A HYBRID LES / SGS-PDF COMPUTATIONAL MODEL FOR TURBULENT PREMIXED COMBUSTION

Transcription

1 A HYBRID LES / -PDF COMPUTATIONAL MODEL FOR TURBULENT PREMIXED COMBUSTION Fernando Oliveira de Andrade, fandrade@aluno.pu-rio.br Luis Fernando Figueira da Silva, luisfer@esp.pu-rio.br Pontifíia Universidade Católia do Rio de Janeiro - Department of Mehanial Engineering - Rua Marquês de São Viente, Rio de Janeiro/RJ, Brazil. Arnaud Mura, arnaud.mura@ld.ensma.fr Laboratoire de Combustion et de Détonique - UPR CNRS ENSMA - BP Futurosope, Frane. Abstrat. A hybrid Large Eddy Simulation / transported Probability Density Funtion (LES-PDF) omputational model is developed to perform the numerial simulation of variable-density low Mah number turbulent reative flows. Transport equations for mass, momentum, and salars are solved together with an equation of state within the LES framework. Turbulene is modeled using the lassial Smagorinsky-Lilly losure whereas hemial reation is first addressed thanks to a global single-step hemistry sheme. The governing equations are disretized using seond order auray spatial and temporal approximations applied to uniform Cartesian meshes within a finite volume framework. The effets of subgrid sale () turbulene on the ombustion proesses are taken into aount thanks to a Lagrangian transported PDF model whih is oupled with the LES solver. The PDF model relies on the use of a Monte Carlo tehnique: Stohasti Differential Equations (SDE), equivalent to the Fokker-Plank equations are onsidered for the progress variable. LES and PDF models are solved simultaneously, exhanging information at eah time step, the veloity field, turbulene frequeny and diffusion oeffiient being provided by LES, whereas the PDF model returns the filtered hemial reation rate. The resulting omputational model is used to perform the numerial simulation of an experimental test ase onsisting of a CH4-air flame established between two streams of fresh and burnt pilot gases in a onstant area square ross setion dut. The equivalene ratio in the fresh mixture is 0.8, and turbulent premixed ombustion takes plae in a regime where a strong interation between hemial reation and turbulene is awaited. The numerial solutions provided by the hybrid LES-PDF approah are assessed in terms of both (i) the influene of turbulene on the flame front and (ii) the effets of ombustion on the veloity field. The global properties of the turbulent flame, suh as the instantaneous and turbulent flame thikness, and the effets of turbulent propagation veloity on the flame stabilization proess are also analyzed. Keywords: turbulent ombustion modeling, Large Eddy Simulation (LES), subgrid sale Probability Density Funtion (PDF), numerial simulation 1. INTRODUCTION Combustion still remains one of the main soures of energy onversion. The ombustion proesses are used to provide energy for domesti heating and transportation. Eletrial energy is also assoiated with ombustion, when generated in thermoeletrial power plants. Industrial proesses, suh as the petroleum refinement, the metal treatment and proessing, and the ement manufaturing depend heavily on ombustion proesses as well. By onsidering these examples, it is lear that the benefits of ombustion to the tehnologial development and to the omfort of people are numerous. However, it is important to observe that ombustion proesses are also assoiated with environmental issues regarding pollution. The maor pollutants resulting from ombustion are the partiulate materials, the unburned or partially burned hydroarbons (UHC), the sulfuri oxides (SO x ), the nitrogen oxides (NO x ), and the arbon monoxides (CO) (Law, 2006). In partiular, an aggravating environmental problem is the anthropogeni global warming, whih is assoiated with the release of CO 2 in the atmosphere from the oil and oal burning, among other proesses. Speial attention has been reently paid to the development of alternative and non pollutant mehanisms of energy onversion, suh as those assoiated to the use of solar and wind energy. However, due to the urrent levels of energy demand and the neessity of large sale prodution, and onsidering that the availability of fossil fuels will remain suffiient for deades to ome, the hanges in the global energeti budget will probably our slowly and the ombustion proesses will remain as one of the most important energy onversion proesses for many years (WEO, 2008). Following this senario, the development of ombustion modeling methodologies is ruial, sine several advantages an be obtained from the use of numerial simulations to assist in the design of industrial equipments, suh as burners, engines and turbines. These advantages inlude, mainly, the optimization of the ombustion proesses effiieny and the assoiated redution of pollutant emissions. Usually, turbulent ombustion modeling relies on the onsideration of partial differential equations that desribe the transport of mass, momentum, hemial speies mass frations and energy, oupled with equations of state. The different methodologies to solve this set of equations are the diret numerial simulations (DNS), the simulations based on Reynolds averaged equations (RANS) and the large eddy simulations (LES). Within the DNS framework, the

