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Linear Eddy Mixing Model Studies of High Karlovitz Number Turbulent Premixed Flames

Flow Turbulence Combust(2014)93:189–219

DOI10.1007/s10494-014-9542-8

Srikant Srinivasan·Suresh Menon

Received:6September2013/Accepted:25April2014/Published online:27May2014

?Springer Science+Business Media Dordrecht2014

Abstract Turbulent premixed flames exhibit different structural and propagation charac-teristics with increasing upstream turbulence intensity starting from thin wrinkled flames in the Corrugated Flamelet regimes to a flame with a thicker preheat zone in the Thin Reaction Zone Regime(TRZ)and finally,becoming more disorganized or broken in the Distributed or Broken Reaction Zone(D/BRZ)regimes under intense turbulence.A single comprehen-sive predictive model that can span all regimes does not currently exist,and in this study we explore the ability of the stand-alone one-dimensional linear-eddy mixing(LEM)model to simulate the flames in all these regimes.Past applications of this1DLEM model have demonstrated reasonable predictions in the flamelet and TRZ regimes and here,new exper-iments in the TRZ regime are specifically addressed to evaluate the predictive capability of this model.Additional simulations in the D/BRZ regimes(where no data is currently available)are performed to determine if the model can be extended to the high turbulence https://www.doczj.com/doc/151888608.html,parison with the data in the TRZ regime shows satisfactory agreement.Anal-ysis suggests varying levels of preheat zone broadening in all the TRZ and D/BRZ cases. While the average heat release distribution for the TRZ cases is nearly identical to the lam-inar unstrained baseline,changes to the species and heat release distribution are observed only at a high Karlovitz Number Ka>103.In the D/BRZ regime it is shown that the tran-sition is related to enhanced turbulent diffusion that dominates molecular diffusion effects causing deviations from the laminar baseline.

Keywords Large eddy simulations·Linear eddy mixing model·Broken reaction zones·Distributed reaction zones

S.Srinivasan( )·S.Menon

School of Aerospace Engineering,Georgia Institute of Technology,Atlanta,GA,USA

e-mail:ssrinivasa7@https://www.doczj.com/doc/151888608.html,

S.Menon

e-mail:suresh.menon@https://www.doczj.com/doc/151888608.html,

1Introduction

Turbulent premixed flames can be classified on the basis of the extent of flame-turbulence interaction.One such classification [22]is shown in Fig.1a.In this commonly used regime

diagram,the parameters used to determine the classifications are the ratios,u /s 0L ,and l 0/δ0L

.The former represents the ratio of the turbulent fluctuation velocity,u to the unstrained

laminar flame speed,s 0L and the latter is the ratio of the integral length scale, 0

to the flame thickness,δ0L .Phenomenologically,at low turbulence intensities,the size of eddies are much

larger than the preheat zone thickness.In this case,the flame structure is merely advected but the flame-surface is corrugated due to straining action of these eddies.In this Corru-gated Flamelet (CF)regime,the flame has a higher propagation speed than the unstretched laminar flame value by virtue of a larger surface area.At higher turbulence intensities,the eddies on the order of the preheat zone thickness (or smaller)can affect the preheat zone structure.Flame broadening is observed,but the reaction zone structure is unaffected.Flames in this Thin Reaction Zone (TRZ)regime,have Karlovitz numbers,Ka >1,where

Ka = (u /s 0L )3δ0L / 0

with Ka =1representing the upper limit of CF regime [22].For still larger Ka (Ka >100)and at high Reynolds number Re =u l 0/ν>1,turbulent diffu-sion by eddies can induce heat loss leading to global quenching [22].This regime is called the Broken Reaction Zone Regime (BRZ)[22].On the other hand,under certain situations [2],turbulent diffusion can affect a continuous burning zone in which,the per-flame heat release is smaller than the unstrained laminar state,but on aggregate basis,have a much larger than the total laminar heat release.The flame,in this case,is deemed to be in the Distributed Reaction Zones (DRZ)regime [1,2,9].

