propagating premixed flame (without flowing motion in the far upstream), it is possible to simplify those equations [8]. To further simplify the problem it can be assumed that the unity Le number is valid for all species and that the chemistry proceeds only through one irreversible reaction. The unstretched laminar flame speed or burning velocity In this way, the integration of the continuity equation leads to: 0 ρu = ρusl = cst (2.31) 0 Where S L is the unstretched laminar flame speed. The integration of the simplified species transport equations leads to: + 0 ρusl( Yib, Yiu, ) = (2.32) ωidx where Y iu, is the species i mass fraction in the unburned gas, Y ib, is the species i mass fraction in the burned gas and ω i is the corresponding reaction rate. Further analysis can provide interesting scaling relation between heat/mass transfer and chemical reactions on the laminar flame speed: 0 S αω ρ (2.33) L p 0

19 11 where α is the thermal diffusion coefficient in the unburned gas, ω p is the average reaction rate of the products in the reaction zone and ρ 0 is the density at the interface between the preheat zone and the reaction zone (from the classical analysis of Zeldovich, Frank Kamenetski and von Karman [10]). Within the flammability limit, the flame speed is sensitive to the temperature and pressure but it is also sensitive to the equivalence ratio of the given mixture defined as: Y F Y F Y (2.34) F φ = s = YO YO YO st where s is the mass stoichiometric ratio. The unstretched laminar flame thickness An approximation for the thickness can also be derived [10]: 0 λ α (2.35) δ L = = 0 0 ρuc psl SL Similarly to Eq. (2.33), a scaling relation can be expressed: 0 δ αρ ω (2.36) L Useful scales in laminar flame From the laminar flame thickness and the laminar flame speed, it is possible to define a characteristic chemical time scale: 0 δ (2.37) L tc = 0 S Introducing the level-set G-equation to study flame propagation Studying the propagation and the dynamics of laminar premixed can be greatly simplified by assuming that the local flame structure is similar to the one dimensional flame. In this way, the flow (Eq. (2.18) and (2.20)) and the chemistry (Eq. 2.24) can be decoupled and solved separately. It is important to point out that no energy equation is mentioned since only adiabatic flames are considered in this work (Sec. 1.3). The main difficulty is then to track/locate the propagating front (and incorporate the one dimensional flame structure). A common way to track propagating surfaces is to introduce a level-set function, G. A G isosurface (isocontour in two dimensions) is selected to represent the front. For simplicity of notations the time evolution of the scalar, given by the well known G-equation [5, 8], is expressed in a vector form as: G (2.38) + u u G = SL G t This equation describes a surface propagating in space with a velocity S L (normal to itself) relative to the local unburned flow velocity, u u. The normal vector to the front is given by, n = G G (2.39) The laminar speed S L is not always equivalent to the unstretched laminar flame speed (as defined in Sec ) as the effect of the local curvature and strain should be considered. The combined effects are often referred as flame stretch, K, defined by the fractional rate of change of a flame surface element A [10]: 1 da (2.40) K = A dt 0 L p

20 12 A general expression of stretch, for a thin flame sheet can be found in [11]. If only the local curvature effects are considered, in the level-set approach, Eq. (2.38) becomes [5], G (2.41) + u 0 u G = SL( 1 Laκ ) G t where L a is the Markstein length [8] and κ is the local curvature which has the following expression: G (2.42) κ = n = G It is straightforward to notice that if the curvature tends to 0, the unstretched laminar flame speed is recovered. Once the front evolves (Eq. (2.41)), the coupling with the one dimensional flame structure requires spatial information (mapping). It is therefore convenient to let the scalar G be defined as a signed distance function. At each point, within a close region of the flame front, referred in this work as narrow band (Sec ), the scalar G corresponds to the closest distance to the front. Additionally by convention the unburned region is defined by negative values (G<0) and the burned region by positive values (G>0). Since Eq. (2.41) does not preserve the distance property, an additional equation is solved, G = 1 (2.43) This step is referred as the re-initialization step. More details on the numerical methods implied when solving the G-equation are given in Sec Fulfilling this requirement, the flame structure can be readily mapped onto the domain through the scalar G values. The density, temperature and viscosity necessary in Eqs. (2.18) and (2.19) are given by a library (Sec ) Steady premixed flamelet library generation Following [7], the steady laminar flamelet library is obtained with the method developed in [12] in which detailed chemicals mechanisms, detailed transport and thermo-chemical data are applied. A counter flow configuration is considered similar to Fig. 1b except that fuel mixture is injected on both sides. In this way, two stretched flames are established. Along the symmetric line, laminar flame speed, temperature, density and species concentration are obtained in a physical coordinate. In this work, the zero level-set (G=0) is defined in the inner layer (Sec ) at one of the radicals maximal concentration e.g CH 2 O. The flame structure, especially details such as the local structure of the CH, CH 2 O and CO layers, can therefore be expressed as a function of G (and K) as shown on Figs. 2 and Premixed flame stabilization Another important aspect of the flame dynamics is the stabilization process especially regarding industrial application. The rim stabilization is the main stabilization mechanism for premixed Bunsen flame. The radicals created in the near region of the burner rim (walls) are quickly recombined (quenching distance) reducing the reaction process. The flame is stabilized in a region where the local flow velocity balances the laminar flame speed. If this equilibrium is strongly disturbed then blow off occurs. The mechanisms for flame stabilization after a bluff-body flame holder, a dump combustor, and a swirl combustor are similar. A recirculation flow zone is generated in these combustors. The flow speed is low in this recirculation zone, and thereby the condition that flame speed and local flow speed balance is met in this zone. If the zone is large enough, hot combustion products will stay in the zone and they continuously ignite the cold fuel/air mixture coming from upstream.

21 Laminar non-premixed flames Laminar non- premixed flame structure As shown on Fig. 1b, non-premixed flames are characterized by a flame front (yellow region) separating the fuel and oxidizer and are often called diffusion flame since diffusion of mass and heat is the rate controlling process. In this counter-flow configuration, the flame develops where fuel and oxidizer are around stoichiometric condition. In this narrow region fuel and oxidizer are consumed, intermediate radicals (CH, CH 2 O, etc.) are formed and quickly terns into products. The products are then diffused away from the front. The structure of diffusion flames is generally described in mixture fraction space (Sec and 2.4.3) Laminar non-premixed flame properties A key variable, the mixture fraction Z, is used to describe diffusion flame (both laminar and turbulent). There are several ways to define the mixture fraction. Following the definition in [8], it can be expressed as: 1 Y F Y (2.44) o Z = φ φ + 1 YF YO where φ is the equivalence ratio defined in Sec.2.3.2, Y is the oxidizer mass fraction in the oxidizer stream (Z=0) and Y 0 F is the fuel mass fraction in the fuel stream (Z=1). The stoichiometric mixture fraction Z st is by definition the value of Z for φ = 1. A transport equation for the mixture fraction can be derived, assuming unity Lewis numbers ( D k = D ), giving: ρz ρuz i Z (2.45) + = ρd t xi xi xi This equation gives valuable information about the mixing process and does not have a reaction rate (mixture fraction is thus often referred as passive scalar) Flamelet description of the laminar non-premixed flame Assuming that the structure of the diffusion flame depends on the mixture fraction Z and time t only, the temperature and species can be written as: T = T ( Z, t) and Yk = Yk ( Z, t) (2.46) A flamelet equation can be derived [8] by changing the reference coordinate of Eq. (2.45) from ( x 1, x2, x3, t) to ( Z, y2, y3, t) where y 2 and y 3 are spatial variables in planes parallel to iso-z surfaces. By further neglecting terms corresponding to gradients along the flame front in comparison to the terms normal to the flame (along the direction of gradient of Z) and assuming that iso-z surfaces remain parallel, this leads to the flamelet equation for diffusion flame: 2 2 Y k Z Z Yk 1 Y (2.47) k ρ = ωk + ρd = ω 2 k + ρχz 2 t xi xi Z 2 Z where χ Z is the scalar dissipation rate, i.e. the rate of molecule mixing defined as: Z Z (2.48) χ z = 2D x i x i 0 O

22 14 In steady configuration Eq. (2.47) becomes: dyk dyk 2ωk ωk = ρχz = (2.49) dz dz ρχz The right hand side of Eq. (2.49) is the ratio between chemical reaction and mixing rate. Fig. 4 presents the effect of infinitely fast chemistry and finite rate chemistry (Eq. (2.49)) on the fuel and air mass fraction in the mixture fraction space (reaction zone at Z st ). Indeed, on the one hand, if ωk ρχz i.e. the chemical reaction is infinitely fast, 2 2 then at the reaction zone, d Yk dz is discontinuous; this leads to the Burke-Schumann [13] 2 2 flame sheet structure (Fig. 4). On the other hand if ωk ρχz is finite then d Yk dz is also finite at the reaction zone, i.e. fuel and air can coexist (Fig. 4). Figure 4. Flame structures with infinite-rate chemistry and finite-rate chemistry. An energy equation can be derived from Eq. (2.25) similarly as Eq. (2.49). Both equations can be solved numerically. For a given scalar dissipation rate, χ Z, one can calculate the flame structures in mixture fraction space. In this way, a steady laminar flamelet library can be built [14, 15]. The effect of the scalar dissipation on temperature is shown in Fig. 5. Figure 5. Schematic illustration of the effect of finite-rate chemistry on the flame structure. For a very small scalar dissipation rate the flame temperature is high (Burke- Schumann flame sheet model). Increasing the scalar dissipation rate leads to a decrease of the flame temperature until a critical condition, χz = χq, at which the flame temperature changes rapidly, dt d χz. This regime may be called flamelet; the flame structure can be rather

23 15 well represented by the flamelet equations, Eq. (2.49). Eventually if χz > χq, the flame is quenched. For flamelet combustion with high scalar dissipation rate, the effect of radiation is not important; the residence time is too short. Therefore, the quenching condition is not affected much by thermal radiation heat transfer Non-premixed flame stabilization As for premixed flames, stabilization mechanisms are of interest. Typically three mechanisms can be responsible for diffusion flame stabilization. Similarly as for premixed flame stabilization, the lift off of a diffusion jet flame could be due to radicals or heat loss to the burner rim (wall). Second as the flow condition may changed (higher inlet speed) the mixing process could be too fast ( χz > χq) and lead to local extinction close the burner rim. As the flow speed increases, this effect can be combined or overcome by the triple flame stabilization described in Sec as the flame becomes partially premixed. 2.5 Laminar partially premixed flames Partially premixed flame definition Ideally the combustion process falls either in a premixed combustion regime identified by YF YO > 0 where fuel ( Y F ) and oxidizer ( Y O ) are well premixed or in a diffusion control combustion regime identified by YF YO < 0 where the fuel and oxidizer are not mixed. Partially premixed combustion, where both premixed and diffusion flames can be observed, is found in many industrial combustion devices, e.g., IC engines and gas turbines, and many other engineering applications. It is also important to state here that the partially premixed combustion should be distinguished from stratified premixed combustion in which only variable fuel-rich or only variable fuel-lean mixture can exist and the diffusion flames will not occur [4]. In the following partially premixed flames will be referred to flames with local stoichiometric mixture in the flow field [16] Partially premixed flame structure and stabilization A typical example of partially premixed flame is the lifted diffusion jet flame as shown on Fig. 6; the stabilization of the flame is ensured by premixed flame propagation while the trailing edge of the flame is of diffusion type with all merging at the triple point (rich and lean premixed flames and diffusion trailing edge). flame front A + P A + P F + P Z=0 air air Z=0 fuel rich rich rich Z=1 air air fuel stream diffusion flame F + P A + P rich premixed flame Z st lean premixed flame lift-off height air fuel stream Figure 6. A schematic of a lifted diffusion flame stabilized by a triple flame.

24 16 Due to the flammability limits (from rich to lean flammability limits), the premixed flames are not very large. They consume only a small portion of the fuel. Most of the fuel will bypass the fuel rich side of the premixed flame and diffuse directly to the reaction zone of the diffusion flame (along the stoichiometric mixture isoline, Z st ) and reacts with the oxidizer there. Therefore the length of the diffusion flame is much larger than the premixed flame. In the analysis of the flame structures one often neglects the triple leading flame. However, when the lift-off height is analyzed, or when the mechanism of flame stabilization at the burner rim is concerned, it is necessary to consider the triple flame structure at the leading edge of the flame. Numerous of experimental investigations e.g. [17-19] and numerical studies e.g. [20-22] have been carried out on triple flames in order to improve understanding and develop modelling approaches for turbulent partially premixed combustion and related stabilization mechanisms (Chapter 5). General modelling is based on combined model tools developed for respectively premixed and non-premixed combustion. Therefore it requires at least two variables for describing each of those regimes e.g. reaction progress variable c or level-set G for premixed combustion and mixture fraction Z for the mixing (diffusion combustion) process.

25 17 3Chapter 3 Basic features of turbulent reacting flows Most flows occurring in industrial processes are turbulent. In some of them, it can be unwanted and purely due to the use of high flow rates, shear, etc. While in others, one wants to use the properties of turbulent flows to improve mixing and the combustion rate, etc. In this section, some basic concept of turbulence relevant to understand Large Eddy Simulation and turbulent combustion approaches will be presented. For more comprehensive information on turbulence, the reader is referred to [23] as one among numerous textbook on turbulence. 3.1 Turbulence Phenomenological description Flows can often be characterized by non-dimensional numbers and thereby scaling laws can be deduced. For viscous flows, the ratio of the convective forces and the viscous forces is of high relevance and expressed as the Reynolds number, u* * Re = l (3.1) ν where u * and l * are characteristic velocity and length scales of a particular flow (Sec ). As an example for canonical pipe flows, if the Re (based on the pipe diameter) is low, typically below 2300, the viscous forces are sufficient to dampen any disturbance in the flow and the flow is laminar. As Re increases, the disturbances tend to grow until the state of the flow changes (unsteady or steady). If the critical Re is reached, instabilities will grow and develop to large unsteady structures, and smaller eddies. The flow becomes turbulent [24, 25]. At this point, very little has been said about turbulence. It is possible to characterize turbulence by a set of features [26]. Namely, turbulence can be described as highly random seemingly chaotic (Irregular). It enhances the transport of momentum, heat and species (Diffusive). As mentioned previously, it arises when the stabilization effect of viscosity on the unsteady motion is too weak compared to the convective forces (Large Reynolds number). The turbulent eddies of various sizes are highly three dimensional. Turbulence requires a continuous feeding of kinetic energy to sustain; otherwise it decays due to energy dissipated in form of heat at the smallest scales (Dissipative). The smallest scales, η, are much larger than the mean free path of the molecules, λ mfp (Continuum). This corresponds to low Knudsen number flow defined as Kn = λmfp η Statistical description of turbulence In a laminar flow, it is possible to calculate or predict any velocity component e.g. u( x, t) at a given location and at a given time from the Navier-Stokes equations (Eqs. (2.18) and (2.20)) and it can also be measured experimentally. Those two values will agree and will be reproducible within small numerical and experimental errors [23]. In a turbulent flow, since u( x, t) is a random variable (unpredictable), instantaneous data are irreproducible and any comparison of instantaneous quantities may be pointless. However, one can define statistical properties to characterize u( x, t) (and other quantities) in

26 18 a given turbulent flow. They converge to unique values in some cases, which depend on the problem conditions. In such cases, the statistics may be reproducible (numerically or experimentally). Some of the statistical quantities and approaches are discussed in the following. Probabilities The behaviour of a random variable U, e.g. a velocity component in a turbulent flow, can be fully characterized by its probability density function (PDF),, [23]. It is defined as the derivative of the cumulative distribution function (CDF), F. The probability P that the velocity component U is less than a value V is given by the CDF: F( V) = P( U < V) (3.2) with, F( ) = 0, F( + ) = 1, F( V + ε ) F( V) (3.3) The PDF is then the CDF derivative with respect to V: df( V) (3.4) ( V ) = dv It determines the probability of finding U in some small value region V U V + dv : P( V U V + dv) = F( V + dv) F( V) = ( V) dv (3.5) Random variables with the same PDF are statistically identical. Averages When the PDF is determined, several other statistical properties can be computed. First the expected value of U is (also known as the mean value or first moment): + (3.6) Ε ( U ) = V ( V) dv ' The fluctuation in U is u = U Ε ( U).The nth central moment is defined as: + n (3.7) μ n = ( V Ε( U) ) ( V) dv with especially the root mean square of the second order moment known as rms. Statistically stationary flow In this thesis, all studied flows are statistically stationary. The PDF (and all statistics) are invariant under a shift in time [23]. Given that, according to the ergodic hypothesis, the ensemble average equals the time average. Hence for time resolved calculations and experiments, the time averages (mean and rms) are: 1 t+ T ' ' (3.8) U mean =< U >= U( t ) dt T t 1 t+ T (3.9) ' ( ) 2 ' Urms = μ2 = Ut ( ) < U> dt T t Here time-averaging is applied a posteriori on the gathered data. It is important to point out that if it is applied a priori on the dependent variables and on the Navier-Stokes equations, one obtains the Reynolds Average Navier-Stokes (RANS) equations, which is a widely used turbulence modelling approach (Fig. 7) in engineering applications. Homogeneous turbulence In homogeneous turbulence, the velocity field, u( x, t), is statistically homogeneous, i.e. all statistics are invariants under a shift in position [23].

27 19 Isotropic turbulence If in addition to homogeneous turbulence, the statistics are invariant under rotations and reflections of the coordinate system, then the turbulence is isotropic [23] Scales in turbulence Intuitively, from the phenomenological description of turbulence, turbulent flow problems and therefore any turbulent reacting flow problems are multi-scale problems in time and space. The energy content of each length scale and the central concept of energy cascade is depicted in a universal form, cf. Fig. 7. Eddies, with various characteristic time and length scales, overlap in space from the largest length scale, determined by the geometry of the flow, down to the smallest scales at which the kinetic energy is dissipated into heat. Although in general, the large eddies are anisotropic (geometry dependant), according to Kolmogorov s first hypothesis for sufficiently large Re, locally isotropic turbulence is a good approximation at small-scales. This constitutes a general assumption in many model developments. It also leads to the definition of various scales and ranges as shown on Fig. 7 and presented below. log E ( k ) modelled in RANS resolved in DNS resolved in LES modelled in LES l 0 energy containing range 1 t λ inertial subrange log k 1 η viscous subrange Universal equilibrium range Figure 7. Energy spectrum E(k) as a function of wavenumber k. Energy containing range and integral scales The large-scale eddies contain most of the kinetic energy and make up the energy containing range. In homogeneous isotropic turbulence with zero mean velocity, a corresponding integral length scale can be defined through the autocorrelation function: ' ' < u ( xu ) ( x+ r) > (3.10) f() r = < ' ' u ( x) u ( x) > where u ' ( x) is the velocity fluctuation at a given point x in space and r is the distance from this point. The integral length scale is then defined as, (3.11) l 0 = f ( r) dr It represents the mean distance at which the velocity fluctuations are correlated. 0

28 20 A time scale can be defined from the length and velocity scales at the large eddies: 0 t 0 = l (3.12) u ' where the velocity scale of the large eddies is on the order of the root mean square of the velocity fluctuation, u rms. Inertial subrange and Taylor length scale In the view of energy cascade, energy is continuously transferred from the large eddies to the small eddies where it is dissipated into heat. If the turbulence Re number is high enough there is a range of scales which depends, according to Kolmogorov s second similarity hypothesis, on the dissipation rate ε and the wavenumber k only, but not on viscosity. The Taylor length scale λ t belongs to this range. This length scale can be defined from the second derivative of the autocorrelation function at r = 0 as: 2 2 (3.13) λt = 2 2 d f() r dr The Taylor scale does not have a clear physical interpretation. However as it lies in the inertial subrange, it is a useful reference scale for characterising grid turbulence in LES [23]. Viscous subrange and Kolmogorov scales Finally in the viscous subrange, the Kolmogorov scales can be uniquely determined by the viscosity ν and the dissipation rate ε according to Kolmogorov s first similarity hypothesis. Indeed at this scale, all the turbulent kinetic energy is converted into heat by the viscous dissipation. Using dimensional arguments the three Kolmogorov scales; length, time and velocity can be expressed: ν ν η (3.14) 14 η, tη, uη = ( νε ) ε ε tη In the direct numerical simulation approach (DNS), all scales down to the Kolmogorov scales are resolved (Fig. 7). Hence, the requirements on the grid resolution are important and it is definitely out of reach under high Re number flow conditions. Universal equilibrium range According to Kolmogorov hypotheses, the inertial subrange and the viscous subrange are statistically similar or universal for all high Re number flows; hence they correspond to the universal equilibrium range Large eddy simulation modelling approach The LES approach takes advantages of the above property i.e. existence of a universal range in turbulent flows. Indeed in LES, the large scale turbulent motions that are dependent on the flow configuration are resolved while the smaller scales, that are universal, are modelled. The various quantities Φ (scalars, density, velocity components, etc.) are therefore filtered in the spectral space (components greater than a given cut-off length are suppressed) or in the physical space (weighted averaging in a given volume). The filtered quantity Φ is defined as: ' ' ' Φ ( x) = Φ( x) g( x x) dx (3.15) where g is a filter function and can take different forms e.g. top hat, Gaussian, etc. r= 0

29 21 For non-constant density flows, as considered in this work, Favre, or mass-weighted, filtered quantity Φ is defined as: ρφ= ρφ (3.16) In practice the grid size often determines the spatial filter cut-off which is comparable to the Taylor scale. At this scale, the local resolved energy spectrum should exhibit the -5/3 curve (Fig. 7). The implication of filtering the governing equations presented in Sec. 2.2 in the LES framework are presented in Sec Filtered transport equations for reacting flows in the LES framework As discussed previously, in LES, a spatial filter is applied to the instantaneous balance equations presented in Sec. 2.2, which leads to the following equations [10]: Transport equations for the momentum p ( (3.17) ρu + ρuu = + ( τ ρ uu uu )) t j x i j ij i j i j i xj xi R where τ ij = uiuj uu i j is the subgrid scale stress tensor. The body forces are neglected in this formulation as it is in the reacting flows studied in this work Conservation of mass and transport equations for the species ρ ρu (3.18) + i = 0 t xi ρy k (3.19) + ( ρuy ) ( ( i k = Vk, iyk ρ uy i k uy i k) ) + ωk for k=1,,n t xi xi where Vk, iy k are the filtered laminar diffusion fluxes, u iyk uy i k are the unresolved species fluxes and ω k is the filtered chemical reaction rate. In practice the filtered laminar diffusion fluxes are either neglected or modelled as, Y (3.20) k Vki, Yk = ρdk for k=1,,n x i As mentioned earlier, due to the adiabatic assumption on the flame considered in this work, no energy equation is needed. Different formulations of the filtered energy equations are presented in [10] Modelling of the subgrid scale stress tensor From the Kolmogorov theory, Fig. 7, we assume that practically all the kinetic energy is dissipated at the smallest scales which are not resolved on the LES grid. Hence, there is a need for a model that can dissipate this energy. One simple model is the Smagorinsky subgrid scale (SGS) model [27]. Based on the Boussinesq assumption [26]: δ R ij R u u (3.21) i j τij τkk = νt + = 2νtS ij 3 xj x i where ν t is the SGS viscosity and S ij is the filtered rate of strain tensor. A dimensional argument is used to model the SGS viscosity as ν = ( C Δ ) 2 S (3.22) t s LES

30 22 where the turbulent length scale is assumed to be of the order of the filter size Δ LES and the characteristic filtered rate of strain is S 2SS ij ij. This model has two major drawbacks. First it is found to be too dissipative [10] and C s depends on the flow configuration. A dynamic model has been developed to remedy this problem [28]. Second, this model (and any other SGS viscosity model with ν t > 0 ), transfers energy everywhere from the resolved (filtered) motion down to the unresolved (residual) motions but there is no backscatter [23], i.e. no influence of the unresolved motion on the resolved motion. This could correspond to some extend to a negative viscosity effect. In this thesis, two different approaches are considered: a mixed model based on the scale-similarity model (SSM) and the implicit LES (ILES). The original SSM [29] assumes that the unresolved stresses are dominated by the largest unresolved eddies, which are in terms similar to the smallest resolved eddies. The contribution of the smallest resolved eddies can be estimated by filtering the resolved scales leading to the following expression: R τ ij = u iuj uu (3.23) i j The model was found not dissipative enough [29] and a mixed model was proposed [29, 30] in which a coupling with the Smagorinsky model is required: R τ = C uu uu ˆˆ 2 CΔ S S (3.24) ( ) ( ) 2 ij l i j i j s LES ij where is a second filter and C l is a model constant. To avoid an excessive dissipative effect due to the Smagorinsky model without using a dynamical estimation, the dissipative character of the numerical scheme is used instead [31, 32]. If the resolution of the LES is fine enough (comparable to the Taylor scale) and the convective terms in the momentum equations are discretized by a high odd-order scheme, the numerically approximated system has a dissipative nature due to the leading order truncation error. Hence if a large part of the energy spectrum is resolved, it is acceptable not to use an explicit SGS model. ILES has been successfully applied in many LES applications [31, 33-36] Modelling of the unresolved scalar transport: species fluxes Unresolved scalar transports is often described using a gradient assumption (similar to RANS modelling) written below for the species fluxes, ν t Y (3.25) k uy i k uy i k= for k=1,,n Sck xi A similar assumption is used in the modelling of unresolved terms in the mixture fraction equation (Sec. 5.2). This is proven to be more suitable than most non-dissipative models since it is important to ensure that the combustion scalars, such as mass fractions and mixture fraction, are not subjected to numerical wiggles. The numerical wiggles can lead to overshooting of mass fractions above 1 or less than 0, thus leading to non-physical results Problem of reaction rate closures The simulation of reacting flows in the LES framework requires the use of models for the filtered reaction rates; since the reaction layer is in most situations thinner than the grid size and therefore can not be resolved. Furthermore a direct approach to model the filtered reaction rates ω k is not always suitable [10]. In fact, specific models are required for each combustion regime due to the turbulence and chemistry interaction. The different turbulent combustion

31 23 regimes for premixed, non-premixed and consequently partially premixed flames are presented in Sec. 3.3 with an emphasis on the flamelet regimes considered in this work. 3.3 Flame and turbulence interaction: combustion regime classifications As discussed in Chapter 2, combustion requires that fuel and oxidizer are mixed at a molecular level. The main effect of turbulence on combustion, as it enhances mixing of species and heat, is to increase the combustion rate [10]. On the modelling point of view adopted here, i.e. in the LES framework, it leads to the unclosed filtered reaction rates with no direct closure (Sec ). The development of models (Chapter 4 and Chapter 5) is based on physical analysis and arguments on the various time and length scales involved in combustion processes. Different regimes can be identified for both premixed and non-premixed flames (and consequently for partially premixed flames). The main focus will be given on regimes where a scale separation is possible, i.e. if the time and length scales of chemical reactions are separated from those of the turbulence. To simplify the analysis, it is common to assume simplified chemistry and equal diffusivities for all reactive species. Additionally, it is assumed that the Lewis number and the Schmidt number are unity Regimes in turbulent premixed combustion The regimes of turbulent premixed flames were developed to help the understanding of the possible interaction between the premixed flame front and the turbulent eddies. They are usually discussed in terms of velocity and length scale ratios in diagrams [5]. Fig. 8 shows five different regimes [5], based on non-dimensional numbers. The Damkhöler number Da, is the ratio between the characteristic time scale of the flow based on the integral length scale (Sec.3.1.3) and the characteristic chemical time scale as: 0 t0 0SL Da = ' 0 t = l (3.26) c uδ L The Karlovitz number Ka, is the ratio between the characteristic chemical time scale to the characteristic time scale of the flow based on the Kolmogorov length scale: (3.27) t ( δ L ) u c η Ka = = = tη η S ( L ) If instead the inner layer characteristic thickness, δ R ( δ 0 R = δl 10 ) is used, cf. Eq. (3.27), a second Ka number is defined as: 2 ( δ ) Ka R Ka (3.28) δ = = 2 η 100 With the current assumptions (Sc=1) and using Eq. (2.35), the turbulent Reynolds number (Eq. (3.1)) can be written as: u ' l (3.29) 0 = 0 0 S δ Re t L L

32 24 u S L ' 0 l l δ δ L L Figure 8. Regime diagram for turbulent premixed combustion. After [5]. Laminar flames When Ret < 1 the flow is essentially laminar and the combustion lies in the laminar flame regime (Sec. 2.3). Wrinkled flamelets ' 0 As Re t increases, the flow becomes turbulent. If u S L < 1, i.e. the turbulent fluctuations are rather small compared to the laminar flame propagation, then turbulence can not compete with the advancement of the flame front. Corrugated flamelets In this regime, Ka < 1, the smallest scales (Kolmogorov eddies) in the flow are larger than the laminar flame thickness and they do not interfere with the local structure of the flame. Therefore the flame front generally remains quasi-laminar. Thin reaction zones If Ka is further increased, the Kolmogorov eddies become smaller than the flame thickness. Since Ka δ < 1, they are still larger than the inner layer (where the most important reactions occur) and therefore they may only interfere with the preheat zone and the oxidation layer (Sec ). This in terms can modify the flame speed. Broken reaction zones ' 0 If u S L is very large and Ka δ > 1, the Kolmogorov eddies in the flow are smaller than the inner layer thickness of the flame. As they can enter the inner layer, they perturb the chemical reactions by increasing the heat and the radicals loss to the preheat zone. This can lead to local extinction of the flame, hence broken reaction zone regime. As stated in Sec. 1.3, this work is limited to configurations where the flames are considered as thin reaction zones or corrugated flamelets.

33 Regimes in turbulent non-premixed combustion Similarly to premixed flames, regimes can be identified for turbulent non-premixed flames using appropriate scale comparison. However, unlike in premixed combustion, non-premixed flames do not exhibit well-defined characteristic scales e.g. a propagation speed and the local flame thickness depends on the flow conditions [10]. Additionally chemical reactions are generally limited by mixing (Sec. 2.4). Nevertheless a regime diagram can be constructed based on the Da number and Ka number which can be utilised as they are defined in Sec (in terms of time scales ratio) as well as the Re t number based on the integral scales (defined in Sec ) as shown on Fig. 9. Re t Figure 9. Regime diagram for turbulent non-premixed combustion. Laminar flames If, Ret < 1, the flame is laminar (Sec. 2.4). Flamelet regime If Ka < 1, the chemical reaction time is shorter than the smallest eddy time, which implies that the chemical reaction is very fast, and the reaction zone is likely very thin. Flamelet assumption may be applicable. The laminar flame structure is rather unaffected by the local flow conditions. However in this regime, it should be pointed out that the reaction is not necessarily infinitely fast (Sec ). Under high scalar dissipation rate conditions, flame quenching may occur. Thick reaction zone For Da<10 and Ka>1, the chemical reaction is slow, the reaction zone is thick. It should be pointed out that most non-premixed turbulent flames are in the flamelet regime: the chemical reaction is fast; the reaction zone is thin Partially premixed combustion: a combination of regimes When partially premixed combustion is encountered, with typically premixed and nonpremixed combustion modes within the same flame, a common modelling approach is to considerer those modes separately (possible under certain assumptions described in Chapter 5). Since this work focuses on low Ka and high Da flames, the development of models for partially premixed flames will be based on flamelet regimes. Further discussions on modelling of partially premixed flame are presented in Chapter 5.

34 26

35 27 4Chapter 4 Modelling of turbulent premixed combustion using LES approach The unclosed filtered reaction rates (Sec ) are the main issue when modelling turbulent reacting flows. Typical models for the closure of these terms or modelling approach to overcome this difficulty are discussed within the corrugated and thin flamelet regimes (Sec ) in the following Sec The specific approach used in this work, the G-equation, will be presented separately in Sec Overview on the combustion models Four types of models may be identified [I] (other classifications are of course possible and mainly the models appearing in this work are described here): models based on turbulent mixing descriptions, [37-39] models based on flame front topology together with flamelet structure modelling [5, 40-45] models based on single-point probability density function (PPDF) of scalar fields and geometrical flame surface analysis [46-49] models based on finite rate chemistry [50-55] Description of turbulence mixing Models based on turbulence mixing are usually a direct extension of existing RANS models and based on the assumption that, at high Da number, turbulent mixing, rather than the chemical reaction rates, controls the combustion process. Models falling into this group are for example the eddy break up (EBU) model [37, 39] and the eddy-dissipation concept (EDC) model [38]. In the LES-EBU, the reaction rate can be expressed as [10]: ω ( ( 1 c = C EBU ρθ θ) ) t (4.1) Δ where C EBU is a model constant, θ is the reduced temperature and t Δ is a subgrid turbulent time scale. θ is defined as: θ = ( T Tu ) ( Tb Tu) (4.2) The subgrid turbulent time scale is estimated as: Δ tδ l (4.3) ' uδ where l Δ is a subgrid length scale. The model constant needs also to be adjusted and is likely to be dependant on the flow conditions and mesh size.

36 Flame front topology This type of models is developed for the flamelet regimes or thin reaction regimes. Progress variable approach In this approach, the flame is described by a unique parameter, the progress variable evolving monotonically from the unburned region to the burned region across the front. Typically the progress variable (or reduced variable) is built from the temperature as in Eq. (4.2) or fuel mass fraction (Le=1) as: YF ( xi, t) Y (4.4) F, u cx ( i, t) = YFb, YFu, Both progress variables evolve from 0 in the unburned gas to 1 in the burned gas. A transport equation can be derived (Sec ) which in its filtered form reads: ρc ( ) ( c ρuc i ρuc i ρuc i ) (4.5) + + = ρd + ωc t xi xi xi xi where the subgrid scale transport is generally modelled with a gradient-diffusion assumption as in Eq. (3.25). The two terms on the right hand side can be written from freely propagating one dimensional flame analysis (c evolution expressed in the flame reference [10]) as: c (4.6) ρd + ωc = ρsd c xi xi where S d is the local displacement of the iso-surface c. Different closures for Eq. (4.6) can be found in the literature e.g. in the form of flame surface density approach [48] or through diffusion and production terms [56], etc. In the flame surface density approach, Eq. (4.6) is modelled as: ρsd c ρuslσ Δ = ρuslξδ c (4.7) where ΣΔ is the flame surface density per unit volume at the subgrid level and Ξ Δ is the subgrid scale wrinkling factor. Algebraic expressions or balance equations can be found in the literature for either term [48] especially for the subgrid scale wrinkling as discussed in Sec G-equation approach This is the approach used in this work and therefore it will be described in details in Sec Probability density function In a statistical approach, the reaction rate at all points can be expressed as: ω = ω Y,..., Y, T ( Y,..., Y, T) dy... dy (4.8) Y1,..., YN, T ( ) 1 N 1 N 1 N Therefore the probability density functions of all variables (species or temperature) entering in Eq. (4.8) have to be expressed from either experimental or DNS data or estimated using presumed probability density function or transported [46].

37 Finite rate chemistry In the finite rate chemistry approach, different methods are used to estimate the filtered reaction rates. All of them require a reduced reaction mechanism (two or more steps). Thickened flame model From the scaling relations in Eqs. (2.35) and (2.36), it is possible to thicken the flame front by a thickening factor F t while preserving the flame speed [57]. Recalling from Eq. (2.16) that the pre-exponential factor A p scales with the reaction rate, the new set of relations is: A (4.9) 0 p 0 2 SL Ftα and FtδL Ft α Ap = Ft α Ap F t 0 Since the thickened flame thickness is Fδ l, the Da number (Eq. (3.26)) is also decreased by a factor F t which means that the flame becomes less sensitive to turbulence motions. To remedy this issue an efficiency function E t is introduced to correct for the unresolved features on the flame speed, which are typically the wrinkling effects [51]. Partially stirred reactor (PaSr) model In the PaSR model, [58], the flow is divided into fine structures (*) and surroundings ( 0 ). The fine structures form topologically complex regions, composed of a muddle of interacting tube-, ribbon- and sheet-like structures, [59], in which most of the dissipation and mixing take place. Since most of the mixing occurs in the fine structures, the reactions also take place here as the reactants are mixed at scales down to the molecular scales. This implies that, * * * * 0 0 ω = ( ρ, T, Y ) ω ρ, T, Y dρdtdy = γ ω ρ, T, Y + (1 γ ) ω ρ, T, Y (4.10) ( ) ( ) ( ) k k k k k k k k k ρ TYk * where γ denotes the reacting subgrid volume fraction. The quantities in the fine structures and surroundings are related, as for the EDC model [I], through subgrid balance equations and characteristic time scales need also to be found at subgrid level. More details are given in [I]. In this approach no a priori assumption is necessary on the combustion regime. 4.2 Level set G-equation model G-equation in the LES framework The direct use of a spatial filter on Eq. (2.38) [60, 61], is not always suitable as the scalar G does not have a definition on the entire field. Different approaches are used in [62, 63]. In this work, the G-equation is derived directly based on physical argument. The underlying assumption is that the local structure of the flame remains similar as the one dimensional flame. Therefore the main effect of the unresolved scales is to wrinkle the flame front. As discussed in Sec , the smallest turbulence scales in LES have been filtered out and have to be modelled. Hence their effect on the filtered flame front have to be considered i.e. the propagation speed of the front has to be corrected accordingly. For convenience, the vector form is preferred in this section. The formulation of the G-equation becomes: G (4.11) + u u G = ST, Δ G t where is the propagation speed including subgrid scale wrinkling effects (Sec.4.2.2) and uu ST, Δ is the unburned filtered velocity at the flame front. For simplicity the same filtering

38 30 symbols are used for the spatially filtered convective velocity and for the scalar G. In fact, this equation can be seen and derived as an equation describing the motion of a smoothed flame front in a spatially filtered flow field [7, 64] as shown in Fig. 10. filtered flame front flame front Figure 10. The smoothed (filtered) flame front and the actual wrinkled flame front on a computational grid. In the following, the smoothed flame front will be referred to as the filtered flame front. As in Sec , the structure of the one dimensional flame is expected to be mapped into a the G ~ - field, defined as a signed distance using the re-initialization equation: ~ G = 1 (4.12) It is shown in [44], that although in this case the G ~ -scalar corresponds to the distance from the filtered front, it can be treated as the distance to the instantaneous flame front when determining the major species and thermophysical properties i.e. density and temperature in the flow field. However, when the minor species concentration e.g. CO and subgrid scale effect has to be considered, the distance function to the instantaneous front has to be explicitly corrected [42, 44] Propagation of the filtered flame front The propagation speed, ST, Δ, or subgrid scale turbulent flame speed, strongly depends on the unresolved flame surface wrinkling. The wrinkling of the flame increases the burning area and thereby increases the propagation of the resolved flame front as shown on Fig. 10. Several models for the turbulent flame speed can be found in the literature; most of them are derived from theoretical arguments in combination with experimental data, and could be divided in three classes. First, by assuming an equilibrium at the subgrid scale level between flame surface and turbulence, the wrinkling factor of the flame surface may be estimated as, Δ (4.13) ΞΔ 1+ c0 σ S 0 t where σ is the tangential subgrid scale strain term, Δ the LES filter size and S 0 t l the unstrained laminar flame speed. Estimating the subgrid scale strain term as the ratio of the ' subgrid scale velocity u Δ and the filter size Δ, the wrinkling factor can be written [51], ' ST, u (4.14) Δ Δ =Ξ 1 c 0 Δ = SL SL This class of models [44, 51, 65] may be generalized as, n 1 n ' S T, u (4.15) Δ Δ =Ξ 1 c 0 Δ = S L S L L

39 31 where the model constant c0 and n have to be evaluated prior to computation or dynamically [60]. A formulation valid for both corrugated and thin flamelet regimes (where the preheat zone may be disturbed), derived from RANS, is first proposed in [62], 2 ' (4.16) Δ c1δ uδδ Ξ Δ = 1 c1 c δl δl δlsl and modified in [45] as, (4.17) b3cs Δ b2csδ b2dt Ξ Δ = b1SctδL 2b1SctδL δlsl where the turbulent diffusion coefficient is: ' CsΔu (4.18) Δ Dt = Sct where the turbulent Schmidt number is set to 0.5. Other models can be found in the literature [66, 67]. The second class of model relies on the fractal theory [68] where the wrinkling factor is expressed as, 2 0 ( (, ' 0,Re D ' 0 (4.19) Ξ Δ = Γ Δ δ L uδ SL Δ) ( uδ SL) ) where D is the fractal dimension and Γ is an efficiency function. Finally the third one corresponds to a power law formulation [51, 69, 70]. In [69], the wrinkling factor is given by: β ' 0 ' 0 u (4.20) Δ Δ Ξ Δ = 1+ min, Γ 0 ( Δ δ L, uδ SL,ReΔ) 0 δ L S L Fig. 11 presents the results from the different formulations in a classical non-dimensional diagram with the turbulence intensity as the horizontal axis and the subgrid scale flame speed as the vertical axis. 0 Figure 11. Subgrid scale turbulent flame speed evaluated by different models for Δ = 4. δ L