2 governing equations are solved up to the smallest time and length sales assoiated with the turbulene and hemial reation, without any resort to losure assumptions. Suh an approah requires the use of high order preision numerial shemes and the availability of tremendous proessing and omputational storage apaities, whih makes its appliation usually restrited to oversimplified geometrial onfigurations and low Reynolds numbers. In RANS a temporal filtering proess is applied to the governing equations, whih suppresses all the turbulent modes and results in transport equations written for the time averaged flow properties. The influene of turbulene on the mean flow field is determined by the turbulene models. At first sight, LES an be thought as an intermediate framework between DNS and RANS: a spatial filtering proess is used to separate the large from the small flow strutures. The former are omputed expliitly whereas the latter is aounted for through the use of sub-filter turbulene models. It is worth mentioning that, in the present study, a onventional approah has been retained i.e. the LES filter is applied diretly in the physial spae and the lassial box (or top-hat) filter has been retained. Sine the behavior of the small sale turbulene is, in general, more isotropi, the LES approah presents the advantage of determining the rate of the energy transfer from the large sale, where the maor part of turbulent kineti energy is ontained, to the smallest sales. Therefore, one an reasonably expet the LES to provide muh more information about the flow field and its dynamis than the RANS strategy. In the ase of reative flows, the diret spatial filtering of the hemial reation rate is often impratiable beause the orresponding harateristi sales are too small to be aptured at the LES level. As a onsequene, the influene of the exothermi hemial reations on the flow field evolution must be determined thanks to physial models. The most important mehanisms assoiated with the hemial reations, suh as moleular diffusion, turbulent miro-mixing and heat release, take plae at sales that are usually several orders of magnitude smaller than the harateristi mesh size retained to perform the numerial simulations of pratial onfigurations (Pith, 2006). Nevertheless, LES is a very promising approah for ombustion studies beause most of reating flows exhibits strong oherent strutures as those enountered for instane when large sale thermo-aoustis instabilities spread out in onfined flows (Besson et al. 1999, Guilbert et al. 2008). LES ould also lead to an improved desription of the turbulene / ombustion interations beause, in LES, large strutures and instantaneous fresh and burnt gases zones, where turbulene properties differ, are expliitly omputed. With these aspets in mind, the present work desribes a hybrid Large Eddy Simulation / transported Probability Density Funtion (LES-PDF) omputational model to perform the numerial simulation of variable-density low Mah number turbulent reative flows. In this model, transport equations for mass, momentum, and salars are solved together with an equation of state within the LES framework. Turbulene is modeled using the lassial Smagorinsky- Lilly losure (Smagorinsky, 1963), whereas hemial reation is addressed by means of a global single-step hemistry sheme. The governing equations are disretized using seond order auray spatial and temporal approximations applied to uniform Cartesian meshes within a Finite Volume (FV) framework. The effets of sub-grid sale turbulene on the ombustion proess are addressed by means of a Lagrangian transported PDF model whih is oupled to the LES solver. The transported PDF model relies on the use of a Monte Carlo tehnique: Stohasti Differential Equations (SDE), equivalent to the original Fokker-Plank equations are onsidered for the progress variable. LES and PDF models are solved simultaneously, exhanging information