LES-PDF SIMULATION OF A HIGHLY SHEARED TURBULENT PILOTED PREMIXED FLAME

Transcription

1 LES-PDF SIMULATION OF A HIGHLY SHEARED TURBULENT PILOTED PREMIXED FLAME Abstract V. N. Prasad, K. H. Luo and W. P. Jones k.h.luo@soton.ac.uk School of Engineering Sciences, University of Southampton, Southampton SO7 BJ, UK Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 AZ, UK The Eulerian stochastic field method in the context of Large Eddy Simulation (LES) is applied to simulate a highly sheared turbulent piloted premixed flame. A gradient diffusion type model for the sub-grid scale (sgs) stresses along with a -step reduced chemistry mechanism involving 9 species to represent the chemical reactions have been employed. Following previous studies, eight stochastic fields have been used to characterize the influence of the sub-grid fluctuations. Prior to the reactive case, the isothermal flow was simulated to assess the chosen grid and inflow conditions. The results of the flame statistics of the temperature and the species concentrations showed a good agreement with the experimental data, though the flame length was under predicted as a result of a premature jet break up. Introduction Lean turbulent premixed combustion has gained increasing importance in the recent years and the majority of the newly built gas turbines operate in this regime, replacing the commonly popular non-premixed operational mode. The advantages are lower formation of pollutants such as CO and lower NO x emissions, thus providing potentially cleaner combustion. The potential drawback, however, is a propagating flame which is difficult to control and may result in instabilities such as a flashback into the nozzle and thus security concerns. Therefore, a comprehensive understanding of the physics involved in premixed turbulent combustion is strongly desirable. The fact that turbulence is present in most combustion devices increases the complexity and a thorough understanding of the underlying physical processes requires the knowledge of the turbulence-chemistry interactions at the smallest scales. Complementing experimental techniques, numerical methods have gained significant importance in the recent decades to analyse complex flame phenomena. The most popular approach, RANS, has become a well established tool in the industry. Whilst offering solutions in a relatively short time with low computational requirements, RANS has the disadvantage of being strongly model-dependent and performs poorly in highly unsteady flows. The most detailed approach, Direct Numerical Simulation (DNS), on the other hand aims to resolve the entire range of turbulent scales. The enormous computational and storage requirements, however, currently limit its usage to academic problems at low to moderate Reynolds numbers. In this context, Large Eddy Simulation (LES), combines the advantages of both approaches. It solves directly for large scale motions and accounts for the unresolved sub-grid scales by means of, often fairly simple, modelling. Especially for highly unsteady flows such as swirl flows this method seems very attractive with the potential to provide a detailed insight into complex phenomena such as precessing vortex cores with an affordable computational requirement. In turbulent reacting flows, the chemical reactions occur on the smallest scales of the flow and the turbulence-chemistry interaction poses a central problem in understanding combustion physics in turbulent flows. A range of models have been proposed in the recent years (see e.g [],[] or [] for recent reviews), with each model predominantly accounting for a particular flame regime (premixed or nonpremixed). A class of models being independent of the flame structure are transported pdf methods, which aim to obtain the statistics of the unknown quantities from a solution of an exact transport equation for the sub-grid probability density function (pdf ) (see [4] for a recent review on pdf methods). The most commonly adopted form of the transported pdf method are Lagrangian particle methods, where notional particles form a system of equivalent stochastic differential equations (sde). Initially proposed in