40 32 As the turbulence level decreases towards 0 (laminar), all wrinkling models converge towards ' ' 0 1 thus recovering the laminar flame speed. In the low u Δ region up to uδ S L < 3, all models predict similar subgrid scale flame speed ( ST, Δ ), with the fractal model giving a slightly lower value and models from Eq. (4.15) giving a slightly higher value. For higher turbulence level, the discrepancies between models become more significant. Models based on Eq. (4.15) giving a linear increase of the subgrid scale flame speed with the subgrid scale turbulence velocity. This discrepancy is partly due to the selection of the model constants (Eq. (4.15)). Models proposed in [62] and [45] behave similarly whereas the fractal model and power law model (with cut-off) give similar wrinkling factor. The sensitivity of the turbulent flame speed to the subgrid scale velocity is relatively ' low for these model at high u Δ. The key issue here is not only the flame speed itself but also ' the prediction of u Δ Subgrid scale turbulence velocity A first approach to estimate the subgrid scale velocity u ' Δ would be from the subgrid scale viscosity provided by the LES SGS model (Sec ). It is however, complicated to extract ' ' u Δ from the SSM model or in case of ILES (Sec ). An explicit model for u Δ is needed. In this work both the Smagorinsky model and the one equation eddy viscosity model (OEEVM) [71] are used in respectively in [II-IV] and in [I]. The subgrid scale velocity estimated from the Smagorinsky model ( static or dynamic) leads to: ' u = C Δ Δ s S (4.21) where Δ is the LES filter size, ũ is the spatial filtered velocity, S is the characteristic rate of strain and C s is a model constant set to 0.17 [23] or determined dynamically [60]. In the OEEVM approach, the subgrid scale velocity is given by u ' = Δ 2 k 3. The transport equation for the subgrid scale kinetic energy is given by: ' ' 2 ' k k ν k (4.22) + u j = + P D 2 t x Sc x j where P and D are respectively the model subgrid scale production and dissipation, modelled as: ' 2 ' ( ) 3/2 (4.23) k P = ck h S k and D = ce h with c k = 0.07 and e 1.03 h = Δx Δy Δ z and the characteristic rate of strain S = 2 SS ij ij. Those terms may suffer from similar issue as classical Smagorinsky model regarding the estimation of the rate of strain (Sec ). Other models can also be found in the literature such as the high pass filter model and model based on similarity assumption. High Pass Filter model c = with the length scale ( ) 1/3 A model based high pass filter (HPF) eddy-viscosity model is investigated in [72] with the resulting expression for the subgrid scale turbulent velocity: HPF u = C Δ Δ s, ω S( H* u ) (4.24) c j

41 33 HPF where H* ũ is the high-pass filtered quantity and Cs, ω is a model constant function of the c cut-off wavenumber. Model based on similarity assumption Finally, a model based on similarity considerations can also be preferred [51]: ' 3 2 uδ = c 1 Δ ( ũ ) (4.25) where c 1 is a model constant ( c 1 = 2 ) Limitations of the method It should be noted that Smagorinsky based models, described in Sec , can in practice ' give an unreasonably high estimation for the subgrid scale velocity, u Δ, in the vicinity of the flame front (due to the rate of strain evaluation). This is caused by the thermal expansion which increases the velocity and hence is interpreted as a region of high subgrid turbulence. The resulting subgrid scale turbulent flame speed, S, is then over estimated. The planar laminar flame speed may not be recovered [51]. This can be avoided by setting an offset between the propagated level-set and the density gradient in the flame [[64], III]. T, Δ

42 34

43 35 5Chapter 5 Modelling of turbulent partially premixed combustion using LES approach Under the scope of this work i.e. low Ka number (high Da number), the modelling approach for turbulent partially premixed flames can be based on flamelet assumptions. Several modelling approaches can be found in the literature depending on, as introduced in Sec. 2.5, the combination of the mixture fraction Z and either a progress variable c or the level-set G. Recalling Fig. 6, Z is used as a tracer for the (partial) mixing of the fuel and the oxidizer and it is used to describe the diffusion controlled part of the flame. The progress variable c or the level-set G allows the description of the flame front responsible for the flame stabilization. An overview of the models based on the progress variable is given in Sec. 5.1 while the model used in this work, based on the G-equation, is described in Sec The reaction progress variable approach based models In the following discussion, the reasoning will be based on the instantaneous equations for simplicity. The spatial filtering (LES) applied to the resulting equations will generate unresolved terms for which closure are necessary as discussed in Sec Reaction progress variable for partially premixed flames Recalling the mixture fraction transport equation derived in Sec In general, the mixing field of partially premixed combustion can be described in terms of the mixture fraction Z( x i, t), Z ρ uiz Z (5.1) + = ρd t xi xi xi The reaction process can be characterized from the evolution of certain species, k, mass fraction, Yk( xi, t ) (Sec ), ρyk ρuiyk Y (5.2) k + = ρd k + ωk t xi x i x for k=1,,n i A first requirement in the choice of the progress variable (species) is that it should evolve monotonically across the flame. For example, a variation of the fuel mass fraction from its initial value to zero indicates a progress of the fuel consumption process. In multiple reaction systems, a single fuel mass fraction or product mass fraction may not be suitable to describe the reaction progress since the fuel is typically converted to intermediates first and then to final products subsequently. Some authors [4, 73] have defined the reaction progress as a linear combination of reactants and product species, e.g. c ( xi, t) = YH ( x, ) (, ) (, ) 2 i t M H + Y 2 H 2O xi t M H 2O + YCO x 2 i t M (5.3) CO2 Other authors have chosen the normalized temperature [74] as the reaction progress variable. Depending on the definition of the progress variable, transport equations for the progress

44 36 variable can be derived from the fundamental governing equations of the reactive flows (e.g. Eq. (5.2)). For convenience, it is conventional to normalize the reaction progress variable to varying from 0 to 1, where c( x i, t) = 0 in the reactants and c( x i, t) = 1 in equilibrium products. With the above assumptions Yk( xi, t ) can be expressed as: Yk( xi,) t = Yk[ c( xi,), t Z( xi,) t ] (5.4) In [75], from Eq. (5.1) and Eq. (5.2), the transport equation can be derived for c( x i, t), ρc c (5.5) + ( ρuc i ) = ρd t xi xi xi Yk Yk Y k + ωk + ρχ 2 c + ρχ 2 Z + ρχz, c Yk c c Z c Z where three scalar dissipation rates have been introduced and written here in a vector form for simplicity: χz = D Z Z, χc = D c c, χz, c = D Z c (5.6) The first two scalar dissipation rates describe respectively the mixing rate of fuel and oxidizer, and the mixing rate of the hot products with unburned mixture. The third scalar dissipation rate describes the rate of transport of reactants across the iso-z surfaces [75]. Recently a multidimensional flamelet-generated manifolds (MFM) approach was introduced [76]. The idea of MFM is to tabulate the chemistry independently of the combustion regime (premixed or non-premixed). The table will then be accessed using a set of 5 parameters (more if the variances are included), Z, c, χ Z, χ c and χ Z, c. This approach presents two major challenges. First, for obvious reason the construction of the table is complex. Second accurate models must be found for the various scalar dissipation rates (Eq. (5.6)). Models for χ Z and χ c can be found in the literature and has been used successfully while models for χ Z, c are not trivial. Usually models are derived based on simplifications of Eq. (5.5) as described below Simplification of the progress variable transport equation Under certain conditions the c-equation can be simplified. If c( x i, t) is a linear function of Y k, 2 2 the second order derivative in c is zero: Yk c = 0 ; thus the second term in the bracket of Eq. (5.5) disappears. If the dominating flame region is of premixed type one may define the progress variable as [75], YF,0Z YF( xi, t) (5.7) cz ( ; xi, t) = Y Z Y ( Z) F,0 F, where Y F is the local fuel mass fraction, Y F, 0 is the mass fraction of fuel at the fuel inlet (in case of air premixing with fuel already at the fuel inlet Y,0 < 1, otherwise it is unity) and YF, ( Z) is the mass fraction of fuel in the downstream of the premixed flame, i.e. around the diffusion flamelet. From Eq. (5.7) and noting that the c( x i, t) is linear function of Y F, Eq. (5.5) can be rewritten as [75], F

45 37 ρc c 2 dy ( Z) + ( ρuc) = ρd + ω + Y ρχ t x x x Y Z Y Z dz F, i c,0,,0, ( ) F Z c i i i F F 2 c dyf, ( Z) ρχ 2 Z YF,0 Z YF, ( Z) dz where the rate of the progress variable is ω (5.9) F ω c = YF,0 Z YF, ( Z) If the flame front is assumed to be thinner than any other flow scales (flamelet hypothesis), the transport of reactants through the thin interface between fresh and burned gases may be assumed one-dimensional in the direction normal to the gradient of the progress variable (other transport phenomena being negligible) leading to χ Z, c 0. This term is not negligible in a diffusion flame where the diffusion of the reactants along the iso-z surfaces is important [75]. Eq. (5.8) is valid in both the premixed flames and at the triple flame point of partially premixed flames. For the trailing diffusion flamelet the reaction progress variable is unity, then Eq. (5.8) and Eq. (5.9) are reduced to, 2 dyf, ( Z) (5.10) ρχz = ω 2 F, dz which is similar to the stationary flamelet equation (Sec ). ω F, denotes the fuel consumption rate at the downstream diffusion flames. Eq. (5.10) provides the governing equation to determine the downstream fuel composition introduced in Eq. (5.7). In the premixed flame part, from Eq. (5.10) it appears that in the flow field outside the diffusion flamelet reaction zones YF, ( Z) is a linear function of Z, then the last term in Eq. (5.8) disappears. Furthermore, if at the triple flame front the gradient of Z is perpendicular to the gradient of the c, i.e. χ Z, c 0, then the third term on the right hand side of Eq. (5.8) is negligible. Eq. (5.8) is then reduced to, ρc c (5.11) + ( ρuc i ) = ρd + ωc t xi xi xi which is identical to the classical transport equation for a reaction progress variable for the premixed flames (Sec ), where similar closure for ω c can be used; accounting for the dependency of the laminar flame speed on the local equivalence ratio (Z) [75, 77]. At the triple flame point, however one may still assume that χ Z, c 0 based on the argument that the gradient of Z is perpendicular to the gradient of the c.the transport equation for c can be written as ρc c c (5.12) + ( ρuc i ) = ρd + ωc + ωf, t xi xi x i YF,0 Z YF, ( Z) The second term on the right hand side of Eq. (5.12) is the rate due to the premixed flame and the last term is due to the diffusion flamelet downstream. This is the term that makes the difference between triple flame and a pure premixed flame. Several authors [20, 75] have argued that under standard conditions, the fuel consumption in a diffusion flame is much smaller than in a premixed flame ω << F, ωf, p and therefore the contribution from the diffusion flame part is negligible. (5.8)

46 38 The description of the diffusion flame part needs the modelling of terms related to Eq. (5.5) and Eq. (5.8) in Eq. (5.4). As mentioned previously a diffusion flamelet approach can be used but other approaches developed for diffusion flames are also possible [75] e.g. Conditional Moment Closure or PDF. A flame index is preferred to combine equation (5.11) with an appropriate description of a diffusion flame [75] instead of solving Eq. (5.8). The flame index is defined to discern the nature of the reaction zones (whether it is diffusion flame or premixed flame) [78]: YF Y (5.13) O FID = YF YO For diffusion flames FID = 1 where as for premixed flames FID = 1. In non-premixed flames with local extinction and re-ignition the structures of the reactions can be of partially premixed type, i.e. the flame index can be both positive and negative unity Modelling partially premixed flames within flammability limit If the flame is assumed to be controlled by premixed combustion (Eq. (5.11)), one may use the premixed flamelet tabulation, also known as flamelet generated manifolds (FGM) [73, 74, 79, 80] or flamelet prolongation of intrinsic low dimensional manifolds (FPI) [81]. Namely, a class of premixed flames is pre-simulated for a given set of equivalence ratio (converted to mixture fraction) and strain rates, then the distribution of mass fractions of species and reaction rates along the one dimensional physical space is obtained. It was shown recently, in LES [73] that this assumption holds (with reasonable accuracy) up to relatively high stoichiometric ratio for methane flame. Of course those methods fail when the diffusion process starts playing an important role [81] (scalar terms in Eq. (5.5)). They are not suitable in this work since the studied flames are found to be diffusion controlled. 5.2 A two-scalar variable flamelet model In this approach, the trailing edge is assumed to be controlled mainly by diffusion combustion. The mixing field and the iso-surfaces of stoichiometric mixture can be computed using the mixture fraction and its transport equation (Eq. (5.1)) which is given in its filtered form as: ρz ρuz i Z (5.14) + = ρd ( ρuz + i ρuz i ) t xi xi xi xi The last terms on the right hand side correspond to the subgrid scale terms and are modelled following Sec Within the flamelet assumption (Sec ), the turbulent flame structure (species, density, etc.) is similar to the corresponding one-dimensional laminar diffusion flame (Sec ). The filtered density, temperature and mass fractions are calculated using steady laminar flamelet library and a presumed density-weighted probability density function (PDF) [III]. As described in Sec , when the scalar dissipation rate is higher than a critical value, the flame cannot sustain; physically it is extinguished. At high Re number, this mechanism only is not likely to be fully responsible for the partially premixed flame stabilization [III, IV, IV]. The stabililization is likely ensured by a premixed propagating flame front which does not influence the overall combustion (similar to Sec ). Following the work in [82], the partially premixed front is modelled using the level-set G-equation (for the triple-flame front tracking) and mixture fraction flamelet chemistry. The formulation of the G-equation is

47 39 almost the same as presented in Sec The only difference resides in the modelling of the turbulent subgrid scale velocity, S. T, Δ In [5, 82], a model for the propagation of time averaged mean triple-flame front is proposed. The model has been used to predict the lift-off height of a jet flame [82]. Extending the model to LES filtered scales, the following model is proposed here: S ( 0 ( ) ' T, Δ = SL Z + uδ) ( 1 aχz χq) (5.15) where a is a model constant of the order of unity and χ q is the scalar dissipation rate at which quenching occurs. Typically, the laminar flame speed at the triple-flame front is higher than freely propagation stoichiometric flame [20]. However, since this quantity is rather small compared ' to u Δ, it is not significant to refine it. Although the premixed flame position needs to be computed at the triple flame only, to make the numerical implementation easier and also to divide the flow field into the burnable part and the inert part, it is desirable to compute the entire zero level-set surface in the flow field. Thus G > 0 denotes the possible burnable domain where the diffusion flamelet model can be applied and G < 0 the domain which the mixture is assumed chemically inert [III].

48 40

49 41 6Chapter 6 Numerical methods The system of non-linear partial differential equations, i.e. the governing equations, presented in Sec. 3.2 lacks analytical solutions except for simplified problems. However they can be approximated by algebraic finite difference equations which can be solved efficiently on a computer (CFD). In the following, the numerical methods used in this work (low Mach number flow in-house code [3]) are briefly introduced. 6.1 Grid system The numerical method is based on finite difference on staggered Cartesian grid. The Cartesian grid is easy to generate and does not require large computational storage. Additionally, high order discretization schemes (Sec. 6.2 and Sec. 6.3) can be achieved and are relatively easy to construct as compared to on an unstructured grid. Both uniform grids [IV,V] and stretched grids [I-III,VI] have been considered in this work. In case of stretched grid, an analytic stretch function has been used. In a staggered grid, the velocity components and scalars (e.g. pressure, density, G, etc.) are defined at different locations respectively at the cell surface and at the cell centre. It prevents the odd-even decoupling between the pressure and the velocity and it does not required boundary conditions (Sec. 6.4) for the pressure. For such a grid, each quantity Φ is indexed using a set of integer i, j, k (for x, y, z directions, respectively) corresponding to either the centre cell (scalars) or the cell surface (velocity). For simplicity in the following Φ will refer to scalar filtered quantities. 6.2 Discretization of the scalar and momentum equations As discussed previously in this thesis, due to the combustion models, the transport of species (Sec ) is reduced to the transport of scalars Z (Sec. 5.2) or/and G (Sec. 4.2 and 5.2). The numerical methods for the G-equation will be described separately in Sec The mixture fraction (Eq. (5.13)) and the momentum transport equations (Eq. (3.17)) require a numerical approximation for the time derivatives, convective and diffusive terms in order to be discretized and then solved. Recalling the above nomenclature at grid point i and time step n, n t, the time derivative is approximated using a second order Adams-Bashforth scheme as [3]: n+ 1/2 n n 1 Φ Φ Φ 3 1 (6.1) n n 1 = L( Φ ) L( Φ ) t Δt 2 2 where L corresponds to the spatial operator, including all the spatial derivatives. The convective terms are discretized using a 5 th order WENO scheme (Sec ) while the diffusion terms are discretized using a 4 th order central difference scheme e.g. in x- direction: 2 Φ Φ i Φi 1 30Φ i + 16Φi+ 1 Φ (6.2) i x 12( Δx) i

50 42 At the first and second grid point near the boundary, the stencil required by these high order schemes are not available. Therefore the schemes are reduced to first order upwind and second order central difference, respectively for the convection and diffusion terms Fifth-order Weighted Essentially Non Oscillatory (WENO) scheme In this work, the momentum and scalar transport equations are expressed in their nonconservative form when solved [3]. Typically the convective terms are written as: C Φ = ũ Φ (6.3) where the first order derivative is then discretized using the fifth order WENO scheme proposed by [83]. Focusing on the x-direction for simplicity, it is written as: ˆ ˆ 2 Φ Φi+ 1 2 Φi 1 2 = with ˆ 1 2 px ( ) (6.4) Φ i+ 12 = px ( ) Δx i x i Δx 24 x where px ( ) is a three-point polynomial interpolant to Φ. Note that ˆΦ here is not a filtered quantity but an approximation of the quantity Φ. As shown in Fig. 12, if the local convection velocity is positive, px ( ), can be constructed from three sets of points including Φi in the upwind direction. ( u i ) > 0 Φ ˆ i+1/2 p 1 ( x):( Φ i 2, Φ i 1, Φ i ), p 2 ( x ):( Φ i 1, Φ i, Φ i + 1 ) p 3 ( x ):( Φ i, Φ i + 1, Φ i + 2 ) Figure 12. Illustration of the stencil selection for building the 5 th order WENO scheme. Three different stencils can therefore be computed for Φ ˆ i+ 1/ 2 using the polynomial interpolants: ˆ (6.5) Φ i+ 1/2,1 = Φi 2 Φ i 1 + Φi ˆ Φ i+ 1/2,2 = Φ i 1 + Φ i + Φi ˆ Φ i+ 1/2,3 = Φi Φ i+ 1 + Φi For each stencils a smoothness indicator IS k is also computed as: (6.6) IS0 = ( Φi 2 4Φ i 1+ 3Φ i) + ( Φi 2 2Φ i 1+Φi) IS1 = ( Φi 1 4Φ i+ 1) + ( Φi 1 2Φ i +Φi+ 1) IS = 3Φ 4Φ +Φ + Φ 2Φ +Φ ( ) ( ) 2 i i+ 1 i+ 2 i i+ 1 i Finally, Φ ˆ i+ 1/ 2 is evaluated as a linear (weighted) combination: Φ ˆ ˆ ˆ ˆ i+ 1/2 = ϖ0φ i+ 1/2,1 + ϖ1φ i+ 1/2,2 + ϖ2φ (6.7) i+ 1/2,3 where the coefficient ϖ k is expressed as: αk C (6.8) k ϖ k = with αk = α + α + α ε + IS ( ) k i+ 12

51 43 The linear optimal weights C k are C 0 = 0.1, C 1 = 0.6 and C 2 = 0.3. ε is a small constant 6 (typically ε = 10 ) set to avoid numerical singularity. A comparison of WENO schemes with other schemes in terms of accuracy, efficiency and resolution properties have been reported in [36]. 6.3 Discretization of the G-equation The implementation of the G-equation requires specific considerations which are addressed below. As discussed in Sec , two equations must be solved in the level-set approach i.e. Eq. (4.11) for the motion of the flame front and Eq. (4.12) for re-initializing the surrounding G-scalar to a signed distance function (re-initialisation). To solve the re-initialization, Eq. (4.12) is expressed as: G (6.9) 0 = sgn( G )( 1 G ), G ( x,0) = G ( x) t * where t is not a physical time. In order to reduce the computational time, a narrow band (cells around the front, Sec ) is defined in which updating and re-initializing processes will be performed. A recent survey of various methods for the re-initialization equations, including the presently used sub-cell-fix method, is given in [84] Spatial derivatives A third order WENO scheme is used for the spatial derivative in the G-equation. As compared to that described in Sec.6.2.1, it combines only a second order central scheme with a third order upwind scheme in Eq. (6.5). Hence the stencil required is smaller (3 grid points) which do not require an extended narrow band (Sec ) and still give satisfactory results [7, 64]. More details on the implementation of this scheme can also be found in [7] Time integration Time integration is performed using a third order total variation diminishing (TVD) type n 1 Runge-Kutta scheme [85]. The optimal method for obtain G + from G t = L ( G ) has a form as [85]: (6.10) 1 n n G = G +ΔtL( G ) 2 3 n G = G + G + ΔtL( G ) n+ 1 1 n G = G + G + ΔtL( G ) This formulation is implemented for time integration of Eq. (4.11) and (6.9) Re-initialization step Since solving Eq. (4.11) does not preserve the distance function, a re-initialization step is needed. Furthermore, it is convenient in the level-set G-equation approach to define a narrow band (Fig. 13) only inside which Eq. (4.11) and (6.9) will be solved. It is possible because the entire G-scalar motion is controlled by the zero level-set propagation, which only requires a number of cells determined by the convective scheme (Sec.6.3.1). The narrow band width is therefore chosen in the range of 3 grid cells at each side of the zero level set, in each direction. The values of G are assigned to large signed values outside the narrow band. It should be pointed out that another requirement of the narrow band width is the laminar flame thickness (mapping, Sec ). The scheme requirements cover,

52 44 in the LES framework, the mapping requirements. By solving Eq. (6.9) within the narrow band, one must ensure that: the scalar G corresponds to the distance to the zero level-set (flame front) the zero level-set of the re-initialized G-field correspond to the original one The risk of altering the position of the zero level-set by directly solving Eq. (6.9) is reduced by dividing the re-initialization process into two sub-steps [7]. In a first sub-step, as shown on Fig. 13, the adjacent cells to the zero level-set are determined in which the value of G is obtained by calculating the normal distance to the front (sub-cell-fix). In a second sub-step, Eq. (6.9) is solved for the rest of the narrow band cells. In this way the requirement on G (Eq. (4.12)) is ensured in the narrow band and the mapping of density and species can be carried out. zero level-set (flame front) cells adjacent to the zero level-set G > 0 G < 0 narrow band region Figure 13. Schematic of the narrow band region around the zero level-set (flame front, solid line). The cells which have a neighbour cell with an opposite sign of G are marked in gray. The numerical methods regarding the G-equation are the same in all approaches considered in this work. 6.4 Boundary conditions Wall boundary condition At the walls, the no-slip boundary condition is applied for velocity while zero gradients are set for other scalars Turbulent inlet condition Inflow condition is one important parameter in LES as the simulated flow is time dependant. Such information is difficult to obtain from experiments since it is not easy to generate 3D time dependent velocity and scalar quantities from experiments. Thereby inflow turbulence needs to be generated numerically. In this work two approaches are considered; the first one is based on a digital filter method [I,II,VI] and the second one is based on inlet library generated from another dedicated LES of the inlet upstream flow [III-V]. Digital filter method The turbulent fluctuations (the time dependent part of the inflow condition) are modelled using a digital filter technique [86]. The choice is motivated by the flexibility and the low-cost of the technique. Artificially generated turbulent fluctuations are prescribed at the inlet (e.g. in the x-direction) as: ' u =< uexp ( x) > + u ( x) F( x, t) (6.11) IN IN

53 45 where < u ( x) exp > is the mean velocity profile (often obtained from experiments) and the last term provides seemingly turbulent fluctuations depending on an estimated inflow turbulence integral length scale L and a given Reynolds stress tensor. In particular it enables specifying (hence varying) not only the second moments but also the characteristic length scales. Generated inflow library If none or little experimental data are available in the region of interest e.g. at the inlet of the conical burner [III-V], it is preferable to simulate the inflow conditions as far upstream as possible from the main computational domain (or in region where approximation can be made). It is particularly important in partially premixed flames where although the mean velocity profile (and fluctuation levels) can be estimated from the mass flow rate, mixture fraction profile is on the other hand difficult to estimate (due to partial mixing). Unfortunately, due to computational cost (domain size and resolution) it is sometimes difficult to extend the domain of interest to cover the whole upstream flow. As discussed in Sec , separate simulations for the inflow conditions can provide relevant information on large scale and time dependent velocities and mixture fraction for the domain of interest. The results are then stored and used later for the simulation of the main computational domain. This method is computationally affordable Outflow condition As for the inflow conditions, one would like the outflow conditions to have less unphysical influence on the region of interest. Convective outflow condition is a good compromise between accuracy and cost as it can convect the large flow structure outside of the domain n 1 (diffusion effects are neglected) [87]. The formulation is given, at time step t + (e.g. in the x- direction), by: * n n u OUT u OUT n u (6.12) OUT = u OUT Δt x * where ũ OUT is an intermediate velocity and the spatial derivative scheme can be discretized n 1 using a first order upwind scheme. u + OUT is obtained by adding a correction velocity, uniform in the entire outlet, to ũ * OUT. The correction velocity is determined from the global mass conservation in the computational domain (continuity constraint). 6.5 Solution algorithm The in-house CFD code [3] used in this thesis follows the low Mach number assumption. In the low-mach number assumption, the acoustic wave is neglected, the physical pressure is spitted into two parts: the thermodynamic pressure P 0 and the dynamic pressure p. The thermodynamic pressure is constant in space and varies only in time, it is this pressure used in the equation of state to determine the mixture density. In this work, all computations are done at atmospheric thermodynamic pressure. The formulation of the Navier-Stokes equations for variable-density flows is expressed, ρ (6.13) + ( ρũ) = 0 t u 1 1 (6.14) + u u = p + t t ρ ρ where p here is the dynamic pressure and t is the total stress tensor (account for the resolved and unresolved terms).

54 46 In this semi-explicit LES code [3], the time step is fixed. In order to ensure the Courant Friedrichs Lewy condition a typical Courant number of C = ( umaxδt) Δ xmin < 0.15 is set. The combustion model (e.g. level-set G-equation) is first advanced in time (e.g. n n 1 G G + n 1 ) to provide the density at t + n+ 1 n+ 1 (e.g. ρ = ρ( G )) giving a constraint on the velocity divergence. In order to solve Eq. (6.14), the following fractional-step method is used. During the first step, the modified momentum equation is explicitly advanced between n t and t * as, * n u u 3 1 (6.15) n n 1 = L( u ) L( u ) Δt 2 2 where L corresponds to the spatial operator, including here the convection and diffusion terms but excluding the dynamic pressure term. A second step is then required to correct the intermediate velocity ũ * with the dynamic pressure gradient, n+ 1 * u u 2 (6.16) n+ 12 = p n+ 1 n Δ t ρ + ρ ρ + n 1 Multiplying Eq. (6.16) by and then apply the divergence operator the variable coefficient Poisson equation is given by: n+ 1 n+ 1 n+ 1 n+ 1 * 2ρ n 12 ( ρ ) ( ρ + u u ) (6.17) p = n+ 1 n ρ + ρ Δt where the continuity constraint at t n+ 1 is obtained from the time-discretized continuity equation is given as: 1 n+ 1 1 n 1 n 1 Π1ρ Π 0ρ +Π 1ρ (6.18) n+ 1 n+ 1 + ( ρ ũ ) = 0 Δt where Π 1 = 32, Π 0 = 2 and Π 1 = 12 are the coefficient for second order interpolation of the time derivative (for fixed time stepping). Finally the Poisson equation Eq. (6.17) can be written, n+ 1 1 n+ 1 1 n 1 n 1 2 ρ n Π1ρ Π 0ρ +Π 1ρ n+ 1 * (6.19) p n 1 n ( ρ + = ũ ) ρ + ρ Δt Δt The variable coefficient Poisson equation is solved iteratively using a Gauss-Seidel method with a multi-grid method to accelerate the convergence. More details can be found in [3]. In the ghost fluid method for finite thickness interface, the solution algorithm defers slightly from e.g. Eq. (6.19). This will be described in Sec Handling high density gradient: ghost fluid method (GFM) As introduced in Sec.3.2.5, in LES, the flame front is in most situations thinner than the grid size or the filter length. The sharp density gradient at the flame interface can lead to numerical instability especially for reacting system with high density ratio. Among different approaches discussed in [II], the flame front can be treated as a propagating discontinuity based on the ghost fluid method (GFM) formalism. The idea is therefore to divide the flow field, using the zero level-set, into two distinct regions, respectively a fresh gas zone with high density and a hot combustion product zone with low density. In each zone, all field variables are smooth and therefore easy for numerical approximation of the derivatives. The pressure and velocity are related across the flame using

55 47 jump discontinuity conditions to ensure momentum and mass conservation as shown on Fig. 14. Φ Φ b Φ [ Φ ] Φ u Figure 14. The ghost fluid method decomposition where and [ Φ ] is the jump at the interface. After [90]. x Φ u and Φ b are the unburned and burned states In this work, two variations of GFM have been considered: the first one tracks infinitely thin (IT) discontinuities [88, 89] and will be referred as GFMIT while the second one tracks interfaces of finite thickness (FT) [90] and will be referred as GFMFT. The later is in fact an extended version of the GFMIT. Both methods rely on the level-set G-equation for the front propagation as described in The GFMIT approach Following [89], conservation of mass and momentum implies the standard Rankine-Hugoniot jump conditions across the interface (for instantaneous quantities): G (6.20) ρ ( un D) = 0 with n = G ρ ( un D) 2 + p = 0 (6.21) where [ Φ ] =Φb Φ u is the jump in a quantity across the interface and n is the flame front normal vector. The normal component of the interface velocity D is defined as: D= ( un ) u + S (6.22) 0 where S is the front propagation speed ( S L, S L or S T, Δ ). In case of moving front, the tangential velocities (to the flame front) are unchanged across the interface. Denoting the mass flux in the moving reference frame (speed S) by: M = ρ un D = ρ un D (6.23) b (( ) ) u ( ) b ( ) This allows writing Eq. (6.20) to as [ M ] = 0 and by substituting D into Eq. (6.23): Starting with [ D ] = 0 and since [ M ] = 0 be written as: Eq. (6.21) can also be rewritten as: u M = ρus (6.24), the jump condition for the velocity components can [ u] 1 = M ρ n p = M ρ [ ] 2 1 (6.25) (6.26)

56 48 The unburned and respectively burned pressure and velocities can therefore be extended across the interface in the burned and unburned region (ghost cells). More details about this method can be found in [89] The GFMFT approach As discussed previously the instantaneous density and momentum can be decomposed: ρ = ρ + α ρ (6.27) u [ ] [ ] ρu= ρuuu + α ρu (6.28) where α is a Heaviside function and the jump conditions are the same as in Sec Following [90], the idea is to consider a finite thickness for the interface. The conservative filtered variables are represented through the flame brush as: ρ = α ρb + ( 1 α ) ρ u (6.29) ρu = α ρbu b + ( 1+ α ) ρuu (6.30) u where the function α varies from 1 in the burned gases to 0 in the unburned gases. The spatial filtering operation applied to Eqs. (6.27) and (6.28) gives: ρ = ρ u + α [ ρ ] (6.31) ρu = ρuu u + α [ ρu] (6.32) Although Eq. (6.31) is equivalent to Eq. (6.27), the momentum decomposition (Eq. (6.32)) is only equivalent to Eq. (6.28) if α [ ρu] α [ ρũ ]. This assumption generates an error localized in region of α cst and is proportional to the fluctuations of the speed in the unburned gases. In the flamelet regime or in the thin reaction zone regime (considered in this thesis), this is negligible with regards to the jump created by the thermal expansion [90]. If the fractional-step algorithm (Sec. 6.5) is used for the burned and unburned and the global quantities through the flame brush are considered then similarly to Eq. (6.29) and (6.30), the pressure gradient can be decomposed to as: P = α P + (1 α) P (6.33) b Eq. (6.33) leads to a modified pressure equation and the jump condition for the pressure is related to the solution algorithm. More details about the method can be found in [90]. Specific issues related to the method implementation, in the in-house CFD code, and its validation are presented in [II]. Not only the GFMIT and GFMFT methods are stable in case of high density ratio but they also allow overcoming some of the limitations for the G-equation presented in Sec For example, the stencil necessary for the rate of strain tensor can be computed with the ghost cells at the front with limited effects of the thermal expansion. u 6.7 Parallel computation Message Passing Interface (MPI) is used to parallelize the computation. At the interface boundary, variables in three grid cells in each Cartesian coordinate direction are exchanged between the neighbouring processors to keep high order discretization schemes applicable. The synchronization of the variables is done at each necessary sub-steps of the computation, e.g. after momentum and scalar transport, at all stages of the multi-grid process and finally after velocity correction.

57 49 7Chapter 7 Results and summary of publications The results of this thesis have been summarized in the eight papers listed in the preface; six of the papers are attached in the appendix of this thesis. In this chapter, a brief summary of the main results is given to highlight the work. The work can be summarized to four groups listed below: Development and validation of LES model for partially premixed flames. An experimental rig has been developed in Lund as a result of collaborations between Lund University and Cairo University. The experimental and numerical results are presented in [III-V]. As discussed in the previous chapters partially premixed flames are typically made up of premixed flames in the leading front and diffusion flames in the main body. Whereas modelling of non-premixed flames have been relatively successful with well validated models such as the flamelet approach, it is still challenging in modelling of premixed flames, especially when it comes to the triple flame fronts. The level-set G-equation approach is adopted for modelling the triple flame propagation. This approach is validated in partially premixed flames and in classical premixed flames since the database is relatively well established in the literature. Papers [I] and [II] presents the model validation and comparison of different SGS modelling approaches. It is well known that the reaction zone and hence the heat release zone in turbulent flames is not resolved in LES. The unresolved thin reaction zone in LES can cause numerical instability, especially when the density ratio is high. If the flames are thickened numerically or explicitly by using thickened flame model or by adjusting the chemistry, the structures and dynamics of the premixed flames may be affected. A ghost fluid method is implemented to investigate this question. There cases are studied, the case of hydrodynamic instability and the case of the bluff body stabilized premixed flames. The numerical evaluation is presented in [II]. LES has shown to be sensitive to the inflow turbulence. Well documented experimental data are desirable for a successful LES. However, most of the experimental data are incomplete and limited by the experimental technique. To study the sensitivity of LES results to inflow turbulence, the flow dynamics and mixing process in the near field of a partially premixed burner has been studied. The results are presented in [VI]. 7.1 Modelling of turbulent premixed flame at high density ratios In LES of premixed and partially premixed flames, the chemical reaction zones are typically thinner than the filter length and the grid size. If a spatial filtering is explicitly employed the reaction zones are typically distributed in a few LES meshes. If no explicit filtering is involved the reaction zone can be as thin as one LES grid cell. As discussed earlier this can cause numerical instability, especially when the density ratio is high. A ghost fluid method for the infinitely thin reaction zones (GFMIT) corresponding to the implicit LES filtering and for the finite thickness reaction zones due to explicit LES filtering (GFMFT) is implemented and tested in [I] and [II]. Furthermore, different modelling

58 50 concepts for premixed flames (hence for the leading fronts of the partially premixed flames) are evaluated by comparison with the experimental data. The results are presented in [I]. The Volvo Validation Rig case (VR-1) is chosen as a test case for validation of the premixed flame models and also to examine the performance of the numerical methods since experimental data including LDV measurements of the velocity [91] and CARS measurements of the temperature [92] are available for two different density ratios Problem set-up The rig is a simple rectangular cross sectional channel with a height of 120 mm and a width of 240 mm. A propane/air mixture, with an equivalence ratio of 0.6 and a mass flow rate of 0.6 kg/s, flows through a honeycomb screen that controls the level of turbulence at the inlet. The channel length from the inlet to the exit of the chamber is L=1000mm. Operating at atmospheric pressure, the flame is stabilized by a prismatic triangular shape bluff body with a side length, h=40mm positioned 320mm downstream of the inlet. More details about the geometry can be found in [I]. Two operating conditions are considered and summarized in Table 1. In particular, the density ratio varies from 3.3 to case T in [K] u in [m/s] Re S l [m/s] ρu ρb c288k c600k Table 1. Conditions of the different VR-1 cases for flame speed model comparison Summary of the corresponding papers Comparison of LES models applied to bluff body stabilized flame The performance of two conceptually different categories of LES combustion models is compared: one flamelet model (the present level-set G-equation) and four finite rate chemistry models: the PaSR, TFM, EDC and PPDF LES models. The finite rate chemistry LES models are all implemented in an unstructured production LES code based on the C++ library OpenFoam. Both codes show good agreement with experimental data for the non-reacting reference case suggesting that the subgrid modelling for the residual stress is acceptable. All LES models predict qualitatively similar density ratio effect on the flame dynamics. Close to the flame holder, in all cases, the unsteady separating vorticity on both sides of the prism rolls up, which generates a flow that wraps the flame around these regions of intense vorticity and therefore wrinkles the flame. This process is dominated by the Kelvin Helmholtz (KH) instabilities. Further downstream in case of low density ratio, the lower frequency Bénard/von-Karman (BVK) instability (observed in cold case simulations and experiments) starts to dominate the KH instabilities and thus modify the flame dynamics (asymmetric flapping of the flame). In case of high density ratio, the BVK instability effects are suppressed and the flame is rather symmetric. Further details of results are given in [I] and [II]. 7.2 Development and validation of LES model for turbulent partially premixed flames Turbulent partially premixed combustion is studied in a conical burner both numerically (model validation) and experimentally. The mixing and flow dynamics in the mixing tube is first studied using LES. A two-scalar flamelet model based on the level-set G-equation for the leading triple flame propagation and mixture fraction for the trailing diffusion is developed and validated. Two different flame stabilization mechanisms, one triple flame propagation and

59 51 one diffusion flamelet stabilization, are evaluated. Furthermore, the fuel effect and cone effect on the flame stabilization and local extinction is studied. The results are published in [III-V], and a brief summary is given below Development of the optical burner Zst x D d d i D = 9.7 mm d i = 6.8 mm d = 8 mm L D fuel air L fuel a. fuel air fuel b. Figure 15. a. Schematic illustration (left) and photo (right) of the conical flame rig. b. Schematic illustration of the concentric tubes (mixing chamber) and a mixing length of L=5D. The present burner is developed as results of collaboration between Lund University and Cairo University. The burner is similar to the conical burner studied in [93-95] with modification of the cone by using a BK7 glass to enable optical access. A schematic illustration of the burner and a photo of the flame and the cone are shown in Fig. 15a. Below the cone, the burner consists of two stainless steel tubes where the inner tube supplies air (red arrow) and the outer tube supplies fuel (blue arrow), cf. Fig. 15b. Mixing starts at the exit of the inner tube and continues within a mixing distance, L, between the exit of the inner tube and the exit of the outer tube (referred as mixing chamber). With L/D=0, the flame is non-premixed. With L/D>50, the flame is considered as fully premixed. The degree of premixing of fuel and the primary air at the exit of the tube is dependent on the mixing process and on the mixing length. 3 For the case studied numerically, the mass flow air supply is kg s and the 3 mass flow fuel (CH 4 ) supply is kg s. This gives an overall equivalence ratio ofφ = 3 and a characteristic cone inlet speed of 20 m/s. More details about the geometry and the computational domain can be found in [III]. Other mixing lengths (level of partial premixing) are studied in [IV] while behaviour of low calorific gas is studied in [V] Modelling of the mixing process in the mixing chamber The mixing process prior to combustion is important in partially premixed combustion as it can influence the stabilization and the dynamics of the flame. The fuel/air mixture can be changed from non-premixed to rich premixed as the mixing length increases [IV]. Therefore, LES is utilized to characterize the mixing process at different mixing length up to 10 D. The simulation is performed on a uniform grid with a resolution of 76μ m. In this configuration, the velocity ratio isη = 0.48 ( η = U1 U0, whereu 1 is the velocity of the annular stream and U 0 is the velocity of the central stream) whereas the diameter ratio is di D= 0.7. Due to the concentric configuration, typical mean velocity profile in annular jets is a nearly top-hat shape [96]. The velocity profile of the inner tube is one of the factors influencing the characteristics of the various flow regions in a coaxial jet and may result in a

60 52 decrease of the inner potential core [96, 97] (faster mixing). Therefore three different conditions are tested on the inflow conditions for the inner tube as presented in Table 2. case velocity profile turbulent fluctuations case1a top-hat yes case1b top-hat no case1c parabolic no Table 2. Inflow boundary conditions for the inner tube. Fig. 16 shows the radial profiles of the mean mixture fraction obtained from numerical simulations and experimental results from planar laser induced fluorescence (PLIF) of acetone taken at the exit of the mixing tube (for different mixing lengths). The influence of the inlet boundary conditions is clearly identified. The shape of the velocity profile, as expected, has a stronger influence on the mixing process further downstream of the tube as seen at L/D=7. The parabolic profile at the inner tube inlet over-predicts the mixing when compared to the experimental data whereas top-hat profiles seem to be in better agreement. The use of fluctuations at the inlet gives slightly better results close to the inlet up to L/D=3. Indeed, it enhances the shear layer instabilities close to the inlet [VI]. Both numerical and experimental results show a low mixture fraction close to the jet centre line. Further downstream, at L/D=5 and L/D=7, both results agree well. At the centre line, the mixture fraction is likely to fluctuate around Z st as it is observed in the cone simulation [III]. L/D=3 L/D=5 L/D=7 Figure 16. Mean mixture fraction radial profiles at different locations downstream of the mixing tube for case1a (- -), case1b (- -), case1c ( ) and experimental PLIF-acetone measurements (x and +). Fig. 17 shows large coherent structures using the λ 2 visualization technique. Close to the inlets, the development of quasi axi-symmetric vortex ring in the shear layer is clearly seen. This is due to Kelvin-Helmholtz instabilities. Figure 17. Instantaneous isosurface of λ2 (yellow) and stoichiometric mixture fraction (red). Around L/D=1, streamwise vortices begin to form in counter-rotating pairs. We can define different steps in the mixing process in the tube. In a short region, close to the inlets, where no

61 53 large structures have formed the mixing process is controlled by molecular diffusion. Then the formation of Kelvin-Helmholtz vortices allows large scale efficient mixing as observed on Fig. 18a. In addition to this process, the formation of streamwise vortices induces ejection of air into the fuel stream characterized by mushroom type structures as shown in Fig. 18b. a. b. Figure 18. a. Instantaneous mixture fraction field (gray scale) in the central plane. b. Instantaneous mixture fraction field (gray scale) isosurface of λ2 (yellow) in the transverse section located at L/D=1.2. From L/D=3, fewer large coherent structures are observed. In fact the velocity profiles (not shown here) quickly exhibit a fully developed turbulent profile. However large pockets of stoichiometric mixture fraction are still observed far downstream, e.g. L/D=10. By saving time dependent u, v, w velocity components and mixture fraction Z at a given plane, for example L/D=5, an inflow boundary condition can be generated for the partially premixed flame case with L/D=5. In this way, turbulent structures and the mixing dynamics (fuel pockets) are preserved for the conical partially premixed flame simulations discussed below Summary of the corresponding papers Experimental and numerical study of a conical turbulent partially premixed flame In the cone configuration described in Sec , the structure and stabilization mechanism were studied using laser diagnostics and LES. Planar laser induced fluorescence (PLIF) of CH showed thin CH layers inside (stabilization region) and outside the cone (trailing edge) which is supportive to the flamelet combustion concept. Inside the cone the flame was found to be stabilized above the nozzle, according to both CH PLIF and chemiluminescence images. This is well understood from LES results. In LES it was found that near the inside wall of the cone there is a reversal flow towards the nozzle. The high speed flow of fuel/air mixture from the nozzle forms a shear layer where large scale vortices exist. These vortices effectively stabilize the triple-flame front. The triple-flame model correctly predicted the flame structures and stabilization inside the cone. It also showed promising capacity in predicting the flame fine details such as the local structure of the CH. Further details are given in [III]. Effect of partial premixing on stabilization and local extinction of turbulent methane/air flames In the cone configuration described in Sec the degree of partial premixing of fuel and air was controlled by varying the length of the mixing chamber (L). It is found that in general partially premixed flames are more stable when the level of partial premixing of air to the fuel stream decreases. With the operating conditions considered in this paper, an optimal level of partial premixing is found where the flame is most stable. It was close to this condition that the flamelet LES model, predicted rather well the mean flow field and the mechanism for flame stabilization could be explained. However this model based on the stationary flamelet tabulation method using the scalar dissipation rate as quenching mechanism was unable to

62 54 predict the local extinction observed experimentally even in stable operating conditions. Experimentally, the local flame extinctions were found in locations where locally high velocity flows impinge to the flame. By increasing premixing air to fuel stream successively, local extinction holes are found to develop leading to eventual flame blow-out. Further details are given in [IV]. Structures and stabilization of low calorific value gas turbulent partially premixed flames in a conical burner The stability characteristics of partially premixed turbulent flames of different fuels in the conical burner and a corresponding jet burner (without the cone) are investigated further for different biomass derived fuels. The fuels are methane (purified biogas from biomass digestion), methane diluted with nitrogen, and mixtures of CH 4, CO, CO 2, H 2 and N 2, with a LHV of about 12 and 24 MJ/Nm 3 simulating typical products derived from biomass gasification. The structures of the conical flames at different overall equivalence ratios and Reynolds numbers are studied using simultaneous PLIF of acetone and OH. The flow field and mixing process are simulated using large eddy simulation. The methane/air and diluted methane/air jet flames show multiple stability regimes, burner rim-attached flames, lifted flames, and flame blowout. There is also a hysteresis regime where the flames can either rim-attached or lifted, depending on the initial state of the flames. The LCV gas jet flames show different characteristics. The flames are more liable to rim-attached, and the blowout jet velocity is higher than the corresponding methane/air flames. The LCV gas flames are more stable than the methane/air flames although the LHV is lower for the LCV gases. This characteristic is attributed to the presence of hydrogen in the mixture. The conical flames exhibit much wider stability domain in terms of overall equivalence ratio and burner exit flow velocity than the corresponding jet flames without the cone. The stability of the conical flames is less sensitive to the gas composition, equivalence ratio and exit flow velocity. This is fundamentally different from the jet flames. PLIF measurements and LES flow structures are used to explain the mechanisms of the enhanced flame stabilization in the conical burner. More details are referred to [V]. 7.3 Influence of inflow boundary in LES of low Re jets In many flow configurations LES results were shown to be sensitive to inflow conditions so that modelling of the inflow boundary conditions is often a vital ingredient for a successful simulation. Large eddy simulations of a turbulent jet flow with rich premixed methane/air mixture discharging to ambient air are carried out. The flow configuration corresponds to a lifted partially premixed flame that has relevance to modern gas turbine and piston engine combustion. The focus of this paper is on the dynamic behaviour and the mixing process of the fuel jet flow with the ambient air in the near field of the jet. The aim of this study is to investigate the sensitivity of the flow dynamics and mixing to inflow conditions. The decay of the core flow, the instabilities in the shear layer, and the different modes of the flow dynamics are systematically analyzed for different inflow conditions. It is concluded that with larger inflow turbulence length scales, the onset of shear layer instability is earlier, the mixing between the fuel flow and the ambient air is enhanced, and the core flow region is decreased. When the inflow turbulence is highly anisotropic, e.g., with one component of the Reynolds stress much higher than the others, the onset of shear layer instability is delayed and the mixing becomes slower. The mean profile of the inflow is shown to significantly affect the flow due to the difference of velocity gradient in the jet shear layer. Further details are given in [VI].