at eah time step, the veloity field, harateristi turbulene frequenies and diffusion oeffiients being provided by LES, whereas the PDF model returns the filtered hemial reation rate. The main obetive of the hybrid approah is to ombine the LES apabilities that allow to evaluate diretly the maor part of the turbulent kineti energy with a transported PDF model to aount for the ontributions of sub-grid sale hemial reations. The final aim of the present work is to be able to simulate with a suffiient level of fidelity the most important phenomena involved in the interation between turbulene and ombustion. The experimental onfiguration studied by Magre et al. (1988) is used to evaluate the quality of the numerial results obtained with the proposed strategy. The experimental test ase onsists of a CH4-air premixed turbulent flame established between two streams of fresh and burnt gases in a onstant area square ross setion hannel. In the experiments, premixed turbulent ombustion takes plae in a regime where a strong interation between hemial reation and turbulene is expeted. The quality of the numerial solutions provided by the hybrid LES-PDF approah is assessed in terms of both (i) the influene of turbulene on the flame front struture and (ii) the effets of ombustion on the veloity field. The global properties of the turbulent flame, suh as the instantaneous and turbulent flame thikness, and the effets of veloity propagation on the flame anhoring loation are also analyzed. 2. MATHEMATICAL FORMULATION 2.1. Eulerian LES Equations The reative mixture is fully desribed by the hemial speies mass frations, the speifi enthalpy, pressure, and the omponents of the veloity vetor. The density is desribed as a funtion of the three former variables. The evolution of the reative flow field is governed by the transport equations of mass, momentum, speies mass frations and enthalpy, oupled with equations of state. In the present work, the following simplifying assumptions are retained:

3 (a) fluid is onsidered as Newtonian, (b) body fores, heat transport by radiation, Soret and Dufour effets are negleted, () assumptions of unity Lewis number and equal moleular diffusion oeffiient for all speies are retained, and (d) the model is developed for low Mah number flows. The assumptions regarding the hemial kinetis mehanisms inlude: (a) global, single-step and irreversible hemial reations, (b) fuel is the defiient reatant, and () null temperature exponent β of the Arrhenius expression. Under the previous set of hypotheses, the speies mass frations and enthalpy transport equations an be written in terms of a single salar transport equation. The orresponding variable is a normalized temperature, known as progress variable, whih is suffiient to evaluate the hemial reation rate. In the present work, the retained LES filter is a box filter assoiated with the uniform Cartesian mesh. Applying a Favre-filtering proess on the transport equations of mass, momentum and progress variable leads to (Pope, 2000), ( ρu ~ ) ρ + t x i i = 0, (1) ρui t ( ρu ~ u ~ ) i + x p = xi τ i τ i +, (2) ( ρu ~ ~ ) ρ ~ ~ Q + = ργ t x x + S (), (3) where u are the veloity vetor omponents, ρ is the density, p is the pressure, is the progress variable, Γ is a diffusion oeffiient and S is the hemial reation rate term. The visous stress tensor, τ i, is given by, u ~ i τ i µ = u ~ 2 u ~ k + µ i 3 k δ, i where µ is the kinemati visosity, and ( u ~ τ = ρ u u ) and Q ( u u ~ ~ ) i i iu ~ the sub-grid progress variable flux, respetively. These terms are modeled aording to, (4) = ρ are the sub-grid visous tensor and τ i u u ~ ~ i 2 = µ + δ i x xi 3 u ~ k k, (5) Q ~ ~, x S x µ = = ρ Γ ρ (6) where S is the sub-grid Shmidt number and µ is the sub-grid visosity, obtained with the use of the lassial Smagorinsky-Lilly model, 2 µ ( C ) S ~ = 2ρ S i, (7) where C s is the Smagorinsky onstant, is the harateristi omputational mesh length sale and S i is the resolved rate of strain tensor Lagrangian PDF Equations The Eulerian transport equation for the sub-grid progress variable PDF, P, an be written as (Colui et al., 1998), ( u ~ u C ) P ρs() ρp ρu ~ P ρ = ρ ργ C P t C ρ P, C (8) where the third term on the left hand side involves the onditional sub-grid sale veloity flutuation, and the first and seond terms on the right hand side are the onditional miro-mixing and the filtered hemial reation rate terms,