2 the RANS framework [] this approach has also been incorporated into LES calculations [6, 7, 8, 9]. The advantage of the method is, that the errors are purely stochastic and do not arise from spatial discretisation errors, which are however partly reintroduced through complex interpolations into the physical space. In recent years, the stochastic field method - a fully Eulerian approach to represent the sub-grid pdf - has emerged [, ]. In LES, a wide range of flame configurations has been simulated with this method, e.g. auto-ignition in lifted flames [], extinction in partially premixed flames [], spark ignition in a non-premixed jet flame [4] and a premixed swirl flame []. The method is promising as it can be easily incorporated into any existing codes using conventional finite-difference schemes. However, the errors stem from the spatial discretisation as well as the stochastic sampling. To further investigate the performance of the stochastic field method in conjunction with LES, a series of recently investigated piloted premixed swirl flames (PM-, PM-, PM- and PM- ) designed at the University of Sydney are investigated [6, 7, 8]. These flames operate at high Reynolds numbers and low Damköhler numbers with the same boundary conditions for each case except for the inlet velocity of the fuel stream, similar to the Sandia Flames. In this paper, we present results for PM- with a detailed insight into the compositional structure of the flame. Governing equations in LES LES involves a direct numerical simulation of the large scale energetic motions with the effects of the unresolved sub-grid scale motions being modelled. The separation of the scales is achieved through a spatial filter, which for a function f = f(x, t) is defined as its convolution with a filter function G according to : f(x, t) = Ω G(x x ; λ(x))f(x, t)dx () The integration is defined over the entire flow domain Ω and the filter function has a characteristic width of λ, which may vary with position. In combusting flows, strong density fluctuations occur which can be treated through the use of a density weighted filter, defined by f(x, t) = ρf/ρ. Applying a density weighted filter to the conservation equations of mass, momentum, α =...N species and enthalpy leads to following set of filtered equations: Continuity: ρ t + ρũ i =, () Momentum: Scalars: ρỹα t ρũ i t + ρũ iũ j x j = p + x j σ ij x j τ ij () ( ) + ρũ jỹα = ρd Ỹα J α,k + ρ ω α (Y α ) (4) x j x k x k x k Enthalpy: ( ) ρ h t + ρũ j h = ρd h J h,k () x j x k x k x k where σ ij is the viscous stress tensor and D is the diffusivity, assumed equal for species and enthalpy. The sub-grid scale stress tensor τ ij = ρ (ũ i u j ũ i ũ j ) is determined via the standard Smagorinsky model [9]: µ sgs = ρ (C S ) S ij (6) where C S is the Smagorinsky constant obtained from a dynamics procedure [] and where S ij is the Frobenius norm S ij Sij of the resolved strain tensor. The filter width is taken as the cubic root of the local grid cell volume. For the sub-grid scalar fluxes J k, it is common to adopt a gradient model of the form J k = µsgs φ σ sgs x k, φ Y α, h, where σ sgs is the sub-grid Schmidt number. This leaves the

3 filtered chemical source term ρ ω α (Y α ), which represents the net rate of formation and consumption of the chemical species α, to be the only unknown quantity. Sub-grid pdf When attempting to close the filtered chemical source term, the important sgs fluctuations need to be considered. Here we introduce a sub grid (or filtered fine grained) PDF P sgs for this purpose. An exact evolution equation for the joint sub-grid scale PDF of the entire set of scalars needed to describe the reaction can be derived by standard methods, e.g. []. Two term remain unknown, and need to be modelled - the sub grid transport and micro-mixing terms. In the present work the Smagorinsky sub-grid viscosity model is used for transport and the Linear Mean Square estimation closure (LMSE) [] is applied for micro-mixing. The final form of the modelled time evolution of the sub-grid pdf is: ρ P sgs (ψ) t P sgs (ψ) + ρũ j x j [( (µ N α= σ + µ sgs σ sgs [ ρ ω α (ψ) ψ P ] sgs (ψ) = α )] ) Psgs (ψ) ρ N τ sgs α= [ (ψ α φ α (x, t)) ψ P ] sgs (ψ) α (7) where σ sgs has been assigned a value.7. reaction. ω α (ψ) is the net species formation rate through chemical Eulerian stochastic field method Equation 7 is solved using the Eulerian stochastic field method. Psgs (ψ) is represented by an ensemble of N s stochastic fields for each of the N scalars, namely ξ n α(x, t) for n N s, α N. Two formulations of the method can be devised depending on whether an Ito or Stratonovich interpretation of the stochastic integral is adopted; for a description of these alternatives see [] and []. In the present work the Ito formulation is adopted and the stochastic fields thus evolve according to: dξ n α = ũ i ξ n α dt + [ ] Γ ξn α dt + ( Γ ) / ξα n dwi n ( ξα n τ φ ) α dt + ω α(ξ n n )dt, (8) sgs where Γ represents the total diffusion coefficient and dwi n represent increments of a (vector) Wiener process, different for each field but independent of the spatial location x. This stochastic term has no influence on the first moments (or mean values) of ξα. n The sub-grid time scale is obtained from τ sgs = µ+µsgs. The stochastic fields given by Equation 8 are not to be mistaken with any particular ρ realization of the real field, but rather form an equivalent stochastic system (both sets have the same one-point PDF, []) smooth on the scale of the filter width. Test case formulation The targeted flame PM- belongs to a series of four pilot premixed flames (PM-, PM-, PM-, PM-) in a vitiated coflow, which have been designed and investigated at the University of Sydney [6, 7, 8]. The burners are geometrically identical and only differ in the pilot inlet velocity, which is ranges from m/s- m/s and results in a significant increase of the turbulence-chemistry interactions. The setup comprises a central annular jet of 4 mm diameter, a pilot with an outer diameter of. mm and a coflow of 9 mm diameter. Through the central jet, a lean premixed CH 4 -air mixture (φ =., T jet = 98 K) is injected. The pilot, providing a stream of a burnt CH 4 -air mixture at stoichiometry and T pilot = 8K can be considered at chemical equilibrium. The burner is embedded in a vitiated coflow consisting of a lean burnt H -air mixture (φ =.4 ) at chemical equilibrium (T co = K). Measurements are available for the isothermal case (air injected through all three inlet