63 Contributions by the candidate to the papers in this thesis The results reported in this thesis are based on experimental data and LES data. The candidate of this thesis took part in some experiments and planning of the experimental cases for the results in [IV, V]. The LES results in [II-VI] are obtained by the candidate. In [I], the LES results from the level-set G-equation model are obtained by the candidate. The results of other models are obtained by Christer Fureby at FOI. The candidate took part in the writing of all papers.

64 56

65 57 8Chapter 8 Concluding remarks and future perspectives Large eddy simulations of turbulent premixed and partially premixed flames are carried out in this thesis work. The main effort has been placed on flames in the flamelet regimes. Level-set G-equation flamelet models are validated on premixed flames stabilized by a bluff body where two different density ratios are investigated. The results show that at different density ratios the flame dynamics are different. At low density ratios, the flame and flow show Bénard/von-Karman (BVK) instability and large-scale low frequency flame oscillations at the downstream which are captured by LES. At high density ratios, this instability is suppressed and only small-scale wrinkling on the flame surface is predicted. This finding is confirmed by the experimental results, thereby proving that the models and LES are readily useful for studying unsteady flame dynamics in the turbulent premixed flames. To capture the high density ratio flames, a numerically robust ghost fluid method is evaluated. Based on this, the effect of flame thickening on the flame dynamics is investigated. The level-set G-equation model is then extended to simulate the leading front (triple flame) in the partially premixed flames. A two-scalar flamelet model, based on the G-equation and the mixture fraction based diffusion flamelet model, is developed for partially premixed flames. The model is evaluated in partially premixed flames stabilized in a conical burner and showed good agreement with the inner reaction zone structures characterized by the CH layer. The model is also shown to be able to simulate the flame stabilization in the conical burner. The experimental results in the conical burner showed that the CH 2 O layers are substantially thick. This thicker CH 2 O layers can not be reproduced by the flamelet models. Furthermore, at high Reynolds numbers, there are local extinctions in the trailing part of the partially premixed flames. The present state-of-art models such as the two-scalar flamelet models failed to predict the flame extinction. The formation of the flame holes and the healing of the flame holes are not well understood and there is a lack of correlation between local extinction and flame hole healing with the local flow quantities such as flame stretch rate under realistic turbulent flame conditions. Research effort is needed to develop reliable models for partially premixed flames under the condition of local flame extinction and reignition. It is shown in this thesis that LES of turbulent mixing in the partially premixed jet configuration is very sensitive to the inflow conditions. Not only the mean flow profiles but also the Reynolds stress and turbulence length scales are shown to alter the dynamics of the shear layer flows and hence the mixing with the ambient air. This calls for the development of numerical inflow turbulence generation methods and also more detailed experimental data at the inlet in a well documented experimental database.

66 58 Acknowledgements This work was carried out at the Division of Fluid Mechanics, Faculty of Engineering at Lund University (Sweden). The work was sponsored by the Swedish Research Council (VR), the Swedish Energy Authority (STEM) and the Center for Combustion Science and Technology (CeCOST). The computations were run through the Swedish National Infrastructure for Computing (SNIC) in the Center for Scientific and Technical Computing for Research at Lund University (LUNARC) and in the High Performance Computing Center North (HPC2N). I am very grateful to my supervisor, Xue-Song Bai, whose encouragements, guidance, patient explanations and support enabled me to develop my understanding in this challenging field of combustion modelling. Also I would like to thank my co-supervisor, Laszlo Fuchs, for his support. I would like to thank Christer Fureby, Mohy Saad Mansour, Zhongshan Li, Sven-Inge Möller, Christophe Duwig and Julien Savre for the nice and fruitful collaborations along the different projects. I also want to express my gratitude to Bo Li, Zhiwei Sun, Andreas Lantz, Robert Collin, Seyed Mohammad Hosseini, Beibei Yan and Changye Liu for their great work in the lab and their contributions to the papers. Special thanks should be given to my friends and colleagues at the division of Fluid mechanics and at the division of combustion physics, for making those years as interesting and fun as they could be throughout collaborations, seminars, after works and Monday soccer rituals. In particular, I would like to mention Kalle for introducing me to the PhD student world, Rixin for sharing his knowledge in numerics with me, Tobias for his friendliness, Robert for giving useful computer tips, Edouard for being French with me, Holger for having better stories than the daily newspaper and Piero for invading my office in a friendly way. To those who I have not mentioned by name, you know who you are, I also express my sincere appreciation. I am very happy for meeting you all as I have learned a lot from you in fluid mechanics, laser diagnostics and not at least life. Je voudrais remercier mes amis, qui se reconnaitront ici, et ma famille pour leur soutien indéfectible dans les mauvais comme dans les bons moments. En particulier, je remercie mon père Christian, mes frères Xavier et Marc ainsi que ma mère Francoise à qui je voudrais dédier cette thèse tant elle a été et restera, pour nous, une grande source d inspiration dans le travail et surtout dans la vie. At last but not least, I would like to thank min sambo, Sara, for always being supportive and for her patience and her love during those years.

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73 Paper I

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75 Comparison of LES Models Applied to a Bluff Body Stabilized Flame E. Baudoin 1,R.Yu 2, K.J. Nogenmyr 3,X.SBai 4, Lund University, SE , Lund, Sweden C. Fureby 5, The Swedish Defense Research Agency FOI, SE Tumba Stockholm, Sweden Present-day demands on combustion equipment are increasing the need for improved understanding and prediction of turbulent combustion. Large Eddy Simulation (LES), in which the large-scale flow is resolved on the grid, leaving only the small-scale flow to be modeled, provides a natural framework for combustion calculations as the transient nature of the flow is resolved. In most situations, however, the flame is thinner than the LES grid, and subgrid modeling is required to handle the turbulence-chemistry interaction. Here, we examine the predictive capabilities and the theoretical links between LES flamelet models, such as the G-equation model (G-LES), and LES finite rate chemistry models, such as the Thickened Flame Model (TFM-LES), the Partially Stirred Reactor model (PaSR-LES), the Eddy Dissipation Concept (EDC-LES) model and a Presumed Probability Density Function (PPDF-LES) model. The models are described, and theoretical links between the models are discussed in terms of the turbulent flame speed and flame thickness. The performance of the different models is explored by applying the models to study a bluff-body stabilized flame and the resulting predictions are compared with experimental data for two operating conditions. B = subgrid stress tensor b E = subgrid energy flux vector b i = subgrid species flux vector c = progress variable Nomenclature 1 PhD Student, Dept. of Energy Sciences, 2 Researcher, Dept. of Energy Sciences 3 Researcher, Dept. of Energy Sciences, Present Address: Dept. of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong 4 Professor, Dept. of Energy Sciences 5 Research Director, Defense Security Systems Technology, Grindsjön Research Center, AIAA Associate Fellow Corresponding author 1

76 C pi = specific heat of specie i c X = model coefficients D = rate-of-strain tensor Da = Damköhler number D i = species (mass) diffusivity E = energy spectra E = total energy F B f = subgrid flux function F C f = convective flux function F D f = diffusive flux function G = kinematic G-field h = enthalpy h = height of the bluff body flame holder h i,f = enthalpy of formation for species i h = heat flux vector j i = species mass-flux k = subgrid kinetic energy Ka = Karlowitz number k f/b,j = forward / backward rate constant of reaction j Ma = Mach number M i = molar mass of specie i n = refraction index p = pressure P ij = stoichiometric matrix Pr = Prandtl number Q = radiative heat loss R = gas constant Re = Reynolds number S = viscous stress tensor Sc i = Schmidt number s u = laminar flame speed T = temperature u = vector of unknown dependent variables v = velocity ẇ i = species reaction rate ẇ j = reaction rate of species i in reaction j Y i = species mass fraction x = axial flow coordinate z = mixture fraction Greek V P = cell volume = LES filter width = efficiency function = exponential factor u = laminar flame thickness f = flame thickness = subgrid dissipation = standard deviation 2 = second eigenvalue of the velocity gradient tensor = subgrid volume fraction = thermal diffusivity = equivalence ratio μ = dynamic viscosity = kinematic viscosity ( )= probability density function i = vector of species = flux limiter 2

77 = density = time scale = velocity scale = length scale = reacting volume fraction = model constant Sub- and superscripts k = subgrid t = turbulence * = fine structures 0 = surroundings K = Kolmogorov I = integral m = mixing c = chemical = subgrid ' = fluctuation rms = root-mean-square ad = adiabatic eff = effective SGS property SGS = subgrid in = inflow max = maximum = Spatial filtering = density weighted spatial filtering < > = time average I. Introduction and Background Few have escaped the recent year s intense debate on climate change. Several reports highlighting how anthropological activities threaten to create a significant change of the global climate has received extensive media attention. This has led to tougher demands from both authorities and consumers for manufacturers to increase energy efficiency and find alternate strategies. For manufacturers of combustion devices, such as gas turbines, ICengines and boilers, predictive modeling of turbulent reacting flows has proven a valuable and cost effective tool for evaluating new designs. For these reasons large efforts has been invested in model development during the recent years. This development has also been supported by the constantly increasing available computational power. Still, however, the interaction of turbulence with physical processes such as chemical kinetics poses a great challenge, [1]. Present-day engineering combustion models usually rely on Reynolds Average Navier Stokes (RANS) models together with mixing-based, flamelet or presumed shaped Probability Density Function (PDF) combustion models, [2-3]. In spite of the simplicity of these models, they have been successful in predicting some gross features of combustion, such as combustor exit temperature profiles, whereas they are unable 3

78 to predict transient phenomena such as flameout and relight in gas turbines, combustion instabilities in gas turbines and afterburners, cycle-to-cycle variations in IC engines and pollutants formation. Significant advances in modeling non-reactive turbulent flows are now possible with the development of Large Eddy Simulation (LES), [4-5], and similar methods, [6]. The philosophy behind LES is to explicitly solve for the large (energetic) scales of the flow, directly affected by boundary conditions, whilst modeling the small (less energetic) scales of motion. This gives LES an advantage over RANS in that the unsteadiness of the (convectively mixing) flow is taken explicitly into account. LES of reacting flows is under intense development, [7-11], but for the resolution applied in typical engineering LES the interaction of turbulent mixing with chemical reactions occurs at the subgrid- and smallest resolved scales. In practice, this means that the reaction rates depend on both the resolved and unresolved scales, with only the resolved scales available for closure modeling. For historical and practical reasons, the development of LES combustion models has to a large extent been based on models and methods from RANS, [2], usually leading to improved predictions, [12]. This is due primarily to that in LES the large flow and mixing scales are resolved, in spite of that the flame cannot usually be resolved on the LES grid. Improved LES combustion models are however needed to deal with new concepts such as synthetic or bio-fuel combustion or plasma-assisted combustion. In this study we aim at investigating the predictive capabilities and the theoretical links between flamelet models, here represented by a G-equation model, [13], and finite rate chemistry models such as the thickened flame model, [14], or the partially stirred reactor model, [11]. Two different codes, with different numerical methods and grid generation capabilities will be used as examples of present-day LES research and production codes. The different LES predictions will be compared with each other, and with detailed laboratory measurement data from an atmospheric lean bluff body stabilized premixed propane-air flame, [15-17]. II. The Volvo Validation Rig Combustor In this investigation we have chosen to use the Volvo Validation Rig, [15-17], as test case due to the wealth of experimental data including high speed imaging, conventional gas analysis, LDV measurements of the velocity, [16], and CARS measurements of the temperature, [17]. The Validation rig is designed to provide experimental data for validation of computational fluid dynamic and combustion models as well as for providing data that will aid in the advancement of understanding turbulent combustion. The rig consists of a rectilinear channel, with a rectangular cross-section, divided into an inlet section and a combustor section, as shown schematically 4

79 in figure 1a. The inlet section is used for flow straightening, turbulence control, fuel injection, mixing and seeding. The air entering the inlet section is distributed over the cross-section by a critical orifice plate that also isolates the combustor acoustically from the air supply system. Gaseous propane (C 3 H 8 ) is injected and premixed with the air just upstream of the critical orifice plate by a multi-orifice injector and the turbulence level is controlled by honeycomb screens throughout the inlet section. The combustor section is of modular design with the walls split into several interchangeable sections with the top and bottom walls being water cooled and the side walls being air cooled to accommodate the quartz windows for optical access as shown in figure 1b. The premixed C 3 H 8 -air flame is stabilized behind a wedge-shaped flameholder of height h=0.04 m, and the rectilinear combustor discharge into a large diameter circular duct. Experimental data are available for two baseline cases characterized by the inflow pressures, velocities and temperatures: Case I, p 0 =101 kpa, v 0 =17.3 m/s and T 0 =288.0 K, and Case II, p 0 =101 kpa, v 0 =34.3m/sandT 0 =600.0 K, corresponding to Re 46,000 and 28,000, respectively. For both cases the inflow turbulence level is about 3% and the equivalence ratio for both cases is The combustor geometry is prone to combustion instabilities as observed experimentally in [15-17] and computationally in [8, 12, 18-22]. (a) (b) Figure 1. A schematic illustration of the Volvo Validation Rig as used in the present study (a) and a photography of the operating rig and experimental devices (b). III. Computational Models The reactive flow equations are the balance equations of mass, momentum and energy describing convection, species diffusion and chemical reactions, [23]. In LES all variables are decomposed into resolved and unresolved (subgrid) components by a spatial filtering operation such that f= f+ f,where f= f/ is the Favré filtered variable. By filtering the reactive flow equations, and rearranging terms we have that, t ( ) + ( ṽ) = 0, t ( Ỹi) + ( ṽỹi) = (ji bi) + ẇi, t ( ṽ) + ( ṽ ṽ) = p + (S B), t ( Ẽ) + ( ṽẽ) = ( pṽ + Sṽ + h b E). (1) 5

80 Here, is the density, v the velocity, p the pressure, S the viscous stress tensor, E=h p/ v 2 the total energy, h the enthalpy, h the heat flux vector, Y i the species mass-fraction, ẇ i the species reaction-rate and j i the species mass-flux. The unresolved transport processes are concealed in the subgrid stress tensor and flux vectors B= (v v ~ ṽ ṽ), bi= (vy ~ i ṽ Ỹ i) and b E= (ve ~ ṽẽ), which result from filtering of the convective terms. Following [24] we assume the gas mixture to be linear viscous with Fourier heat conduction, Fickian diffusion and Arrhenius chemistry, and following [11] we neglect the presumably small subgrid contributions to the constitutive equations, so that j i D i Ỹi, p R T, S 2μ D D and h T, where R is the (composition dependent) gas constant and D= 1 2 ( ṽ+ ṽ T ) the rate-of-strain tensor. Here, Stokes assumption is used, with μ being the molecular viscosity following Southerland s law. The species and thermal diffusivities are D i=μ/sc i and =μ/pr, respectively, where Sc i and Pr are the Schmidt and Prandtl numbers, whereas the temperature, T, results from inverting the thermal equation-of-state, Ẽ h p/ + 1 2ṽ2,inwhich h= M i=1 Ỹ ih i,f + i=1 M Ỹ i C P,i (T) dt. The filtered reaction rate of specie i is ẇ i=m i N J=1 P ijẇ j,wherem i is the molar mass, P ij the stoichiometric coefficients, ẇ j =k f,j N i=1 ( Y i ) Pij k b,j N i=1 ( Y i ) Pij the reaction rate of the j th reaction step and k f/b,j(t) the forward and backward rate constants, respectively. In order to close the LES equations we need to provide models for the subgrid stress tensor B and flux vectors b i and b E, and for the filtered species reaction rate, ẇ i, taking into account the turbulence chemistry interactions manifested by the non-linearity of ẇ i. Concerning the subgrid stress tensor and flux vectors, we observe that these terms are not unique for reactive flows and closure models can be acquired from the plethora of subgrid models available for non-reactive flows, [4]. Due to the well-known difficulty of modeling the filtered species reaction rates, [2-3], together with the issue of finding sufficiently accurate reduced reaction mechanisms, [1], turbulent combustion models are often based on a different set of equations, with the species equation replaced by modeled equations for passive and/or reactive scalars. Four types of models may be identified: (i) models based on turbulent mixing descriptions, [25-26], (ii) models based on flame front topology together with flamelet modeling, [27-28], (iii) models based on single-point Probability Density Function (PDF) of scalar fields and geometrical flame surface analysis, [7, 29], and various finite rate chemistry models, [8, 10-11, 14, 20, 24]. Here, we will examine both flamelet and finite rate chemistry models, differing primarily in the way that the turbulence-chemistry interactions are represented. In the flamelet models the flame is considered thin compared to the length scales of the flow, and the flame is then viewed as an interface between fuel and oxidizer T 6

81 (for non-premixed combustion), and between reactants and products (for premixed combustion). Because of the separation of scales, it is convenient to decouple the treatment of the flow and interface from that of the chemistry, which can be represented by the unstretched laminar flame speed, s 0 u, and by mapping the structure of a 1D laminar flame onto the normal of the interface, closure of the flamelet model is achieved, [13, 30]. In the finite rate chemistry approach, different methods are used to estimate ẇ j, which, due to the exponential temperature dependence, are non-linear, and thus hard to model. However, once this problem is dealt with the prospects of the finite rate chemistry approaches are excellent since they can handle not only non-unity Le-number cases but also different types of flames (diffusion, premixed and non-premixed) in the same way with identical submodels. A. Flamelet Level-Set G-Equation LES Model In this flamelet approach, the motion of the reaction zone is captured using a level-set approach, [13, 31], with the zero level-set representing the flame, governed by the G-equation, t (G) + (Gṽ) = s sgs G, (2) where s sgs is the propagation velocity of the zero level-set on the resolved LES scales. In order to consistently use the G-field as flame coordinate, the G field must determine the position of and the normal distance to the flame. This is performed in a re-normalization step, where G =1 is enforced under the boundary condition that the zero level-set should be left undisturbed. A recent survey of different re-initialization schemes can be found in [32]. In order to close the problem, an explicit expression for s sgs is needed as a function of available data, [33]. The effects of the unresolved scales must be taken into account in s sgs, so that this modeling must include the wrinkling by all scales down to the flame thickness, and the effects occurring at scales smaller than the flame thickness, such as the increase in flame speed by increased diffusivity of heat and mass in the flame, [27]. These intertwined processes have proven difficult to model, with modeling based on the unstretched laminar flame speed, s u 0, together with an estimate of the unresolved scales, [13, 21]. A wrinkling factor based model is used so that s sgs= s u 0. Following the work of Charlette et al., [33], the wrinkling factor is modeled using, s sgs /s 0 u = ( / l, v /s 0 u,re ) = (1 + min[ / l, v /s 0 u ]), (3) where is the efficiency function determined according to [34] with =0.80. The laminar flame thickness l is respectively 0.50 mm and 0.25 mm for case I and case II and is therefore much smaller than the grid size. Due to the constant equivalence ratio, the laminar flame speed s u 0 isassumedtobeconstantandis0.12m/sand0.67 7

82 m/s for Cases I and II, respectively, as determined from a 1D freely propagating flame configuration using the GRI mechanism and the FlameMaster code, [35]. The temperature, density and species mass fractions are tabulated in flamelet libraries as functions of the distance function, G, to the inner-layer defined at the position where CH radicals are maximum. The subgrid scale velocity fluctuation is deduced from the subgrid scale kinetic energy with v = 2k/3. The subgrid scale kinetic energy, k, is obtained from the One Equation Eddy Viscosity Model (OEEVM), [36], in which a modelled transport equation is solved for k. The density coupling and the handling of the sharp density gradient at the flame interface needs specific care: in this investigation we have adopted the approach of Nguyen et al, [37], and divided the flow, using G=0, into two regions: a reactant zone with high density and a product zone with low density. Pressure and velocity are modeled across the flame using jump discontinuity conditions to ensure momentum and mass conservation. The closure for the unresolved transport terms, i.e. the subgrid stress tensor and flux vector in the momentum and energy equations, are provided implicitly by the numerical algorithm and therefore no explicit model has been applied. This approach has been studied extensively recently, [38], and is generally acceptable if the flow is reasonably well-resolved, and only a limited fraction of the total turbulent kinetic energy is left unresolved. The requirement on the subgrid model is hence mainly to drain kinetic energy from the re-solved scales at the correct rate; a constraint fulfilled by the odd-ordered weakly dissipative discretization of the convective term in the momentum and energy equations, as discussed in more detail in [5]. B. Finite Rate Chemistry LES Modeling In the finite chemistry approach different techniques are employed to model the low-pass filtered reaction rates ẇ i =M i N j=1 P ijẇ j in the species equations (1 2 ), which are highly non-linear, and thus difficult to model. All finite rate chemistry models will require a reaction mechanism and for this study, involving CH 4 -air combustion, we use a reduced two-step mechanism with the rate parameters tuned to mimic the GRI 2.1 mechanism, C 3H O2 3CO + 4H2O 2 with CO + 1 O2 CO2 2 with ẇ1 = 1.5 A 1e Ta1 /T Y1 C3H8 Y0.5 O2, ẇ2 = 1.5 A 2e Ta2 /T Y 1 CO Y0.5 O2, (4) where A 1= m 1.5 s 1 kg 1.5 mol, T a1=10072 K, A 2= m 1.5 s 1 kg 1.5 mol and T a2=6047 K. Associated with the reaction mechanism is the treatment of molecular diffusion. Most chemistry packages use polynomial fits for the diffusion coefficients, D i. This technique is accurate but expensive and can be replaced by a 8

83 simple model based on that the Schmidt numbers, Sc i=μ/d i, are virtually constant. The diffusion coefficients are thus defined as D i=μ/sc i. This model results in that s 0 u 0.13 and 0.64 for the conditions of Cases I and II, respectively. Based on the results of previous studies, [10-11], four finite rate chemistry models will be used: the Thickened Flame Model (TFM), the Eddy Dissipation Concept (EDC), the Partially Stirred Reactor (PaSR) model and a Presumed Probability Density Function (PPDF) model. O Rourke & Bracco, [39], were the first to notice that the unstretched laminar flame speed, s 0 u Dẇ/ 2, and flame thickness, l D/ /s 0 u,whered is an appropriate diffusivity and ẇ an appropriate reaction rate, can be used to rescale the flame thickness whilst preserving the laminar flame speed. This idea can then be used to artificially thicken the flame so that it can be resolved on the LES grid whilst maintaining s 0 u.moreprecisely, by reducing ẇ by a factor F, the laminar flame speed remains unchanged if D is increased by a factor F such that the laminar flame thickness becomes increased by a factor F, and from (2 1 ) it follows that, t ( Ỹi) + ( ṽỹi) = (FDi Ỹi) + F 1 M i M j=1 P ijẇ j(,ỹk, T). (5) The wrinkling of the thickened flame is underestimated by a factor of E= ( l)/ ( lf ),where is the wrinkling and lf is the resolved flame thickness. Following Colin et al, [14], this can be remedied by increasing the flame speed by a factor E since s t0 =Es u 0, which in turn implies that both D and ẇ should be multiplied with E so that, t ( Ỹi) + ( ṽỹi) = (FEDi Ỹi) + F 1 EM i M j=1 P ijẇ j(,ỹk, T). (6) For the purpose of this study we use the fractal flame wrinkling model of Fureby, [40], to model at both scales l0 and lf. Not that the diffusivity ED in (6) may be decomposed as ED=D+(E 1)D which corresponds to the sum of the molecular diffusivity D and a subgrid diffusivity (E 1)D. More recently, an improved version of the TFM have been developed by Legier et al, [41], who proposed to evaluate F dynamically. In the EDC model, [42], the flow is divided into fine structures (*) in which mixing and reactions are assumed to take place and surroundings ( O ) dominated by large-scale flow structures. The conditions in the fine structures and surroundings are related through the (subgrid) balance equations (Y i * Y i0 )/ * ẇ i (,Y i *,T * ) and N i=1 (Y i *h* Y i i0 h i0 )/ * = N i=1 h i,f ẇ i (,Y i *,T * ),inwhich * is the subgrid time. By defining the resolved fields ( ~ ) as Ỹi= * Y i *+(1 * )Y i 0 and T= * T * +(1 * )T 0, with * being the reacting volume fraction, (Y* i Ỹi) = (1 * ) * ẇ i (,Y i *,T * ), N i=1 (Y i *h*(t * i ) Ỹi h i ( T)) = (1 * ) * N i=1 h i,f ẇ i (,Y i *,T * ). (7) Here, * and * are estimated from a model of the energy cascade (based on the K41 hypothesis]) involving the 9

84 fine structure velocity v * =( ) 1/4 v K and length scales l * =(125 2 ) 1/4 l K, and the fine structure dissipation * =6 2 (v * ) 3 /l *,wherev K and l K are the Kolmogorov velocity and length scales and a model constant, [42]. The value of can be obtained from equating the subgrid dissipation and the dissipation at the first level of the cascade process, c k 3/2 / =6 2 ( v) 3 /,sothat 0.57resultinginthat v * v K and l * 2.52l K. The subgrid residence time and internal intermittency factor are defined in the energy cascade model as * = 1 2 l* /v * and * =(v * / v) 3, respectively, so that * 1.23( μ/ k 3/2 ) 1/2 and * 1.02(μ/ k 1/2 ) 3/4, respectively. The filtered species transport equations can then be expressed as, t ( Ỹi)+ ( ṽỹi)= (D i Ỹi b i )+M i M j=1 P ij [ * ẇ j (,Y i *,T * )+(1 * )ẇ j (,Y i0,t 0 )], (8) in which the second reaction rate term usually can be neglected. In the PaSR model, [43], the flow is also divided into fine structures (*) and surroundings ( 0 ). The finestructures form topologically complex regions, composed of a muddle of interacting tube-, ribbon- and sheetlike structures, [44], in which most of the viscous dissipation and mixing take place. Since most of the mixing occurs in the fine-structures, the reactions also take place here as the reactants are mixed at scales down to the molecular scales. This implies that, ẇ j(,t,y i) = (,T,Y i)ẇ j(,t,y i) d dtdy i = * ẇ j(,t *,Y * i ) + (1 * )ẇ j(,t 0,Y i0 ), (9) T Yi in which * is the reacting subgrid volume fraction. The conditions in the fine structures and surroundings are related, as for the EDC model, through the subgrid balance equations (Y i * Y i0 )/ * =ẇ i (,Y i *,T * ) and N i=1 (Y i *h* Y i i0 h i0 )/ * = i=1 N h i,f ẇ i (,Y i *,T * ),which,using Ỹi= * Y i *+(1 * )Y i 0 becomes, (Y* i Ỹi) = (1 * ) * ẇ i (,Y i *,T * ), N i=1 (Y * i h * i (T * ) Ỹi h i ( T)) = (1 * ) * N i=1 h i,f ẇ i (,Y * i,t * ). (10) The reacting volume fraction, *, is, following [44], assumed to be identical to the intermittency, and thus modeled as the ratio of the reacting volume to the local cell volume * = c / c / *,inwhich * is the sum of the chemical and mixing time scales * = c+ m. The chemical time-scale, *, scales with the laminar flame thickness and flame speed such that, * u /s u D/s 2 u,inwhich D= /Pr. The mixing time scale needs to represent all time scale between the integral time-scale, = / v, and the Kolmogorov time-scale, K=( / ) 1/2, with =c k 3/2 /, which can be re-expressed as K= 1 1/2 1/2 v 3/2 with =c1/2 ( 3 2 )3/ Combining these time scales the results in that m= K 1/4 3/4 v 5/4 so that * =( / * )( 3/4 v 5/4 )/(s 2 u 3/4 + 3/4 v 5/4 ). The filtered species transport equations can then finally be expressed as, 10

85 t ( Ỹi)+ ( ṽỹi)= (D i Ỹi b i )+M i M j=1 P ij [ * ẇ j (,Y i *,T * )+(1 * )ẇ j (,Y i0,t 0 )], (11) in which the second reaction rate term usually can be neglected. In the PPDF model the filtered reaction rate is expressed in terms of a presumed joint PDF (,T,Y),in which Y=[Y 1, Y N] T,sothat ẇ i=m i j=1 M P ij (,T,Y)ẇ j(,t,y) d dtdy. The choice of PDF is critical since it needs to represent very different flow and mixing states and at the same time allow analytical manipulations. To simplify the evaluation of the integrals, statistical independence of density, temperature and species mass fractions is assumed so that, ẇ i=m i M j=1 P ij 0 ( ) 0 T (T) 0 Y (Y)ẇ j(,t,y) d dtdy, which often is further simplified by assuming that ( )= ( ). The selection of the PDFs is not obvious and in this study a multivariate -PDF, [45], and a clipped Gaussian PDF, [46], are used. More precisely, 1 Y (Y 1,,Y N) = ( p=1 N p ) [ (1 N N p=1 ( p ) p=1y p) N p=1 Y p 1 p ], T (T) = 1 Tin T erfc( 2 2 ) (Tin T) T 2 2 T e 1 2 ( T T T )2 [H(T T in) H(T T ad )]+ 1 Tmax T erfc( ) (T ad T), (12) T in which, and H are the Gamma, Dirac and Heavyside functions, respectively, T ad the adiabatic flame temperature, and p=ỹp[ N i=1ỹ i (1 Ỹi)/ Y 2 1], p 0, Y2 = N p=1 Y p 2 and T2 = T 2. The multi-variate -PDF (12 1 )is fully defined by the species mass fractions, Ỹ i, and the sum of the subgrid mass-fraction variances, Y2,andis known to represent a spectrum of PDF-shapes depending on the parameters p= p (Ỹp, Y 2 ). The clipped Gaussian PDF (12 1 ) is composed of a Gaussian PDF, characterized by the temperature, T, and its variance, T2,that is combined with two Dirac functions dealing with the tails of the distributions that exceeds the physical limits T in and T ad, respectively. For a reaction rate of the standard form ẇ j= m A je Ta,j /T N P k=1 Y kj k the evaluation of the 1 Y k P kj integrals Y(Y) N Tad 0 k=1 dy and T (T)e T in Ta,j /T Ta,j used in integrating the e /T term, can be performed analytically to result in, Erf( t ( Ỹi) + ( ṽỹi) = (Di Ỹi bi) + T ) eta /2 T Erfc( T N m=1 Pmj k=1 [( i=1 dt, with the mean value theorem of integral calculus T ) 2 2 i=1 N ( k=1 T N i )+k 1] N Pkj k=1 Ỹ k Pkj ( i +k 1)) M i M j=1p ijẇ j(, T,Ỹk ), (13) where Erf denotes the error function and Erfc=1 Erf the complementary error function. To close the unresolved transport terms in the momentum, energy and species equations in the described four finite rate chemistry models, the Mixed Model (MM), [47], defined as a linear combination of a scale similarity and a subgrid viscosity/diffusivity term have been used. This model is a simple extension of the Bardina et al model, [48], in which B= (ṽ ṽ ṽ ṽ) 2μ k DD and b f= (ṽ f ṽ f) D k f, for all scalar fields f. where the subgrid viscosity and diffusivities are μ k= c k k 1/2 and D k=μ k / f and where f is the turbulent Schmidt or 11

86 Prandtl number. Here, k is the subgrid kinetic energy, k= 1 trb, that is obtained by the One Equation Eddy Vis- 2 cosity Model (OEEVM), [36], in which a modeled transport equation is solved for k. C. Turbulent Flame Speed Estimation and Comparison The effective flame speed, s f, and the associated flame thickness, f, are interesting quantities to compare when assessing the different LES models. Following e.g. [2], the flame-speed is the eigenvalue to a representative one-dimensional (1D) steady-state reaction-diffusion equation, and generally takes the form s f= D effẇ eff / 2, where D eff is the effective thermal diffusivity and ẇ eff a representative reaction rate. Similarly, the effective flame thickness, f, can be estimated from its definition f=d eff / s f, so that for a laminar flame, s f=s u 0 and f=d/ s u 0 = l. For the G-equation model the flame wrinkling is a model parameter so that s f= s u 0, whereas the effective flame thickness becomes f=f l,inwhich F= / l. For the TFM model the effective flame thickness is f=f l, with F prescribed, so that the effective flame speed becomes s f=es u 0, in which E is the efficiency factor. For the EDC, PaSR and PPDF models, all being of similar form, in which the effective laminar flame speed is s f= s u 0,where = * (1+D k /D) or = (1+D k /D) is the flame-wrinkling factor, resulting in an effective flame thickness of f=f l in which F= (1+D k /D)/ * or F= (1+D k /D)/, with given by equation (13) is the thickening factor. The Quasi Laminar (QL) model, [24], may be interpreted as a special case of the PaSR model with * =1 and D k =0. From Table 1, summarizing the different effective flame speed and flame thickness estimates, it is evident that all FRC LES models are of the same form, with the effective turbulent diffusivity, D k, and the parameters of the turbulence chemistry interaction model, or *, respectively, or equivalently F and E in the TFM model, determining the turbulent flame speed and flame thickness. Compared to the flamelet model, essentially assuming single step chemistry, and F are instead modeled, but resulting in similar expressions for the turbulent flame speed and flame thickness. Table 1. Turbulent flame speed estimates. Model s f f F G equation s0 u F l (1+min[ / l, v /s u0 ]) / l TFM s0 u F l E / l =prescribed PaSR s0 u F l * (1+D k /D) (1+D k /D)/ * EDC s0 u F l * (1+D k /D) (1+D k /D)/ * PPDF s0 u F l (1+D k /D) (1+D k /D)/ QL s0 u l

87 IV. Numerical Methods, Boundary Conditions and Summary of Simulations In this study we apply two different LES codes, an in-house code developed at Lund University, [49], and OpenFoam, [50], to perform the simulations. The first code is a university research code whereas the second code is a multi-purpose code capable of handling arbitrary unstructured and moving grids etc. In this Section we briefly describe the two codes and the numerical methods employed together with the computational set-ups, and initial and boundary conditions used in the different simulations. The in-house code is based on a uniform staggered Cartesian grid permitting the usage of high order Finite Difference (FD) discretization schemes is used for the G-LES (see Table 2). It is an incompressible solver in which the momentum and continuity equations are discretized using a 5 th order WENO scheme for the convective terms and a 4 th order central difference for the remaining terms. The time integration is performed by a 2 nd order Adams-Bashforth scheme. The level set G equation (2) is discretized using a 3 rd order WENO scheme andintegratedintimebya3 rd order TVD Runge-Kutta method. Details on the numerical scheme and the validation of the code are presented in [51]. The OpenFoam code, [50], which is an unstructured finite volume multi-purpose code for solving partial differential equations, is used for the finite rate chemistry LES (see Table 2). This code consists of a C++ library, providing discretization, parallelization, operator and solver features on top of which different application codes are written. The C++ library is extensively verified, and includes among other features dimensional checking as an aid to code and model development. From the C++ library different families of LES solvers for incompressible, compressible and reactive flows have been used to study different applications, [11, 52-53]. The reactive LES code used here is verified using the method of manufactured solutions and validated against different DNS and experimental data sets. The LES equations are discretized using an unstructured collocated Finite Volume (FV) method using Gauss theorem so that, t (u P) + 1 V P f[f fc (u) F fd (u) + F fb (u)] = s P (u), (14) where u=[,ỹi,ṽ,ẽ]t is the vector of unknowns, and F fc (u), F fd (u), F fb (u) and s P (u) are the convective, diffusive, subgrid fluxes and the source terms, respectively. The flux-reconstruction scheme for F fc (u) is based on a2 nd order monotonicity preserving non-linear interpolation algorithm, using compact stenciles, capable of eliminating unphysical overshoots in the scalar fields. To minimize the non-orthogonality errors in the viscous and subgrid fluxes, F fd (u) and F fb (u), are split into orthogonal and non-orthogonal parts. 2 nd order central dif- 13

88 ference operators are used for the orthogonal part whilst 2 nd order face interpolation of the gradients of the variables is used for the non-orthogonal parts. Time-integration is performed by a 2 nd order semi-implicit Crank- Nicholson scheme, together with a PISO procedure in the spirit of Rhie & Chow for cell-center-ed data storage structure, [54]. Stability is imposed using compact numerical stenciles and by enforcing conservation of kinetic energy. The equations are solved sequentially, with iteration over s P (u) to obtain rapid convergence, with a fixed CFL number of 0.4. Different computational domains are used for the two codes as summarized in Table 2. The computational domains used consists of only the combustor section, extending m upstream of the flameholder trailing edge and m downstream of the flameholder trailing edge for the FD model, and 0.32 m upstream of the flameholder trailing edge and 0.68 m downstream of the flameholder trailing edge for the FV model. Both computational models employ periodic boundary conditions, 0.12 m apart, in the spanwise direction, and the FD model use a uniform structured grid with 8.4 Mcells, whereas the FV model use an unstructured grid of 2.2 Mcells, refined towards the upper and lower walls and towards the flameholder and in the shear layers after the flameholder. The characteristic spacing in the FD model is 1.0 mm whereas for the FV model it ranges from 4.0 mm to 0.4 mm. For all LES reported steady inflow conditions are used at the inlet for the velocity, temperature and combustion variables, whereas wave-transmissive outflow boundary conditions are used. For the FD model adiabatic slip wall boundary conditions are used on the upper and lower walls, as well as on the flameholder, whereas for the FV model no-slip isothermal wall boundary conditions are used on the upper and lower walls and adiabatic no-slip wall boundary conditions are used on the flameholder. In case of the FV model the wall temperature is fixed at 288 K for Case I, and 600 K for Case II, respectively. Table 2. Summary of combustion simulations. Run Case Combustion Subgrid flow Discretization Code Width Grid T wall Model Model 1 I G-Eq. Implicit FD In-house 3h 8.4 Mcells adiabatic 2 I PaSR OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcell I EDC OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcells I TFM OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcells I PPDF OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcells I PaSR OEEVM 2 nd order FV OpenFoam 3h 7.4 Mcells II G-Eq. Implicit FD In-house 3h 8.4 Mcells adiabatic 8 II PaSR OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcells II EDC OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcells II TFM OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcells II PPDF OEEVM 2 nd order FV OpenFoam 3h 2.2 Mcells II PaSR OEEVM 2 nd order FV OpenFoam 3h 7.4 Mcells