4 respetively. The filtered hemial rate appears in a one-point losed form and does not require any speial modeling beause the progress variable PDF is suffiient to obtain it. Models are required for the two other terms. The sub-grid sale veloity flutuation is modeled by using the lassial gradient diffusion hypothesis (Pope, 1985), ρ ( u C u ~ ) P = Γ P, and the onditional miro-mixing term is modeled using the IEM model as (Villermaux e Falk, 1994), ~ C ρ Γ = ργ [ Ωm ( )], (10) x C where Ω m is the turbulent frequeny provided by LES. Thus, the final form of the Eulerian progress variable PDF transport equation is, (9) ρp t u ~ P + = C ( Γ + Γ ) + [ Ω ( ) P ] P m [ S() P]. C (11) From a general point of view, for a system desribed by N thermohemial variables evolving in three spatial dimensions with temporal variations, the oint salar PDF is a funtion of (N +3) variables and time (Pope, 1985). As a onsequene, the use of either a standard finite differene (FD) or finite-volume (FV) numerial sheme is found to be impratiable. Therefore, to solve the Eulerian PDF transport equation an equivalent partile system is defined suh that the partiles evolve using SDE in time and spae. The evolution in physial spae is desribed by transport equations that use the filtered flow field from the LES solver (Pope, 1985), ( Γ + Γ ) u ~ 1/ 2 dxi = i + dt + [ 2( Γ + Γ )] dwi, (12) i where x i is the instantaneous partile position in Cartesian oordinates and dw i is the orresponding Wiener proess, haraterized by a Gaussian distribution with zero mean and variane dt, where dt is the urrent value of the omputational time step. Along the progress variable sample spae diretion, the partiles are affeted by both mixing and hemial reation proesses (Pope, 1985), [ Ω ( ) S( ) ] dt. d = m + (13) For a suffiiently large number of stohasti partiles, the resolution of the system of Eqs. (12) and (13) is statistially equivalent to the solution of the PDF transport equation (11) Coupling of the Hydrid Model Solvers The LES and PDF solvers are oupled by means of a feedbak mehanism that exhange information at eah integration time step. At the beginning of the simulations the flow field is initialized and the stohasti partiles are uniformly distributed in the whole 3-D uniform Cartesian mesh. Balane equations for the mass and momentum, i.e. Eqs. (1) and (2), are first solved within the finite volume framework. The filtered veloity omponents, the turbulene frequeny and the diffusion oeffiients are provided to the Lagrangian solver, so that the partiles evolution an be evaluated aording to the SDE Eqs. (12) and (13). The filtered reation rate term, evaluated using the Lagrangian representation, then feeds the Eulerian balane equation for the progress variable Eq. (3). In the next step, the transport equation of the progress variable, Eq.(3), is solved, so the temperature field is determined, and the new density field is alulated by means of an equation of state. This proedure is repeated for eah integration time step until the end of the simulations. 3. RESULTS AND DISCUSSIONS 3.1. Experimental Test Case The experimental set-up of Magre et al. (1988) is used to evaluate the numerial results obtained in this work. In this experiment, preheated air enters an admission dut at temperature of 600 K, being later separated into the main and

5 auxiliary duts. In the main dut, CH4 is ineted immediately upstream of a tabulator, a two-dimensional diaphragm used to homogenize the mixture and generate turbulene, leading to an overall equivalene ratio of 0.8. In the auxiliary dut, CH4 is ineted and the mixture is subsequently burned in a ombustion hamber, resulting in a pilot flow at temperature of the order of 2,000 K. Both fresh mixture and the burnt gases enter the test setion, whih onsists of a 1,300 mm long hannel, with a onstant area square ross setion of 100 x 100 mm. The fresh mixture enters the upper 80 x 100 mm region, with a mean longitudinal veloity of 65 m/s, and the burnt gases enter the lower 20 x 100 mm with a mean longitudinal veloity of 130 m/s. Even without ombustion, this experimental onfiguration provides very high turbulene intensity, produed by the development of a spatial shear layer and also by the inoming flow veloity flutuations. In fat, the measured turbulene intensity ranges from 5 to 20%. The estimated turbulene and hemistry harateristi times are of the order of 10µs and 1ms, respetively, leading to an approximate value of the Damköhler number, Da, of The Kolmogorov length sale and the CH4-air laminar flame thikness are about 0,10 and 0.25 mm, respetively, resulting in a value of the Karlovitz number, Ka, of approximately 10. As a onsequene, turbulent premixed ombustion is expeted to take plae in the thikened