Recent interest in recreating D/BRZ flames in the laboratory or numerical simulations is motivated,in part,by the low emissions (NOx )benefits that lean premixed combustion offers.Lean premixed combustion systems are likely to fall under the D/BRZ regimes under

highly turbulent conditions,where u /s 0L

>100can occur [9].In their investigation involv-ing a piloted premixed jet burner,Dunn et al.[9]studied premixed flames for a range of Ka ～100?3500and Re ～720?5200.Here,a compressed natural gas/Air mixture at an equivalence ratio,φ=0.5was investigated and the D/BRZ traits of flame broadening,and de-correlation between the location of peak OH and temperature were observed only at Ka ～O (103).For Ka =100,flamelet-like flame was observed with locations of the temperature maxima well correlated with the peak in OH.

In a DNS study of stoichiometric hydrogen-air flames at a much lower Re ～17[26]and Ka ～87reaction zone was not effected but in another DNS study of hydrogen-air premixed flames at Ka ～10?1562and Re ～2?54[2]it was found that the distributed burning char-acter is attained for progressively richer mixtures.The seemingly counter-intuitive behavior can be explained by thermo-diffusive instabilities associated with lean hydrogen flames [2].For a given inflow composition,combustion in hydrogen mixtures occurs at a higher than the global φdue to preferential diffusion of H 2across the flame.At low Karlovitz numbers,molecular diffusion (non-unity Lewis number)effects are important,but at higher Karlovitz numbers (Ka ～1562)turbulent diffusion appears to dominate and a distributed burning behavior is observed.

Numerical simulations of high Re turbulent premixed flames will require methods other than DNS,and currently,large-eddy simulation (LES)appears to be the only possible solu-tion method.However,a key requirement for LES is accurate subgrid modeling to account for the effect of small-scale processes on the resolved motion.In the past,many closures have been developed based on specific regime(s)of focus.For example,in the flamelet

Fig.1Premixed regime diagram showing flames simulated using the stand-alone 1DLEM model.Figure (a)shows the Flames F and B studied earlier [29,33],and in Fig.(b),are the new

experimental flames [10,40]and other numerically defined flames investigated in this study

1010

101

103

l 0/δ0

u ′/s 0L

(a) Flames simulated in the past

1010

101

103

l 0/δ

u ′/s

(b) Flames simulated in the current study

regime,the G-equation model [22,23]models the wrinkled flame as an infinitely thin interface propagating with an effective flame speed.The closure effort is focussed on the interpretation of the flame brush and in the subgrid model for effective flame speed.Such models have been extended into the TRZ regime [23]and have also been applied to prob-lems spanning CF and TRZ regimes [19].Alternate models based on the progress variable can also be used in the CF regime where the reaction source term involves a flame surface density term ,defined as the flame surface area per unit volume inside the subgrid.Both algebraic [3]and transport equation based closure [12]have been used to compute .

A key advantage of these methods is that finite-rate kinetics are not included directly,and thus the methodologies are computationally very efficient.However,these models are typically limited to operating in either purely premixed or non-premixed regimes.Special treatment is often required to extend these for multi-regime configurations (or stratified premixed configurations)and as such,have problems predicting pollutant formation [15].Alternate approaches that directly include species transport and finite-rate kinetics in the flame zone have also been developed.For example,the dynamically thickened flame [7]solves for a resolvable flame structure by adjusting the transport and the reaction kinet-ics so that the correct flame structure and propagation characteristics are captured on the LES grid.Another approach is based on the subgrid implementation of the linear-eddy model (LEM)in which the reaction-diffusion processes are solved inside each LES cell

on a one-dimensional(1D)domain that is aligned locally to the flame normal direction [5].Within this1D flame-normal domain detailed kinetics and multi-component transport can be included.This approach(called LEMLES hereafter)has been employed in LES for premixed flames in both canonical[30]and complex reacting flow configurations[21,37].