4 U jet [m/s] U pilot [m/s] U co [m/s] Isothermal.7.8 Reactive. 4. Table : Velocity boundary conditions for the isothermal and reactive case for PM- streams) and the reactive including velocity, temperature and species concentration data. The velocity inlet conditions for the isothermal and reactive cases are summarised in Tab.. Computational parameters For the simulation, an in-house finite volume LES code with a collocated grid arrangement (BOFFIN [4]) has been used. The discretisation of all gradients is of second order, except for the convective part of the scalar equations, which was discretised with a TVD-scheme to ensure boundedness. More details regarding the numerics, especially regarding the implementation of the stochastic field equations can be obtained from []. The utilised computational mesh with. M cells corresponds to an inverted pyramid with a square duct inlet of 8D 8D ( grid points), an outlet square of D D and extends D ( grid points) in axial direction, similar to the grid used in [] to simulate Sandia Flame D - a piloted non-premixed jet flame. The aspect ratio of this grid, i.e. the ratio of the cell extensions in the respective coordinate directions, is shown in Figure along with two extracted line profiles at x/d = and x/d =. It is evident, that an aspect ratio close to is maintained throughout the entire domain, which results in a quasi-uniformity of the mesh. To account for the entrainment at the narrow lateral boundaries, inflow conditions using the values of the coflow have been imposed. A non-reflective convective outflow condition is applied at the exit plane. At the inflow, a fully developed pipe profile for the velocity has been imposed in the jet region and flat top hat profiles in the pilot and coflow region. Turbulent fluctuations were generated with the synthetic inflow generator by [6] and top hat species and enthalpy profiles in the jet, pilot and coflow have been prescribed. As in previous works (see e.g. [, 4, ]), eight stochastic fields are utilised to characterise the sub-grid fluctuations along with an IEM mixing model using a micro-mixing constant of C D =.. and the chemistry was represented by a -step ARM reduced GRI. mechanism [7]. D x/d= D x/d= x/d= x/d= 8D Figure : Aspect ratio of the utilised mesh. Two line profiles at x/d = and x/d = are shown along with the overall extension of the computational domain. 4