flame regime (Borghi, 1988), whih is haraterized by a strong interation between hemial reations and turbulene Computational Meshes and Boundary Conditions The omputational domain retained to perform the numerial simulations inludes the test setion entrane up to 800 mm downstream. Three different uniform Cartesian meshes are employed, denoted 500k, 800k and 1600k, onsisting of 250x40x50, 320x50x50 and 320x50x100 ontrol volumes in the x, y and z diretions, respetively. The governing equations are disretized using a seond order entral differenes sheme and the temporal disretization is ahieved by a three time-level sheme (Ferziger e Peri, 2002). The seleted time step value is 10 µs, so that the Courant Friedrih Levy (CFL) riteria do not exeed 0.5. The total simulation time is 250 ms, whih orresponds to 25,000 numerial iterations. The veloity-pressure oupling is managed by a lassial SIMPLEC algorithm. Two different kinds of inflow veloity boundary onditions have been used: (a) mean veloity transverse evolution imposed using a hyperboli tangent funtion, and (b) mean veloity transverse evolution typial of a fully developed turbulent hannel flow. On both types, normally distributed random flutuations have been added. Outflow boundary onditions are presribed by imposing a null longitudinal veloity omponent derivative and null transverse veloity omponents. No slip and adiabati onditions are onsidered at the hannel walls Simulations of Inert Cases Before running the reative omputations, numerial simulations of inert ases have been performed with three main obetives in mind: (1) preliminary hek of the statistial equivalene between the Eulerian and Lagrangian PDF formulations, (2) analysis of both the omputational mesh influene and sensitivity to inflow boundary onditions, (3) analysis of the struture of the inert flow, with speial emphasis on the turbulene properties. The equivalene between the Eulerian and Lagrangian PDF formulations are analyzed in terms of the first and seond statistial moments of an inert passive salar. The omparisons are performed at four different ross setions downstream of the test setion inlet. The obtained results, not reported here for the sake of brevity, display a quite good agreement between the Lagrangian and the Eulerian formulations: maximum observed disrepanies are lower than 8% in the shear layer region. Sine these differenes an be related to the number of partiles initially distributed in eah ontrol volume, used by the Monte Carlo solver, a partile number sensitivity analysis has been also performed using suessively 100, 200 and 300 partiles initially distributed in eah ontrol volume. It is observed that, for this partile number interval, the results obtained for the first and seond statistial moments of the inert passive salar remains almost idential and the agreement is not signifiantly improved by inreasing the number of partile per ell. As a onsequene, to redue the omputational osts, the simulations of reative ases are based on 100 partiles initially distributed in eah ontrol volume. The inert flow field analysis demonstrates that, for both inflow boundary onditions, the simplified tehnique used in this work to generate the inflow turbulent flutuations results in underestimated levels of turbulene ativity ust downstream of the test setion inlet. The maor proportion of the turbulent kineti energy develops 200 mm downstream from the hannel entrane, due to the strong transverse gradients of the mean longitudinal veloity omponent produed in the shear layer region. It is also found that, outside this region, the alulated Reynolds stresses are onsiderably lower than those evaluated from the experiments. The mesh sensitivity analysis shows that the results obtained by the three meshes are pratially idential, in terms of the mean longitudinal veloity omponent and the Reynolds stresses omponents distribution. Small differenes are noted when the spatial development of the shear layer is onsidered. The mesh 1600k seems to reprodue the beginning of the shear layer growth a little bit further upstream than the other two meshes. Figures 1(a) to () show the iso-surfae of vortiity magnitude, for ω =8000 s -1, as obtained using the meshes 500k, 800k and 1600k, respetively. Sine the inflow boundary onditions desribed by a typial mean veloity transverse evolution of fully developed turbulent hannel flow resulted in a slightly better agreement with the experimental data in terms of the Reynolds