89 V. Non-Reacting Flows: Validation and Comparison Non-reacting LES have been performed for Case I in order to further validate and compare code behavior against non-reacting data, [16]. Figure 2 shows the time averaged axial velocity, ṽ x, and corresponding axial rms velocity fluctuations, ṽ rms x, at different streamwise location downstream of the bluff body. Comparing the axial distribution of ṽ x on the centerline (not shown) reveals a short recirculation region, as also indicated in the first cross-sectional profile at x/h= Further downstream, at x/h=3.75, no negative axial velocity is observed in the centre region and the flow have started to recover. Even further downstream x/h=9.40, the axial velocity exhibits a profile close to a typical developed channel flow. The rms velocity fluctuations, ṽ rms x,also present good qualitative and quantitative agreement between experimental data and the LES predictions. Two peaks are observed at x/h=3.75 corresponding to the developing shear layer. They merge further downstream and flatter ṽ rms x profiles are obtained further downstream (at x/h=3.75 and 9.40). These non-reacting reference Case I simulations support the predictive capabilities of both codes although being fundamentally different, predictions are in good agreement with the data and with each other. This is necessary for further comparison involving combustion models in reacting flow conditions. The transverse velocity signal at the centerline at x/h=0.95 exhibits a shedding frequency of 110 Hz using the FD model and 104 Hz using the FV model, whilst the experimental LDV data shows a shedding frequency of 105 Hz. Figure 2. Time averaged axial velocity and rms-fluctuation profiles at x/h=0.95, 3.75 and 9.40 for the non-reacting reference Case I, see Table 1 for additional details. Legend ( ) experimental data [16], ( ) In-house FD LES model and ( ) OpenFoam FV LES model. VI. Reacting Flows: Flow Physics, Validation and Comparison In figure 3 some illustrative results from the different LES computations are presented for Cases I and II to provide an overview of the two cases and how they are predicted by the different LES models. The flow is illus- 15

90 trated by iso-surfaces of the second-largest eigenvalue, 2, of the tensor D 2 + W 2,where D and W are the symmetric and skew-symmetric components of ṽ, respectively, [55], the axial velocity, ṽ x, and the temperature, T. The flame anchors behind the flameholder due mainly to the recirculation of hot combustion products in the wake behind the wedge flameholder, and spanwise vortices are continuously shed-off the upper and lower edges of the wedge. Compared to the non-reacting LES (not shown) and the corresponding laboratory measurement data, [15], both reacting cases results in a longer and somewhat wider region of reversed flow followed by an acceleration of the flow due to the volumetric expansion resulting from the exothermicity. For Case I all LES models predict a symmetric shedding of spanwise 3-vortices of the upper and lower edges of the wedge, resulting in a wrinkled flame that does not attach to the combustor walls. For Case II all LES models again predict a symmetric shedding of spanwise 3-vortices of the upper and lower edges of the wedge that, however, develops a staggered pattern further downstream due to vortex-vortex and vortex-flame interactions. A noteworthy delay in this process is however observed for the G-equation LES model for which the transition into a staggered pattern occurs significantly later. This affects the length of the recirculation zone and the flame-wall interactions. Indeed the higher flame speed of Case II results in a wider flame that attach to the combustor wall at x/h 10 in the finite chemistry simulations and at x/h 15 in the G-equation model. Longitudinal 12-vortices develop in both cases together with undulations of recently shed 3-vortices in regions of high strain, and subsequently both these vortex structures are modified by volumetric expansion, baroclinic torque production, temperature dependent molecular diffusion and subgrid diffusion due to small-scale turbulence. The upper and lower shear-layers are accountable for the large-scale mixing between cold reactants and hot products, forming a combustible composition, affected by the aforementioned unsteady vortical pattern that primarily governs the large-scale mixing. The high-temperature region is associated with exothermicity, and burning occurs at the fuel-rich side of the shear-layers, where also most of the mixing between cold reactants and hot products takes place. As the flame develops downstream it propagates normal to itself, causing negatively curved wrinkles to contract and positively curved wrinkles to expand being further affected by pressure fluctuations, p. As the flame oscillates, it interacts with the pressure thereby increasing p, which, in turn, affects the flame by further perturbing it. This process continues until equilibrium is reached at which acoustic instabilities may or may not have developed. Such acoustic instabilities develop if the heat release, Q, is in phase with p, such that p continues to increase. By analyzing the transverse velocity signal at the centerline at x/h=0.95 we find that the shedding (at 105 Hz) observed experimentally, [15], and computationally, [18], in the 16

91 non-reacting cases is virtually canceled in both reacting cases. However, in the shear layers, shedding occurs at ~140 Hz, which is consistent with data from Sanquer et al, [56]. The predicted dynamics of both Cases I and II are well documented experimentally, [15-17], and the agreement between the LES results shown here and data is acceptable to good. The delay observed in the dynamics development for case II using G-equation is expected to affect the statistical agreement with experimental data. The most noteworthy feature of LES is that all models have at least the potential to accurately discriminate between the observed modes of operation of Cases I and II. (a) (b) (c) (d) Figure 3. Images of instantaneous axial velocity, 2 iso-surfaces and T iso-surfaces for (a) PaSR LES of Case I, (b) PaSR LES of Case II, (c) G-Eq. LES of Case I and G-Eq. LES of Case II. As seen in figure 3 most LES distinguish between Case I and II, with Case II resulting in a wider and less wrinkled flame that partially attaches to the cooler combustor walls and develops a staggered pattern. This change in behavior, with increased flow velocity and reactant temperature, is found experimentally, and found to affect the mean flow, concentration and temperature, [15-17], as will be evident later. Similar observations have been made previously, in other bluff-body stabilized combustors, as reviewed by Shanbhogue et al, [57]. In both cases, the flame lies almost parallel to the flow just downstream of the wedge and, thus, almost directly in the wedge shear layer, as seen in figure 4, which shows a gray-scale map of 2 together with iso-lines of T and T at 1200 K, representing the instantaneous and time-averaged flame. The unsteady separating vorticity on both sides of the wedge rolls up, which generates a flow that wraps the flame around these regions of intense vorticity and therefore wrinkles the flame. This process is in both cases dominated by convective Kelvin Helmholtz (KH) instabilities, with a frequency of f KH 140 to 160 Hz, and essentially generating the 3-vortices be- 17

92 tween which 12-vortices subsequently develops. As the shear layer mixing continues to feed the corrugated flame with fresh cold reactants the lower frequency Bénard/von-Karman (BVK) absolute instability starts to dominate the KH instabilities and thus modifying the flame. Depending on the dilatation ratio ( u / b or T b/t u) different equilibria is reached between the KH and BVK instabilities resulting in either Case I, with suppressed BVK instabilities and a symmetric flame for which T b/t u 5.6, or Case II, with a stronger relative influence of the BVK instabilities and hence a more asymmetric flapping flame for which T b/t u 3.2. These observations are in line with the theoretical results of Erickson et al, [58], and the complied experimental data summarized by Shanbhogue et al, [57]. (a) Figure 4. Gray-scale maps of 2 together with iso-lines of T and T at 1200 K, representing the instantaneous and time-averaged flame, for (a) PaSR LES of Case I and (b) PaSR-LES of Case II. (b) The combustion dynamics was further studied using the Proper Orthogonal Decomposition (POD) method, which is a technique for extracting the dynamically significant flow features in terms of POD modes, [59-60]. The POD technique allows the reconstruction of the instantaneous flow from the POD modes, and thus permits an examination of how the POD modes contribute to the combustor dynamics. Given the velocity field ṽ=ṽ(x,t) and a vector base i (x), the POD results in that ṽ(x,t) ṽ N (x,t)= N i=0 i (t) i (x) such that mode i=0 is the time-averaged ṽ-field and that the approximation ṽ N ofthedataset ṽ converges to ṽ when N goes to infinity. The base vectors i (x) are derived from the eigenvalue problem (AA T ) i (x)= i (t) i (x), where A=(ṽ ṽ ) (ṽ ṽ ). This eigenvalue problem is of the size M M, where M is the grid size, and is too expensive to compute and instead we use Sirovich s method of snap-shots, [61], by which the adjoint eigenvalue problem (A T A) i (x)= i (t) i (x) is solved from which the POD modes are i=a i/ i. One should note that the POD are here applied to the velocity and therefore overweights the coherent structures in the burnt gases, where velocity is higher. 18

93 In figure 5a and 5b are the four lowest POD modes from Cases I and II, respectively, presented as predicted by the PsSR-LES model. For Case I the four first modes, i (x), i=1,, 4, account for 40%, 7%, 6% and 4% of the kinetic energy of the fluctuating velocity, respectively, and for Case II the corresponding modes account for 18%, 13%, 11% and 4% of the kinetic energy of the fluctuating velocity, respectively. For both cases 100 snapshots have been used, and for both cases the four lowest order POD modes contain 50% of the kinetic energy, whereas recovering 90% of the total kinetic energy will require about 50 modes. The kinetic energy distributions found here are well in line with previous POD investigations of other turbulent flows, [62-63]. ForCaseIthe1 st mode is symmetric and represents the acceleration due to the volumetric expansion, which primarily occurs within the mean flame. The 2 nd mode shows a cross-over pattern with acceleration due to volumetric expansion occurring around the average flame. The 3 rd mode shows two essentially stationary vortex structures behind the flameholder downstream of which no well-defined vortex structures can be identified. The 4 th mode is similar to the 2 nd mode but is more intermittent. Here the 2 nd,3 rd and 4 th modes are all associated with the KH instability dominating this case. For Case II the 1 st mode is initially symmetric due to the initial KH instability, but develops into an asymmetric mode, as a consequence of the BVK instability, after about 1h. This mode incorporates the acceleration due to the volumetric expansion, which primarily occurs within the average flame. The 2 nd and 3 rd modes are both asymmetric, with two counter-rotating vortex structures dominating the 2 nd mode, whereas the 3 rd mode complements the (asymmetric) shedding pattern due to the BVK instability. The 4 th mode is essentially symmetric and similar to the 2 nd mode of Case I thus being responsible for the KH instabilities. 19

94 (a) (b) Figure 5. The four lowest order POD modes from the PaSR LES model for (a) Case I and (b) Case II in a region close to the flameholder. The vector-fields show the POD modes whereas the background (in grayscale) shows the POD mode energy distribution. Superimposed is also a contour of the average temperature field at T=1200 K. In figure 6a and 6b the time histories of the POD time coefficients, i(t), of each mode are presented for Cases I and II, respectively. From time-series like this we can compute the characteristic frequencies of each mode and relate those to the different types of instabilities described earlier. Here it is found that the convective KH instabilities, dominating essentially all modes of Case I, have frequencies between 140 and 160 Hz, but extending down to 50 Hz and up to 1000 Hz. The BVK instabilities dominating the 1 st,2 nd and 3 rd modes of Case II, and weakly present in the 2 nd mode of Case 1 are associated with frequencies between 10 and 200 Hz. This observation is in line with the overall characteristics of KH and BVK instabilities, [57]. (a) (b) Figure 6. Time history of the POD time coefficients for (a) Case I and (b) Case II. Legend: ( ) mode 1, ( ) mode 2, ( ) mode 3 and ( ) mode 4. Figure 7 presents comparisons of the time-averaged axial velocity, ṽ x, at x/h=0.95, 3.75 and 9.40 downstream of the flameholder from the G equation LES, PaSR-LES, TFM-LES, EDC-LES and PPDF-LES models for Case I (upper panel) and Case II (lower panel). We first observe that both reacting cases have a longer and wider recirculation region, and hence a more gradual dissipation of momentum in the wake, than the corresponding non-reacting cases. The agreement between the predicted and measured ṽ x profiles is reasonable for both Cases I and II, with the largest discrepancies occurring in the recovery region, at x/h=3.75 and at x/h=9.40, especially for the G-equation LES in Case II. In particular, it should be observed that all LES success- 20

95 fully capture the differences in ṽ x between Cases I and II, with Case I resulting in the largest recirculation region, as also observed experimentally. In the near wake, characterized by the profiles at x/h=0.95, all LES models show similar results, with the predictions for Case I resulting in slightly sharper profiles compared to the experiments. In the recovery region, characterized by the profiles at x/h=3.75, all finite chemistry models overpredict ṽ x in the wake with between 5% and 10% of the inflow velocity v 0. Contrary to that, the G-equation LES shows very good agreement with the experimental data for Case I whereas it overpredicts the wake of case II with some 10%. Further downstream the velocity profiles parallel those of fully developed turbulent channel flow but having a bulk velocity about twice the inlet bulk velocity due to the volumetric expansion as a result of the exothermicity. At x/h=9.40 good agreement is again obtained, but with some finite chemistry models (TFM- LES and EDC-LES) resulting in slightly more peaked profiles for Case I, whereas some finite chemistry models (TFM-LES, PPDF-LES and PaSR-LES) underpredict ṽ x for Case II. Both G-equation LES significantly underpredict ṽ x, particularly in the wake region, which is significantly longer than for the finite rate chemistry LES, and what is observed in the experimental data. The discrepancies in the G-equation LES is believed to be caused by a too low turbulent flame speed and thus to a too low flame wrinkling factor. Figure 7. Time averaged axial velocity profiles at x/h=0.95, 3.75 and 9.40 from Case I (upper row) and Case II (bottom row). Legend ( ) experimental data [16], ( ) G-equation flamelet LES, ( ) PaSRLES,( )EDCLES,( ) TFMLESand( ) PPDFLES. Figure 8 shows comparisons of the axial rms-velocity fluctuations, v rms x, at x/h=0.95, 3.75 and 9.40 from the G equation LES, PaSR-LES, TFM-LES, EDC-LES and PPDF-LES models for Case I (upper panel) and Case II (lower panel). The general agreement between the predicted and measured fluctuations, v rms x, is generally acceptable but with larger deviations than for the time-averaged axial velocity ṽ x. One particular reason 21

96 for this may be that v rms x takes longer time to converge both experimentally and numerically. At x/h=0.95 the bi-modal shear layer profile is evident in both cases and reasonably well predicted by the LES models, but with the G-equation LES model underpredicting both the shear layer peaks and the freestream value. This is currently believed to be the result of the combustion model assumption that tends to delay the instability development in the close region behind the flame holder. At x/h=3.75 the experimental data of Case I develop into a single peaked profile, whereas the experimental data of Case II shows a weak but distinct bi-modal profile. The scatter in the LES results of the rms velocity fluctuations are higher at this location, and it is only the PaSR-LES and PPDF-LES that may distinguish between the two Cases I and II. Further downstream, at x/h=9.40 good agreement is obtained for Case I for all LES, whereas for Case II only the PaSR-LES and PPDF-LES models show good agreement with the experimental data. The G-equation LES underpredict v rms x whereas the EDC-LES and TFM-LES overpredict v rms x significantly in the core of the flow. Figure 8. Time averaged axial rms velocity fluctuation profiles at x/h=0.95, 3.75 and 9.40 from Case I (upper row) and Case II (bottom row). Legend ( ) experimental data [16], ( ) G-equation flamelet LES, ( ) PaSRLES,( )EDCLES,( ) TFM LES and ( ) PPDF LES. Figure 9 shows comparisons of the time-averaged temperature, T, at x/h=0.95, 3.75 and 9.40 between the G equation LES, PaSR-LES, TFM-LES, PPDF-LES and EDC-LES models for Case I (upper panel) and Case II (lower panel). Here we compare with both CARS measurement data, [17], at x/h=0.95, 3.75 and 9.40 and with gas analysis data, [15], at x/h=0.95, 3.75 and Note that the experimental data at the far downstream location from CARS and gas analysis are not collected at the same positions, with the CARS data obtained at the same axial position as the velocity and rms-velocity fluctuations of figures 5 and 6, respectively. At x/h=0.95 satisfactory agreement is obtained between both experimental profiles and all five LES profiles, with 22

97 the G-equation LES profile being marginally sharper and predicting a slightly higher temperature. At x/h=3.75 considerable differences are observed between the two experimental data sets, with the CARS profile being significantly narrower than the gas-analysis profile. In [17] it is argued that the CARS data is more accurate than the gas analysis data since the samples employed in the gas analysis temperature measurements continues to react in the sampling tubes, thus overpredicting T. All LES predictions agrees well with the CARS data, with the EDC LES showing virtually perfect agreement whilst the other models, particularly the G equation LES in case II, overpredicts the temperature, T, by about 10%. Further downstream, at x/h=9.40 the CARS and gas analysis based T profiles agree well with each other for Case I whereas for Case II they differ. For Case I all LES computations predict the T profiles within 5% accuracy, with the G-equation model predicting a narrower flame. This is coherent with the velocity profiles discrepancy found in figure 5. For Case II large discrepancies can be observed. In particular does the G equation LES overpredict the CARS T profiles, with which the finite rate chemistry LES predictions agree well. The reason for this is probably the difference in vortex shedding, illustrated in figure 2, between the G-equation LES and the finite rate chemistry LES. Figure 9. Time averaged temperature profiles at x/h=0.95, 3.75 and 9.40 from Case I (upper row) and Case II (bottom row). Legend ( ) CARS experimental data, [17], at x/h=0.95, 3.75 and 9.40, (+) experimental gas analysis data, [15], x/h=0.95, 3.75 and 8.75, ( ) G-equation flamelet LES, ( ) PaSRLES,( )EDCLES,( ) TFMLESand( ) PPDFLES. Figure 10 shows comparisons of the time-averaged CO mass fraction, ỸCO, at x/h=0.95, 3.75 and 8.75, between the G equation LES, PaSR-LES, PPDF-LES, EDC-LES and TFM-LES models for Case I (upper panel) and Case II (lower panel). The ỸCO profiles for G-equation model are reconstructed from time sampling of G over 0.3 s and 0.1 s for respectively Case I and Case II and includes subgrid scale effects following [21]. At the 23

98 first location, at x/h=0.95, good agreement between the experimentally measured and predicted ỸCO profiles can be observed for all LES models, with only small differences in peak values between the model predictions. At x/h=3.75 the discrepancies between the predictions and the measurements are larger, with also larger differences between the model predictions. Here, the PaSR-LES, EDC-LES, PPDF-LES and G-equation LES prediction agree well with each other, whereas the TFM-LES shows a flatter profile with less pronounced peak values in the extended shear layers. Comparing with the T profiles of figure 7, taking into consideration that he CARS profiles are the most accurate, it appears as if the LES profiles are more accurate than the measured profiles, but it is interesting to note that the predicted and measured peak values are in good agreement with each other. At x/h=8.75 excellent agreement between all LES and the measurement data is found for Case I (which also show excellent T agreement) whereas for Case II the agreement is generally poor, as indicated by the results for T in figure 7. Figure 10. Time averaged CO mass fraction profiles at x/h=0.95, 3.75 and 8.75 from Case I (upper row) and Case II (bottom row). Legend ( ) gas analysis experimental data [15], ( ) G- equation flamelet LES, ( ) PaSRLES,( ) EDC LES, ( ) TFM LES and ( ) PPDF LES. In figure 11 we compare predicted and measured PDF s of T at three locations in the upper shear-layer downstream of the flameholder (x/h, y/h)= (0.95, 0.50), (3.75, 0.50) and (9.40, 0.50). The predicted PDF s for Cases I and II are calculated from 20,000 and 40,000 samples, respectively, during a time interval of 0.2 s for the finite rate chemistry models whilst G-LES sampling time is equivalent to the one used for the gas analysis. The experimental PDFs are obtained from 1000 samples collected over 3 minutes. At the centerline (not shown) a single-mode PDF is obtained both experimentally and computationally for both cases, with the predicted PDFs being sharper than the measured ones. In the shear-layers, single-mode PDF dominates in the recirculation re- 24

99 gion (x/h=0.95) in both cases, but centered at different average temperatures due to the different flame temperatures. Far downstream, at x/h=3.75, the PDFs become bimodal in Case II as captured by all LES models, indicating entrainment of cold reactants into the reaction region; succession of cold and hot gas in the probe volume. In this case, large vortical structures develop closer to the bluff body and stagger when propagating downstream as seen in figure 3. Similar to the previous measurement point the mean temperature is centered at different values although the ratio between the two peaks seems to be captured by all LES. For Case I, the EDC PaSR and PPDF models seems in better agreement with the experimental data compared to the TFM and G-equation models, the bi-modal PDF of which suggests a larger spreading of the flame as seen in figure 9. Due to the G- equation model formulation, i.e. thin flamelet, sharper profiles of the PDFs are observed in general as compared to finite-rate chemistry models. The measured shear-layer PDFs result in broader temperature distributions, suggesting that the shear-layer reaction zones are fluctuating widely, resulting in temperatures even beyond the adiabatic flame temperature, which however appears unphysical. Figure 11. Probability Density Functions (PDF) of the temperature at three locations at (x/h, y/h)= (0.95, 0.50), (3.75, 0.50) and (9.40, 0.50) in the upper shear layer from Case I (top row) and Case II (bottom row). Legend ( ) CARS experimental data, ( ) G-equation flamelet LES, ( ) PaSR LES, ( )EDCLES,( ) TFMLESand( ) PPDF LES. VII. Concluding Remarks In this study we have examined the performance of two conceptually different categories of LES combustion models: one flamelet model and four finite rate chemistry models: the PaSR, TFM, EDC and PPDF LES models. The G-equation LES model is implemented in a structured LES in-house code whereas the finite rate chemistry LES models are all implemented in an unstructured production LES code based on the C++ library Open- 25

100 Foam. Both codes show good agreement with experimental data for a non-reacting reference case, suggesting that the subgrid modeling is acceptable. The focus of the study has been on a well-documented lean premixed bluff body stabilized flame, operating in two different regimes with respectively low and high density ratios. The flamelet model used is based on the level-set G-equation, in which the level-set is used to track the flame front. This model assumes infinitely thin flame thickness at all positions. The key input and closure parameter of the G-equation model is the propagation speed modeled using a power law expression for the flame wrinkling. The used finite rate chemistry models represent the current span of modeling sophistication in providing closure for the filtered reaction rates and physical insight in terms of the turbulence chemistry interactions of multi-step kinetics. A theoretical study suggests that all these models are comparable in terms of global parameters, such as the turbulent flame thickness and flame speed despite their intrinsic differences. Comparing predictions from the different models with experimental data for the first and second order statistical moments of velocity, temperature and species concentrations, as well as with temperature probability density functions, suggests that the finite chemistry models gives surprisingly similar results, in spite of the significant differences between the models, and very good agreement with the experimental data. Concerning the flamelet G-equation model we find larger deviations with the experimental data that may be attributed to the novel state of this modeling compared to that of the finite rate chemistry, but also some important differences in terms of the vorticity dynamics that is more likely to be attributed to the formulation of the G-equation model. In spite of these differences, all LES models examined show reasonable results when compared together, and with experimental data for such a complicated flame. This supports the recognized robustness of LES when applied to complicated flows and flames, which, in turn, is a result of the fact that the motion of the most energetic flow structures is captured in LES. Acknowledgement This work was supported by the Swedish research funding organizations (the Swedish Energy Agency, the Swedish Foundation for Strategic Research, the Centre for Combustion Science and Technology, and the Swedish Research Council), the European Union through the Lund University Combustion Center Large Scale Facility, and the Swedish Armed Forces. References [1] Bray, K.N.C., The Challenge of Turbulent Combustion, 26 th Int. Symp. on Comb., The Combustion Institute, Pittsburgh, USA, pp 1-26,

101 [2] Poinsot, T. and Veynante, D., Theoretical and Numerical Combustion, Edwards, [3] Veynante, D. and Vervisch, L., Turbulent Combustion Modeling, Prog. Energy and Comb. Sci., 28,, Issue 3, pp , [4] Sagaut, P., Large Eddy Simulation for Incompressible Flows, Springer Verlag, Heidelberg, [5] Grinstein, F.F., Margolin, L.G. & Rider, W.J. (Eds), Implicit Large Eddy Simulation Computing Turbulent Fluid Dynamics, Cambridge University Press. See Chapters 1, 11 and [6] Spalart, P.R., Strategies for Turbulence Modeling and Simulations, Int. J. Heat & Fluid Flow, 21, Issue 3, pp , [7] Givi, P., Filtered Density Function of Subgrid Scale Modeling of Turbulent Combustion, AIAA.J., 44, Issue 1, pp 16-23, [8] Porumbel, I. and Menon, S., Large-Eddy Simulation of Bluff Body Stabilized Premixed Flames, AIAA , [9] Pitsch H., Large Eddy Simulation of Turbulent Combustion, Annu. Rev Fluid Mech., 38, pp , [10] Fureby, C., Comparison of Flamelet and Finite Rate Chemistry LES for Premixed Turbulent Combustion, AIAA , [11] Fureby, C., LES Modeling of Combustion for Propulsion Applications, Phil. Trans. R. Soc. A, 367, pp , doi: /rsta [12] Möller, S-I., Lundgren, E. and Fureby, C., Large Eddy Simulation of Unsteady Combustion, 26 th Int. Symp. On Comb, pp , [13] Peters, N., Turbulent Combustion, Cambridge Press, U.K., [14] Colin, O., Ducros, F., Veynante, D. and Poinsot, T., A Thickened Flame Model for Large Eddy Simualtions of Turbulent Premixed Combustion,Phys. Fluids, 12, Issue 7, pp , [15] Sjunnesson, A., Olovsson, S. and Sjöblom, B., Validation Rig A tool for Flame Studies, VOLVO Aero AB, S , Trollhättan, Sweden, [16] Sjunnesson, A., Nelson, C. and Max, E., LDA Measurements of Velocities and Turbulence in a Bluff Body Stabilized Flame, Laser Anemometry 3, ASME, [17] Sjunnesson, A., Henriksson, P and Löfström C., CARS Measurements and Visualization of Reacting Flows in a Bluff Body Stabilized Flame, AIAA , [18] Fureby, C., Large Eddy Simulation of Combustion Instabilities in a Jet-Engine Afterburner Model, Comb. Sci & Tech, 161, pp , [19] Fureby, C., A Computational Study of Combustion Instability Related to Vortex Shedding, 28 th Int. Symp. on Comb, pp , [20] Giacomazzi, E., Battaglia, V. & Bruno, C., The Coupling of Turbulence and Chemistry in a Premixed Bluff-body Flame as Studied by LES, Comb. Flame, 138, pp , [21] Wang, P. & Bai, X.S., Large Eddy Simulation of Turbulent Premixed Flames using Level-set G-Equation, Proc. Comb. Inst. 30, pp , [22] Düsing, K.M., Large Eddy Simulation turbulenter Vormischflammen, PhD Thesis Darmstadt University of Technology,

102 [23] Oran, E.S. and Boris, J.P., Numerical Simulation of Reactive Flow, Elsevier, New York, [24] Grinstein, F.F. and Kailasanath, K.K, Three Dimensional Numerical Simulations of Unsteady Reactive Square Jets, Comb. & Flame, 100, pp 2-10, [25] Magnussen, B. and Hjertager, B.H., On Mathematical Modeling of Turbulent Combustion, 16 th Int. Symp on Comb., pp , [26] Bray K.N.C and Moss J.B., A Unified Statistical Model of the Premixed Turbulent Flame, Acta Astronautica, 4, pp , [27] Pitsch H., A Consistent Level Set Formulation for Large Eddy Simulation of Premixed Turbulent Combustion, Comb. Flame, 143, p , [28] Weller, H.G., Tabor, G., Gosman, A.D. and Fureby, C., Application of a Flame-Wrinkling LES Combustion Model to a Turbulent Shear Layer Formed at a Rearward Facing Step, 27 th Int. Symp. on Comb, pp , [29] Gao, F. and O Brien E.E, A Large Eddy Simulation Scheme for Turbulent Reacting Flows, Physics of Fluids A, 5, Issue 6, pp , [30] Nilsson, P. and Bai, X.S., Modeling Turbulent Premixed Combustion Using a Level-Set Flamelet Approach, Experimental Thermal and Fluid Science, 21, Issue 1-3, pp 87-98, [31] Im, H.G., Lund, T.S. and Ferziger, J.H., Large Eddy Simulation of Turbulent Front Propagation with Dynamic Subgrid Models, Phys. Fluids, 9, Issue 12, pp , [32] Sun, M.B., Wang, Z.G., Bai, X.S., Assessment and Modification of Sub-cell-fix Method for Re-initialization of Level-set Distance Function, Int. J. Numer. Meth. Fluids, To appear, [33] Pocheau, A., Front Propagation in a Turbulent Medium, Europhys. Lett., 20, Issue 5, pp , [34] Charlette, F., Meneveau, C. & Veynante, D., A Power-Law Flame Wrinkling Model for LES of Premixed Turbulent Combustion; Part I: Non-Dynamic Formulation and Initial Tests. Comb. Flame, 131, pp , [35] Pitsch H.; FlameMaster, A C++ Computer Program for 0D Combustion and 1D Laminar Flame Calculations. [36] Schumann, U., Subgrid Scale Model for Finite Difference Simulation of Turbulent Flows in Plane Channels and Annuli, J. Comp. Phys., 18, Issue 4, pp , [37] Nguyen, D.Q., Fedkiw, R.P. and Kang, M., A Boundary Condition Capturing Method for Incompressible Flame Discontinuities, J Comp. Phys. 172, Issue 1, pp 71-98, [38] Fureby, C. and Grinstein, F.F., Large Eddy Simulation of High-Reynolds-Number Free and Wall-bounded Flows, J. Comp. Phys., 181, p , [39] O Rourke, P.J. and Bracco, F.V., Two Scaling Transformations for the Numerical Computation of Multidimensional Unsteady Laminar Flames, J. Comp. Phys., 33, Issue 2, p , [40] Fureby C., A Fractal Flame Wrinkling Large Eddy Simulation Model for Premixed Turbulent Combustion, Proc. Comb. Inst. 30, pp , [41] Legier, J.P., Poinsot, T. & Veynante, D., Dynamically Thickened Flame LES Model for Premixed and Non- Premixed Turbulent Combustion, Proc. of the 2000 CTR Summer Program, p 157, [42] Magnussen, B., On the Structure of Turbulence and a Generalized Eddy Dissipation Concept for Chemical Reaction in Turbulent Flow, 19th AIAA Aerospace Sience Meeting, St Louis, USA,

103 [43] Berglund, M., Fedina, E., Fureby, C., Sabel nikov, V. and Tegnér J., On the Influence of Finite Rate Chemistry in LES of Self-Ignition in Hot Confined Supersonic Airflow, To appear in a special issue of AIAA J on advanced scramjet simulations, DOI: / [44] Meneveau, C. and Sreenivasan, K.R., The Multifractal Nature of Turbulent Energy Dissipation, J. Fluid Mech., 224, pp , [45] Girimaji S.S., Assumed -PDF Model for Turbulent Mixing: Validation and extension to Multiple Scalar Mixing, Comb.Sci.&Tech., 78, Issue 4-6, pp , [46] Gaffney, R.L., White, J.A., Girimaji, S.S. & Drummond, J.P., Modeling Turbulence / Chemistry Interactions Using Assumed PDF Methods, AIAA , 1992 [47] Bensow, R. and Fureby, C., On the Justification and Extension of Mixed Models in LES, J. Turb. 8, Issue 54, pp 1-17, [48] Bardina, J., Ferziger, J.H. and Reynolds, W.C., Improved Subgrid Scale Models for Large Eddy Simulations, AIAA , [49] Li, B., Baudoin, E., Yu, R., Sun, Z.W, Li, Z.S, Bai, X.S, Aldèn, M. and Mansour, M.S.; 2009, Experimental and Numerical Study of a Conical Turbulent Partially Premixed Flame, Proc. Comb. Inst. 32, pp [50] Weller, H.G., Tabor, G., Jasak, H. and Fureby, C., A Tensorial Approach to CFD using Object Oriented Techniques, Comp. in Physics, 12, Issue 6, pp , [51] Yu, R., Nogenmyr, K.J. and Bai, X.S., Large Eddy Simulation of Swirling Turbulent Methane/Air Flames, EC- COMAS Thematic Conf. Comput. Comb., Lisbon, [52] Fureby, C. & Bensow, R, LES at Work: Quality Management in Practical LES, In Quality and Reliability of Large Eddy Simulations, Eds. Meyers, J., Geurts, B & Sagaut, P., Springer Verlag, pp , [53] Fureby, C., Towards Large Eddy Simulation in Engineering, Prog. Aerospace Science, 44, Issue 6, pp , [54] Issa, R.I., Solution of the Implicitly Discretized Fluid Flow Equations by Operator Splitting, J. Comp. Phys., 62, Issue 1, pp 40-65, [55] Jeong, J. and Hussain, F., On the Identification of a Vortex, J. Fluid Mech., 285, pp 69-94, [56] Sanquer, S., Bruel, P. and Deshaies, B., Some Specific Characteristics of Turbulence in the Reactive Wakes of Bluff Bodies, AIAA.J. 36, Issue 6, pp , 1998 [57] Shanbhogue, S.J., Husain, S. and Lieuwen, T., Lean Blowoff of Bluff Body Stabilized Flames: Scaling and Dynamics, Progress in Energy and Comb. Sci., 35, issue 1, pp , [58] Erickson, RR., Soteriou, M.C., and Mehta, P.G., The Influence of Temperature Ratio on the Dynamics of the Bluff Body Stabilized Flames, AIAA , [59] Berkooz, G., Holmes, P. and Lumley, J. L., The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows, Annu. Rev. Fluid Mech., 25, pp , [60] Smith, T., Moehlis, J. aand Holmes, P., Low-dimensional Modeling of Turbulence using the Proper Orthogonal Decomposition: A tutorial, Nonlinear Dyn., 41, Issue 1-3, pp , [61] Sirovich, L., Turbulence and the Dynamics of Coherent Structures, Part I, Quarterly Appl. Math., 45, Issue3,pp ,

104 [62] Couplet, M., Basdevant, C. and Sagaut P,, Calibrated Reduced-order POD-Galerkin System for Fluid Flow Modelling, J. Comp. Phys. 207, Issue 1, pp , [63] Duwig, C. and Fureby, C., LES of Unsteady Lean Stratified Premixed Combustion, Comb Flame, 151, Issue 1-2, pp ,

105 Paper II

106

107 Numerical simulation of premixed flame/turbulence interaction at high density ratio conditions E. Baudoin, R. Yu, X.S. Bai Division of Fluid Mechanics, Dept. of Energy Sciences Lund University, Lund, Sweden Abstract In large eddy simulations of premixed turbulent flames the reaction zones are typically thinner than the sub-grid scales. There are different numerical and modelling approaches handing these unresolved reaction zones. This paper presents a ghost-fluid numerical approach coupled with the level-set G-equation flamelet model for the sub-grid scale phenomenon. The method is particularly useful for flames with high density ratios. The numerical model is tested on three cases, (a) two dimensional laminar premixed flames interaction with vortices under different flame thickness and density ratio conditions, (b) hydrodynamics instability of two dimensional laminar premixed flames, (c) three dimensional turbulent premixed propane/air flame stabilized by a bluff body. It is found that the ghost-fluid method can ensure reliable numerical stability for very thin flames at high density ratios. The thickening of the reaction zones generally lead to damping of hydrodynamic instability and as a result a slower growth rate of flame wrinkling and a slower propagation of the flame front. The growth rate of the flame wrinkling is sensitive to the length scale of the initial disturbance to the flame front. The scale of flame front wrinkling is shown to remain similar to the length scale of the initial disturbance, whereas the amplitude of the wrinkling increases with time. It is found that density ratio has a significant impact on the flame structures and instability. With higher density ratio the hydrodynamic instability is enhanced, and in the bluff body stabilized flames the lower frequency Bénard/von-Karman instability is suppressed. The structures and dynamics of bluff-body stabilized flames are therefore significantly different at different density ratios. This numerical result is confirmed in earlier experimental observations. 1. Introduction Modelling of premixed turbulent combustion using large eddy simulation (LES) has become a viable approach for many engineering combustion processes. Yet, there are several modelling and numerical difficulties remained unresolved. Focusing on the flamelet regime of turbulent premixed flames, where the entire reaction zone is thinner than the smallest turbulence eddies, the Kolmogorov eddies, it is realized that in LES the smallest resolved scale is still larger than the reaction thickness. Certain numerical algorithm and modelling are needed to handle the reaction zones that are embedded in the sub-grid scales. For premixed flames with high density ratios between the unburned and burned mixtures, there is a sudden jump in the density, velocity and pressure within one LES grid if top-hat filter is used. Indeed, conventional fraction step methods used to solve the low Mach number flows are known to 1

108 suffer from numerical instability at high density ratio (e.g. ρu ρ b > 3) [1-3]. This is caused by the dispersion error in the momentum fluxes and in the continuity equation [4]. A predictorcorrector method [5-7] can be employed to enhance numerical stability within moderate density ratio ( ρu ρ b < 7 ), in which the conventional fraction step method is used in the predictor step followed by a corrector step. In both cases, low mesh resolution (less than 5 cells for the reaction zones) increases the risk of numerical instabilities. To improve numerical stability special focus has to be placed on the sudden jump in the density field. In some cases, the combustion model itself can overcome the difficulty. This is the case when using the thickened-flame model [8], in which the reaction zone is artificially thickened by a factor of the order of 20. Relevant quantities are corrected in order to recover the flame propagation velocity. In other models such as the flame surface density approaches [9], the reaction zones are thicker due to the use of transport equations for modelling the reaction zones explicitly. The numerical difficulties in these approaches are less severe. For models that imply thin reaction zones, e.g. the level-set G-equation flamelet approach, certain filtering and smoothening on the density field is needed [10,11], however, complete stability can not be guaranteed for high density ratio flames. For the models employing spatial filtering with large filter length, or the thickened flame model and the flame surface density based models there is a question regarding the impact of flame thickening on the flame-turbulence interaction, although key properties such as the propagation velocity of the reaction zone are often ensured in the different models under idealized conditions. A useful numerical method has been proposed for high density ratio flames [12-15]. The continuity equation is re-formulated to the form of u = ( Dρ/ Dt) ρ. The idea is to calculate the material derivative of the density from DT / Dt and DYi / Dt which are determined using the energy conservation and the species transport equations. When level-set G-equation flamelet models [16] are used, it is not straightforward to compute the material derivatives of species mass fractions since the species transport equations are not explicitly solved in the model. Instead, other approaches such as the ghost fluid method (GFM) [17], can be used. GFM was developed to track discontinuities typically found in multiphase flows. It was later used to study laminar flame front in [18] where the flame was assumed to be infinitely thin. The method was recently extended to allow for finite flame thickness without explicit filtering of the dependent variable [19]. There are several issues that are unclear with the method. Most important, it is not clear how does the thickening of the reaction zone affect numerical stability as well as the dynamics and propagation of premixed flames. The aim of this paper is to evaluate the behaviour of this method in simulation high density ratio premixed flames. The development of flame surface wrinkling due to hydrodynamic instability and due to flame/vortex interactions is considered to evaluate performance of GFM and to study the development of flame instabilities at different density ratio and flame thickening conditions. Finally, the GFM approach is used to simulate turbulent premixed propane/air flames stabilized by a bluff body under different density ratio conditions. 2

109 2. Ghost fluid method for the level-set G-equation flamelet model For laminar premixed flame propagation, or turbulent premixed flames in the flamelet regime, level-set G-equation based flamelet model [16] has been used by many researchers to study various flames including flames in gas turbine combustors [20], swirling flames [21] and bluff-body stabilized flames [10,22]. The model has also been used to study hydrodynamic instabilities of laminar premixed flames [23,24]. The governing equations employed in the model include the Navier-Stokes equations for variable-density flows is expressed as follow, Dρ Dt ( u) 0 + ρ = (1) u ρ + ρu u = p + t (2) t where u is velocity vector, p is pressure, ρ is density, and t is the viscous stress tensor. The flame front is modelled using a level-set G-equation that is a kinematic equation describing the propagation of premixed flame in a flow field, G + u G = sd G (3) t where G is a level-set function with G=0 denote the flame front and G>0 the burned side and G<0 the unburned side of the flame. s d is the propagation speed of the flame front. The G- function can be converted to a distance function if a re-initialization process is applied, i.e. by solving the following equation, G = 1 (4) In the infinitely thin reaction zone limit, the density field is a step function of the level-set function, ρu, G < 0 ρ( G) = (5) ρb, G > 0 The jump of density across the flame is therefore, The momentum jump is obtained from the continuity equation, ρ = ρb ρu (6) ρu = ρ uu sd G G where u is the velocity component normal to the flame front, and u u is the normal velocity component on the unburned side of the flame. From (6, 7) it appears that the jump condition for the normal velocity can be written, ρ u u = u = s b u d ρb G G (7) (8) 3