In this study,we focus specifically on the features of the subgrid LEM model but using a stand-alone model.The stand-alone LEM approach has been used for studying soot-radiation interaction in non-premixed combustion[43],investigate the effect of thermal stratification on autoignition for Homogeneous-charge compression engines[20],investi-gate Type Ia supernova flames[39]and develop closures for premixed and non-premixed combustion[28,32].Pertinent to our investigation,the standalone model has been used extensively in the past in the flamelet and thin-reaction zone regimes[29,33].This approach called1DLEM hereafter is an extension of the stand-alone1D LEM model developed ear-lier[33]and its capability for scalar mixing and premixed flame modeling has been reported extensively elsewhere[14,18,33].Extension of the1DLEM model to the D/BRZ regimes is the focus of this study.Extension of the LEMLES approach to the D/BRZ regime will be considered in a future effort.

The1DLEM model has been described in literature and its strengths and limitations are well known[18,29,33].In particular,1DLEM models the effect of small-scale(isotropic)turbulence on the reaction-diffusion processes by using a stochas-tic stirring process along the1D line.Without the stirring process,the1DLEM reduces to the classical laminar premixed flame simulation method(e.g.,CHEMKIN PREMIX [13])well known in the literature.Thus,detailed kinetics and transport can be used as in the laminar method and the resolution is refined to resolve all the scalar variations across the flame.

The stirring process is implemented concurrently with the reaction-diffusion processes and allows small-scale turbulent eddies to interact with the combustion process.This makes 1DLEM an inherently unsteady simulation model.Also,since stirring is stochastic,only statistical data can be obtained from these simulations.Each stirring event is designed to account for the action of an individual eddy on the scalar gradient[14].The eddy size and the frequency of the stirring events are determined using Kolmogorov scaling laws.Thus, these eddies can instantaneously wrinkle the flame and the distance between adjacent flame crossing(on the1D line)can be used to estimate the local radius of curvature[33].Since all eddies in the inertial range can interact with the flame structure,over a period of simulation time statistics of flame-turbulence can be obtained.It has been demonstrated in the literature [29,33]that although any single stirring event is stochastic,statistical convergence of the simulation is achieved and thus its results can be compared to data,as done in the past for the flamelet[33]and the TRZ[29]regimes.We revisit these observations using data from recent experiments.

The1DLEM model is also limited in its capability.1DLEM model cannot capture the effect of large-scale coherent structures and geometry dependent features of a problem. Thus,large-scale straining and3D flame structure cannot be captured(however,note that LEMLES can capture these effects).On the other hand,if we focus only on the local flame structure we can recover some of the key physics at the small-scales.Increase in the turbu-lent Re can be accounted for by ensuring all the relevant inertial range eddies are included in the stirring process.The1DLEM approach is also cost-effective(when compared to LES) when relatively detailed kinetics and transport are included.This is especially relevant in this study since detailed kinetics and transport are necessary to study D/BRZ regimes where radical formation and transport can be important when high small-scale strain(intense turbulence)interacts with the flame.Thus,within its limited ability,1DLEM model is a

good candidate to investigate these flames and to carry out a systematic sensitivity analy-sis for a range of equivalence ratios and Ka numbers.It has been suggested[11,27],that in high Ka number flames,the flame-structure is influenced significantly by the smaller isotropic eddies rather than the large scale structures.The1DLEM model’s limitation to small-scale turbulence becomes an advantage in this case as long as the local structure is only statistically investigated.

Various premixed flames have been investigated using the1DLEM in the past and few of these are shown in Fig.1a.In the current study,in addition to some of these flames,we also compare flame statistics against data from a piloted Bunsen burner[40]and a weak swirl burner[10]operating in the CF/TRZ regime.The operation-regimes of these flames are shown in Fig.1b.Finally,in the D/BRZ regimes we simulate flames under conditions that have not yet been studied either experimentally or numerically(using DNS or LES)to assess LEM’s predictions in this regime.