6 stresses distribution, and the mesh 1600k provided a better desription of the shear layer growth, these parameters are seleted for the simulations of reative ases. (a) (b) () Figure 1. Iso-surfae of vortiity magnitude, for ω =8000 s -1 : (a) mesh 500k, (b) mesh 800k, () mesh 1600k Simulations of Reative Cases The simulations of inert ases were ruial to determine the mesh size and the number of partiles to be employed in the Monte Carlo solver. Also, the inflow boundary onditions were seleted, whih onsist of typial mean veloity transverse evolution of fully developed turbulent hannel flow, superposed by normally distributed random flutuations. The main dut inoming flow is haraterized by a maximum longitudinal veloity of 65 m/s, being the inoming turbulene presribed by uniform distributed random flutuations orresponding to 10% of the mean veloity transversal evolution. The auxiliary dut inoming flow presents a maximum longitudinal veloity of 130 m/s, and the inoming turbulene is introdued by uniform distributed random flutuations orresponding to 20% of the mean veloity transversal evolution. The splitter plate that separates both inoming flows is also represented at the entrane of the omputational domain. The inlet progress variable transverse evolution is desribed by a uniform distribution, where = 0 orresponds to the fresh mixture at temperature T=T u = 600 K and =1 represents the burnt gases at T=T b =2000 K. Figure 2 illustrates the transverse evolution of the mean longitudinal veloity and the progress variable used as the inlet boundary onditions for the simulations of reative ases h(mm) h(mm) _ u(m/s) Figure 2. Inlet boundary onditions used in the simulations of reative ases. Left: transverse evolution of the mean longitudinal omponent of veloity. Right: transverse evolution of progress variable. In the reative ases, the fresh mixture is haraterized by a equivalene ratio of 0.8, so that ombustion is initiated by the ontat of the fresh mixture with the burnt gases at the region immediately downstream of the hannel entrane, and develops along the hannel length, reahing an anhoring point at the upper hannel wall about 400 mm

7 downstream. The hemial reation rate is determined by the Arrhenius empirial expression, using an ativation temperature of 10,000 K, the fresh mixture temperature of 600 K and the burnt gases temperature of 2,000 K, resulting in, 3 10,000 S() = 2ρ A10 ( 1 ) exp (14) where A is the pre-exponential onstant, whih is inversely proportional to the hemistry harateristi time sale. Figure 3(a) shows a shadowgraph image of the turbulent flame (Magre et al., 1988), represented by the diagonally distributed darker strutures, obtained by the use of high speed film reording at 4,000 frames per seond. This figure presents a region between 150 to 350 mm downstream of the test setion entrane. It an be seen that the flame strutures are wrinkled and distributed by the effets of turbulene on hemial reations. The flame appears to be in the thikened regime, and aording to Magre et al. (1988), features a thikness larger than 10 m in the diretion parallel to the stream lines. It is important to observe that the shadowgraph method provides spanwise integrated information, so the visual thikness is possibly enlarged by the flame wrinkling in the spanwise diretion. Figure 3(b) shows an instantaneous image of the filtered hemial reation rate alulated by the LES-PDF model, whih represents the instantaneous turbulent flame for a region omprehending 100 to 350 mm downstream of the test setion entrane. Differently from the spanwise integrated experimental visualization, this image orresponds to a longitudinal plane obtained in the enter of the test setion. Diret omparisons between the two images an be performed only qualitatively. (a) (b) Figure 3. (a) Instantaneous experimental visualization of the flame front (ourtesy of Dr. Roland Borghi), (b) Instantaneous visualization of the flame front, orresponding to the filtered reation rate alulated by LES- PDF. It an be seen that the LES-PDF model is apable of representing some effets of turbulene upon the hemial reations, in partiular the wrinkling and strething of the flame front. In fat, in some regions, pokets of reating gases are displaed away from the flame front, possibly in the loations of higher turbulent mixing intensity. Some disontinuities in the flame struture are also apparent, whih indiate the possibility of loal reation extintion. The thikness of the instantaneous flame ranges from 5 to 10 m, resulting in a good agreement with the experiments onduted by Magre et al. (1988). Figure 4. Time averaged hemial reation rate (turbulent flame brush thikness).