110 By using the jump conditions the reactive flow field is essentially divided into two parts with two different densities; in each part the density change is small and conventional approaches such as the fractional step methods can be used. The above governing equations are valid for both laminar and turbulent premixed flames with infinitely thin reaction zone. In finite thickness flames, the density changes from the unburned to the burned ones within certain spatial distance. In LES, when applying spatial filtering of the governing equations with a filter size larger than the grid size, the density would also change within the filter size, e.g. over a few grids. In these situations, a filter function can be introduced to account for the smooth transition of density from the unburned to the burned ones, ρ = ρ + α( G) ρ, ρu = ρ u + β( G) ρu (9) u u u Both α(g) and β(g) are monotonic increasing function of G varying from zero on the far unburned side to unity at far burned side. In the infinitely thin flame limit, α(g) and β(g) are simply the Heaviside function. If the thickness of the reaction (filtered reaction zone in LES) is not very large, α(g) and β(g) may be prescribed with an identical function [19]. 3. Numerical procedure The governing equations are numerically solved using a high order finite difference scheme on a staggered Cartesian grid system. The momentum equations are discretized using a fifth order WENO scheme [25] for the convective terms and fourth order central difference scheme for the diffusion terms and the pressure gradient. The level-set G-equation is discretized using a third order WENO scheme. A predictor-corrector approach is used for time integration. When the density gradient in the burned and unburned mixtures is not large, only the predictor step is needed. The code has been used to simulate turbulent swirling non-premixed flames [25], flows in internal combustion engines [26] and turbulent partially flames stabilized in a conical burner [27,28] where the numerical results were shown to agree well with the experimental data. 4. Results and discussions 4.1 Simulation of 2D flame/vortex interaction Simulations of flame-vortex interaction allow for evaluating combustion modelling in a single scale flow with well controlled size and time scales of the vortex. The results are of relevance to turbulent flames in the flamelet regimes. In the following, a pair of vortices interacting with stoichiometric methane/air flames at the temperature of 300K and pressure of 1 atm is considered to evaluate the present numerical method and to investigate the performance of the level-set G-equation approach. This test case is similar to the experimental case of Mueller et al. [29], and it has been used previously to evaluate the ghost fluid method for finite thickness flames [19]. 4

111 The computational domain is 0 x* 100, 0 y* 200, where the domain lengths have been normalized using a reference length L, defined as L = α sd (α is the heat diffusion coefficient in the fresh gas). L is mm. The size of the vortex is r=4l. The velocity of the vortex is 10.6 s d. Under one-dimensional freely propagating conditions the laminar burning velocity of the mixture is s d = m/s, and the thickness of the flame is δ = 0.33 mm. The flow velocity is set as s d to allow for the flame remaining in the computational domain during the period of early propagation. These parameters are taken from previous numerical simulations of similar problems [7]. The reference density profile across the reaction zone is taken from one dimensional numerical simulations using detailed chemistry. A typical representation of the density field for a coarse grid configuration is shown on Fig. 1 and the different density profiles are shown on Fig. 3a. Figure 2 shows the flame front positions (G=0) and the vorticity fields at different (nondimensional) time (t*) obtained using different grid resolutions. t* is defined as t* = tsd / δ. The coloured lines correspond to G=0 for the different grids while the black lines correspond to the vorticity iso-lines. Initially, the flame is not disturbed by the vortices. At t*=10, the flame front starts to wrinkle as the vortices approach the reaction zone. During this process the vorticity decays. In the initial stage, t*<15, the flame fronts simulated on the three grids coincide one another. As the vortices are convected towards the flame front, t*>10, the velocity between the vortices is enhanced by the counter rotating pair of vortices whereas it is suppressed in the outer region by the vortices. The flame responds to the change of velocity field by flashing back to upstream in the low speed region and by blowing downstream in the high speed region, Figs. 2b-2d. As the wrinkling amplitude increases with time an unburned gas pocket is formed in the central region after t*=20. This pocket is separated from the main flame shortly afterwards and it is burned out quickly by the surrounding hot gas (not shown here), cf. Fig. 2d. It is interesting to note that the flame-generated vorticity in the burned side of the flame appears counter rotating with respect to the upstream vorticity; this phenomenon has been observed in previous numerical [19,7] and experimental studies [29]. Although the overall solutions from the three grids are very similar it is evident that there is significant difference between the wrinkling structures from the coarse grid and those from the two finer grids, cf. Figs. 2c, 2d. The flame front from the coarse grid ( ) is a little asymmetric. This is due to that the resolution for the pair of vortices is not sufficient thus the staggered grids cause a little asymmetry of the vortices and in the flow field, which eventually leads to a slight asymmetry of the flame front. The difference between the flame front structures from the medium grid ( ) and the fine grid ( ) agree each other better and the flame fronts are more symmetric. To examine the effect of the flame thickening on the flame vortex-interaction, simulations with several flame thicknesses are carried out. The baseline case is the one with the laminar flame thickness. About 6 grid points are used within the flame thickness of 0.33 mm with the medium grid ( ). Additionally, two thicker flames and two thinner flames are studied. In one of the thinner flame cases the thickness is set to zero, i.e. the density jump occurs in one grid cell. The results are presented in Fig. 3 where the G=0 iso-lines are shown for 5

112 different flame thickness cases. It is observed that the thickening of the flame has substantial impact on the development of flame wrinkling. The infinitely thin flame case has the fastest growth in the flame surface wrinkling, whereas the thickest flame case (with two times the thickness of the laminar flame, and thus 12 grid points within the flame thickness) has the slowest growth of the flame wrinkling. In the study reported in [8], it was found that the effect of thickening became more apparent when the flame was thickened 10 times (thickened flame model). 4.2 Simulation of hydrodynamic instability In thin premixed flames due to the sudden change of velocity across the flame, the streamlines are deflected across the flame front if the flame front is oblique to the velocity vectors. As a result, a small wrinkling of the flame front can grow larger. This phenomenon is related to the well-known hydrodynamic instability (also known as the Landau-Darrieus instability) [30]. The development of flame wrinkling is studied here to examine the effect of flame thickening on the numerical simulation of hydrodynamic instability. The computational domain and flame parameters are set to the same as that in the previous section. The initial flow velocity field is approximated using a planar flame profile approximation; the inflow boundary is set at the low boundary y*=0. The inflow velocity is equal to s d. In the flow field a planar flame front is perturbed to have a sine wave shape with a wave length λ and a small amplitude of a, 2 π x * y* = asin + π λ where a is taken as the grid cell size. Several wave lengths, density ratios and reaction zone thicknesses were tested to examine their impact on the hydrodynamic instability. Fig. 4 shows the velocity field and the flame front at two different times, t*=1 and t*=3. The initial flame front (at t*=0) is also shown in the figure. It is clear that the initially nearly planar flame front grows to strongly wrinkled flame. Similar to the simulations of flame-vortex interaction, a grid sensitivity is performed around the fine grid ( ) and very little discrepancies were found within a ± 25% grid refinement (not shown here). The grid is used in the following simulations, which gives a grid size of a In Fig. 4, the density ratio was set to be ρu / ρ b = 7.27 and the initial disturbance has a wave length of λ = 8.2δ. The flame thickness is set to that of the physical flame thickness δ. The wave length is substantially larger than the flame thickness. It has been known that hydrodynamic instability is related to the thickness of the reaction zone and curvature of the wrinkling scale. To investigate this issue, simulations with four different wave lengths of the initial disturbance are carried out. Fig. 5 shows the development of flame wrinkling under different initial disturbance conditions for density ratio of The thickness of the reaction zone has been kept to the laminar flame thickness). It is seen that with shorter initial wave length ( λ = 4.1δ, Fig. 5a), the small initial disturbance (with an amplitude of one grid cell size, a) grows faster; the 6

113 initial sine waves are distorted as they evolves in time, with the leading fronts (low y*) remain smooth and the trailing front (high y*) develop to cusps. The flame front propagates to low y* direction, indicating a flashback of the flame to the unburned mixture due to the enhanced propagation speed by the flame front wrinkling. On the other hand, with larger wave lengths in the initial flame front ( λ = 5.5δ,8.2δ and 16.1δ, Figs. 5b-5d) the disturbance grows also quickly. It appears that although the initial sine wave shape of the flame front is quickly distorted, the periodicity of the flame wrinkling is kept. The longer the initial wave length the larger the wrinkling scales in the direction parallel to the initial flame front. For this reason with larger wave length in the initial flame front, e.g. λ = 16.1δ (Fig. 5d) the growth of the flame surface area is slower and the overall propagation speed is slower. Fig. 6 show the effect of density ratio on the hydrodynamic instability. The initial disturbance to the flame front has a short wave length, λ = 4.1δ and the reaction zone thickness is kept to be the same as the laminar flame thickness. With density ratio of 1.27, the flame front shape does not change substantially in time, although there is tendency of forming cusps in the trailing front of the flame (high y*). Increasing the density ratio leads to enhanced hydrodynamic instability as the evolution of flame surface wrinkling increases more rapidly as the density ratio increases. Fig. 7 shows further the effect of density ratio on the flame instability with initial distance of larger wave length, λ = 8.2δ. The same trend of flame instability is seen here as the shorter wave length case shown in Fig. 6. It is clear that with small density ratio of 1.27, the flame is more stable. When the density ratio is 3.27, the flame is instable with wrinkling grows rapidly. Further increase the density ratio the growth rate of flame wrinkling is enhanced. Fig. 8 shows the effect of flame thickening on the hydrodynamic instability. The initial flame surface wrinkling has a wave length of 4.1δ. The wave length of the wrinkled flame surface remains constant as the flame propagates. The wrinkling amplitude develops, however, larger when the flame thickness is smaller. The hydrodynamic instability is enhanced when the flame is thin, where it is weakened when the flame is thickened. This is true for both low and density ratio cases (Fig. 8) and for both low and high initial wrinkling wave length (not shown here). 4.3 Simulation of turbulent premixed propane/air flames stabilized by a bluff body In three dimensional turbulent premixed flames the flame front is highly wrinkled. The flame front wrinkling is likely due to several different mechanisms, flame/turbulence eddy interaction, hydrodynamic instability, and other intrinsic instability mechanisms or mechanisms related to the combustor configuration [30]. For a particular flame, the interplay of several different mechanisms will determine the outcome. In the following turbulent propane/air premixed flames stabilized by a bluff body are simulated using G-equation flamelet model with different thickness of the filtered reaction zone. Detailed description of the model and the modelling for the SGS stress are referred to [10,22]. The rig is a simple rectangular cross sectional channel with a height of 120 mm and a width of 240 mm. A propane/air mixture, with an equivalence ratio of 0.6 and a mass flow rate of 0.6 kg/s, flows 7

114 through a honeycomb screen that controls the level of turbulence at the inlet. The channel length from the inlet to the exit of the chamber is 1000 mm. Operating at atmospheric pressure, the flame is stabilized by a prismatic triangular shape bluff body with a side length, h=40 mm positioned 320 mm downstream of the inlet. More details about the geometry can be found in [31,32]. Three operating conditions are considered and summarized in Table 1. In particular, the density ratio varies from 1 for the non-reactive flow case (C3) to 6.3 for the high density ratio reactive flow case, C1. Table 1. Flow conditions for the three VR-1 cases [31,32]. T 0 and U 0 are respectively the inflow temperature and the bulk flow velocity at the inlet. s d is determined from one dimensional freely propagating flame simulations. case T 0 [K] U 0 [m/s] Re s d [m/s] ρu ρb C , C , C ,000 non-reactive 1 LES is performed using 768x128x128 cells in the axial, cross flow, and spanwise directions, respectively. The cell size is 0.94 mm in all three directions. In the spanwise direction only half of the channel was simulated by assuming periodic condition in the spanwise direction. This has been tested and proven to be adequate [22]. This grid resolution is verified after grid sensitivity test with coarser grids. Two filter sizes were employed to investigate the flame thickening on the results; one is two times the grid size and one is four times the grid size. The computations are done with 48 processors. Fig. 9 shows ensemble averaged mean axial velocity components from LES and LDV measurements along the cross flow direction at three different axial positions downstream of the bluff-body flame holder (here x is the distance to the trailing edge of the bluff body; a is the height of the bluff body). LES results obtained for the non-reactive flow case C3 are in good agreement with the LDV data. The LES results for the low density ratio case C2 from the two different flame filtering lengths agree each other very well, and they are also in reasonably good agreement with the LDV data. For the high density ratio case C1, there are significant differences between the results from the two different filtering lengths, especially at the downstream location x/h=9.4. The LES results are in reasonable agreement with the LDV data. Fig. 10 shows the rms axial velocity from LES and LDV measurements along the cross flow direction at the same three different axial positions downstream of the bluff-body flame holder as in Fig. 9. Again, for the non-reactive flows the LES results and the LDV data agree each other very well. For the high density ratio case C1, the LES results from the thinner filtered reaction zone with the filter length of 2 grid size gives higher rms axial velocity at the two downstream locations than the thicker reaction zone LES. The thinner reaction zone results are in better agreement with the experimental data. At x/h=0.95 the rms axial velocity 8

115 from LES is significantly lower than the LDV data. Since this axial position is very close to the trailing edge of the flame holder, the production of turbulence in the shear layer of wake flow is not significant. Turbulence at this position is likely inherited from upstream, e.g. in the wall boundary layer around the flame holder and near the channel wall, which is difficult to capture due to the relative coarser grid used in the LES. Note that for the non-reactive flow case the rms axial velocity at x/h=1.525, which is a little further downstream than the axial position of x/h=0.95, the LES results agree very well with the experiments. This indicates that the current grid resolution for the shear layer of the wake flow is sufficient. For the low density ratio case the LES results from the two different filter lengths agree each other well (the thin filter results are not shown in the figure). At the location of x/h=0.95 the discrepancy between the LES data and the LDV data is large; this is similar to the high density ratio case. Fig. 11 shows the mean temperature profiles along the cross flow direction at three downstream axial positions. For all cases, the temperature profiles are in fairly good agreement with each other and also with the experiments. It is seen that for the high density ratio case the thin filtered flame has a slightly wide spreading of the temperature profiles in the cross flow direction. For the low density ratio, the temperature profiles from the two different flame thickening filters are in fairly good agreement with each other. The level-set G-equation flamelet model can capture the Kelvin-Helmholtz instability close to the flame holder as shown in Fig. 12. The results are obtained from LES with the reaction zone filtered into two grid cells. In near flame holder region, a nearly two dimensional wrinkling of the flame surface is seen in both the case C1 and case C2. The wrinkling is rather weak due to the weak turbulence in this region. The flame wrinkling is likely due to the interaction between the large scale vortices generated in the shear layer interacting with the flame front. Further downstream, e.g. x/h>3, the flame wrinkling becomes three dimensional. For the low density ratio case C2, there is a large variation of wrinkling scales; of particular importance is the large wrinkling structure in the axial direction which is largely semi twodimensional, and another important one is the small local wrinkling structure. The large scale wrinkling leads to an asymmetric behaviour of the flame, and this is caused by the low frequency flow shedding originated from Bénard/von-Karman (BVK) instability, which interacts with the flame front. For the high density ratio case C1, however, the flame wrinkling at further downstream of the flame holder is essentially localized at relatively smaller scales. These small scale wrinkling structures are likely to be due to hydrodynamic instability. The Bénard/von-Karman (BVK) instability found at low density ratios, including the non-reactive flow case, Fig.13, is suppressed at the high density ratios. Fig. 13 shows the instantaneous streamlines and the axial velocity distribution in the middle spanwise plane for the three flow cases, C1, C2, and C3. The vortex shedding due to Bénard/von-Karman (BVK) instability is clearly seen in the non-reaction flow case and the low density ratio case, whereas in the high density ratio case, it is suppressed completely. This result is consistent with the high speed video in the VR-1 experiments [31,32]. 9

116 5. Conclusions In large eddy simulations of turbulent premixed flames the reaction zones are not resolved. If top-hat filter is used, the density and velocity jump across one LES grid cell; if an explicit filter with filter length wider than the grid cell is used the reaction zones are filtered in the filter width that is typically a few grid cells wide. These different filtering widths can yield significant impact on the numerical results especially on the development of flame surface wrinkling. In this paper, two different mechanisms leading to flame wrinkling are simulated, one is the wrinkling of flames due to flow vortex/flame interaction and another is the hydrodynamic instability. Level-set G-equation flamelet approach is used in the numerical simulation, for methane/air premixed flame with unity Lewis number. To simulate the high density ratio flames, a numerically highly robust method for high density ratio conditions, the ghost fluid method, is implemented to take into account flames with explicitly spatial filtering. Then, numerical simulations are carried out for two dimensional laminar premixed methane/air flame interacting with a pair of vortices convected by the flow, and for two dimensional hydrodynamic instability of the laminar premixed methane/air flame. The flame vortex interaction is substantially affected by the flame thickness. Thinner reaction zone yields faster development of flame wrinkling. The flame wrinkling due to hydrodynamic instability is however affected by the flame thickening and density ratio in a rather complex manner. For low density ratio flames, in general, the flame tends to be rather stable; the initial disturbances at the flame front do not develop to high wrinkling of the flame. For high density ratio flames the hydrodynamic instability is important. It is seen that the length scale of initial disturbance at the flame front in the direction parallel to the flame surface is kept during the development of flame instability, although the amplitude of the wrinkling increases with time. The leading front of the flame facing the unburned mixture remains smooth, whereas the trailing fronts of the flame surface develops to cusps. Thickening of the flame exerts significant impact on the development of flame wrinkling. The thinner the flame the faster the growth of the flame surface wrinkling. The G-equation flamelet model and the ghost fluid method are applied to simulate turbulent premixed propane/air flames stabilized by a bluff-body flame holder. The simulation results are in fairly good agreement with the LDV data and the mean temperature field. In case of low density ratio, the LES results from different filter lengths, are shown to not significantly affect the mean axial velocity and mean temperature field. It does show, however, significant effects on the rms velocity field and thereby on the dynamics of the flames at high density ratio conditions. Acknowledgments This work is supported by CeCOST (Swedish national center for combustion science and technology) and VR (Swedish Research Council). 10

117 References 1 Kim J., Moin P., Application of a fractional-step method to incompressible Navier-Stokes equations, Journal of Computational Physics 59, Ferziger J.H., Peric M., Computational Methods for Fluid Dynamics, Springer-Verlag, Cook A.W., Riley J.J., Direct Numerical Simulation of a Turbulent Reactive Plume on a Parallel Computer, Journal of Computational Physics 129, , Pierce C.D., Progress-variable Approach for large Eddy Simulation of turbulent combustion, Doctoral Thesis, Stanford, Najim, H.N.; Wyckoff, P.S.; Knio, O.M., A semi-implicit numerical scheme for reacting flow. I. Stiff chemistry, Journal of Computational Physics, 143, , Knio, O.M.; Najm, H.N.; Wyckoff, P.S., A semi-implicit numerical scheme for reacting flow. II. Stiff, operator-split formulation, Journal of Computational Physics, 154, , Lessani B., Papalexandris M.V., Time-accurate calculation of variable density flows with strong temperature gradients and combustion, J. Comput. Phys. 212 (2006) Colin O., Ducros F., Veynante D., Poinsot T., A thickened flame model for large-eddy simulations of turbulent premixed combustion, Phys. Fluids 12(7) (2000) M. Boger, D. Veynante, H. Boughanem, A. Trouve. Direct numerical simulation analysis of flame surface density concept for large eddy simulation of turbulent premixed combustion. 27th Int. Symp. on Comb., , P. Wang, X.S. Bai, Large eddy simulation of turbulent premixed flames using level-set G-equation, Proc. Combust. Inst. 30, , Vreman A.W., Albrecht B.A.,van Oijen J.A., de Goey L.P.H., Bastiaans R.J.M., Premixed and nonpremixed generated manifolds in large-eddy simulation of Sandia flame D and F, Combust. Flame, 153 (2008) Nicoud F., Conservative High-Order Finite Difference Schemes for Low-Mach number Flows, Journal of Computational Physics, 158, 71-97, Almgren A.S., Bell J.B., Colella P., Howell L.H., Welcome M.L., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, Journal of Computational Physics, 142, 1-46, Day M.S., Bell J.B., Numerical simulation of laminar reacting flows with complex chemistry, Combustion, Theory and Modelling, 4, , Bell J.B., Day M.S., Rendleman C.A., Woosley S.E., Zingale M.A., Adaptive low Mach number simulations of nuclear flame microphysics, Journal of Computational Physics, 195, , Peters N., Turbulent combustion, Cambridge University Press, Cambridge, UK (2000) 17 Fedkiw R.P., Aslam T., Merriman B., Osher S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys. 152 (1999) Nguyen D.Q, Fedkiw R.P., Kang M., A boundary condition capturing method for incompressible flame discontinuities, J. Comput. Phys. 172 (2001)

118 19 Moureau V., Minot P., Pitsch H., Berat C., A ghost-fluid method for large-eddy simulations of premixed combustion in complex geometries, Journal of Computational Physics 221 (2007) H. Pitsch, Ann. Rev. Fluid Mech. 38 (2006) K.J. Nogenmyr, C. Fureby, X.S. Bai, P. Petersson, R. Collin, M. Linne, Combust. Flame 156 (2009) E. Baudoin, R. Yu, K.J. Nogenmyr, X.S. Bai, C. Fureby. Comparison of LES models applied to bluff body stabilized flame. Proceedings of 47th AIAA Aerospace Sciences Meeting, AIAA , Orlando, Florida, Helenbrook B.T, Law C.K., The role of Landau-Darrieus instability in large scale flows, Combust. Flame 169 (1999) Helenbrook B.T, Martinelli L., Law C.K., A numerical method for solving incompressible flow problems with surface of discontinuity, J. Comput. Phys. 148 (1999) R. Yu, K.J. Nogenmyr, X.S. Bai, Large eddy simulation of swirling turbulent methane/air flames, ECCOMAS Thematic Conf. on Comput. Comb., June, Lisbon, Yu R., Large Eddy Simulation of Turbulent Flow and Combustion in HCCI Engines, Doctoral Thesis, Lund University, B. Li, E. Baudoin, R. Yu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Aldén, M.S. Mansour. Experimental and numerical study of a conical turbulent partially premixed flame. Proc. Combust. Inst. 32: , B. Yan, B. Li, E. Baudoin, C. Liu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Aldén, G. Chen, M.S. Mansour. Structures and stabilization of low calorific value gas turbulent partially premixed flames in a conical burner. Experimental Thermal and Fluid Science 34: , Mueller C.J., Driscoll J.F., Reuss D.L., Drake M.C., Rosalik M.E., Vorticity generation and attenuation as vortices convect through a premixed flame, Combust. Flame 112 (1998) Liñán A., Williams F. A., Fundamental Aspects of Combustion, Oxford University Press (1993) A. Sjunnesson, C. Nelson, E. Max. LDA Measurements of Velocities and Turbulence in a Bluff Body Stabilized Flame. Fourth International Conference on Laser Anemometry ASME, A. Sjunnesson, P. Henriksson, C Löfström. CARS Measurements and Visualization of Reacting Flows in a Bluff Body Stabilized Flame. 28 th Joint Propulsion AIAA conference, AIAA ,

119 ρ Figure 1. A sketch of typical density field across the flame on a coarse grid with a density ratio of

120 (a) (b) (c) (d) Figure 2. Instantaneous flame front (G=0 iso-lines) computed on three grids: 128x64 (green, coarse grid), 192x96 (blue, medium grid) and 256x128 (red, fine grid). Also shown in the figure are the instantaneous iso-lines of positive vorticity (solid lines) and negative vorticity (dashed lines) on the finest grid at different times: (a) t*=0, (b) t*=15, (c) t*=20, (d) t*=25. 14

121 (a) (b) (c) (d) Figure 3. Flame-vortex interaction simulated on 96x192 grids under different reaction zone thickness conditions. The reaction zone thickness functions are shown in (a); the flame fronts G=0 isolines at different instant are shown in (b) t*=15, (c) t*=20, and (d) t*=25. 15

122 (a) (b) (c) (d) Figure 4. Landau-Darrieus instability for λ/δ=8.2 and ρ u /ρ b =7.27. Axial velocity field (color bar), isolines of G=0 (black) at t*=1 (a,b) and t*=3 (c,d) and isolines of G=0 (red) at t*=0. 16

123 (a) (b) (c) (d) Figure 5. Time sequence of Landau-Darrieus instability for ρ u /ρ b =7.27 and various λ/δ. Isolines of G=0 for t*=0 ( ), t*=1.18 (--), t*=2.36 (- -), t*=3.54 ( ), t*=4.72 ( ), t*=5.90 (- -) with λ/δ=4.1 (a), 5.5 (b), 8.2 (c) and 16.1 (d). 17

124 (a) (b) (c) (d) Figure 6. Time sequence of Landau-Darrieus instability for λ/δ=4.1 and various ρ u /ρ b. Isolines of G=0 for t*=0 ( ), t*=1.18 (--), t*=2.36 (- -), t*=3.54 ( ), t*=4.72 ( ), t*=5.90 (- -) with ρ u /ρ b =1.27 (a), 3.27 (b), 5.27 (c) and 7.27 (d). 18

125 (a) (b) (c) (d) Figure 7. Time sequence of Landau-Darrieus instability for λ/δ=8.2 and various ρ u /ρ b. Isolines of G=0 for t*=0 ( ), t*=1.18 (--), t*=2.36 (- -), t*=3.54 ( ), t*=4.72 ( ), t*=5.90 (- -) with ρ u /ρ b =1.27 (a), 3.27 (b), 5.27 (c) and 7.27 (d). 19

126 (a) (b) (c) (d) Figure 8. Time sequence of Landau-Darrieus instability for λ/δ =4.1 and various flame thickness and ρ u /ρ b. Isolines of G=0 for t*=0 ( ), t*=1.18 (--), t*=2.36 (- -), t*=3.54 ( ), t*=4.72 ( ), t*=5.90 (- -) with ρ u /ρ b =3.27 and flame thickness 0.5δ (a), ρ u /ρ b =3.27 and flame thickness 2δ (b), ρ u /ρ b =5.27 and flame thickness 0.5δ (c) and ρ u /ρ b =7.27 and flame thickness 2δ (d). 20

127 Figure 9 Mean axial velocity at different axial positions along the cross flow direction. Top row, Case C3, middle row, case C1, and bottom row, Case C2. Symbols (+) represent experimental data [31,32]; solid lines represent LES results with the reaction zone filtered to 2 grid cells, dashed lines represent LES results with the reaction zones filtered to 4 grid cells. 21

128 Figure 10 RMS axial velocity at different axial position along the cross flow direction. Top row, Case C3, middle row, Case C1, and bottom row, Case C2. Symbols (+) represent experimental data [31,32]; solid lines represent LES results with the reaction zone filtered to 2 grid cells, dashed lines represent LES results with the reaction zones filtered to 4 grid cells. 22

129 Figure 11 Mean temperature at different axial position along the cross flow direction. Top row, Case C1, and bottom row, Case C2. Symbols (+) represent experimental data [31,32]; solid lines represent LES results with the reaction zone filtered to 2 grid cells, dashed lines represent LES results with the reaction zones filtered to 4 grid cells. 23

130 ρ ũ (a) Case C1 ρ ũ (b) Case C2 Figure 12. Instantaneous flame surface (G=0) and instantaneous axial velocity field calculated from LES. The reaction zone is filtered to 2 grid cells. 24

131 ũũ (a) Case C3 ũũ (b) Case C1 ũũ (c) Case C2 Figure 13. Instantaneous streamlines and instantaneous axial velocity field for the nonreactive flow Case C3 (a). Instantaneous flame surface (G=0), streamlines and axial velocity calculated from LES for reactive Case C1 (b) and C2 (c). The reaction zone is filtered to 2 grid cells. 25

132

133 Paper III

134

135 Available online at Proceedings of the Combustion Institute 32 (2009) Proceedings of the Combustion Institute Experimental and numerical study of a conical turbulent partially premixed flame B. Li a, E. Baudoin b,r.yu b, Z.W. Sun a, Z.S. Li a, X.S. Bai b, *, M. Aldén a, M.S. Mansour c a Division of Combustion Physics, Lund University, P.O. Box 118, S Lund, Sweden b Division of Fluid Mechanics, Lund University, P.O. Box 118, S Lund, Sweden c National Institute of Laser Enhanced Sciences, Cairo University, Egypt Abstract The structure and dynamics of a turbulent partially premixed methane/air flame in a conical burner were investigated using laser diagnostics and large-eddy simulations (LES). The flame structure inside the cone was characterized in detail using LES based on a two-scalar flamelet model, with the mixture fraction for the mixing field and level-set G-function for the partially premixed flame front propagation. In addition, planar laser induced florescence (PLIF) of CH and chemiluminescence imaging with high speed video were performed through a glass cone. CH and CH 2 O PLIF were also used to examine the flame structures above the cone. It is shown that in the entire flame the CH layer remains very thin, whereas the CH 2 O layer is rather thick. The flame is stabilized inside the cone a short distance above the nozzle. The stabilization of the flame can be simulated by the triple-flame model but not the flamelet-quenching model. The results show that flame stabilization in the cone is a result of premixed flame front propagation and flow reversal near the wall of the cone which is deemed to be dependent on the cone angle. Flamelet based LES is shown to capture the measured CH structures whereas the predicted CH 2 O structure is somewhat thinner than the experiments. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Partially premixed flame; Conical burner; Laser diagnostics; Large-eddy simulation 1. Introduction Partially premixed combustion is found in many industrial combustion devices, e.g., IC engines and gas turbines, and many other engineering applications. To understand the structure, dynamics, and statistics of partially premixed flames, theoretical, numerical, and experimental * Corresponding author. address: Xue-Song.Bai@energy.lth.se (X.S. Bai). studies have been carried out on various flame configurations, e.g., highly strained turbulent rich methane/air bunsen flames stabilized by hot combustion products [1], lifted jet flames [2,3], laminar partially premixed co-flows [4], laminar partially premixed counter-flow flames [5,6], and low-swirl stratified premixed turbulent flames [7,8]. The lift-off height and stabilization mechanisms in partially premixed flames are of particular practical interest. For example, the lift-off height of spray flames is an important parameter in diesel engines. Several mechanisms of flame stabilization have been proposed e.g., diffusion flamelet quenching /$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi: /j.proci

136 1812 B. Li et al. / Proceedings of the Combustion Institute 32 (2009) at high scalar dissipation rate [9,10] and propagation of premixed flame [11,12]. Recently, Mansour [13 15] has developed a concentric flow conical nozzle burner that consists of a mixing tube and a cone mounted on top of the tube. This conical burner provides an excellent opportunity to study partially premixed flames since different degrees of partial mixing can be conveniently generated in the mixing tube ranging from perfectly premixed to non-premixed conditions. The flame serves as a good test case for developing and validating simulation models for partially premixed flames. Previous experimental studies on this conical flame using sampling technique have shown that the flame is rather sensitive to the level of partial mixing and the nozzle cone angle. An optimal level of partial mixing has been found in which the flame is more stable with respect to changes in the overall equivalence ratio. Tracer smoke images revealed that a reversal flow exists near the exit of the cone and this was expected to be responsible for the stabilization of the conical flame [15]. However, precise information about the flow and flame structures in the cone was not available due to the limit of optical access in the metal cone experiments in [13 15]. The objective of this study is to investigate the details of flame structures and the mechanisms of the flame stabilization. A glass cone has been developed, which allows laser diagnostics of the flame inside the cone. Planar laser induced florescence (PLIF) of CH and chemiluminescence video imaging are used to characterize the structures of the leading flame front inside the cone. CH and CH 2 O PLIF are also carried out above the cone to identify the instantaneous reaction zone structures. In the entire flame, the instantaneous CH layers are continuous and thin, which indicates that the chemical reactions involving CH radicals are in very thin zones. This is consistent with the flamelet assumptions in turbulent flames and provides a basis for the development of flamelet library approach [16,17]. To gain further information about the mixing process in the mixing tube and to examine two flame-stabilization mechanisms for the partially premixed flame inside the cone, large-eddy simulation (LES) based on a two-scalar variable flamelet model is carried out. Numerical simulation of partially premixed flames has recently become a scientific focus. Several modeling approaches have been developed based on flamelet assumption. Peters et al. [3,17] developed a mixture fraction level-set G-function formulation for lifted jet flame where partially premixing occurs in front of the leading flame front. The model considered two flame-stabilization mechanisms, the tripleflame propagation, and the local extinction of diffusion flamelet at high scalar dissipation rate, to describe the lift-off phenomenon near the jet nozzle. Several other groups have adopted mixture fraction progress variable formulation to model the partial mixing field and the reaction fronts [18,19]. In the present study, the mixture fraction level-set G-function flamelet formulation is adopted. The level-set G-equation is to capture the propagation of the flame front which has been visualized with heat-release contours [20] and the propagation speed of the triple edge has been analyzed systematically by several authors [21,22]. This information is adopted in the current model in the sub-grid scale flame propagation. 2. Experimental setup The present burner is similar to the conical burner developed by Mansour [15] with modification of the cone by using a BK7 glass to enable optical access. A schematic illustration of the burner and a photo of the flame and the cone are shown in Fig. 1. The burner consists of two stainless steel tubes where the inner tube supplies air and the outer tube supplies fuel. Mixing starts at the exit of the inner tube and continues with a premixing distance of L between the exit of the inner tube and the exit of the outer tube. With L =0, the flame is non-premixed. With L > 0 the flame is partially premixed since at the exit of the tube certain degree of premixing of fuel and the primary air is already achieved. It has been shown experimentally that, at L/D = 5 (D is the inner diameter of the outer tube), the best flame stability is obtained, i.e., the overall equivalence ratio of the fuel/air mixture is minimum to allow a stable combustion. In the current study, L/D = 5 burner geometry is used, where the burner diameter D is 9.7 mm; the half-cone angle is 26 ; the length of the cone is 64 mm, and the diameter of cone exit is 73 mm. The overall equivalence ratio is 3 and L Zst x D fuel air fuel Fig. 1. Schematic illustration of the conical flame rig (left), and a direct photo of the burner with flames (right).

137 B. Li et al. / Proceedings of the Combustion Institute 32 (2009) the characteristic inlet speed at the cone inlet is 20 m/s. Chemiluminescence imaging was performed through a fused silica lens employing an ICCD camera (Princeton PI-MAX, pixels). An exposure time of 30 ls was utilized to capture the unsteady turbulent motions. Chemiluminescence from CH * and C 2, etc. populated by chemical reactions was collected through the BK7 glass cone. Regarding the CH detection there are two electronic transitions, A 2 D X 2 P and B 2 R X 2 P, which are usually excited for LIF applications in the near ultraviolet and visible spectral range. A short summary of different excitation detection schemes can be found in [23]. In this work excitation of the B X(0, 0) R-branch band-head was made near nm and broadband fluorescence from the B X(0, 1), A X(1, 1), and A X(0, 0) bands overlapping at around 431 nm was collected. The fluorescence emission caused by population of the A state is due to electronic energy transfer from the B state [24]. This approach allows sufficient suppression of the elastically scattered light by a simple color glass filter (Schott GG-400) and the collection of an ample amount of signal. A second harmonic of a flash lamp pumped, ring-cavity, tunable, pulsed alexandrite laser (PAL/PRO TM, Light Age Inc.) was utilized. By operating in multimode, broad band output as around 387 nm was obtained after frequency doubling. More details of the alexandrite laser systems have been described in [25]. The laser beam was formed into a thin laser sheet and cross the interrogated region in the flame. The fluorescence was collected perpendicularly to the laser sheet by the same ICCD camera as the one used for chemiluminescence imaging. CH PLIF has been measured at positions both above and inside the cone for different flame conditions. For the measurements inside the cone, a laser sheet with 17 mm width and 15 mj per pulse was utilized. For the measurement outside the cone, a laser sheet with 30 mm width and pulse energy of 35 mj per pulse was adopted. For formaldehyde PLIF measurements, the excitation and detection scheme reported by Brackmann et al. [26] was adopted. A Nd:YAG laser (Brilliant B, Quantel) was employed to provide the required 355 nm UV excitation beam with 115 mj per pulse, 10 Hz the repetition rate and 10 ns pulse duration. Through an optical system, the laser beam was formed into a 30 mm width thin laser sheet and sent to cross the flame at the interrogated region. The same ICCD camera used for CH PLIF was aligned perpendicularly to the laser sheet and PLIF images were recorded through a long pass Schott filter WG385. The spatial resolution is about 0.06 mm. The measurements were only performed at the position above the cone. 3. Modeling Numerical and experimental studies have shown that the reaction zones of partially premixed flames are at close to stoichiometric mixture [3,17,20,21]. The mixing field and the isosurfaces of stoichiometric mixture can be computed using a conserved scalar, mixture fraction. By assuming unity Lewis number, the transport equation for mixture fraction can be written as (in the LES spatial filtered form): oqez þ oq~u jz e ¼ o ot ox j ox j qd o ez ox j! þ o q~u j ez qu j Z ; ox j ð1þ where Z is the mixture fraction; q is the density of the gas mixture; u j is the velocity component in Cartesian coordinate x j direction; D is the diffusion coefficient of the gas mixture. In Eq. (1) over-bars denote spatially filtered quantity whereas over-tildes denote density-weighted spatial filtering. The last term on the r.h.s. of Eq. (1) is the sub-grid scale mixing effect that is modeled using the Smagorinsky Lilly model. A stoichiometric mixture cannot always be ignited in a turbulent flowfield. Asymptotic analysis of laminar flames shows that when the scalar dissipation rate is higher than a critical value, the flame cannot sustain; physically it is extinguished. The quenching scalar dissipation rate is about v q = s 1 for a methane/air non-premixed laminar flame at atmospheric condition, and partially premixed flames are easier to quench when the fuel stream is premixed with air [5,6]. If local extinction at high scalar dissipation rate is assumed to be the mechanism causing the lift-off of the partially premixed flame, the filtered density, temperature and mass fractions are calculated using laminar flamelet library and a presumed density-weighted probability density function (pdf) [16,27 29], Z Z Z 1 }ðz; v; x; tþ qðx; tþ ¼ dz dv ; q L ðz; vþ Z Z Z ey j ðx; tþ ¼ Y j;l ðz; vþ}ðz; v; x; tþdz dv; Z Z Z et ðx; tþ ¼ T L ðz; vþ}ðz; v; x; tþdz dv; ð2þ where q L (Z,v), Y j,l (Z,v), and T L (Z,v) are density, mass fractions of species j and temperature that are obtained from laminar flamelet library, calculated using counter-flow configuration and detailed chemical kinetics [30]. The density and species mass fractions are a function of the mixture fraction (Z) and scalar dissipation rate (v). At local extinction conditions (v > v q ), the inert mixture composition is used in the flamelet library [16].

138 1814 B. Li et al. / Proceedings of the Combustion Institute 32 (2009) The density-weighted joint probability density function (pdf) }(Z,v;x,t) is presumed to be a product of the pdf of mixture fraction and the pdf of scalar dissipation. The pdf of mixture fraction is presumed to be clipped Gaussian function that depends on two parameters, the filtered mixture fraction at the resolved scale and the SGS variance of mixture fraction. The pdf of scalar dissipation rate is presumed to be a log-normal distribution. Details about the model can be found in [8,27]. Pitts [31] has tested a similar quenching flamelet model for lifted jet flame and found that the model cannot predict the lift-off height properly. This leads to the development of triple-flame propagation model of Müller et al. [3]. In their formulation the partially premixed flame front is simulated using level-set G-equation (for the triple-flame front tracking) and mixture fraction flamelet chemistry. The G-equation can be written as! 1=2 o eg ot þ o eg o eg o eg ~uc j ¼ S sgs ; ð3þ ox j ox j ox j where ~u c j is the spatial filtered (density-weighted) velocity conditioned at the iso-surface of eg ¼ 0. Introducing the re-initialization operation on G function, one has jr egj ¼1; then eg is the signed distance to the flame front. In Eq. (3) s sgs is the propagation speed of the triple-flame front on the filtered scale. As the filter size in LES is typically larger than the laminar flamelet thickness, flamelet wrinkling on sub-grid scales needs to be considered in s sgs. Peters et al. [3,17] introduced a model for the propagation of time averaged mean triple-flame front. The model has been used to predict the lift-off height of a jet flame [17]. Extending the model to LES filtered scales, the following model is proposed here: S sgs ¼½s L ðzþþu sgs Šð1 av=v q Þ ð4þ where a is a model constant of the order of unity. As will be discussed later, in the current flame v < v q the LES results are not sensitive to a. Typically, the laminar flame speed at the triple-flame front is higher than freely propagation stoichiometric flame [21]. However, since this quantity is rather small compared to u sgs, it is not significant to refine it. It should be pointed out that one needs to compute the flame position at the triple flame only. However, to make the numerical implementation easier and also to divide the flow field into the burnable part and the inert part, it is desirable to compute the entire egðx; y; z; tþ ¼0 surface. eg > 0 denotes the possible burnable domain where the diffusion flamelet model can be applied. In eg < 0 domain, the mixture is assumed chemically inert. The spatially filtered continuity, momentum and energy equations as well as the mixture fraction transport equation are discretized and solved on a staggered Cartesian grid using a finite difference algorithm with multi-grid acceleration. A 5th order WENO scheme is employed for the convective terms, whereas all other spatial derivatives are discretized with a 4th order central scheme, and the time integration is performed by a second order implicit scheme. The level-set G-equation is solved with a 3rd order WENO scheme in combination with a 3rd order TVD Runge Kutta time integration scheme. Details about the numerical scheme and validation of the code are presented in [32]. The computational domain includes both the flow inside and outside the cone. To obtain information about the flow condition at the exit of the mixing tube, LES and acetone (as a fuel tracer) PLIF are carried out. The mixing process inside the mixing tube is simulated using fine uniform grid with about mm mesh size in both axial (x, Fig. 1) and cross flow directions. The velocity and fuel/air mass fractions at the exit of the mixing tube are stored for each time step in a database which is used as inflow condition for the simulation inside and outside the cone. The time averaged mean fuel mass fraction is compared with the acetone PLIF measurements at the exit of the mixing tube. The two profiles agree each other well when both are normalized with their respective maximal values at the exit of the mixing tube. The flow domain outside the cone extends to 5.26D 0 in the axial direction and 3.08D 0 in the cross flow directions, respectively, where D 0 (=73 mm) is the diameter of the cone exit. Convective outflow boundary condition is used at the far downstream outside the cone whereas at other far field zero normal gradient of dependent variables are assumed. To optimize the meshes, grid stretching in the cross flow direction is used. The stretching function is defined as an arctangent function which allows resolution up to 0.35 mm in the cross flow directions while it is fixed to 0.75 mm in the axial direction. The time step is s for a CFL number of The number of meshes is millions assigned to four processors working in parallel. 4. Results and discussion Figure 2 shows the flame structure inside the cone. The CH PLIF shows two lines from the thin reaction zones in the flame crossed by the laser sheet. They are wrinkled and one side of the reaction zone is higher than the other side, which indicates a dynamic oscillatory motion of the flame front. The CH PLIF image corresponds well with the LES results from the triple-flame propagation model where two highly wrinkled thin CH layers are shown. The chemiluminescence image shown in Fig. 2a is likely correlated with the CH radicals. They also show a two-layer structure.