5 Isothermal case Prior to the analysis of the flame, a cold flow simulation has been performed to assess the quality of the chosen grid and the inflow conditions. The velocity conditions for this case are summarised in Tab.. The mean and rms axial velocities along the centreline for the isothermal case are shown in Fig.. The comparison with the experiments show, that the initial velocity decay corresponding to the jet breakup is slightly over predicted. The tendency overall is in good agreement. The rms velocity is over predicted up to the jet break up point at about 8 jet diameters, where it decays rapidly and then follows the data of the experiments. In general the behaviour of the flow along the centreline in terms of mean and rms is captured well. 4 Case exp Umean [m/s] 8 6 Urms [m/s] 4 x/d x/d Figure : Mean axial velocity along the centreline for the PM- non-reacting case The radial profiles of the mean and rms axial velocities are shown in Fig. at three axial downstream locations (x/d =,, ). The profile closest to the inlet shows an insufficient amount of spreading which may be related to the under prediction of the jet break up seen in the t profile in Fig.. At the other two downstream positions, the spreading of the jet is over predicted and this behaviour is also reproduced in the plots of the rms velocities. A possible reason may stem from the usage of the inflow generator, for which a length scale needs to be provided in addition to the mean fluctuations. Though the inflow data does not seem to influence the initial few diameters, an interaction of the inflowing large scale vortices with the shear generated turbulence further downstream may lead to excessive mixing. A further improvement may also be achieved by further refining the area around the nozzle exit and jet/pilot shear layer. The overall results, however, are reasonable and suggest that the quality of the inflow conditions and the mesh are sufficiently well chosen for the study of the reacting flow. Reactive case Detailed species and temperature measurements are available from the experimental data [7] and will be compared with the simulation results in this section. To gain a better impression of the instantaneous flame structure, snapshots of the temperature and the heat release contours computed from the product of OH CH O are shown in Fig. 4. The pilot, which stabilises the flame, extends about diameters downstream. The contours of the heat release, representing the reaction zone, are very thin, of the order of mm. The flame front region is mainly located in the mixing layer of the jet and the pilot and the downstream vicinity of the fuel jet break up point at about diameters. Mean radial profiles of the temperature and mass fractions of CH 4, O, H O, CO, CO, OH and H are shown in Figures to 8 at four axial downstream locations (x/d = 7.,, and 4). The reaction zone is located in the jet/pilot mixing layer (. ). At the location closest to the nozzle exit (x/d = 7.), the flame front temperature profile is overall captured to a good extent, although the spreading and thus the flame speed is slightly over predicted in the simulations. In the vicinity of the pilot/coflow mixing layer ( ), the temperature profile is over predicted. This indicates,

6 Umean [m/s] z/d= Sim Exp 6 4 z/d= z/d= 8 Urms [m/s] Figure : Radial profiles of the mean axial velocity for the PM- non-reacting case Figure 4: Instantaneous snapshots of the temperature field and the heat release of the flame that there may be strong heat losses across the pilot/coflow wall in the experiments and the adiabatic top-hat temperature profile prescribed at the inflow should be reconsidered in future calculations. Some analysis has been performed by [8], however concluding, that these seen heat losses do not affect the flame behaviour further downstream. The profile of CH 4 shows a similar level of agreement, though the centreline decay is more pronounced. The profiles of the reaction products CO and H O and the oxidiser O closely follow the trend of the temperature, though the prediction quality of water appears to be slightly better. The same holds for the CO, OH and H, though the deviation of theses species appears to be greater. The temperature profiles at x/d = are considerably over predicted in the jet core region and this may be associated to the wrong prediction of the flame length and a premature fuel jet breakup. This was partly seen in the cold flow simulation in the rapid decay of the centreline velocity and indicates a premature fuel jet break up. The effect may stem from the imposed length scales of turbulence inflow generator, which may interact excessively with the shear generated vortices and lead to increases instabilities and thus enhanced mixing. This is also visible in the profiles of CH 4 by the over prediction of the fuel consumption. In fact, fuel is fully consumed at x/d =, whereas some amount of fuel, though 6

7 to a very small extent, is still present in the experiments. At x/d = 4 the conditions are that of the vitiated coflow with a small amount of burnt gases which also has been captured by the simulation, seen in the temperature profiles and the low level of carbon based species CO and CO. Overall, the basic characteristics of the flame have been captured to a reasonable extent. It is notable, that even minor species such as OH and H (see Fig. 8) are predicted well. This confirms the trend seen in other flame configurations using the LES-stochastic field method with the same ARM reduced mechanism [, ]. x/d=7. x/d= Temperature[K] Temperature[K] (a) Temperature. x/d=7.. x/d=.... CH4 CH (b) CH 4 Figure : Mean radial profiles of the temperature (top) and CH 4 mass fraction (bottom) Conclusion and Outlook 7