8 Figure 4 shows the time averaged hemial reation rate in kgm -3 s -1, obtained in a longitudinal plane at the enter of the test setion, and for a region omprising the test setion entrane up to 500 mm downstream. Here, the averaged hemial reation rate orresponds to the mean turbulent flame thikness, or the turbulent flame brush thikness, δ T. In the experiments of Magre et. al. (1988) it is mentioned that δ T is approximately 10 m, measured in the diretion parallel to the stream lines. In the present work, δ T is around 10 m in h=50 mm, that is, in the middle of the hannel height. Higher values are observed toward the upper hannel wall and a gradually derease of the values our in the diretion of the hannel entrane. (a) t = 35 ms, A=2000, i.e. φ=0.50 (b) t = 37 ms, A=2270, i.e. φ=0.54 () t = 42 ms, A=2940, i.e. φ=0.64 (d) t = 45 ms, A=3400, i.e. φ=0.71 (e) t = 46 ms, A=3500, i.e. φ=0.72 (f) t = 47 ms, A=3600, i.e. φ=0.74 (g) t = 52 ms, A=4000, i.e. φ=0.80 Figure 5. Instantaneous flame front as a funtion of the equivalene ratio for a longitudinal ross setion loated in the middle of the hannel width.

9 The interation between turbulene and hemial reations are also investigated for different values of the equivalene ratio, φ. The simulations start with φ= 0, that is, with the inert flow. At t=35ms a reative mixture haraterized by φ= 0.5 is ineted. This value is inreased along the simulation up to φ= 0.8 at t=50 ms, and kept onstant until the end of the simulations. This proedure is ahieved by linearly inreasing the pre-exponential onstant of the Arrhenius expression: for A = 2000, the equivalene ratio is approximately 0.5 and for A = 4000 the equivalene ratio is about 0.8. Figures 5(a) to (g) show the instantaneous flame front, represented by the instantaneous hemial reation rate, obtained for a longitudinal plane loated in the middle of the hannel width, as a funtion of the equivalene ratio. Figure 5(a) and (b) show that, for low values of equivalene ratio, the flame is positioned pratially parallel to the inoming flow. In these ases, the hemial reations our on the shear layer developing region, where the turbulene intensity is the highest. For this reason, it is observed that the flames are haraterized by large thiknesses, with the maximum values ranging approximately from 15 to 20 m. It is also seen that the flames are strongly wrinkled and strethed by the turbulent movements. With the inrease of the equivalene ratio, the turbulent flame veloity propagation also inreases. Figure 5() shows that the flame front bends toward the fresh mixture, advaning firstly in the region near the upper hannel wall, where the flow longitudinal veloities are lower when ompared to the shear layer region. It an be seen that during this flame front movement, turbulene ats by displaing pokets of reating gases along the downstream flow. Figure 5(d) shows a larger inlination of the flame front toward the fresh mixture. It an be seen that the flame is now loated ompletely out of the shear layer influene region. Sine in this region the turbulent intensity is lower, the flame front harateristi thikness dereases. However, the strong wrinkling and strething indued by turbulene are still evident. Figures 5(e) and (f) show that the flame front angle keeps inreasing with respet to the longitudinal hannel walls, aording to the equivalene ratio inrement, and Figure 5(g) illustrates the stabilized premixed turbulent flame, whih has an anhor loation at the upper wall between 300 and 400 mm downstream of the hannel entrane. In the ase of low Mah number flows, most of the influenes of ombustion on the turbulene properties are due to gases density variations assoiated to thermal expansion. The energy release from the hemial reations inreases the gases temperature in the flame front region, whih leads to volumetri expansions and to onsequent dereases of speifi mass. In the present work, the temperature gradients at the flame front region result in a density variation ratio between fresh and burnt gases of approximately 3, whih may strongly influene the behavior of the mean and flutuating harateristis of the veloity field. Figure 6. Distribution of the rms of the longitudinal flutuating veloity omponent. Figure 6 shows the rms of the longitudinal flutuating veloity omponent, obtained for a longitudinal plane loated in the middle of the hannel width, and for a region omprising the test setion entrane up to 500 mm downstream. The blak line indiates the position of the mean flame brush, orresponding to the time averaged reation rate. This figure indiates a strong inrease of the longitudinal veloity flutuations in the flame front region, partiularly on the upper half of the hannel. This inrease an be assoiated with the gases aeleration through the flame front, leading to the rise of the longitudinal omponents of the flutuating veloity. It is also observed that the ombustion influenes on the veloity flutuations are propagated to the region downstream of the flame front, and smaller flutuation inreases are experiened in the region immediately upstream of the flame front. Figures 7(a) and (b) show instantaneous images of the lower and upper region of the flame front presented in Figure 3(b), respetively, obtained for a longitudinal plane loated in the middle of the hannel width, where the flame front is represented by the blak lines. The phenomenon known as flame generated vortiity (Peters, 2000) is investigated. The positive vortiity is represented in red olor and is assoiated with a ounter-lokwise rotation. The negative vortiity is represented in blue olor and is assoiated to the lokwise rotation relative to the longitudinal ross setion.