139 B. Li et al. / Proceedings of the Combustion Institute 32 (2009) Fig. 2. A chemiluminescence image taken with 30 ls exposure time (a); a snap-shot of CH PLIF (b); an instantaneous CH (color/bright contours) and instantaneous stoichiometric mixture fraction (white lines) from LES (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) There are two stoichiometric iso-lines shown in the two-dimensional (2D) cut-plane in Fig. 2c. In three-dimension, they are iso-surfaces. The thin CH layer is associated with the outer stoichiometric surface only, indicating that the outer stoichiometric surface is burnable whereas the inner stoichiometric surface is not burnable. This can be understood from the distribution of zeroth level-set ð eg ¼ 0Þ shown in Fig. 3a. Due to the high speed flow from the nozzle (at the low boundary of the cone) the partially premixed triple flame is blown upwards, see the inner eg ¼ 0, Fig. 3a. To verify this triple-flame structure, we show results from the quenching flamelet model in Fig. 3b and c. The predicted scalar dissipation rate is high in the low part near the nozzle in the neighborhood of the stoichiometric surfaces (Fig. 3f). The scalar dissipation rate in the current case is about 5 s 1 (with the highest one about 15 s 1 ), which is lower than the quenching scalar dissipation rate of s 1 for a methane/air laminar flame in atmospheric conditions [5,6]. As a result, the quenching flamelet model predicted a burnable inner stoichiometric surface, Fig. 3b. Accordingly, four CH lines are predicted in the 2D cutplane as shown in Fig. 3c. This is contradictory to the CH PLIF and the chemiluminescence images shown in Fig. 2a and b. It appears that the triple-flame propagation model predicts results that are more consistent with the experimental observations, and this is also consistent with the previous speculations from the sampling flame analysis [15]. Fig. 3. Results from LES calculations. (a) Instantaneous temperature field (color/bright contours) and zeroth level-set (white lines) with the triple-flame model; (b) instantaneous temperature field (color/bright contours) and stoichiometric mixture fraction (white lines) with the quenching diffusion flamelet model; (c) instantaneous CH field (color/bright contours) and stoichiometric mixture fraction (white lines) with the quenching diffusion flamelet model; (d) instantaneous temperature field (color/bright contours) and streamlines (white lines) with the triple-flame model; (e) instantaneous mixture fraction field (dark gray: Z 6 Z st, light yellow/light gray: Z P Z st ) and zeroth level-set; (f) instantaneous scalar dissipation rate field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

140 1816 B. Li et al. / Proceedings of the Combustion Institute 32 (2009) Fig. 4. Instantaneous CH distribution above the cone. Left, LES results; right, PLIF results (x, distance to the cone exit plane). Fig. 5. Instantaneous CH 2 O distribution above the cone. Left, LES results; right, PLIF results. The triple-flame-stabilization mechanism can be visualized by the velocity field and temperature field. Illustrated in Fig. 3d are the instantaneous streamlines from LES showing the flow direction (not the magnitude of the speed). Near the wall of the cone, the flow stream is flowing upstream towards the nozzle. This generates shear layers with the flow from the nozzle and due to Kelvin Helmholtz instability vortices of large-scales are formed. In between the reversal air flow and the rich fuel/air mixture there is a low speed zone which is favorable for the triple-flame front to be stabilized, as shown in Fig. 3d. From Fig. 3e, it is seen that the lowest flame front is located near but not always exactly at the stoichiometry mixture fraction. The lowest front position is determined by the triple-flame propagation and the local flow velocity. The flame structures above the cone are presented in Figs. 4 and 5 where instantaneous CH and CH 2 O distributions from both LES and PLIF are shown. Two wrinkled thin CH layers are seen in Fig. 4. These layers are associated with the outer stoichiometric surfaces shown in Figs. 2 and 3. The flame structures above the cone from LES are similar with and without the use of triple-flame propagation model, since once the flame is stabilized inside the cone the main structure of the flame outside the cone is of non-premixed diffusion controlled type. The CH structures from LES resemble very well the CH PLIF images, although a little thicker CH layer is predicted in LES (about 1.7 mm) than that in the PLIF (about 1.25 mm). This is due to the fact that the LES spatial resolution is coarser than PLIF resolution (0.35 mm for LES and 0.06 mm for PLIF). The fact that CH PLIF layers are thin and the flamelet based LES results are similar to the PLIF images is a supportive evidence that the flamelet concept is acceptable for this type of flames. The CH 2 O layers from PLIF are rather thick; this may correspond to two effects. One is that the CH 2 O radicals are associated with chemical reactions at lower temperature fuel rich zones as an intermediate species with relatively longer life time than other radicals such as CH. Numerical calculations using detailed chemistry of 1D counter-flow laminar flame show that both the maximum of CH 2 O mass fraction and the width of the CH 2 O in mixture fraction space increase rapidly as the flow strain rate increases. This corresponds to the thicker CH 2 O layer in the current flamelet based LES results shown in Fig. 5. Another effect is that turbulent eddies may enter into the reaction zones associated with CH 2 O radicals since the life time of CH 2 O radicals are relatively long and CH 2 O layer in the laminar flamelet is relatively thick. This would transport CH 2 O radicals to wider zone and thereby it would broaden the distribution of CH 2 O radicals as indicated in the PLIF measurement. This effect is partly taken into account in the laminar flamelet based LES models by means of smoothening of the mixture fraction field by turbulence (cf. Eq. 2). However, since the life time of CH 2 O may be comparable to the turbulence-eddy time, more dynamic interaction between the flow and CH 2 O related reactions may play a role. This may explain the discrepancy that the thin peaks of

141 B. Li et al. / Proceedings of the Combustion Institute 32 (2009) CH 2 O are found in the LES result but not in the PLIF results. In this regard, interactive flamelet based LES models would perform better. It should be pointed out that comparing instantaneous LES and PLIF images are possible only in a qualitative sense. One can compare the thickness of the instantaneous CH and CH 2 O layers, and the degree of flame surface wrinkling. However, the exact distribution of these species in space cannot be quantitatively compared since they are not taken at the exact instant and the flame is unsteady. For example, in the CH images, the LES shows more structures in the inner part of the flame than that in the PLIF image, e.g., at 0.01 m < x < 0.04 m (Fig. 4). This is likely due to the three-dimensional effect of the imaging. At the particular instant of the LES, turbulence eddies pushed the flame front in the direction normal to the shown plane, thereby more structures of the CH zones are shown in the inner part of the shown plane. To evaluate the LES and PLIF in a quantitative way we consider comparing statistically averaged quantity, e.g., flame surface density (FSD). The mean reaction zone structure of a non-premixed flame may be characterized using FSD [33]. Following the numerical procedure of Donbar et al. [33] the flame surface density above the cone from the CH surfaces in the PLIF image and LES is computed and shown in Fig. 6. The FSD was determined using ensemble average of 200 CH PLIF images, and for LES it is from time average of 10,000 time steps which corresponds to 0.05 s of the flow time. Since the PLIF images are two-dimensional, the LES two-dimensional results in the corresponding cross section were used in the FSD calculation. The CH layer corresponds to the fuel-consumption reactions. From Fig. 6, it appears that the thickness of the mean CH (and mean fuel-consumption) zones at x = m is about 15 mm. At downstream axial positions (e.g., x = m, Fig. 6) the mean CH layer becomes thicker, and the two mean flame brushes merge. The maximum value of the flame surface density is about 0.2 mm 1 at x = m and m, and at far downstream the peak flame surface density decreases. This trend continues until the mean flame tip is reached. The LES and PLIF imaging show good agreement with each other in both the thickness of the mean flame brush and also the shape of the FSD profiles. 5. Conclusions The structure and stabilization mechanism of a partially premixed methane/air flame held on a conical jet burner were studied using laser diagnostics and LES. Thin CH layers in the cone as well as above the cone were detected using PLIF which is supportive to the flamelet combustion concept. Inside the cone the flame was found to be stabilized above the nozzle, according to both CH PLIF and chemiluminescence images. This is well understood from LES results. In LES it was found that near the inside wall of the cone there is a reversal flow towards the nozzle. The high speed flow of fuel/air mixture from the nozzle forms a shear layer where large scale vortices exist. These vortices effectively stabilize the triple-flame front. LES was carried out using two different flamelet models, one considered local flame quenching at high scalar dissipation rate, and one considered the triple-flame front propagation. The former model predicted a flame attachment to the nozzle with four CH layers in the two-dimension axisymmetric plane, which contradicted to the CH PLIF and chemiluminescence results. The triple-flame model correctly predicted the flame structures and stabilization inside the cone. Similar results for the flame structures above the cone were given by both models. Acknowledgments Fig. 6. Flame surface density (normalized by its maximum) along radial direction at (a) x = m, (b) x = m. This work was sponsored by the Swedish Research Council VR, SSF and STEM through CeCOST.

142 1818 B. Li et al. / Proceedings of the Combustion Institute 32 (2009) References [1] M.S. Mansour, Y.C. Chen, N. Peters, Combust. Flame 116 (1999) [2] T. Plessing, P. Terhoeven, N. Peters, M.S. Mansour, Combust. Flame 115 (1998) [3] C.M. Müller, M. Breitbach, N. Peters, Proc. Combust. Inst. 25 (1994) [4] B.A.V. Bennett, C.S. McEnally, L.D. Peefferle, M.D. Smooke, Combust. Flame 123 (2000) [5] R. Seiser, L. Truett, K. Seshadri, Proc. Combust. Inst. 29 (2002) [6] K. Seshadri, X.S. Bai, Proc. Combust. Inst. 31 (2007) [7] P. Petersson, J. Olofsson, C. Brackman, et al., Appl. Opt. 46 (19) (2007) [8] K.J. Nogenmyr, P. Petersson, X.S. Bai, et al., Proc. Combust. Inst. 31 (2007) [9] N. Peters, F.A. Williams, AIAA J. 21 (3) (1983) [10] J.P.H. Sanders, A.P.G.G. Lamers, Combust. Flame 96 (1-2) (1994) [11] L. Vanquickenborne, A.V. Tiggelen, Combust. Flame 10 (1) (1966) [12] D. Bradley, P.H. Gaskell, X.J. Gu, Combust. Flame 96 (3) (1994) [13] M.S. Mansour, Combust. Sci. Technol. 152 (2000) [14] M.S. Mansour, Combust. Sci. Technol. 174 (2002) [15] F. El-Mahallawy, A. Abdelhafez, M.S. Mansour, Combust. Sci. Technol. 179 (2007) [16] N. Peters, Prog. Energy Combust. Sci. 10 (1984) [17] N. Peters, Turbulent Combustion, Cambridge University Press, [18] D. Bradley, P.H. Gaskell, X.J. Gu, Proc. Combust. Inst. 27 (1998) [19] K. Bray, P. Domingo, L. Vervisch, Combust. Flame 141 (2005) [20] P. Domingo, L. Vervisch, Proc. Combust. Inst. 26 (1996) [21] G.R. Reutsch, L. Vervisch, A. Linan, Phys. Fluids 7 (6) (1995) [22] G. Amantini, J.H. Frank, M.D. Smooke, A. Gomez, Combust. Flame 147 (2006) [23] Z.S. Li, J. Kiefer, J. Zetterberg, et al., Proc. Combust. Inst. 31 (2007) [24] C.J. Randall, C. Murray, K.G. McKendrick, Phys. Chem. Chem. Phys. 2 (2000) [25] Z.S. Li, M. Afzelius, J. Zetterberg, M. Aldén, Rev. Sci. Inst. 75 (2004) [26] C. Brackmann, Z.S. Li, M. Rupinski, N. Docquier, G. Pengloan, M. Aldén, Appl. Spectrosc. 59 (6) (2005) [27] F.di. Mare, W.P. Jones, K.R. Menzies, Combust. Flame 137 (2004) [28] X.S. Bai, M. Balthasar, F. Mauss, L. Fuchs, Proc. Combust. Inst. 27 (1998) [29] X.S. Bai, L. Fuchs, F. Mauss, Combust. Flame 120 (2000) [30] N. Peters, in: N. Peters, B. Rogg (Eds.), Reduced Kinetic Mechanisms for Applications in Combustion Systems, Lecture Notes in Physics, vol. m15, Springer-Verlag, Heidelberg, 1993, pp (Chapter 1). [31] W.M. Pitts, Proc. Combust. Inst. 22 (1988) [32] R. Yu, K.J. Nogenmyr, X.S. Bai, Large eddy simulation of swirling turbulent methane/air flames, in: ECCOMAS Thematic Conf. Comput. Comb., June, Lisbon, [33] J.M. Donbar, J.F. Driscoll, C.D. Carter, Combust. Flame 122 (2000) 1 19.

143 Paper IV

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145 Effect of partial premixing on stabilization and local extinction of turbulent methane/air flames E. Baudoin 1, X.S. Bai 1*, B. Yan 1,2, C. Liu 1,2, A. Lantz 3, S.M. Hosseini 1, B. Li 3, A. Elbz 4, M. Sami 5, Z.S. Li 3, R. Collin 3, G. Chen 2, L. Fuchs 1, M. Aldén 3, M.S. Mansour 5 1 Division of Fluid Mechanics, Lund University, P.O. Box 118, S Lund, Sweden 2 Faculty of Environmental Science and Engineering, Tianjin University, , China 3 Division of Combustion Physics, Lund University, P.O. Box 118, S Lund, Sweden 4 Faculty of Engineering, Helwan University, Egypt 5 Cairo University, Natl. Inst. Laser Enhanced Sci., Cairo, Egypt *Corresponding author: Xue-Song Bai Division of Fluid Mechanics, Lund University, P.O. Box 118, S Lund, Sweden Xue-Song.Bai@energy.lth.se Key words: partially premixed flame, conical burner, flame stabilization, local extinction. 1

146 Abstract The stabilization characteristics and local extinction structures of partially premixed methane/air flames were studied using simultaneous OH-PLIF/PIV techniques, and large eddy simulations employing a two-scalar flamelet model. Partial premixing was made in a mixing chamber comprised of two concentric tubes, where the degree of partial premixing of fuel and air was controlled by varying the mixing length of the chamber. At the exit of the mixing chamber a cone was mounted to stabilize the flames at high turbulence intensities. The stability regime of flames was determined for different degree of partial premixing and Reynolds numbers. It was found that in general partially premixed flames are more stable when the level of partial premixing of air to the fuel stream decreases. For the studied burner configuration at high Reynolds numbers there is an optimal partial premixing level of air to the fuel stream at which the flame is most stable. OH-PLIF images revealed that for the stable flames not very close to the blowout regime, significant local extinction holes appear already. By increasing premixing air to fuel stream successively, local extinction holes develop leading to eventual flame blowout. Local flame extinction was found to frequently attain to locations where locally high velocity flows impinging to the flame. The local flame extinction poses a future challenge for model simulations and the present flames provide a possible test case for such study. 2

147 1. Introduction Partially premixed flames, defined as flames where the compositions of the mixture vary from fuel-rich to stoichiometry and fuel-lean [1], are found in many engineering applications. In modern internal combustion engines, using multiple injections of fuel to control emissions, partially premixed charge are typically formed before ignition. Due to the presence of rich, lean and stoichiometric mixtures in partially premixed flames, lean and rich premixed flame fronts exist in the leading front followed by the main diffusion flame [2-4]. The combustion characteristics of partially premixed flames are not well understood and modeling of partially premixed flames is challenging [1,2,5]. Local flame quenching and re-ignition in laminar and turbulent flames is an issue that has attracted the attention of recent research. Once a non-premixed flame is locally quenched, edge flame exists at the extremity of the reactions, where the flame progressively evolves to partially premixed flames [6]. The critical strain rate for quenching of partially premixed flames is influenced by the degree of partial premixing. For laminar flames, experimental and theoretical studies [7,8] using a counter flow configuration showed that partial premixing of fuel to the air stream can increase the quenching strain rate, whereas partial premixing of air to the fuel stream can decrease the critical quenching strain rate. In high Reynolds number turbulent flames, the quenching process is more complex. In an experimental study of partially premixed flames [9], it was shown that premixing of air to the fuel stream can decrease the flame stability regime, which is consistent with the laminar flame results of [7,8], however, a moderate partial premixing of air to the fuel stream can also improve the combustion stability. The fundamental physics behind this requires further investigation. The main objectives of this study are to investigate the stability characteristics of turbulent partially premixed flames for different level of partial premixing, and to examine the structures of local extinction of flames at different conditions. To generate turbulent partially 3

148 premixed flames at high turbulent intensities, the concentric flow conical flame burner described in [9-11] is adopted. Previous studies, at a particular partial premixing condition, have shown that the flame is stabilized in the cone due to the presence of a triple flame at the leading flame front in the recirculation region induced by the entrainment of ambient air to the cone [12,13]. The core of the flame is of diffusion flame type. Experiments showed that the flame is very stable as compared with the corresponding jet burner (removal of the cone) and that the stabilization position of the leading flame front in the cone was found to be rather independent of the Reynolds numbers and the fuels [13]. In the present study we focus on the effect of the partial premixing on the flame structure and the flame dynamic using experimental measurements of simultaneous OH-PLIF and 2D- PIV for methane/air mixtures at different degree of partial premixing and an overall equivalence ratio of 3. The statistical mean flame is measured using a 3D-PIV technique. The results are compared with numerical simulations using a two-scalar flamelet large eddy simulation model (LES) described in [12]. 2. Experimental setup The burner used in this study consists of two parts, a variable size mixing chamber and a quartz-glass conical nozzle, Fig.1. The mixing chamber is composed of two concentric tubes of diameters d=6.8 mm and D=9.7 mm for the inner tube and the outer tube, respectively. Their respective lip-thicknesses are 1.2 and 2.3 mm. By adjusting the inner tube position, one can vary the size of the chamber (the mixing length L) to determine the level of partial premixing in the flames. A quartz-glass conical nozzle, with a half cone angle of 26, is mounted at the exit of the mixing chamber. Detailed information about the burner can be found in [9]. 4

149 In order to characterize the flame structure and the dynamics of the leading flame fronts, simultaneous OH-PLIF and PIV is carried out inside the cone. OH radicals are excited through the Q 1 (8) transition near 283 nm and the resulting fluorescence emission is collected around 308 nm. A frequency doubled dye laser (Continuum ND60) with Rhodamine 590 as dye solution is used to provide the required output energy pulse at 283 nm. The dye laser is pumped by the second harmonic of a Nd:YAG laser system (Continuum NY82). A laser sheet with 50 mm height and pulse energy of 12 mj per pulse are utilized. The fluorescence signal was collected perpendicularly to the laser sheet employing an ICCD camera (Princeton Instruments PI-MAX, pixels) equipped with a UV-lens (UV-Nikkor, f=105 mm) after being filtered through a UG 11 filter (Schott, 3 mm) combined with a long pass filter at 295 nm. The spatial resolution was 0.13 mm/pixel. For the 2D-PIV, seeding is exited twice at 20 Hz (for each measurement) using a doublepulsed Nd-YAG Class 4 Gemini-PIV laser (New Wave Research, Inc.) operated with 100 mj double pulse (25mJ per pulse) and at 532 nm. The laser sheet thickness of each pulse is about 1~2 mm. The signal is collected perpendicularly to the laser sheet employing a CCD camera (LaVision FM3S Double Shutter, pixels) equipped with a bandpass filter (532 nm) with transmission up to 95% in order to minimize the noise effects of background light. Seeding of Titanium Dioxide powder particles of 20μm diameter (ρ=4.23 g/cm 3 ) is continuously injected in the air stream in the upstream of the mixing chamber. The data is processed using LaVision software (Davis 7) with pixels interrogation windows and 50% area overlap leading to a spatial resolution of 1.2 mm. About 1000 images are recorded, simultaneously for the 2D OH-PLIF and PIV, for each L/D case at a rate of 2.5 Hz. All samples are used to compute statistics. The maximum intensity is observed for case L/D=7 with a value less than 10% higher than for L/D=3 and 5

150 L/D=5. All OH signal intensities (instantaneous, mean and rms), after background noise subtraction, are normalized with respect to this overall maximum for consistency. The 2D-PIV was tuned to best capture the seeding particles in the reaction zones. A 3D- PIV technique is used to accurately capture the overall high speed flow inside the cone. The 3D-PIV setup is a Dantec model with two CCD cameras and double pulse, two-head Nd:YAG laser with pulse energy of 50 mj at the second harmonic 532 nm. The cameras are HiSense MkII PIV CCD cameras (model C CP) with 1280 x 1024 CCD light sensitive array and equal number of storage cells. The objectives of the cameras are covered with interference filters at 532 nm with a bandwidth of 10 nm. The laser pulse duration is 6 ns and the pulse delay is controlled according to the flow velocity with a minimum of 0.2 μs. 3. Numerical modeling A two-scalar flamelet model described in [12] is employed in the LES. The triple flame front propagation is modeled using a level-set G equation approach while the core of the flame is modeled using a diffusion flamelet approach. Local extinction is taken into account in flamelet tabulation using the scalar dissipation rate. The spatial filtered continuity, momentum and mixture fraction transport equations are discretized and solved on a staggered Cartesian grid using a fifth order WENO schemes for the convective terms and forth order central difference scheme for the diffusion terms. Time integration is made using a second implicit scheme. The level-set G-equation is solved using the third order WENO scheme. Further details about the method and boundary conditions are referred to [12]. The computational domain outside the cone extends to 5.26D 0 in the axial direction and 3.08D 0 in the cross flow directions, respectively, where D 0 (=73 mm) is the diameter of the cone exit. In order to ensure good resolution inside the cone grid stretching in the cross flow direction is 6

151 used. This allows a resolution up to 0.35 mm in the cross flow directions while it is fixed to 0.75 mm in the axial direction. 4. Results and discussions First, the stability behavior of the flame is studied by varying the mixing length (L/D) and the overall equivalence ratio successively for different Reynolds numbers. Figure 2 shows three lines defining the critical boundary between stable flames and blowout for the Reynolds numbers (Re, defined based on the bulk flow at the exit of the mixing chamber) of 4000, 8000, and 12000, respectively. At each point the equivalence ratio (Φ) is the overall equivalence ratio of the fuel and air supplied to the inlets of the mixing chamber. The mixing length (L/D) determines the concentration gradient in the mixture at the exit of the mixing chamber, and as a result the local degree of partial premixing in the reaction zone. Stable flames occur when the overall equivalence ratio is above a critical equivalence, whereas below this critical equivalence ratio blowout occurs. It appears that when more air is premixed to the fuel stream while keeping the same Reynolds number, i.e. decreasing the overall equivalence ratio, the flame changes from the stable state to blowout. This phenomenon is consistent with the experimental and theoretical results for laminar counter flow flames [7,8], where it was shown that flame extinction becomes easier when air is premixed to the fuel stream, since it helps the oxygen leakage through the reaction zone to the fuel stream thereby decreasing the temperature and the radical pool in the reaction zone. Furthermore, one can notice that with decreasing Reynolds numbers, the critical equivalence ratio for flame stabilization decreases. The flame stability behavior is affected by the composition gradient in the fuel/air mixture. As shown in Fig.2, for the Reynolds number flames, with L/D=0 (high composition gradient) and L/D=10 (low gradient), the critical equivalence ratio for flame extinction is 7

152 relatively higher than the cases in between. It appears that with certain composition gradient in the mixture the stability of flame can be improved, whereas with too high composition gradient the flames become less stable. For the Reynolds number 4000 flames, however, increasing the composition gradient in the mixture leads to an increase of the stability domain, i.e., the larger the composition gradient in the mixture the larger the stability domain. To improve the understanding of the above flame stability behavior we examine the OH PLIF images taken at L/D=5 and Φ=1.5 and 2, Fig.3. At Φ=1.5 and Re=4000, the OH radicals are found in narrow and smooth zones. The flame is essentially laminar flamelet type. With Φ =1.5 and increasing the Reynolds number from 4000 to 8000, the OH radicals distribute in a much wider zone, and some part of the OH zone is disconnected indicating local flame extinction. Increasing the Reynolds number to with Φ=1.5 the flame is blown out, as indicated in Fig.2, where Φ=1.5 and L/D=5 is below the critical boundary of Re=12000 flows. Similar trend can be seen for the Φ=2 flames (Fig.3c-e). For a given overall equivalence ratio and L/D, increasing the Reynolds number leads to moving up of the critical boundary to higher overall equivalence ratios and the studied flames in Fig.3 are closer to blowout conditions. As the flames approach to blowout conditions, the OH radicals are found in thicker zones. Figure 4 shows the effect of L/D on the flame structures. At Re=12000 and Φ=3, the three flame cases (L/D=3, 5, 7) are in the stable flame regime as indicated in Fig.2. Despite this, one can notice the existence of local flame extinction holes in all these flames, Fig.4. It is typical when local extinction is encountered, the flow velocity is locally large as indicated by the small window at the low-right corner in each figure, e.g., Fig.4f. The local flow is typically impinging to the flame (shown in the figure as the OH layer), Fig.4a,c,f,i. It appears that the locally high strain rate causes the local extinction of the flame. Direct numerical 8

153 simulations of partially premixed flames [14] showed the same observation that local quenching is more often attained at the high strain rate regions. Comparing the snap-shot of OH for the case of L/D=5 shown in Fig.4d-f and the corresponding one shown in Fig.3e, which has a lower equivalence ratio of Φ=2 and L/D=5, Re=12000, one can identify the effect of partially premixing on OH broadening in the flames. Since the two flames have the same Reynolds number the intensities of turbulence in the two flames are on the same order; with high Φ and the same L/D, the composition gradient in the flames shown in Fig.4d-f is higher than that in Fig.3e. It can therefore be concluded that the OH layer is thinner with higher composition gradient. The same conclusion can be drawn from the three L/D flames shown in Fig.4. With smaller L/D, thus higher composition gradient, the OH layer is thinner, and the flame is more stable. One may note that the leading flame fronts are stabilized in the recirculation zones introduced by the entrained air flow near the wall of cone. The recirculation zones oscillate in the cone causing the flame fronts to oscillate, as seen in Figs.4d,e. The flame is flashing back to lower positions if the local flow is along downward direction (Fig.4d), or pushed upwards if the local flow is along upwards direction (Fig.4e). LES using the level-set and mixture fraction formulation showed good agreement with the overall flame structures, especially the leading flame front stabilization in the recirculation zones introduced by the entrained ambient air for the case L/D=5 and Re=12000 [12]. Further comparison of the model results with the local extinction structures discussed above show that the model, which employs a stationary flamelet library approach with local extinction modeled using scalar dissipation rate, is not accurate in simulating the flame holes. It has been observed in DNS studies that once a flame hole is formed, the required scalar dissipation rate for developing the flame holes further is lower than the quenching scalar dissipation of stationary flamelets [6]. To take into account such unsteady effect, potential models, e.g. 9

154 those involving reaction progress variables [15-17] need to be developed and validated. Recently the progress variable approach of Bray et al. [15] has been applied to simulation of lifted jet flames and the approach shows promising features [18]. To quantitatively present the measurements and simulations, we examine the mean statistical properties of the flames. Figure 5 shows the normalized mean (a-c) and rms (d-e) of OH-PLIF signal. The maximum intensity is almost the same in the studied L/D cases. As an average OH signal is detected as early as x/d=1.5 for all cases. This shows that the mean leading flame front is stabilized in nearly the same position in the cone, independent of the degree of partial premixing. This observation is consistent with the previous experiments using different fuels and Reynolds numbers [13]. This is owing to the fact that the entrained air flow and recirculation zone structures are insensitive to the fuels and the degree of partial premixing when the Reynolds number is high enough so that the flow is developed to fully turbulent. The effect of L/D is shown in the broadening of the mean OH layers. The mean and rms profile of OH signal is widened as the level of partial premixing increases. The mean flame thickness increases as x/d increases in all the cases. For the present studied flames that have the same Reynolds numbers, the broadening of the OH layer as L/D increases is mainly an effect of the finite-rate chemistry. The broadening of OH layer is an indication of flames subjecting to flame extinction as seen in the snap-shot OH images in Figs.3 and 4. Figure 6a shows the probability density function (PDF) of stabilizing the leading flame fronts at a given height. As seen the highest PDF of flame front position is around x/d=1.5. The flames are seen to distribute between wide regions, 1<x/D<5. Most frequently the flames are found in a region 1.5<x/D<2. The three flames show similar distribution of the flame front position. Figure 6b shows the PDF of number of OH signal extinction sites (i.e. of finding flame holes) on the right branches in the OH images for the three L/D cases. As seen, 10

155 most frequently 2 flame holes are found in these flames. The PDF of number of OH extinction sites in the L/D=7 flame is shifted slightly to having more flame holes, corresponding to the fact that the L/D=7 flame is closer to the critical boundary of flame blowout, Fig.2. The mean axial velocity along the axis and along radical direction at two axial positions from the 3D-PIV and LES are shown in Fig.7. The velocity is normalized based on the bulk flow velocity at the exit of the mixing chamber (U 0 =20 m/s). The axial velocity decreases along the burner axis, due to the expansion of the cone. The radial profile from LES shows negative axial velocity at x/d=2, indicating the flow recirculation. The PIV could not capture the air entrainment properly due to the lack of seeding in the entrained air streams. The LES mean OH distribution is compared with the PLIF measurements. Although they are two different quantities, the shapes of the OH profiles from PLIF and LES can be suitable for comparison. The inner boundary of the mean OH layer is predicted well by LES. The thickness of the OH layer predicted from the stationary flamelet model is larger than the measured one. Despite such discrepancy in the flame structure, the mean flow velocity predicted by the flamelet LES model is in reasonably good agreement with experimental data, showing the relatively low sensitivity of the mean flow field prediction to combustion models. 5. Conclusions The effect of partial premixing on the stabilization and local extinction of partially premixed methane/air flames is studied in a concentric flow conical burner using 3D-PIV, and simultaneous OH-PLIF and 2D-PIV. The experimental results are compared with model prediction using a two-scalar flamelet LES model. It is found that for a given Reynolds number partial premixing of air to the fuel stream decreases the flame stability. At high Reynolds numbers with a moderate gradient in the composition the flame is more stable, 11

156 whereas at low Reynolds numbers, the lower the composition gradient the less stable the flame. The stability behavior is related to the OH-radical layer. When the flame is close to the critical condition of flame quenching the OH layer broadens and flame holes appear. The onset of flame holes is shown to be directly coupled to the local flow velocity and thereby the local strain rate. Due to the triple-flame propagation in the recirculation zones the leading flame front position is found to be insensitive to the partial premixing, and the highest PDF of leading flame position is found near the location where the measured mean axial velocity is negative. Although the flamelet LES model predicted rather well the mean flow field and the flame stabilization, it failed to predict the local extinction structures based on the stationary flamelet tabulation method using the scalar dissipation rate. The required critical strain rate for local extinction is likely lower than the one calculated from one dimension stationary laminar flames. 6. Acknowledgement This work was partly sponsored by the Swedish Research Council VR, SSF and STEM through CeCOST and the Swedish International Development Agency (SIDA) fund for the MENA countries through the joint project between Egypt and Sweden, and partly sponsored by China Natural Science Foundation (grant no ) and CSC (China Scholarship Council). 12

157 References 1 R.W. Bilger, S.B. Pope, K.N.C. Bray, J.F. Driscoll, Proc. Combust. Inst. 30 (2005) C.M. Müller, M. Breitbach, N. Peters, Proc. Combust. Inst. 25 (1994) G.R. Reutsch, L. Vervisch, A. Linan, Phys. Fluids 7 (6) (1995) P. Domingo, L. Vervisch, Proc. Combust. Inst. 26 (1996) N. Peters, Turbulent Combustion, Cambridge University Press, V. Favier, L. Vervisch, Combust. Flame 125 (2001) R. Seiser, L. Truett, K. Seshadri, Proc. Combust. Inst. 29 (2002) K. Seshadri, X.S. Bai, Proc. Combust. Inst. 31 (2007) F. El-Mahallawy, A. Abdelhafez, M. S. Mansour, Combust. Sci. and Tech. 197 (2007) M.S. Mansour, Combust. Sci. Technol. 174 (2002) M.S. Mansour, Combust. Sci. Technol. 152 (2000) B. Li, E. Baudoin, R. Yu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Aldén, M.S. Mansour, Proc. Combust. Inst. 23 (2009) B. Yan, B. Li, E. Baudoin, C. Liu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Aldén, G. Chen, M.S. Mansour, Exp. Therm. Fluid Sci. (2009), doi: /j.expthermflusci M. Yaldizli, K. Mehravaran, H. Mohammad, F. A. Jaberi, Combust. Flame 154 (2008) K. Bray, P. Domingo, L. Vervisch, Combust. Flame 141 (2005) C. D. Pierce, P. Moin, J. Fluid Mech. 504 (2004) M. Ihme, C. M. Cha, H. Pitsch, Proc. Combust. Inst., 30 (2005) S. A. Ferraris, J.X. Wen, Combust. Flame 150 (2007)

158 LIST OF FIGURES Figure 1. Schematic illustration of the burner and the PIV/OH-PLIF windows. Figure 2. Stability regime of the flames at different Reynolds numbers, solid: Re=12000, dashed line: Re=8000, dot line: Re=4000. Symbols correspond to conditions of PIV and OH- PLIF measurements. Figure 3. Snap-shots of OH-PLIF at different equivalence ratios and Reynolds numbers for L/D=5, (a) Φ=1.5, Re=4000; (b) Φ=1.5, Re=8000; (c) Φ=2, Re=4000; (d) Φ=2, Re=8000; (e) Φ=2, Re= Figure 4. Normalized simultaneous OH-PLIF signal, instantaneous streamlines and velocity vectors from PIV (in the zoomed windows) for flames with Φ=3, Re=12000 and different mixing lengths, L/D=3 (a,b,c), L/D=5 (d,e,f) and L/D=7 (g,h,i). Zoomed windows correspond to the marked regions (dashed line square). Figure 5. Normalized mean of OH-PLIF signal (a-c) and its rms (d-f) inside the cone for flames with Φ=3, Re=12000 and different mixing lengths, L/D=3 (a,d), L/D=5 (b,e) and L/D=7 (c,f). Figure 6. PDF of flame front position (a) and PDF of OH signal extinction holes (b) inside the cone for flames with Φ=3, Re=12000 and different mixing lengths. The OH signal threshold is taken as 5% of the maximum intensity, below which the flames are considered locally quenched. Figure 7. Mean axial velocity along the centre line (a), along radial direction (b), normalized OH-PLIF intensity, and normalized OH mole fraction from LES along radial direction (c), for flames with Φ=3, Re=12000 and L/D=5. Symbol/line notations at x/d=2: Exp (o), LES (solid line), at x/d=4: Exp ( ), LES (dashed line). 14

159 PIV window: 96*77mm PLIF window: 66*66mm 70mm D=9.7mm 19mm 8.0mm 6.8mm L Figure 1. Figure 2. 15

160 Figure 3. Figure 4. 16

161 Figure 5. Figure 6. 17

162 Figure 7. 18

163 Paper V

164

165 Experimental Thermal and Fluid Science 34 (2010) Contents lists available at ScienceDirect Experimental Thermal and Fluid Science journal homepage: Structures and stabilization of low calorific value gas turbulent partially premixed flames in a conical burner B. Yan a,b,b.li c, E. Baudoin b, C. Liu a,b, Z.W. Sun c, Z.S. Li c, X.S. Bai b, *, M. Aldén c, G. Chen a, M.S. Mansour d a Faculty of Environmental Science and Engineering, Tianjin University, Tianjin, China b Division of Fluid Mechanics, Lund University, Lund, Sweden c Division of Combustion Physics, Lund University, Lund, Sweden d Cairo University, Natl Inst Laser Enhanced Sci., Cairo, Egypt article info abstract Article history: Received 14 October 2009 Accepted 16 October 2009 Keywords: Low calorific gas flames Partially premixed flames Conical burner Flame stability Local flame extinction Experiments are carried out on partially premixed turbulent flames stabilized in a conical burner. The investigated gaseous fuels are methane, methane diluted with nitrogen, and mixtures of CH4, CO, CO2, H 2 and N 2, simulating typical products from gasification of biomass, and co-firing of gasification gas with methane. The fuel and air are partially premixed in concentric tubes. Flame stabilization behavior is investigated and significantly different stabilization characteristics are observed in flames with and without the cone. Planar laser induced fluorescence (LIF) imaging of a fuel-tracer species, acetone, and OH radicals is carried out to characterize the flame structures. Large eddy simulations of the conical flames are carried out to gain further understanding of the flame/flow interaction in the cone. The data show that the flames with the cone are more stable than those without the cone. Without the cone (i.e. jet burner) the critical jet velocities for blowoff and liftoff of biomass derived gases are higher than that for methane/ nitrogen mixture with the same heating values, indicating the enhanced flame stabilization by hydrogen in the mixture. With the cone the stability of flames is not sensitive to the compositions of the fuels, owing to the different flame stabilization mechanism in the conical flames than that in the jet flames. From the PLIF images it is shown that in the conical burner, the flame is stabilized by the cone at nearly the same position for different fuels. From large eddy simulations, the flames are shown to be controlled by the recirculation flows inside cone, which depends on the cone angle, but less sensitive to the fuel compositions and flow speed. The flames tend to be hold in the recirculation zones even at very high flow speed. Flame blowoff occurs when significant local extinction in the main body of the flame appears at high turbulence intensities. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Biomass derived fuels are important ingredients for heat and power production owing to the renewable nature and sustainable availability of this energy source. There are considerable interests from industry to utilize biomass gasification and pyrolysis gases in gas engine and gas turbine applications. These gases are often called low calorific value (LCV) gases and they are typically comprised of H 2, CO, CH 4 and a small amount of higher hydrocarbons [1]. There may be large amount of N 2 in gases from air-blown gasification, and CO 2 in gases from landfills. These inert gases lead to lower calorific value of the mixture gas than pure H 2 or natural gas. Combustion characteristics of LCV gases, such as the laminar burning velocity and NO x formation have been reported in the literature. * Corresponding author. Tel.: ; fax: address: xue-song.bai@energy.lth.se (X.S. Bai). From a practical point of view, it is important to known the stability behavior of LCV gas flames in different type of burners such as jet/bunsen burners. The stability of lifted jet flames with model LCV fuels has been studied recently [2], where the fuels utilized were mixtures of CH 4 and C 2H 4 diluted by N 2. The relationships between the liftoff velocity and dilution, liftoff height behavior as well as reattachment conditions were described. The stabilization mechanism of lifted jet flames of CH 4 diluted with N 2 was investigated in [3]. It was reported that stationary lifted flames were only observed in the near field of coflow jets. The stability behavior of LCV gases in a jet flame burner is investigated in this study. Recently, experimental and numerical studies of a conical burner were carried out for different fuels [4,5]. It was shown that the cone enhances significantly the stability of flames. In this study the stability behavior of LCV gases in the conical burner is examined. The considered LCV gases correspond to the biomass gasification gases co-firing with natural gas with a low heating value (LHV) of MJ/N m 3. Another LCV gas considered is a mixture of methane and nitrogen, also having a LHV of 24 MJ/N m /$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi: /j.expthermflusci

166 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) It is shown that in the conical burner the flame stability is insensitive to the fuels and heating values; however, when the cone is removed, i.e. with jet flame burner, the flame liftoff and blowoff conditions are very sensitive to the fuel compositions. To investigate the mechanisms of flame stabilization in the conical burner, planar laser induced fluorescence (PLIF) technique is used to detect the OH radicals and a fuel-tracer species (acetone) inside the cone. Large eddy simulation (LES) is carried out to investigate the flow field in the cone. Table 1 The compositions (vol.%) and low heating values of studied fuels, and laminar burning velocity of stoichiometric fuel/air mixture at atmospheric and room temperature conditions (300 K). The laminar burning velocity was calculated using Cantera code and GRI 3.0 mechanism. Test fuels CO H 2 CH 4 CO 2 N 2 S L (m/s) LHV (MJ/N m 3 ) LCV gas LCV gas MN Methane Experimental setup and conditions The burner consists of two vertical concentric tubes. The inner and outer tubes are 6.8 mm and 9.7 mm in diameter and with a lip-thickness of 1.2 mm and 2.3 mm, respectively (cf. Fig. 1). A mixing distance L, which is the distance between the inner tube exit and the outer tube exit, can be varied to generate different degree of partial premixing. The fuel flow discharges through the outer tube while the air flow through the inner one. A quartz glass conical nozzle, which makes optical access and observation possible, with half cone angle of 26 is installed at the outer tube exit and used to stabilize the flames. Detailed information about the burner can be seen elsewhere [4,5]. The LCV gases considered in this work are shown in Table 1. LCV gases 1 and 2 are mixtures of typical biomass derived gases (BDG) and methane. Mixing of these gases can improve the LHV from about 5 MJ/N m 3 to above 10 MJ/N m 3. The compositions of the BDG are taken from Ref. [6], which are typical compositions of gases produced in a gasification demonstration plant in Sweden. The LCV gas 1 is mixed by 60% CH 4 and 40% BDG by volume; the LCV gas 2 is mixed by 20% CH 4 and 80% BDG by volume. Fuel MN is a mixture of only CH 4 and N 2 having the same LHV as LCV gas 1. The uncertainty of each gas composition is within ±2.0%. For comparison with the LCV gases, methane/air flames in the corresponding conditions are also measured. To characterize the fuel distribution in the flow field, a tracer of fuel (acetone) was introduced to the fuel stream in the concentric tubes. The gaseous acetone was introduced into the burner by passing the fuel through liquid acetone, which gives about 1% acetone by volume in the fuel mixture. The mass flow rates of the fuel and air streams were monitored using Bronkhorst thermal mass meters and controllers installed in their respective feed lines. The flow meters were calibrated at conditions of 0 C and 1 atm. The flow meters have an accuracy of ±0.9% of the measured values. The Q 1(8), F 2 R + X 2 R + (1, 0) transition at 283 nm was excited using the second harmonic radiation from a Nd:YAG pumped dye laser (Rhodamine 590) and OH fluorescence was detected at about 308 nm. The laser pulse energy is 20 mj. This setup has been shown to be satisfactory in measuring OH distributions in partially premixed methane/air jet flames [7]. Acetone was excited at the same wavelength as OH LIF, which enables the simultaneous measurements of them. A laser sheet with 40 mm height was formed inside the cone of the burner by a cylindrical/spherical lens combination. The OH and/or acetone PLIF signal was collected vertically by an f/4.5 objective (Nikon UV, f = 100 mm), and then captured by an intensified charge coupled device (ICCD) camera (Princeton PI-MAX, pixels). Three different sets of optical glass filters (Scott, 3 mm) were employed in front of the camera to filter the laser scatterings and unwanted fluorescence. For OH-PLIF signal recording a long-pass WG305 and a short-pass UG11 worked together to collect the OH DC 70mm Laser window 26 degree Filter ICCD SL CL 9mm D=9.7mm 45mm Nd :YAG Pumped Dye Laser PB 19mm 8.0mm 6.8mm L=72.75mm L/D=7.5 mm x Top 27.1mm Middle 18.7mm Low 9.2mm r mm Fig. 1. Burner details and experimental setup for simultaneous OH/acetone PLIF measurement: PB, Pellin Broca prism; DC, dichroic mirror; SL, spherical lens; CL, cylindrical lens.