8 . x/d=7.. x/d=.... O.... O (a) O. x/d=7.. x/d=.. HO HO (b) H O Figure 6: Mean radial profiles of O (top) and H O mass fractions (bottom) 8

9 x/d=7. x/d= CO CO (a) CO x/d=7. x/d=.. CO CO (b) CO Figure 7: Mean radial profiles of CO (top) and CO mass fractions (bottom) 9

10 x x/d=7. x x/d= OH 4 x 4 x OH 4 4 (a) OH x 4 x/d=7. x 4 x/d= H x x 4 H 4 4 (b) H Figure 8: Mean radial profiles of OH (top) and H mass fractions (bottom)

11 A highly sheared turbulent piloted premixed flame was simulated using the stochastic field method in LES. Simple gradient diffusion type models were employed for the unresolved sgs fluxes and eight stochastic field along with an IEM mixing model were employed to characterise the sub-grid fluctuations. The chemistry was represented by a -step ARM reduced GRI. mechanism, previously used for flames with extinction with the same model approach. Prior to the study of the flame structure, a cold flow simulation was performed to assess the chosen grid and inflow conditions. The results were satisfactory, although an excessive spreading of the jet was identified, which is possibly related to the sensitivity of the synthetic turbulence inflow generator. The simulation of the flame reproduced basic characteristics, predicting a thin flame front in the vicinity of the stabilised pilot. The flame spreading and thus the flame speed was slightly over predicted and the flame length was under predicted substantially, in accordance with the observation made in the cold flow simulations. The species concentrations were captured to a similar extent as the temperature, including minor species like OH and H. Though the results were encouraging, future work requires some more attention regarding the correct prediction of the flame length. The imposed turbulence at the inlet plane may be very influential and a more detailed study regarding prescribed length scales and fluctuations should aim to eliminate this uncertainty. To obtain a more complete picture of the flame behaviour and better assess the model performance in situations with significant turbulence-chemistry interactions, the simulation of the other three flames of the series is planned as future work. References [] J. Janicka, A. Sadiki, Large eddy simulaton of turbulent combustion systems, Proceedings of the Combustion Institute () [] H. Pitsch, Large-eddy simulation of turbulent combustion, Annual Review of Fluid Mechanics 8 (6) [] T. Echekki, E. Mastorakos (Eds.), Turbulent Combustion Modelling, Springer Verlag,. [4] D. C. Haworth, Progress in probability density function methods for turbulent reacting flows, Progress In Energy and Combustion Science 6 () () [] S. B. Pope, Pdf methods for turbulent reactive flows, Progress In Energy and Combustion Science () (98) 9 9. [6] P. Colucci, F. Jaberi, P. Givi, S. Pope, Filtered density function for large eddy simulation of turbulent reacting flows, Physics of Fluids () (998) 499. [7] F. Jaberi, P. Colucci, S. James, P. Givi, S. Pope, Filtered mass density function for large-eddy simulation of turbulent reacting flows, Journal of Fluid Mechanics 4 (999) 8. [8] M. Sheikhi, T. Drozda, P. Givi, F. Jaberi, S. Pope, Large eddy simulation of a turbulent nonpremixed piloted methane jet flame (sandia flame d), Proceedings of the Combustion Institute () () [9] V. Raman, H. Pitsch, A consistent les/filtered-density function formulation for the simulation of turbulent flames with detailed chemistry, Proceedings of the Combustion Institute (7) [] L. Valiño, A field monte carlo formulation for calculating the probability density function of a single scalar in a turbulent flow, Flow Turbulence and Combustion 6 () (998) 7 7. [] V. Sabel nikov, O. Soulard, Rapidly decorrelating velocity-field model as a tool for solving onepoint fokker-planck equations for probability density functions of turbulent reactive scalars, Physical Review E 7 (). [] W. P. Jones, S. Navarro-Martinez, Numerical study of n-heptane auto-ignition using les-pdf methods, Flow Turbulence and Combustion 8 () (9) [] W. P. Jones, V. N. Prasad, Large eddy simulation of the sandia flame series (d-f) using the eulerian stochastic field method, Combustion and Flame 7 (9) () 6 66.

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