10 Both figures suggest the generation of intense vortiity in the flame front region, whih an be assoiated with the gas expansion effets indued by exothermi hemial reations. The vorties also seem to influene the flame front struture: the superimposition of the blak lines that delineates the flame front with the vortiity field suggests that the strong ounter-lokwise vorties align with the flame front towards the burnt gases, see Figure 7. On the other hand, the strong lokwise vorties appear to distort the flame front to the fresh mixture diretion. Therefore, the ombination of these effets ontributes to the overall flame strething, wrinkling and distribution proesses. However, it must be emphasized that this analysis is not fully onlusive, sine Figures 7(a) and (b) orrespond to a single instantaneous image. A omplete investigation of the flame generation of vorties and the mutual effets on the hemial reations would require a set of instantaneous images reorded in sequene in order to identify the oming vorties, the passage of them through the flame front and the resulting effets of the interation. (a) (b) Figure 7. Vortiity field for the (a) lower and (b) upper instantaneous flame front region. 4. FINAL REMARKS AND FUTURE RECOMMENDATIONS In the present study, a hybrid large eddy simulation / transported probability density funtion (LES-PDF) approah has been developed to simulate a ase of variable-density low Mah number premixed turbulent ombustion, haraterized by the existene of intense interations between turbulene and ombustion. The qualitative analysis of the turbulent flame global properties suggests that the methodology is apable of prediting effets of flame strething, wrinkling and distribution by turbulene. The oupling between turbulent mixing and hemial reations is desribed by the probability density funtion transport equation, whih used in the LES framework permits a detailed representation of the interation between mixing and hemial reation. The effets of inreasing the mixture equivalene ratio are showed by inrementing the Arrhenius pre-exponential onstant and the resulting flame stabilization loation is in good agreement with the experiments. The alulated instantaneous flame thikness, as well as their average ounterparts (i.e. the flame brush thikness), are also well predited. The effets of ombustion on turbulene are evidened by the high levels of flutuating longitudinal veloity in the flame front region and by the flame indued prodution of vortiity. However, further investigations are still required to study quantitatively the influene of ombustion on turbulene. To improve the quality of the results obtained in this work, some modifiations in the model are suggested, suh as: (a) the use of more sophistiated methods for generating inflow turbulent boundary onditions, suh as the ones that take into onsideration a more realisti turbulent kineti energy spetrum, and (b) the use of dynami type of models to alulate the sub-grid visosity. For the further steps, it is suggested to inlude a detailed hemial kinetis desription and also to test different miro-mixing models able to desribe thikened flame as well as flamelet regimes. For the latter, the strong oupling that exists between moleular diffusion and hemial reation must be taken into aount and this requires to improve the evaluation of both (i) the sub-grid mixing time sale, see for instane Mura et al. (2007) and Domingo et al. (2008) and (ii) the funtional form of the miro-mixing model itself (Pope and Anand, 1984 ; Mura et al. 2003). 5. ACKNOWLEDGEMENTS Fernando Oliveira de Andrade would like to thank CAPES and CNPq for the finanial support during his dotorate ativities at PUC-Rio and at the Laboratoire de Combustion et de Détonique de l Université de Poitiers, Frane. This work was performed while Luis Fernando Figueira da Silva was a Visiting Professor, funded by the National Petroleum Ageny (Brazil), on leave from Centre National de la Reherhe Sientifique, Frane.

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