167 414 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) fluorescence at around 308 nm, while a WG345 filter was chosen to suppress OH-PLIF and Rayleigh signals and select the broadband emissions from 350 to 500 nm when acetone PLIF was recorded. Only a WG305 filter was used for simultaneous OH and acetone PLIF measurements. 3. Numerical simulation models The conical flames are simulated numerically using a large eddy simulation model described previously [8 10]. The combustion model is based on a two-scalar flamelet approach where the mixing process is described using a conserved scalar, the mixture fraction, and the triple flame front propagation is modeled using a level-set G-function. Flamelet chemistry is coupled to the flow field simulations using a diffusion flamelet library tabulated as a function of mixture fraction and scalar dissipation rate, from numerical simulations in a counter-flow partially premixed flame configuration. The scale-similarity model (SSM) [11] is used for the sub-grid stresses (SGS) in the spatially filtered momentum equations, and Smagorinsky model is used for the SGS transport fluxes in the scalar transport equations. The spatially filtered continuity, momentum and energy equations as well as the mixture fraction transport equation are discretized and solved on a staggered Cartesian grid using a finite difference algorithm with multi-grid acceleration [12]. A 5th order WENO scheme [13] is employed for the convective terms, whereas all other spatial derivatives are discretized with a 4th order central scheme, and the time integration is performed by a second order implicit scheme. The level-set G-equation is solved with a 3rd order WENO scheme in combination with a 3rd order TVD Runge Kutta time integration scheme. The flow domain outside the cone extends to 5.26D 0 in the axial direction and 3.08D 0 in the cross flow directions, respectively, where D 0 (=73 mm) is the diameter of the cone exit. The number of meshes used in the LES is about 8 millions assigned to four processors working in parallel. The time step in the LES is 5 ls. Further details about the grid, implementation of the inflow and outflow conditions are referred to [5]. 4. Results and discussion 4.1. Regimes of flame stability Fig. 2 shows the stability behavior of the flames with different fuels, with and without the cone. All cases are at the same mixing condition of L/D = 7.5 (cf. Fig. 1). First, one can observe that without the cone (i.e. jet burner) the partially premixed flames can be in different states depending on the overall equivalence ratio and jet velocity (or equivalently Reynolds number based on the flow conditions at the tube exit). For the methane/air flames, and the diluted methane (MN)/air flames, the flames can be attached to the burner rim, regime A, or lifted from the burner rim, regime C, or blowout (total flame extinction), regime above C. The attached flame (regime A) occurs at low jet speed condition; for the MN/ air flames the jet Reynolds number is below 2800 which yields typically laminar flows. For methane/air flames the jet Reynolds number is below These critical Reynolds numbers are fairly independent of the overall equivalence ratio when the equivalence ratio is larger than 6. Blowout happens at much higher jet velocity, and this depends on the overall equivalence ratio. Within regime B the flames can either be attached or lifted; this depends on the initial state of ignition. If the flame is initially attached to the burner rim, increasing the jet velocity gradually will keep the flames attached within regime B. On the other hand, if the initial flame is in regime C, i.e. in lifted flame mode, decreasing the jet velocity gradually to regime B will also keep the flame lifted. Reducing the velocity further to regime A will lead the flame to being re-attached to the burner rim. This path-dependent phenomenon is known as hysteresis [14 16]. The mode of the flames depends on the initial state and the path at which the condition is changed. The hysteresis of flame states is attributed to the significant difference in the distance between the nozzle exit and location of laminar to turbulent transition [17], when the jet flow is varied from low speed to high speed and from high speed to low speed. One can observe that the hysteresis regime (B) of the methane/ air flames is larger than that of MN/air flames, showing the sensitivity of the flame stabilization to heating values of the fuel. Turbulent non-premixed jet flames with methane and ethylene diluted with nitrogen show similar three-regime stability behavior [2]. The liftoff and reattachment behavior were shown to be rather sensitive to the level of nitrogen dilution to the fuel stream; in the liftoff mode, for a given liftoff height the stabilization Reynolds number decreases rapidly with nitrogen dilution. Won et al. [3] showed that the liftoff behavior of jet flames is strongly correlated to buoyancy effect. According to their numerical study, in microgravity conditions only the blowoff and attached flame modes were possible. For the LCV gases 1 and 2 jet flames and with the equivalence ratio less than 8, the flames are in two modes only, attached to the burner rim, or blowoff. Higher equivalence ratio cases for the LCV gas flames were not measurable due to the limitation of the current flow meter systems. By blowoff it is meant that the flame is totally quenched when starting from the attached mode and (a) (b) Reynolds number MN reattached no cone MN lifted no cone MN blow out no cone LCV 1 blow off no cone LCV 2 blow off no cone MN blow off with cone LCV 1 blow off with cone LCV 2 blow off with cone Reynolds number C CH 4 reattached no cone CH 4 lifted no cone CH 4 blow out/off no cone CH 4 blow off with cone C B A 4000 B A Equivalence ratio Equivalence ratio Fig. 2. Flame stability behavior of partially premixed jet flames and conical flames with different fuels at L/D = 7.5.

168 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) gradually increasing the jet velocity, whereas blowout refers to quenched flame starting from lifted mode and gradually increasing the jet velocity [18]. The above discussion of stabilization behavior of the methane, MN and LCV gases flames shows that for jet flames the flame stability is very sensitive to the fuel type. The LCV gas flames are more stable than the MN flames although the LHV of MN is higher. With equivalence ratio less than 4, the LCV gas 2 flames are more stable than the LCV gas 1 flames. The LHV of LCV gas 2 is only half of that in LCV gas 1, but the hydrogen mole fraction in LCV gas 2 is higher than that in LCV gas 1. It appears that with LCV gases where hydrogen is present, the flames are more stable and tend to attach to the burner rim. Similar observations have been reported in [3,14,15,19]. For the flames with the cone, one can observe that the stability characteristics of the flames with methane, MN, and LCV gases are fairly similar: the flame is stabilized inside the cone for large range of equivalence ratio and jet Reynolds numbers, and the blowoff limit is nearly linearly dependent of the overall equivalence ratio. For the flames with overall equivalence ratio higher than 5, it is difficult to observe the blowoff mode in the current rig due to the limited mass flow meters. The flames are more stable than the corresponding ones without the cone. The LCV gas 1 flames are somewhat more stable than that of MN fuel, although they have the same LHV. The difference is however much smaller than that without the cone. Table 2 PLIF experimental conditions for the partially premixed conical flames and notations of the flames. Flames Fuels Q F (l/min) Q A (l/min) u Exit velocity (m/s) L1F1V1 LCV gas L1F2V1 LCV gas L1F1V2 LCV gas L1F2V2 LCV gas L1F1V3 LCV gas ,600 L1F2V3 LCV gas ,520 L2F1V1 LCV gas L2F2V1 LCV gas L2F1V2 LCV gas L2F2V2 LCV gas MF1V1 Methane MF2V1 Methane MF1V2 Methane MF2V2 Methane MF1V3 Methane ,530 MF2V3 Methane ,410 Re 4.2. Structure of the partially premixed conical flames To understand the stability behavior of the conical flames, the reaction zone structures of the flames are studied using simultaneous OH/acetone PLIF imaging. Table 2 lists the experimental cases studied using PLIF of acetone and OH. Three burner exit velocity conditions from the mixing tube (10, 15 and 18 m/s) were considered. The burner exit velocity is defined as the bulk flow velocity (with uniform velocity profile) at the exit of the mixing tube (x = 0, Fig. 1). Two overall equivalence ratio conditions, 2.6 and 4, were considered in the experiments. All these test flames have a mixing length L/D of 7.5. From 1000 single-shots we obtain the ensemble average of distributions of acetone and OH radicals in the LCV gas 1 flames at different tube exit velocity and equivalence ratios, Fig. 3. As seen, the lowest locations of OH distribution for all four flames are almost at the same height. These locations correspond to the leading flame fronts where the flame is stabilized. This result indicates that the flame stabilization of the conical flames is not sensitive to the exit flow Reynolds number and equivalence ratio. This is contrary to the jet flames, where the liftoff height of jet flame increases with increasing jet exit velocity [14 16,19]. Figs. 3c and d shows a slight asymmetry of the leading flame front position at the high speed condition. This is likely a result of a slight asymmetry of the cone and the concentric tubes. Figs. 4 6 show snap-shots of OH signals, respectively for the methane/air flames and LCV gas flames, under different flow rates and at equivalence ratio of 4. For the low speed flames (exit velocity 10 m/s) the instantaneous OH images are fairly smooth. For the higher speed flames (exit velocity 18 m/s) the flames are more wrinkled, due to the elevated level of flame/turbulence interaction. The thicknesses of OH layers in the high speed flames are similar to that in the low speed flames. At low flow speed (10 m/s) the reaction zones (OH layers) in the LCV gas 2 flame shows flame holes, indicating possible local extinction. Although not seen in Figs. 4a and 5a, in general the reaction zones in the methane flames and the LCV gas 1 flames could also contain flame holes. To quantify the flame holes we defined a threshold value of 5% of peak OH signal in the experimental window, below which the reaction rates are considered to be low. A reaction zone disconnected by a pocket where the OH signal is lower than the threshold is considered to have a flame hole. Further, we define a probability of having flame holes in a flame as follows: if there is at least one flame hole in the entire experimental window, the flame is considered to have flame holes. The probability of having flame holes is the number of samples that have at least one flame hole in the experimental window divided by the total number of samples. Fig. 3. Ensemble averaged simultaneous PLIF signals of acetone (light/red) and OH (dark/black) for various conditions: (a) L1F1V1 flame; (b) L1F2V1 flame; (c) L1F1V2 flame and (d) L1F2V2 flame. See Table 2 for flame notations (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

169 416 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) Fig. 4. Distribution of OH radicals in partially premixed methane/air conical flames with equivalence ratio 4 and different exit velocities: (a) 10 m/s; (b) 15 m/s and (c) 18 m/s. light/red/blue region: high OH signals, dark region: low OH signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Distribution of OH radicals in partially premixed LCV gas 1 conical flames with equivalence ratio 4 and different exit velocities: (a) 10 m/s; (b) 15 m/s and (c) 18 m/s. light/red/blue region: high OH signals, dark region: low OH signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 6. Distribution of OH radicals in partially premixed LCV gas 2 conical flames with equivalence ratio 4 and different exit velocities: (a) 10 m/s; (b) 15 m/s and (c) 18 m/s. light/red/blue region: high OH signals, dark region: low OH signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Statistics based on 1000 sample images show that the probability of having flame holes in the methane flames at equivalence ratio 4 and exit velocity of 10 m/s is 0.40, whereas for the LCV gas 1 flames and LCV gas 2 flames under the same exit flow and equivalence ratio the probabilities of having flame holes are respectively 0.50 and LCV gas 2 has a rather low LHV of 12 MJ/N m 3, about half of that of LCV gas 1 and one third of methane. The flame temperature of LCV gas 2 is lower than that of LCV gas 1 and methane flames; thereby it is more liable to local extinction. Although the LHV of LCV gas 1 is much lower than that of methane, the hydrogen presence in the LCV gas 1 seems to help preventing the flame from local extinction, and to off-balance the reduced heating value. As the exit velocity increases the OH layer becomes more frequently disconnected which is a result of stronger flame/turbulence interaction. Under the same exit velocity of 18 m/s and equivalence ratio of 4, the methane flame has a probability of extinction holes about 0.85, whereas for the LCV gas 1 flames the probability of having flame holes is 0.77, showing a strong hydrogen effect. LCV gas 2 has the lowest LHV and also the lowest laminar burning velocity; the flame is totally quenched at 18 m/s exit flow speed.

170 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) Fig. 7. Snap-shots of simultaneous OH radicals and acetone for (a) L1F1V2, (b) L2F1V2, and (c) MF1V2; dark/red region, OH; light grey/green region, acetone. See Table 2 for flame notations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) MF1V1 MF1V2 MF1V MF2V1 MF2V2 MF2V3 Probability 0.02 Probability Height above cone bottom (mm) Height above cone bottom (mm) L1F1V1 L1F1V2 L1F1V L1F2V1 L1F2V2 L1F2V3 Probability 0.02 Probability Height above cone bottom (mm) Height above cone bottom (mm) L2F1V1 L2F1V L2F2V1 L2F2V2 Probability Probability Height above cone bottom (mm) Height above cone bottom (mm) Fig. 8. Probability density function of leading flame front positions in the conical burner.

171 418 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) To further examine the structures of local flame holes we plot snap-shots of simultaneous OH and acetone distribution in Fig. 7. The iso-contours of OH and acetone have a threshold value of 5% of its respective maximal signal intensity; below this value the signal is filtered out in the figure. A smaller threshold value of 1% of the maximal signal intensity was tested and the results are qualitatively similar to that shown in Fig. 7, with however some difficulty of distinguishing the acetone signals with background noise. Figs. 7a and c shows that acetone exists in the flame holes, indicating a leakage of fuel through the flame holes; on the other hand, Fig. 7b shows situations that acetone is very low at the flame holes, indicating a leakage of oxygen through the flame holes to the fuelrich side of the flame. One may postulate the following explanation for the observed flame holes. The rates of engulfment of fuel pockets and air pockets by turbulence eddy to the reaction zones determine whether flame holes would exist. If the engulfment rates are low, mixture of the fuel and air pockets would be sufficient, and chemical reactions would have sufficient time to consume the mixture. However, if the engulfment rates are too high, chemical reactions would have too little time to consume the reactants. Increasing exit flow velocity increases the turbulence kinetic energy and the engulfment rates, thereby leading to more flame holes Stabilization of the conical flames To identify where the leading flame front is stabilized in the cone, we compute the probability density function (pdf) of the lowest position where the OH signal is above a threshold value of 5% of its maximum intensity. Fig. 8 shows the pdf of the leading flame position inside the cone. As seen for all the flames, the leading flame fronts are already seen at 2 mm above the exit of the fuel/ air mixing tube. The leading flame front position oscillates in the cone, corresponding to the spreading in the profile of pdf distribution in Fig. 8. For the low velocity flames (10 m/s) the flame fronts are found to oscillate between 2 and 15 mm above the exit of the mixing tube for most flames except the LCV gas 2 flames, L2F1V1, which has an overall equivalence ratio 2.6 and LHV of MJ/N m 3. For the high speed flames, the flame front position is seen to distribute over wider spatial domain. Instantaneous OH PLIF images and statistics show that in the high speed flames local flame extinction occurs more frequently. The probability of finding flame local extinction in the OH images (among 1000 samples) is more than 0.68 for all the flames with exit flow velocity above 15 m/s. The LCV gas 2 flames with overall equivalence ratio 2.6 and exit flow velocity 10 m/s have also high probability (about 0.79) of finding local flame extinction. The wide spreading of the leading flame fronts is a result of both flow recirculation/flame interaction and flame front quenching and re-ignition. Corresponding to Fig. 6c, the LCV gas 2 flames at tube exit velocity of 18 m/s are totally quenched, thus no pdf profiles are shown in Fig. 8. To investigate the flow/flame interaction, large eddy simulations of the conical flames are carried out. Fig. 9 shows instantaneous flow streamlines and vortex structures superimposed to the iso-surface of stoichiometric mixture fraction. The iso-surface of the stoichiometric mixture fraction is where the chemical reactions occur [5]. The streamlines clearly show the entrainment of the ambient air into the cone, which pushes the fuel/air mixture from the concentric tubes to the center region of the cone. The entrained air flow interacts with the fuel/air flow from the mixing tubes, forming recirculation zones near the exit of the concentric mixing tube. It has been shown in methane/air partially premixed conical flames [5] that the leading flame front stabilization in the conical burner is by triple-flame propagation in the recirculation zones. The unsteady recirculation zones formed inside the cone Fig. 9. Instantaneous flow streamlines, iso-surface of stoichiometric mixture fraction (pink/grey), and vortex structures (yellow/light grey, worm-like filament structures) visualized using the second eigen-value of the velocity gradient tensors (k2), from large eddy simulation of methane/air partially premixed flame in the conical burner, with overall equivalence ratio of 3, mixing length L/D of 5, and tube exit velocity of 20 m/s. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) push/draw the flame up and down. This mechanism is responsible for the observed leading flame pdf distribution in the cone. Since the large scale recirculation zone structure inside cone depends mainly on the cone angle, and less on the flow speed and the fuel properties, it is clear that in the conical burner the flames are more stable, and the stability is less sensitive to the fuel properties. Increasing the flow speed of the fuel/air mixture would not affect significantly the large-scales and therefore the leading flame positions, however, it greatly affects the production of turbulence in the shear layers as demonstrated by the filament worm-like vortex structures in Fig. 9. The turbulence eddies interact with the reactions around the stoichiometric mixture fraction surface, causing local flame extinction if the turbulence intensity is high enough. At extremely high burner exit flow velocity the turbulence eddies are very energetic, which leads to quenching of the reaction zones, and eventually overall flame extinction. The flame extinction is due to strong flame/turbulence interaction, not by blowing off the leading flame front to downstream as that in jet flames. Thus, the stability domain of conical flames is larger than the jet flames. 5. Conclusions The stability characteristics of partially premixed turbulent flames of different fuels in a conical burner and a corresponding jet burner (without the cone) are investigated. The fuels are methane, methane diluted with nitrogen, and mixtures of CH 4, CO, CO 2, H 2 and N 2, with a LHV of about 12 and 24 MJ/N m 3 simulating typical products derived from biomass gasification. The structures of the conical flames at different overall equivalence ratios and Reynolds numbers are studied using simultaneous PLIF of acetone and OH. The flow field and mixing process are simulated using large eddy simulation. The methane/air and diluted methane/air jet flames show multiple stability regimes, burner rim-attached flames, lifted flames, and flame blowout. There is also a hysteresis regime where the flames can either rim-attached or lifted, depending on the initial

172 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) state of the flames. The LCV gas jet flames show different characteristics. The flames are more liable to rim-attached, and the blowoff jet velocity is higher than the corresponding methane/air flames. The LCV gas flames are more stable than the methane/air flames although the LHV is lower for the LCV gases. This characteristic is attributed to the presence of hydrogen in the mixture. The conical flames exhibit much wider stability domain in terms of overall equivalence ratio and burner exit flow velocity than the corresponding jet flames without the cone. The stability of the conical flames is less sensitive to the gas composition, equivalence ratio and exit flow velocity. This is fundamentally different from the jet flames. PLIF images of OH radicals show that the leading flame fronts (where the chemical reactions start and the flame is stabilized) of the conical flames for low to moderate exit flow speed is fairly independent of the equivalence ratios and the burner exit velocity for the tested LCV gas flames and methane/air flames. The leading flame fronts oscillate in a narrow spatial domain 2 15 mm above the exit of the mixing tube. This can be attributed to the unsteady flow recirculation zones generated inside the cones which enhance the flame stabilization. At high flow speed close to blowoff conditions, the leading flame fronts can still be found in regions close to the exit of the mixing tube due to the cone-induced large scale recirculation flow structure. However, the reaction zones become more frequently disconnected and with higher probability of local extinction. Further increase of the flow speed leads to flame blowoff due to overall flame extinction by strong turbulence interaction. Simultaneous OH and acetone images show that the local extinction is a result of fast engulfment of the fuel pockets and air pockets to the reaction zones, which competes with the chemical reaction rates leading to local extinction and leakage of the fuel pockets and air pockets through the reaction zones. Acknowledgements This work was partly sponsored by the Swedish Research Council VR, SSF and STEM through CeCOST and the Swedish International Development Agency (SIDA) fund for the MENA countries through the joint project between Egypt and Sweden, and partly sponsored by China Natural Science Foundation(Grant No ) and CSC (China Scholarship Council). References [1] H. Goyal, D. Seal, C. Saxena, Bio-fuels from thermochemical conversion of renewable resources: a review, Renew. Sustain. Energy Rev. 12 (2008) [2] D. Wilson, K. Lyons, Effects of dilution and co-flow on the stability of lifted non-premixed biogas-like flames, Fuel 87 (2008) [3] S. Won, J. Kim, K. Hong, M. Cha, S. Chung, Stabilization mechanism of lifted flame edge in the near field of coflow jets for diluted methane, Proc. Combust. Inst. 30 (2005) [4] F. El-Mahallawy, A. Abdelhaffz, M. Mansour, Mixing and nozzle geometry effects on flame structure and stability, Combust. Sci. Technol. 179 (2007) [5] B. Li, E. Baudoin, R. Yu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Alden, M. Mansour, Experimental and numerical study of a conical turbulent partially premixed flame, Proc. Combust. Inst. 32 (2009) [6] K. Stahl, L. Waldheim, M. Morris, U. Johansson, L. Gardmark, Biomass IGCC at Värnamo, Sweden past and future, GCEP Energy Workshop, Frances C. Arrillaga Alumni Center, Stanford University, CA, USA, pp [7] J. Kiefer, Z.S. Li, J. Zetterberg, X.S. Bai, M. Alden, Investigation of local flame structures and statistics in partially premixed turbulent jet flames using simultaneous single-shot CH and OH planar laser-induced fluorescence imaging, Combust. Flame 154 (2008) [8] P. Wang, X.S. Bai, Large eddy simulation of premixed turbulent flames using Flamelet approach, Proc. Combust. Inst. 30 (2004) [9] K.J. Nogenmyr, P. Petersson, X.S. Bai, A. Nauert, J. Olofsson, H. Seyfried, J. Zetterberg, Z.S. Li, A. Dreizler, M. Linne, M. Alden, Large eddy simulation and experiments of stratified lean premixed methane/air turbulent flames, Proc. Combust. Inst. 31 (2007) [10] K.J. Nogenmyr, C. Fureby, X.S. Bai, P. Petersson, R. Collin, M. Linne, Large eddy simulation and laser diagnostic studies on a low swirl stratified premixed flame, Combust. Flame 156 (2009) [11] J. Gullbrand, X.S. Bai, L. Fuchs, High order cartesian grid method for calculation of incompressible turbulent flows, Int. J. Numer. Methods Fluids 36 (2001) [12] S. Liu, C. Meneveau, J. Katz, On the properties of similarity subgrid scale models as deduced from measurements in a turbulent jet, J. Fluid Mech. 275 (1994) [13] G.S. Jiang, C.W. Shu, Efficient Implementation of weighted ENO schemes, J. Comput. 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173 Experimental Thermal and Fluid Science 34 (2010) Contents lists available at ScienceDirect Experimental Thermal and Fluid Science journal homepage: Structures and stabilization of low calorific value gas turbulent partially premixed flames in a conical burner B. Yan a,b,b.li c, E. Baudoin b, C. Liu a,b, Z.W. Sun c, Z.S. Li c, X.S. Bai b, *, M. Aldén c, G. Chen a, M.S. Mansour d a Faculty of Environmental Science and Engineering, Tianjin University, Tianjin, China b Division of Fluid Mechanics, Lund University, Lund, Sweden c Division of Combustion Physics, Lund University, Lund, Sweden d Cairo University, Natl Inst Laser Enhanced Sci., Cairo, Egypt article info abstract Article history: Received 14 October 2009 Accepted 16 October 2009 Keywords: Low calorific gas flames Partially premixed flames Conical burner Flame stability Local flame extinction Experiments are carried out on partially premixed turbulent flames stabilized in a conical burner. The investigated gaseous fuels are methane, methane diluted with nitrogen, and mixtures of CH4, CO, CO2, H 2 and N 2, simulating typical products from gasification of biomass, and co-firing of gasification gas with methane. The fuel and air are partially premixed in concentric tubes. Flame stabilization behavior is investigated and significantly different stabilization characteristics are observed in flames with and without the cone. Planar laser induced fluorescence (LIF) imaging of a fuel-tracer species, acetone, and OH radicals is carried out to characterize the flame structures. Large eddy simulations of the conical flames are carried out to gain further understanding of the flame/flow interaction in the cone. The data show that the flames with the cone are more stable than those without the cone. Without the cone (i.e. jet burner) the critical jet velocities for blowoff and liftoff of biomass derived gases are higher than that for methane/ nitrogen mixture with the same heating values, indicating the enhanced flame stabilization by hydrogen in the mixture. With the cone the stability of flames is not sensitive to the compositions of the fuels, owing to the different flame stabilization mechanism in the conical flames than that in the jet flames. From the PLIF images it is shown that in the conical burner, the flame is stabilized by the cone at nearly the same position for different fuels. From large eddy simulations, the flames are shown to be controlled by the recirculation flows inside cone, which depends on the cone angle, but less sensitive to the fuel compositions and flow speed. The flames tend to be hold in the recirculation zones even at very high flow speed. Flame blowoff occurs when significant local extinction in the main body of the flame appears at high turbulence intensities. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Biomass derived fuels are important ingredients for heat and power production owing to the renewable nature and sustainable availability of this energy source. There are considerable interests from industry to utilize biomass gasification and pyrolysis gases in gas engine and gas turbine applications. These gases are often called low calorific value (LCV) gases and they are typically comprised of H 2, CO, CH 4 and a small amount of higher hydrocarbons [1]. There may be large amount of N 2 in gases from air-blown gasification, and CO 2 in gases from landfills. These inert gases lead to lower calorific value of the mixture gas than pure H 2 or natural gas. Combustion characteristics of LCV gases, such as the laminar burning velocity and NO x formation have been reported in the literature. * Corresponding author. Tel.: ; fax: address: xue-song.bai@energy.lth.se (X.S. Bai). From a practical point of view, it is important to known the stability behavior of LCV gas flames in different type of burners such as jet/bunsen burners. The stability of lifted jet flames with model LCV fuels has been studied recently [2], where the fuels utilized were mixtures of CH 4 and C 2H 4 diluted by N 2. The relationships between the liftoff velocity and dilution, liftoff height behavior as well as reattachment conditions were described. The stabilization mechanism of lifted jet flames of CH 4 diluted with N 2 was investigated in [3]. It was reported that stationary lifted flames were only observed in the near field of coflow jets. The stability behavior of LCV gases in a jet flame burner is investigated in this study. Recently, experimental and numerical studies of a conical burner were carried out for different fuels [4,5]. It was shown that the cone enhances significantly the stability of flames. In this study the stability behavior of LCV gases in the conical burner is examined. The considered LCV gases correspond to the biomass gasification gases co-firing with natural gas with a low heating value (LHV) of MJ/N m 3. Another LCV gas considered is a mixture of methane and nitrogen, also having a LHV of 24 MJ/N m /$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi: /j.expthermflusci

174 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) It is shown that in the conical burner the flame stability is insensitive to the fuels and heating values; however, when the cone is removed, i.e. with jet flame burner, the flame liftoff and blowoff conditions are very sensitive to the fuel compositions. To investigate the mechanisms of flame stabilization in the conical burner, planar laser induced fluorescence (PLIF) technique is used to detect the OH radicals and a fuel-tracer species (acetone) inside the cone. Large eddy simulation (LES) is carried out to investigate the flow field in the cone. Table 1 The compositions (vol.%) and low heating values of studied fuels, and laminar burning velocity of stoichiometric fuel/air mixture at atmospheric and room temperature conditions (300 K). The laminar burning velocity was calculated using Cantera code and GRI 3.0 mechanism. Test fuels CO H 2 CH 4 CO 2 N 2 S L (m/s) LHV (MJ/N m 3 ) LCV gas LCV gas MN Methane Experimental setup and conditions The burner consists of two vertical concentric tubes. The inner and outer tubes are 6.8 mm and 9.7 mm in diameter and with a lip-thickness of 1.2 mm and 2.3 mm, respectively (cf. Fig. 1). A mixing distance L, which is the distance between the inner tube exit and the outer tube exit, can be varied to generate different degree of partial premixing. The fuel flow discharges through the outer tube while the air flow through the inner one. A quartz glass conical nozzle, which makes optical access and observation possible, with half cone angle of 26 is installed at the outer tube exit and used to stabilize the flames. Detailed information about the burner can be seen elsewhere [4,5]. The LCV gases considered in this work are shown in Table 1. LCV gases 1 and 2 are mixtures of typical biomass derived gases (BDG) and methane. Mixing of these gases can improve the LHV from about 5 MJ/N m 3 to above 10 MJ/N m 3. The compositions of the BDG are taken from Ref. [6], which are typical compositions of gases produced in a gasification demonstration plant in Sweden. The LCV gas 1 is mixed by 60% CH 4 and 40% BDG by volume; the LCV gas 2 is mixed by 20% CH 4 and 80% BDG by volume. Fuel MN is a mixture of only CH 4 and N 2 having the same LHV as LCV gas 1. The uncertainty of each gas composition is within ±2.0%. For comparison with the LCV gases, methane/air flames in the corresponding conditions are also measured. To characterize the fuel distribution in the flow field, a tracer of fuel (acetone) was introduced to the fuel stream in the concentric tubes. The gaseous acetone was introduced into the burner by passing the fuel through liquid acetone, which gives about 1% acetone by volume in the fuel mixture. The mass flow rates of the fuel and air streams were monitored using Bronkhorst thermal mass meters and controllers installed in their respective feed lines. The flow meters were calibrated at conditions of 0 C and 1 atm. The flow meters have an accuracy of ±0.9% of the measured values. The Q 1(8), F 2 R + X 2 R + (1, 0) transition at 283 nm was excited using the second harmonic radiation from a Nd:YAG pumped dye laser (Rhodamine 590) and OH fluorescence was detected at about 308 nm. The laser pulse energy is 20 mj. This setup has been shown to be satisfactory in measuring OH distributions in partially premixed methane/air jet flames [7]. Acetone was excited at the same wavelength as OH LIF, which enables the simultaneous measurements of them. A laser sheet with 40 mm height was formed inside the cone of the burner by a cylindrical/spherical lens combination. The OH and/or acetone PLIF signal was collected vertically by an f/4.5 objective (Nikon UV, f = 100 mm), and then captured by an intensified charge coupled device (ICCD) camera (Princeton PI-MAX, pixels). Three different sets of optical glass filters (Scott, 3 mm) were employed in front of the camera to filter the laser scatterings and unwanted fluorescence. For OH-PLIF signal recording a long-pass WG305 and a short-pass UG11 worked together to collect the OH DC 70mm Laser window 26 degree Filter ICCD SL CL 9mm D=9.7mm 45mm Nd :YAG Pumped Dye Laser PB 19mm 8.0mm 6.8mm L=72.75mm L/D=7.5 mm x Top 27.1mm Middle 18.7mm Low 9.2mm r mm Fig. 1. Burner details and experimental setup for simultaneous OH/acetone PLIF measurement: PB, Pellin Broca prism; DC, dichroic mirror; SL, spherical lens; CL, cylindrical lens.

175 414 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) fluorescence at around 308 nm, while a WG345 filter was chosen to suppress OH-PLIF and Rayleigh signals and select the broadband emissions from 350 to 500 nm when acetone PLIF was recorded. Only a WG305 filter was used for simultaneous OH and acetone PLIF measurements. 3. Numerical simulation models The conical flames are simulated numerically using a large eddy simulation model described previously [8 10]. The combustion model is based on a two-scalar flamelet approach where the mixing process is described using a conserved scalar, the mixture fraction, and the triple flame front propagation is modeled using a level-set G-function. Flamelet chemistry is coupled to the flow field simulations using a diffusion flamelet library tabulated as a function of mixture fraction and scalar dissipation rate, from numerical simulations in a counter-flow partially premixed flame configuration. The scale-similarity model (SSM) [11] is used for the sub-grid stresses (SGS) in the spatially filtered momentum equations, and Smagorinsky model is used for the SGS transport fluxes in the scalar transport equations. The spatially filtered continuity, momentum and energy equations as well as the mixture fraction transport equation are discretized and solved on a staggered Cartesian grid using a finite difference algorithm with multi-grid acceleration [12]. A 5th order WENO scheme [13] is employed for the convective terms, whereas all other spatial derivatives are discretized with a 4th order central scheme, and the time integration is performed by a second order implicit scheme. The level-set G-equation is solved with a 3rd order WENO scheme in combination with a 3rd order TVD Runge Kutta time integration scheme. The flow domain outside the cone extends to 5.26D 0 in the axial direction and 3.08D 0 in the cross flow directions, respectively, where D 0 (=73 mm) is the diameter of the cone exit. The number of meshes used in the LES is about 8 millions assigned to four processors working in parallel. The time step in the LES is 5 ls. Further details about the grid, implementation of the inflow and outflow conditions are referred to [5]. 4. Results and discussion 4.1. Regimes of flame stability Fig. 2 shows the stability behavior of the flames with different fuels, with and without the cone. All cases are at the same mixing condition of L/D = 7.5 (cf. Fig. 1). First, one can observe that without the cone (i.e. jet burner) the partially premixed flames can be in different states depending on the overall equivalence ratio and jet velocity (or equivalently Reynolds number based on the flow conditions at the tube exit). For the methane/air flames, and the diluted methane (MN)/air flames, the flames can be attached to the burner rim, regime A, or lifted from the burner rim, regime C, or blowout (total flame extinction), regime above C. The attached flame (regime A) occurs at low jet speed condition; for the MN/ air flames the jet Reynolds number is below 2800 which yields typically laminar flows. For methane/air flames the jet Reynolds number is below These critical Reynolds numbers are fairly independent of the overall equivalence ratio when the equivalence ratio is larger than 6. Blowout happens at much higher jet velocity, and this depends on the overall equivalence ratio. Within regime B the flames can either be attached or lifted; this depends on the initial state of ignition. If the flame is initially attached to the burner rim, increasing the jet velocity gradually will keep the flames attached within regime B. On the other hand, if the initial flame is in regime C, i.e. in lifted flame mode, decreasing the jet velocity gradually to regime B will also keep the flame lifted. Reducing the velocity further to regime A will lead the flame to being re-attached to the burner rim. This path-dependent phenomenon is known as hysteresis [14 16]. The mode of the flames depends on the initial state and the path at which the condition is changed. The hysteresis of flame states is attributed to the significant difference in the distance between the nozzle exit and location of laminar to turbulent transition [17], when the jet flow is varied from low speed to high speed and from high speed to low speed. One can observe that the hysteresis regime (B) of the methane/ air flames is larger than that of MN/air flames, showing the sensitivity of the flame stabilization to heating values of the fuel. Turbulent non-premixed jet flames with methane and ethylene diluted with nitrogen show similar three-regime stability behavior [2]. The liftoff and reattachment behavior were shown to be rather sensitive to the level of nitrogen dilution to the fuel stream; in the liftoff mode, for a given liftoff height the stabilization Reynolds number decreases rapidly with nitrogen dilution. Won et al. [3] showed that the liftoff behavior of jet flames is strongly correlated to buoyancy effect. According to their numerical study, in microgravity conditions only the blowoff and attached flame modes were possible. For the LCV gases 1 and 2 jet flames and with the equivalence ratio less than 8, the flames are in two modes only, attached to the burner rim, or blowoff. Higher equivalence ratio cases for the LCV gas flames were not measurable due to the limitation of the current flow meter systems. By blowoff it is meant that the flame is totally quenched when starting from the attached mode and (a) (b) Reynolds number MN reattached no cone MN lifted no cone MN blow out no cone LCV 1 blow off no cone LCV 2 blow off no cone MN blow off with cone LCV 1 blow off with cone LCV 2 blow off with cone Reynolds number C CH 4 reattached no cone CH 4 lifted no cone CH 4 blow out/off no cone CH 4 blow off with cone C B A 4000 B A Equivalence ratio Equivalence ratio Fig. 2. Flame stability behavior of partially premixed jet flames and conical flames with different fuels at L/D = 7.5.

176 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) gradually increasing the jet velocity, whereas blowout refers to quenched flame starting from lifted mode and gradually increasing the jet velocity [18]. The above discussion of stabilization behavior of the methane, MN and LCV gases flames shows that for jet flames the flame stability is very sensitive to the fuel type. The LCV gas flames are more stable than the MN flames although the LHV of MN is higher. With equivalence ratio less than 4, the LCV gas 2 flames are more stable than the LCV gas 1 flames. The LHV of LCV gas 2 is only half of that in LCV gas 1, but the hydrogen mole fraction in LCV gas 2 is higher than that in LCV gas 1. It appears that with LCV gases where hydrogen is present, the flames are more stable and tend to attach to the burner rim. Similar observations have been reported in [3,14,15,19]. For the flames with the cone, one can observe that the stability characteristics of the flames with methane, MN, and LCV gases are fairly similar: the flame is stabilized inside the cone for large range of equivalence ratio and jet Reynolds numbers, and the blowoff limit is nearly linearly dependent of the overall equivalence ratio. For the flames with overall equivalence ratio higher than 5, it is difficult to observe the blowoff mode in the current rig due to the limited mass flow meters. The flames are more stable than the corresponding ones without the cone. The LCV gas 1 flames are somewhat more stable than that of MN fuel, although they have the same LHV. The difference is however much smaller than that without the cone. Table 2 PLIF experimental conditions for the partially premixed conical flames and notations of the flames. Flames Fuels Q F (l/min) Q A (l/min) u Exit velocity (m/s) L1F1V1 LCV gas L1F2V1 LCV gas L1F1V2 LCV gas L1F2V2 LCV gas L1F1V3 LCV gas ,600 L1F2V3 LCV gas ,520 L2F1V1 LCV gas L2F2V1 LCV gas L2F1V2 LCV gas L2F2V2 LCV gas MF1V1 Methane MF2V1 Methane MF1V2 Methane MF2V2 Methane MF1V3 Methane ,530 MF2V3 Methane ,410 Re 4.2. Structure of the partially premixed conical flames To understand the stability behavior of the conical flames, the reaction zone structures of the flames are studied using simultaneous OH/acetone PLIF imaging. Table 2 lists the experimental cases studied using PLIF of acetone and OH. Three burner exit velocity conditions from the mixing tube (10, 15 and 18 m/s) were considered. The burner exit velocity is defined as the bulk flow velocity (with uniform velocity profile) at the exit of the mixing tube (x = 0, Fig. 1). Two overall equivalence ratio conditions, 2.6 and 4, were considered in the experiments. All these test flames have a mixing length L/D of 7.5. From 1000 single-shots we obtain the ensemble average of distributions of acetone and OH radicals in the LCV gas 1 flames at different tube exit velocity and equivalence ratios, Fig. 3. As seen, the lowest locations of OH distribution for all four flames are almost at the same height. These locations correspond to the leading flame fronts where the flame is stabilized. This result indicates that the flame stabilization of the conical flames is not sensitive to the exit flow Reynolds number and equivalence ratio. This is contrary to the jet flames, where the liftoff height of jet flame increases with increasing jet exit velocity [14 16,19]. Figs. 3c and d shows a slight asymmetry of the leading flame front position at the high speed condition. This is likely a result of a slight asymmetry of the cone and the concentric tubes. Figs. 4 6 show snap-shots of OH signals, respectively for the methane/air flames and LCV gas flames, under different flow rates and at equivalence ratio of 4. For the low speed flames (exit velocity 10 m/s) the instantaneous OH images are fairly smooth. For the higher speed flames (exit velocity 18 m/s) the flames are more wrinkled, due to the elevated level of flame/turbulence interaction. The thicknesses of OH layers in the high speed flames are similar to that in the low speed flames. At low flow speed (10 m/s) the reaction zones (OH layers) in the LCV gas 2 flame shows flame holes, indicating possible local extinction. Although not seen in Figs. 4a and 5a, in general the reaction zones in the methane flames and the LCV gas 1 flames could also contain flame holes. To quantify the flame holes we defined a threshold value of 5% of peak OH signal in the experimental window, below which the reaction rates are considered to be low. A reaction zone disconnected by a pocket where the OH signal is lower than the threshold is considered to have a flame hole. Further, we define a probability of having flame holes in a flame as follows: if there is at least one flame hole in the entire experimental window, the flame is considered to have flame holes. The probability of having flame holes is the number of samples that have at least one flame hole in the experimental window divided by the total number of samples. Fig. 3. Ensemble averaged simultaneous PLIF signals of acetone (light/red) and OH (dark/black) for various conditions: (a) L1F1V1 flame; (b) L1F2V1 flame; (c) L1F1V2 flame and (d) L1F2V2 flame. See Table 2 for flame notations (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).

177 416 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) Fig. 4. Distribution of OH radicals in partially premixed methane/air conical flames with equivalence ratio 4 and different exit velocities: (a) 10 m/s; (b) 15 m/s and (c) 18 m/s. light/red/blue region: high OH signals, dark region: low OH signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Distribution of OH radicals in partially premixed LCV gas 1 conical flames with equivalence ratio 4 and different exit velocities: (a) 10 m/s; (b) 15 m/s and (c) 18 m/s. light/red/blue region: high OH signals, dark region: low OH signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 6. Distribution of OH radicals in partially premixed LCV gas 2 conical flames with equivalence ratio 4 and different exit velocities: (a) 10 m/s; (b) 15 m/s and (c) 18 m/s. light/red/blue region: high OH signals, dark region: low OH signals. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Statistics based on 1000 sample images show that the probability of having flame holes in the methane flames at equivalence ratio 4 and exit velocity of 10 m/s is 0.40, whereas for the LCV gas 1 flames and LCV gas 2 flames under the same exit flow and equivalence ratio the probabilities of having flame holes are respectively 0.50 and LCV gas 2 has a rather low LHV of 12 MJ/N m 3, about half of that of LCV gas 1 and one third of methane. The flame temperature of LCV gas 2 is lower than that of LCV gas 1 and methane flames; thereby it is more liable to local extinction. Although the LHV of LCV gas 1 is much lower than that of methane, the hydrogen presence in the LCV gas 1 seems to help preventing the flame from local extinction, and to off-balance the reduced heating value. As the exit velocity increases the OH layer becomes more frequently disconnected which is a result of stronger flame/turbulence interaction. Under the same exit velocity of 18 m/s and equivalence ratio of 4, the methane flame has a probability of extinction holes about 0.85, whereas for the LCV gas 1 flames the probability of having flame holes is 0.77, showing a strong hydrogen effect. LCV gas 2 has the lowest LHV and also the lowest laminar burning velocity; the flame is totally quenched at 18 m/s exit flow speed.

178 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) Fig. 7. Snap-shots of simultaneous OH radicals and acetone for (a) L1F1V2, (b) L2F1V2, and (c) MF1V2; dark/red region, OH; light grey/green region, acetone. See Table 2 for flame notations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) MF1V1 MF1V2 MF1V MF2V1 MF2V2 MF2V3 Probability 0.02 Probability Height above cone bottom (mm) Height above cone bottom (mm) L1F1V1 L1F1V2 L1F1V L1F2V1 L1F2V2 L1F2V3 Probability 0.02 Probability Height above cone bottom (mm) Height above cone bottom (mm) L2F1V1 L2F1V L2F2V1 L2F2V2 Probability Probability Height above cone bottom (mm) Height above cone bottom (mm) Fig. 8. Probability density function of leading flame front positions in the conical burner.

179 418 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) To further examine the structures of local flame holes we plot snap-shots of simultaneous OH and acetone distribution in Fig. 7. The iso-contours of OH and acetone have a threshold value of 5% of its respective maximal signal intensity; below this value the signal is filtered out in the figure. A smaller threshold value of 1% of the maximal signal intensity was tested and the results are qualitatively similar to that shown in Fig. 7, with however some difficulty of distinguishing the acetone signals with background noise. Figs. 7a and c shows that acetone exists in the flame holes, indicating a leakage of fuel through the flame holes; on the other hand, Fig. 7b shows situations that acetone is very low at the flame holes, indicating a leakage of oxygen through the flame holes to the fuelrich side of the flame. One may postulate the following explanation for the observed flame holes. The rates of engulfment of fuel pockets and air pockets by turbulence eddy to the reaction zones determine whether flame holes would exist. If the engulfment rates are low, mixture of the fuel and air pockets would be sufficient, and chemical reactions would have sufficient time to consume the mixture. However, if the engulfment rates are too high, chemical reactions would have too little time to consume the reactants. Increasing exit flow velocity increases the turbulence kinetic energy and the engulfment rates, thereby leading to more flame holes Stabilization of the conical flames To identify where the leading flame front is stabilized in the cone, we compute the probability density function (pdf) of the lowest position where the OH signal is above a threshold value of 5% of its maximum intensity. Fig. 8 shows the pdf of the leading flame position inside the cone. As seen for all the flames, the leading flame fronts are already seen at 2 mm above the exit of the fuel/ air mixing tube. The leading flame front position oscillates in the cone, corresponding to the spreading in the profile of pdf distribution in Fig. 8. For the low velocity flames (10 m/s) the flame fronts are found to oscillate between 2 and 15 mm above the exit of the mixing tube for most flames except the LCV gas 2 flames, L2F1V1, which has an overall equivalence ratio 2.6 and LHV of MJ/N m 3. For the high speed flames, the flame front position is seen to distribute over wider spatial domain. Instantaneous OH PLIF images and statistics show that in the high speed flames local flame extinction occurs more frequently. The probability of finding flame local extinction in the OH images (among 1000 samples) is more than 0.68 for all the flames with exit flow velocity above 15 m/s. The LCV gas 2 flames with overall equivalence ratio 2.6 and exit flow velocity 10 m/s have also high probability (about 0.79) of finding local flame extinction. The wide spreading of the leading flame fronts is a result of both flow recirculation/flame interaction and flame front quenching and re-ignition. Corresponding to Fig. 6c, the LCV gas 2 flames at tube exit velocity of 18 m/s are totally quenched, thus no pdf profiles are shown in Fig. 8. To investigate the flow/flame interaction, large eddy simulations of the conical flames are carried out. Fig. 9 shows instantaneous flow streamlines and vortex structures superimposed to the iso-surface of stoichiometric mixture fraction. The iso-surface of the stoichiometric mixture fraction is where the chemical reactions occur [5]. The streamlines clearly show the entrainment of the ambient air into the cone, which pushes the fuel/air mixture from the concentric tubes to the center region of the cone. The entrained air flow interacts with the fuel/air flow from the mixing tubes, forming recirculation zones near the exit of the concentric mixing tube. It has been shown in methane/air partially premixed conical flames [5] that the leading flame front stabilization in the conical burner is by triple-flame propagation in the recirculation zones. The unsteady recirculation zones formed inside the cone Fig. 9. Instantaneous flow streamlines, iso-surface of stoichiometric mixture fraction (pink/grey), and vortex structures (yellow/light grey, worm-like filament structures) visualized using the second eigen-value of the velocity gradient tensors (k2), from large eddy simulation of methane/air partially premixed flame in the conical burner, with overall equivalence ratio of 3, mixing length L/D of 5, and tube exit velocity of 20 m/s. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) push/draw the flame up and down. This mechanism is responsible for the observed leading flame pdf distribution in the cone. Since the large scale recirculation zone structure inside cone depends mainly on the cone angle, and less on the flow speed and the fuel properties, it is clear that in the conical burner the flames are more stable, and the stability is less sensitive to the fuel properties. Increasing the flow speed of the fuel/air mixture would not affect significantly the large-scales and therefore the leading flame positions, however, it greatly affects the production of turbulence in the shear layers as demonstrated by the filament worm-like vortex structures in Fig. 9. The turbulence eddies interact with the reactions around the stoichiometric mixture fraction surface, causing local flame extinction if the turbulence intensity is high enough. At extremely high burner exit flow velocity the turbulence eddies are very energetic, which leads to quenching of the reaction zones, and eventually overall flame extinction. The flame extinction is due to strong flame/turbulence interaction, not by blowing off the leading flame front to downstream as that in jet flames. Thus, the stability domain of conical flames is larger than the jet flames. 5. Conclusions The stability characteristics of partially premixed turbulent flames of different fuels in a conical burner and a corresponding jet burner (without the cone) are investigated. The fuels are methane, methane diluted with nitrogen, and mixtures of CH 4, CO, CO 2, H 2 and N 2, with a LHV of about 12 and 24 MJ/N m 3 simulating typical products derived from biomass gasification. The structures of the conical flames at different overall equivalence ratios and Reynolds numbers are studied using simultaneous PLIF of acetone and OH. The flow field and mixing process are simulated using large eddy simulation. The methane/air and diluted methane/air jet flames show multiple stability regimes, burner rim-attached flames, lifted flames, and flame blowout. There is also a hysteresis regime where the flames can either rim-attached or lifted, depending on the initial

180 B. Yan et al. / Experimental Thermal and Fluid Science 34 (2010) state of the flames. The LCV gas jet flames show different characteristics. The flames are more liable to rim-attached, and the blowoff jet velocity is higher than the corresponding methane/air flames. The LCV gas flames are more stable than the methane/air flames although the LHV is lower for the LCV gases. This characteristic is attributed to the presence of hydrogen in the mixture. The conical flames exhibit much wider stability domain in terms of overall equivalence ratio and burner exit flow velocity than the corresponding jet flames without the cone. The stability of the conical flames is less sensitive to the gas composition, equivalence ratio and exit flow velocity. This is fundamentally different from the jet flames. PLIF images of OH radicals show that the leading flame fronts (where the chemical reactions start and the flame is stabilized) of the conical flames for low to moderate exit flow speed is fairly independent of the equivalence ratios and the burner exit velocity for the tested LCV gas flames and methane/air flames. The leading flame fronts oscillate in a narrow spatial domain 2 15 mm above the exit of the mixing tube. This can be attributed to the unsteady flow recirculation zones generated inside the cones which enhance the flame stabilization. At high flow speed close to blowoff conditions, the leading flame fronts can still be found in regions close to the exit of the mixing tube due to the cone-induced large scale recirculation flow structure. However, the reaction zones become more frequently disconnected and with higher probability of local extinction. Further increase of the flow speed leads to flame blowoff due to overall flame extinction by strong turbulence interaction. Simultaneous OH and acetone images show that the local extinction is a result of fast engulfment of the fuel pockets and air pockets to the reaction zones, which competes with the chemical reaction rates leading to local extinction and leakage of the fuel pockets and air pockets through the reaction zones. Acknowledgements This work was partly sponsored by the Swedish Research Council VR, SSF and STEM through CeCOST and the Swedish International Development Agency (SIDA) fund for the MENA countries through the joint project between Egypt and Sweden, and partly sponsored by China Natural Science Foundation(Grant No ) and CSC (China Scholarship Council). References [1] H. Goyal, D. Seal, C. Saxena, Bio-fuels from thermochemical conversion of renewable resources: a review, Renew. Sustain. Energy Rev. 12 (2008) [2] D. Wilson, K. Lyons, Effects of dilution and co-flow on the stability of lifted non-premixed biogas-like flames, Fuel 87 (2008) [3] S. Won, J. Kim, K. Hong, M. Cha, S. Chung, Stabilization mechanism of lifted flame edge in the near field of coflow jets for diluted methane, Proc. Combust. Inst. 30 (2005) [4] F. El-Mahallawy, A. Abdelhaffz, M. Mansour, Mixing and nozzle geometry effects on flame structure and stability, Combust. Sci. Technol. 179 (2007) [5] B. Li, E. Baudoin, R. Yu, Z.W. Sun, Z.S. Li, X.S. Bai, M. Alden, M. Mansour, Experimental and numerical study of a conical turbulent partially premixed flame, Proc. Combust. Inst. 32 (2009) [6] K. Stahl, L. Waldheim, M. Morris, U. Johansson, L. Gardmark, Biomass IGCC at Värnamo, Sweden past and future, GCEP Energy Workshop, Frances C. Arrillaga Alumni Center, Stanford University, CA, USA, pp [7] J. Kiefer, Z.S. Li, J. Zetterberg, X.S. Bai, M. Alden, Investigation of local flame structures and statistics in partially premixed turbulent jet flames using simultaneous single-shot CH and OH planar laser-induced fluorescence imaging, Combust. Flame 154 (2008) [8] P. Wang, X.S. Bai, Large eddy simulation of premixed turbulent flames using Flamelet approach, Proc. Combust. Inst. 30 (2004) [9] K.J. Nogenmyr, P. Petersson, X.S. Bai, A. Nauert, J. Olofsson, H. Seyfried, J. Zetterberg, Z.S. Li, A. Dreizler, M. Linne, M. Alden, Large eddy simulation and experiments of stratified lean premixed methane/air turbulent flames, Proc. Combust. Inst. 31 (2007) [10] K.J. Nogenmyr, C. Fureby, X.S. Bai, P. Petersson, R. Collin, M. Linne, Large eddy simulation and laser diagnostic studies on a low swirl stratified premixed flame, Combust. Flame 156 (2009) [11] J. Gullbrand, X.S. Bai, L. Fuchs, High order cartesian grid method for calculation of incompressible turbulent flows, Int. J. Numer. Methods Fluids 36 (2001) [12] S. Liu, C. Meneveau, J. Katz, On the properties of similarity subgrid scale models as deduced from measurements in a turbulent jet, J. Fluid Mech. 275 (1994) [13] G.S. Jiang, C.W. Shu, Efficient Implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) [14] S. Chung, B. Lee, On the characteristics of laminar lifted flames in a nonpremixed jet, Combust. Flame 86 (1991) [15] B. Lee, S. Chung, Stabilization of lifted tribrachial flames in a laminar nonpremixed jet, Combust. Flame 109 (1997) [16] M.S. Mansour, Stability characteristics of lifted turbulent partially premixed jet flames, Combust. Flame 133 (2003) [17] M. Karbasi, I. Wierzba, The effects of hydrogen addition on the stability limits of methane jet diffusion flames, Int. J. Hydrogen Energy 23 (1998) [18] K. Wohl, N. Kapp, C. Gazley, The stability of open flames, Symp. Combust. Flame Explosion Phenomena 3 (1949) [19] R. Chen, A. Kothawala, M. Chaos, Schmidt number effects on laminar jet diffusion flame liftoff, Combust. Flame 141 (2005)

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183 Large eddy simulation of turbulent jet flows: a sensitivity study of inflow boundary conditions E. Baudoin*, C. Duwig 1, X.S. Bai Division of Fluid Mechanics, Lund University, Lund, Sweden 1 also with Haldor Topsøe A/S, DK-2800 Lyngby *Corresponding author: Eric Baudoin: Division of Fluid Mechanics, Department of Energy Sciences, Lund University, Box 118, SE Lund, Sweden eric.baudoin@energy.lth.se Key-words: Turbulent jets, Large Eddy Simulation, fuel/air mixing, azimuthal modes. 1/31

184 Abstract Time resolved numerical simulations, such as large eddy simulations, have the capability of simulating the large energetic flow structures and the unsteady dynamic behaviour. However, the results in many flow configurations were shown to be sensitive to inflow conditions so that modelling of the inflow boundary conditions is often a vital ingredient for a successful simulation. This paper presents large eddy simulations of a turbulent jet flow with rich premixed methane/air mixture discharging to ambient air. The flow configuration corresponds to a lifted partially premixed flame that has relevance to modern gas turbine and piston engine combustion. The focus of this paper is on the dynamic behaviour and the mixing process of the fuel jet flow with the ambient air in the near field of the jet. The aim of this study is to investigate the sensitivity of the flow dynamics and mixing to inflow conditions. The decay of the core flow, the instabilities in the shear layer, and the different modes of the flow dynamics are systematically analyzed for different inflow conditions. It is concluded that with larger inflow turbulence length scales, the onset of shear layer instability is earlier, the mixing between the fuel flow and the ambient is enhanced, and the core flow region is decreased. When the inflow turbulence is highly anisotropic, e.g., with one component of the Reynolds stress much higher than the others, the onset of shear layer instability is delayed and the mixing becomes slower. The mean profile of the inflow is shown to significantly affect the flow due to the difference of velocity gradient in the jet shear layer. 2/31

185 1. Introduction One major challenge in numerical simulations of turbulent flows lies in the non-linear and chaotic nature of the problem. Turbulent jet flows in combustion applications, e.g. lifted jet flames, where fuel and air are injected separately and mix before reaction, exemplify this challenge very well [1,2]. Despite the apparent simplicity of the setup (e.g. a round pipe discharging fuel into ambient air) lifted jet flames exhibit puzzling and complex features in particular when it comes to turbulence/chemistry interaction. For instance, the lift-off height fluctuates in time [3-6] and for a given set of operating conditions both burner-attached flame and lifted flame can be observed depending on the initial conditions [3,4]. One may distinguish different flame states according to the stabilisation location [2]: a ring flame around the jet core or a partially premixed flame located downstream of the nozzle. Furthermore, the flames are rather sensitive to the variation of the fuel and the oxidizer. A phenomenological description of the various lifted jet flames has been presented in the literature in e.g. [1, 2, 7]. There are several mechanisms that can lead to flow and flame instability in lifted jet flames. Based on constant density flow studies, a scenario of the jet flow instability process has been summarized as follows [8,9]. First, axi-symmetric vortices appear in the jet shearlayer due to Kelvin-Helmholtz instability. These ring-type vortices correspond to an azimuthal Fourier mode of m=0, where m denotes the wave number. They travel in the shearlayer around the jet core and undergo bending and pairing into helical structures. As a consequence, the jet core ends with a wrinkled tip. The tip of the core flow region is engulfed around helical vortex tubes (corresponding to the wave numbers of m =1). In between these large scale structures (with a size of about the jet diameter D), smaller streamwise vorticity tubes (braids) have been reported [10] resulting from secondary instabilities (often with 4 m 6). The breakdown and pairing of the rings into helices and braids affect the turbulent 3/31

186 scales distribution by promoting small-scale mixing. Small-scale mixing [10,11] exerts vital impact on lifted jet flame stabilization. It should be pointed out the extreme sensitivity of jet shear-layers to external perturbations [12]. The jet entrainment rate (hence the fuel/air mixing for a lifted flame) relies highly on imperfection in the setup, size and intensity of the coflow and turbulent structures upstream of the nozzle. It is particularly important for LES to have adequate inflow conditions to properly capture the main modes that are sensitive to the inflows. Ideally, one would set the boundary conditions far enough from the region of interest trying to minimize their influence. However, increasing the domain size without reducing the resolution increases the computational cost. Besides, in most experiments the inlet flows are typically not perfectly fully developed or measurable due to the limitations in the experimental apparatus. Together, it causes great difficulty and controversy in LES of flows which are very sensitive to the inflow conditions [12-15]. In practice, an exact reconstruction of the inflow boundary condition from experiments is not feasible. Instead, one finds a compromise so that the modelling of the inflow boundary condition ensures a reasonable approximation of the experimental setup. A common procedure when modelling inflow conditions of lifted jet flame is to use experimentally measured quantities such as axial velocity profile and mixture fraction profile (indicator of mixing) as close as possible to the rim exit. A randomly generated noise with little or no information on the length scale is then superimposed with an amplitude similar to the measured data [16,17]. Numerical simulation can be very sensitive to the location at which the information is prescribed [16]. Recently, the high sensitivity of nonreacting jet flows to nozzle exit conditions in terms of momentum thickness or disturbance amplitude in numerical simulation has been studied in [17,18]. Under the condition of moderate Reynolds numbers (Re 3600), Kim et al. [17] reported that large-scale coherent structures were generated at an earlier axial location as the momentum thickness decreases. 4/31

187 The background disturbances were found to have a strong influence on the shear layer development although the tested turbulence levels were less than 5% of the jet inflow velocity. It was also found that the amplitude of the background disturbance has a weak effect on the evolution of the vertical structures and therefore weak effect on the jet statistics as the momentum thickness decreases (towards top-hat) [17]. In case of thin shear layer a small amount of disturbance is enough to trigger the shear layer evolution. It is important to point out that in most of these studies, the maximum initial momentum thickness remained small and the level of turbulence relatively low (<5%) corresponding to perfectly developed turbulent profiles. Additionally, little information about the effect of turbulence length scales was reported. In practical burner or combustion devices, these ideal conditions are not reachable, with either high intensity disturbance or high momentum thickness [3, 4]. It advocates for further studies with more realistic conditions aiming at understanding the effect of boundary condition on the flow structures and the dynamic behaviour. For lifted jet flames, the fuel/air mixing as well as the flow structures dictates the behaviour of lifted jet flames. State-of-the-art combustion models, accounting for turbulence and chemistry interaction, are built on resolved turbulent velocity and mixing quantities (often mixture fraction) and therefore they rely on accurate prediction of all those quantities. Many LES studies have been reported for jet flames; however, few have presented the sensitivity of results to the inflow conditions. In this paper, a detailed account of the sensitivity analysis is presented for the flow dynamics and fuel/air mixing in the proximal region of a turbulent mixing jet. The jet configuration is chosen as the lifted flame studied experimentally by Mansour [3,4], which is a typical burner found in the literature. The flame is lifted above the burner and stabilized in the proximal region of the jet, i.e. as a ring flame around the core flow region. It was hypothesized that the level of the mixture fraction and its fluctuations 5/31

188 control the stabilization region [3], and the flame stabilization depends on premixed flame propagation and its interaction with the vortex structures in the shear layer [4]. It is therefore essential to properly predict the dynamics of the shear-layer flow to predict accurately the flame stabilization process. To focus on the discussions only the isothermal flows upstream of the flame is considered in this paper with an emphasis on the fuel/air premixing prior to the flame front. 2. Presentation of the simulations 2.1 Large eddy simulation method The basic partial differential equations (PDE) describing the motion of a fluid are based on the conservation of momentum, mass and energy. When several species are involved, one has to employ transport equations for the species mass fractions to account for the mixing process. In the present paper, we restrict to a low-mach number isothermal fuel/air mixing problem and neglect differential diffusion effects so that the species mixing is described by a single scalar, namely the mixture fraction Z (defined as 1 in the pure fuel stream and 0 in the pure air stream). The mixture fraction is used to compute the density which has a rather low influence on the results since the ratio between nozzle flow mixture and the ambient air is 0.9, fairly close to unity. The basic equations are spatially filtered, and the resulting equations can be written as: ρ + ( ρu ) = 0 (1) t ρu + = + + (2) t ( ρuu) P ( ρuu ρuu τ) ρ Z + ( ρu Z ) = ( ρu Z ρuz + ρdz Z ) (3) t PM P0 ρ = RT RT 0 M ( Z ) (4) 6/31

189 where u is the velocity vector, P the pressure, P 0 the reference pressure, T 0 the reference temperature, ρ the density, τ the viscous tensor, R the ideal gas constant, D Z the diffusivity and M the molar weight of the mixture. The bar denotes the averaging operation and the tilde the density weighted averaging. In LES, the averaging operation corresponds to a low pass spatial filtering and it leads to unclosed terms in the PDE presented above, e.g. ρuu ρuu in (2). The main role of the unclosed or Sub-Grid Scale (SGS) terms is to account for the interaction between the resolved and the unresolved scales. In the momentum equations, the SGS terms should account for the dissipative character of turbulence on the small (unresolved) scales as well as for the transfer of energy among the resolved and unresolved scales, including back-scattering of energy from the small unresolved scales to the resolved scales. Corresponding dissipation and backscattering processes should be considered in the SGS terms in the Z-equation. In this study the scale-similarity model of Liu et al. [19] was used for the SGS terms in the momentum equations and the Smagorinsky model was used for the scalar transport equation. The models have been validated in our previous studies of swirling flows [20] and combustion applications [21]. The governing equations are solved numerically using an inhouse code [22], in which a fifth-order Weighted Essentially Non Oscillatory (WENO) scheme [23] has been used for the convective terms that ensures numerical stability but does not affect significantly the resolved scales (i.e. low dissipation and dispersion characteristics). A fourth order central difference is used for the remaining terms of the governing equations. The second order Adams-Bashforth method is used for time stepping. The governing equations are discretized on a Cartesian staggered grid with multi-grid acceleration. Details about the numerical scheme and validation of the code are presented in [22]. 7/31

190 2.2 The burner, computational domain and numerical meshes The jet burner studied experimentally in [3,4] is considered here. The measured velocity and mixture fraction will be used to compare with the numerical results. The diameter of the jet nozzle is D=8 mm. At the nozzle exit, a premixed methane/air mixture has a mean bulk velocity of U b =4.74 m/s and a mean mixture fraction of A coflow of air of 0.2 m/s is supplied along with the jet flow. Both the fuel/air flow and the ambient air temperatures are about 288K, and the pressure is 1 atm. In the experiments the lift-off height of flame is fluctuating in the range x/d=6-7 [3,4]. From the jet nozzle up to a few diameters upstream of the flame stabilization position the flow structure is essentially undisturbed by the heat release [4]. The computational domain starts at the exit of the jet as in typical LES studies. The domain is made up of a cubic box of 10.5D 15D 15D. Computations with a domain size of 20D 15D 15D have shown that the results in the region of interest are not affected by the present shorter domain length. In order to ensure high resolution in the jet and in the mixing layer with sufficiently large domain to not disturb the region of interest by the boundary conditions, a stretching function is used in the cross-flow directions, y- and z-direction, perpendicular to the x-axis (x-direction being the streamwise direction). The following stretching function is considered in y-direction, 1 * tan ( A ( e( y ) 1)) F( y ) = 1+ (5) 1 tan ( A) where A is the stretching coefficient and ey ( ) = 2 ( y y 0.5) is used for the partition * * * max with * y being the position on a uniform grid. The same function is used in z-direction. A minimum grid size of 0.15mm (D/53) is achieved for the baseline grid (BG) in the central jet region while a maximum grid size of 2.3mm (D/3) close to the side boundary conditions, away from the jet, with a total of grid points. Two additional grids, using the same 8/31

191 stretching function (finer resolution along the central jet region) are also tested, i.e., a finer grid (FG) and a coarser grid (CG) presented in Table Boundary conditions First, it is noted that modelling of the inflow boundary using detailed experimental data [24] offers great prospects but it is very seldom applicable since few experiments report detailed data and other avenues are to be explored. Instead, one may model the turbulent fluctuations (time dependent part of the inflow condition) using Klein et al. [25] digital filter technique based on limited experimental data. The approach enables specifying (hence varying) the first two statistical moments and the characteristic length scales. Due to this, the inflow boundary condition constructed using digital filter approach [25] is considered in this study. The mean velocity profile Ui () r at the jet exit is fitted to match experimental data [4] using a 6th order polynomial approximation. As a sensitivity test a top-hat profile based on the mean inflow velocity is also studied below. High level of turbulence was measured at the exit of the burner with the rms velocity of about 21% of the mean axial velocity [4]. However, similar to most experimental studies, in ref. [4] only the axial velocity component has been measured. The radial and azimuthal velocity components have to be approximated in the numerical simulations. A plausible choice is to assume the mean radial and azimuthal velocity components to be zero; the radial and azimuthal rms velocity components are assumed to be the same as that in the axial direction. Based on this, numerically generated turbulent fluctuations can be prescribed at the inlet following [25], in which turbulent fluctuations are generated numerically for a given inflow turbulence integral length scale L and a given Reynolds stress. 9/31

192 In many numerical studies, it is often unclear how inlet turbulence fluctuations are generated. The length scale and time scale are often not specified and their effect on the numerical results is often not known. To investigate this effect, different sets of inlet conditions are used as listed in Table 1. To minimize the number of parameters the turbulent kinetic energy is kept unchanged, whereas the integral length scale L is varied. In addition, the three normal components of the Reynolds stress are varied to take into account both isotropic and anisotropic turbulence conditions. The co-flowing air is assumed to be laminar with a uniform axial velocity of 0.2 m/s. Mixture fraction was measured to be of top-hat shaped profile with its rms fluctuation of about 10% of the mean value [3]. Instantaneous mixture fraction at the inflow boundary is generated in a similar way as the velocity fluctuations. At the outlet, convective outflow condition is applied for the velocity and remaining variables are assumed to have zero gradient. At the far-field side-boundary the flow velocity components are set to be zero, with zero gradient assumed for the mixture fraction. 2.4 Post-processing The velocity component together with mixture fraction and density are sampled every time step in several cross flow planes (x/d<6) in order to construct the mean and the rootmean-square (rms) statistical quantities. The sampling times for all cases are at least 500τ with τ = D U 0.002s being the characteristic flow residence time. It is shown to be enough b to achieve converged ensemble averages in the region of interest. Since we are interested in the dynamics of the mixing process, temporal analysis of major quantities and Fourier Transform (FT) in space are carried out. About 1000 frames are used for the spatial FT, with each frame being taken every 50 time steps; this is found to be sufficient. Increasing further the number of frames did not change the results. 10/31

193 The spatial FT is performed as follow. First, the ensemble averaged velocity < u > is computed from the instantaneous field (where < > denotes ensemble averaged mean quantity). Then, a reduced instantaneous velocity vector is defined as ( ) u = u < u > Ub. The planes are mapped from Cartesian coordinate (stretched grid) to uniform polar coordinate ( y, z) u ( r, θ ) u ( r, θ ) ( rm, ). Finally, the reduced velocity is transformed into the Fourier space, u. The Fourier transform was limited to m<20 since these are the u f dominant modes. The number of points in the radial direction is set to 50 with a radius of 14 mm while it is set to 80 in the azimuthal direction. Two additional grids in ( r, θ ) space have been tested with 60 and 42 points in the azimuthal direction, respectively. The FT results were shown to be insensitive to the grids. The turbulent kinetic energy related to the FT modes is E( rm, ) = 0.5 u ( rm, ) u ( rm, ), where u f is the complex conjugate of u f. Note that the f f above procedure is well in line with recent experimental works [26] and it ensures a proper description of the turbulent kinetic energy distribution in the azimuthal wave number space. 3. Results & discussions 3.1 Grid sensitivity First, the dependence of results on grid resolution has been examined with the top-hat inflow profile for the mean and rms of axial velocity. Isotropic turbulence inflow (i.e. with all normal Reynolds stress components of the same value and all shear stress components zero) has been assumed at the inlet with L=D/6. Figure 1 shows the radial profiles of the mean and rms axial velocity component at different locations downstream of the jet exit obtained on three different grids described in Table 2. The results show that the simulated flow field is not sensitive to the grid used. The mean mixture fraction and the mean axial velocity profiles as well as the rms profiles obtained on the three different grids show very little discrepancy (less 11/31

194 than 5%). As will be discussed further below, instabilities in the shear layer leads to the formation of large turbulent structures and the spreading of mass and momentum radially outwards (the jet spreading). All three grids yield proper resolution of the large velocity gradient in the shear-layer as shown in the axial velocity profile, and as a result, the peak R uu values from the three grids are nearly identical at x/d=2.5 around r/d=0.5. Further downstream (e.g. x/d=5), results from all three grids show the tendency of merging of the peaks of Reynolds stress, corresponding to the end of the core flow region. In the following only results from baseline grid will be presented. 3.2 Sensitivity to mean inflow profile To examine the sensitivity of results to the inflow profiles, LES is performed with two different mean inflow profiles, the top-hat profile and the profile fitted to the experimental data using a 6 th order polynomial function. Results shown in Figures 2-4 were obtained with the isotropic turbulence inflow and L=D/6. Typically, the top-hat inflow may be used when no experimental data are available, whereas polynomials fitted to experimental data can be a natural choice when experimental data are accessible. In Figure 2, the radial profiles of the mean axial velocity for x/d 5 (top figures) show higher axial velocity at large radial position with the top-hap inflow than that with polynomial inflow profile. This indicates that the top-hat inflow results in faster jet spreading due to the higher velocity gradient in the shear layer than with polynomial profiles. For the same reason, it appears that a higher mixing rate (faster spread of mixture fraction) is obtained with the tophat inflow than with the smoother profiles. To understand the mixing process, it is useful to examine the position of the longitudinal vortices (braids), which is due to the secondary instabilities and can be identified by the azimuthal Fourier mode 2 m 6 [11]. These braids are believed to be responsible for the 12/31

195 mixing process as shown in Figure 3. With the top-hat inflow, the azimuthal vortices at the early stage of the jet are located in the mixing layer, which improves the mixing between the core zone and the ambient. The top-hat velocity profiles introduce high velocity gradient which affects not only the position of the shear layer close to the inlet but also the thickness of the mixing layer. The mixing layer is thicker (governed by the size of the azimuthal vortices) and it is less sensitive to inlet excitations. This observation is consistent with the finding of Demare and Baillot [27]. With the polynomial experimental inflow velocity profiles at the inlet, the velocity gradients are lower and the azimuthal vortices are located closer to the jet core region. Thus, the shear layer vortices and the mixing layer are spread over a relatively narrower area. This is clearly observed in Figure 3 at x/d=1, although the differences become smaller further downstream. The above observations are supported by the Reynolds stress profiles shown in Figure 4. The peak R uu at x/d=2.5 are shifted from r/d=0.5 to r/d=0.3, when the top-hat inflow profile is replaced by the polynomial profile, indicating a faster transfer of momentum in the radial direction (jet spreading) with the top-hat inflow. Consistent with this, the rms of mixture fraction become flatter, distributed in a wider radial range when the inflow is of top-hat profile. 3.3 Sensitivity to inflow Reynolds stress To investigate the effect of the inflow Reynolds stress on the flow field, simulations with an anisotropic turbulent inflow (with R vv =R ww =0) and the polynomial inflow profile were performed. Figure 5 presents instantaneous snapshots of velocity vectors, mixture fractions and iso-surfaces of axial vorticity for cases C6a and C6b with respectively isotropic and anisotropic inflow conditions and an integral length scale of L=D/18. The axial vorticity isosurface suggests that the isotropic inflow injects directly axially rotational structures 13/31

196 ( ωx 0 ), while anisotropic excitation enforces ω x = 0 at the inlet. For the isotropic turbulence case C6a, these axial rotational structures are visible in the jet core region; the shear layer was not excited in the early stage of the jet. In the cross section planes at different downstream locations, the mixture fraction field is undisturbed up to x/d=3 in case C6a. The effect of streamwise vorticity tubes (braids) on the mixing process is noticeable at downstream, e.g. at x/d=5. For case C6b the flow in the near field of the jet behaves nearly as a laminar one as very little disturbance could be observed in the mixture fraction field. 3.4 Sensitivity to inflow turbulence length scales The inflow turbulence integral length scale is an input parameter in the numerical representation of inflow turbulence and it is often unknown in most experimental data. Figure 6 shows the mean axial velocity and mixture fraction along the central line simulated using different inflow turbulence integral length scales. The isotropic inflow turbulence condition was used. The results with smaller turbulence integral length scales agree better with the experimental data. In particular, case C6a with L=D/18 matches fairly well the mean velocity and the mixture fraction profiles along the central line. With increasing integral length scale the length of the core flow region (with nearly constant mean axial velocity) decreases. A decay of 50% in the axial velocity (as compared with the inlet velocity) is seen at x/d=3 in case C1a (which has the largest inflow integral length scale, L=D/2), whereas the 50% decay in the axial velocity is reached at x/d=5.8 for case C4a (which has a smaller inflow turbulence length scale, L=D/9). The effect of the inflow turbulence integral length scale on the flow structures can be seen already in Figures 3 & 5 where snapshots of longitudinal vortices were shown for respectively case C3a (L=D/6) and case C6a (L=D/18). These longitudinal vortices appeared earlier in the shear layer in C3a (x/d~3) than in C6a (x/d~5). This corresponds to the stronger decay of mean mixture fraction profile in case C3a than in C6a, Figure 6. 14/31

197 To identify the flow structures in the shear layer, Figure 7 shows the distribution of the turbulent kinetic energy in different azimuthal FT modes. The results shown are for three axial positions in the shear layer (at r/d=0.4 and x/d=1, 3 and 5). In the near field of the jet (x/d=1), with smaller inflow integral length scale the distribution of turbulent kinetic energy in the azimuthal FT modes is wider and the intensity in each mode is lower, corresponding to lower turbulence kinetic energy at the position. With larger inflow turbulence integral length scale, smaller streamwise vorticity tubes (braids) are formed, resulting from secondary instabilities as observed and discussed previously. For cases C1a and C3a, as the tip of core flow region is eventually reached at the x/d=5 plane, the FT mode m=0 becomes dominant. As the integral length scale increases, the transfer of energy from higher modes to the lower modes (e.g. m=0 and m=1) occurs earlier in the upstream. This corresponds to earlier secondary shear layer instability with larger inflow turbulence length, as already seen in Figures 3 and 5 and the related discussions. Similar behaviour has been observed for anisotropic turbulence inflows. As the inflow turbulence integral length scale increases the jet core zone is smaller and the decay of the axial velocity and mixture fraction is faster, Figure 8. Comparing the anisotropic inflow cases with the isotropic inflow cases (Figures 6 & 8), the decay of axial velocity and mixture fraction in the anisotropic cases is slower than in the corresponding isotropic cases. Rapid decrease of the mean mixture fraction occurs further downstream with anisotropic turbulent inflows as compared with the corresponding isotropic turbulence cases with the same integral length scales. An azimuthal FT analysis for cases C2b, C3b and C4b is performed in the shear layer region. The results at two axial planes, x/d=3 and 5, are shown in Figure 9. Profiles at x/d=1 (not shown here) exhibit the same shape as for the isotropic cases (with lower energy content), i.e., as the integral length scale decreases, the modes are distributed wider. The 15/31

198 evolution of the modes in the shear layer and thus the underlying physics downstream of the jet exit is similar to the one observed for isotropic cases. However, the shear layer development is much slower than that in the isotropic turbulent inflow cases. With large inflow turbulence integral length scale (e.g. L=D/2 and L=D/3) the shear layer is not excited directly after the jet exit but rather downstream. This highlights the important effect of the R vv and R ww components in the inflow turbulence. 4. Conclusions This paper presents a numerical study of the sensitivity of flow dynamics and fuel air mixing to the inflow boundary conditions in large eddy simulation of turbulent jet flow. The dynamics of the jet flow, the mixing process and the development of the core flow region are shown to be highly sensitive to the mean velocity profiles of the inflow. With a top-hat velocity profile, the excitation of the shear layer instability is earlier than inflows with a polynomial velocity profile. The integral length scales and the anisotropy of inflow turbulence are shown to affect the excitation of the instabilities and thereby the mixing and momentum transfer in the proximity of the jet. Isotropic turbulence inflow introduces structures similar to the braids that excite efficiently the shear-layer instabilities. The turbulent kinetic energy distribution over the various Fourier modes becomes narrower (m=0 and m=1) in the case with larger integral length scale. With a larger integral length scale at the inflow, the intensity in m=0 and m=1 modes is higher, which results in a rapid growth of energetic large scale structures. The core flow region is therefore reduced and the mixing of jet flow with the ambient is enhanced. Sensitivity study shows that anisotropic turbulent inflows (assuming the normal Reynolds stress component R uu being dominant) attenuate the onset of shear layer instabilities. The shear layer instabilities are not excited as early as for isotropic turbulence 16/31

199 inflows. The anisotropic Reynolds stress components undergo redistribution within the components before exciting effectively the shear layer instabilities. The results of this paper implies that to properly predict the dynamics and mixing process in a jet configuration, detailed information about the mean and Reynolds stresses is necessary but not sufficient; information regarding the length scales of the inflow turbulence is also important. Experimental data at the jet exit containing the first and second moments of velocity and scalars, as well as turbulence length scales are desirable. Acknowledgments This work was supported by the competence centre for combustion CeCoST. The computations were run on HPC2N and LUNARC facilities within the allocation program SNAC. References [1] K.M. Lyons. (2007) Toward an understanding of the stabilization mechanisms of lifted turbulent jet flames: Experiments, Progress in Energy and Combustion Science, 33( 2), [2] C.J. Lawn. (2009) Lifted flames on fuel jets in co-flowing air, Progress in Energy and Combustion Science, 35(1), [3] M.S. Mansour. (2003) Stability characteristics of lifted turbulent partially premixed jet flames, Combustion and Flame 133(3) [4] M.S. Mansour. (2004) The flow field structure at the base of lifted turbulent partially premixed jet flames, Experimental Thermal and Fluid Science, 28 (7), /31

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203 LIST OF FIGURES Figure 1. Radial profiles of the mean axial velocity and the mean mixture fraction, as well as the axial component of Reynolds stress R uu at x/d=2.5 and x/d=5. Results are obtained for three grids, respectively the coarser grid (CG, red dotted line), the baseline grid (BG, black solid line) and the fine grid (FG, blue dashed line). Figure 2. Radial profiles of the mean axial velocity and the mean mixture fraction at x/d=2.5 and x/d=5 for case C3a. Results are obtained with two different mean inflow profiles; red dashed line: inlet profile polynomial fitted to experiments, black solid line: a top-hat inlet profile. Figure 3. Instantaneous 2D snapshots of mixture fraction and velocity vectors in three cross sections perpendicular to the jet axis, and instantaneous 3D iso-surfaces of the axial vorticity ( ω = 2 τ ). Results are obtained for case C3a with two different mean inflow profiles; right x figure: inlet profile polynomial fitted to experiments, left figure: a top-hat inflow profile. Figure 4. Radial profiles of R uu and rms of mixture fraction in the near field for case C3a. Results are obtained with two different mean inflow profiles. Figure 5. Instantaneous 2D snapshots of mixture fraction and velocity vectors in three cross sections perpendicular to the jet axis, and instantaneous 3D iso-surfaces of the axial vorticity ( ω = 2 τ ). Results are obtained with mean inflow polynomial fitted to experiments and two x different inflow turbulence conditions, C6a (left) and C6b (right). Figure 6. Mean axial velocity and mean mixture fraction along the central line for isotropic turbulence inflow with different inflow integral length scales from LES (lines), and experimental data [3,4] (symbols). Figure 7. Distribution of turbulent kinetic energy in different azimuthal FT modes in the shear layer (r/d=0.4) at x/d=1 (top), x/d=3 (middle) and at x/d=5 (bottom) for cases C6a (left), C3a (middle) and C1a (right). Figure 8. Mean axial velocity and mean mixture fraction along the central line for the anisotropic turbulence inflow (R vv =R ww =0) with different inflow integral length scales from LES (lines), and experimental data [3,4] (symbols). Figure 9. Distribution of turbulent kinetic energy in different azimuthal FT modes in the shear layer (r/d=0.4) at x/d=3 (top), and at x/d=5 (bottom) for cases C4b (left), C3b (middle) and C2b (right). 21/31

204 (a) (b) (c) Figure 1. 22/31

205 Figure 2. 23/31

206 Top-hap inflow Polynomial inflow Figure 3. 24/31

207 Figure 4. 25/31

208 Isotropic inflow Anisotropic inflow Figure 5. 26/31