This paper is organized as follows.The1D-LEM formulation and numerical approach is briefly discussed in Section2.The numerical and algorithmic details along with the con-figuration of the numerical and experimental flames are discussed in Section3.Results are discussed in Section4wherein characterization of flames in the CF-TRZ-D/BRZ regimes are performed in addition to model-parameter sensitivity analyses and comparisons to recent data.Conclusions are summarized in Section5with outlines for future studies.

2Formulation and Numerical Approach

The details of the LEM formulation are given elsewhere[6,29,33]and in a recent review [18],and therefore,only the salient features of the model are highlighted for completeness and for subsequent discussion.As noted earlier,the LEM domain is a1D statistical ray through the local3D flame-brush,oriented along the mean propagation direction(or flame normal direction).Thus,this line can be considered a flame normal line across the flame in a typical Bunsen burner experiment such as the one considered in this work[40,41]. Figure2shows schematically some possible instantaneous ways to interpret the1D LEM domain.The1DLEM line is not necessarily a straight line,but a curved line that varies spatio-temporally.The boxed insets in Fig.2also show the computational domain and the temperature distribution along this1D line.The two temperature distributions shown here correspond to different flame conditions(varying axially)and hence,may correspond to two different1DLEM simulations.As shown here,the flame may be perturbed by the actions of stochastic eddies at different locations along the1DLEM line for a given axial location on the Bunsen burner.The conventional1D reaction-diffusion equations are solved along this line,interrupted by instantaneous stirring events that represent the action of turbulent eddies on the flame structure.

The reaction-diffusion equations for1DLEM are:

?Y k ?t =F k,stir?1

(ρY k V k)+

˙ωk W k

(1)

and,the temperature equation is given by:

?T ?t =F T,stir?1

c p

k=1

c p,k Y k V k

ρc p

k=1

h k˙ωk W k(2)

Fig.2A schematic of the Bunsen flame structure is shown in the main along with the interpretations of the 1DLEM domain in the insets.See text for details

In Eq.1,ρis the density,N is the total number of species,and Y k,h k,˙ωk,W k,c p,k and V k represent respectively,the k th species mass-fraction,enthalpy,reaction rate,molec-ular weight,specific heat at constant pressure and species diffusion velocity.Here,V k is computed using the Hirschfelder and Curtiss approximation with a velocity correction for ensuring mass conservation[25].The stochastic term,F k,stir and F T,stir represent respec-tively,the turbulent stirring of the species and the temperature fields.In the above equations, c p andκindicate mixture averaged specific heat and thermal conductivity,respectively.

The1D domain direction s is along the flame propagation direction and the terms, F k,stir and F T,stir in Eqs.1and2are modeled using a triplet mapping process to mimic the rearrangement of a scalar field due to the motion of an eddy[14].Figure3shows schematically the arrangements for the current premixed flame setup.Previously,it was shown that this mapping reproduces the compressive straining of the flame front,recovers multiple flame-crossings in high Re flows(reflecting an increase in the surface area of a Fig.3Initial conditions used in the study.In Fig.(b),L p represents the extent of the product region

3D flame brush),and realistically reproduces turbulent diffusion associated with high Re inertial range turbulence[17,18].

To implement the stirring process three parameters are needed.The eddy size range f(l) and the stirring frequency are first determined[33]and then,the location of the mapping event is randomly chosen with every location on the1D line having an equal probability.In this procedure,eddy sizes are chosen randomly from a distribution based on inertial range scaling[14].From a size range,η< < 0,the selection probability of eddy size, is

f( )=5

?8/3

η?5/3? ?5/3

(3)

where, 0is the integral length scale andη～ 0Re?3/4,the Kolmogorov scale.In the model,ηe=Nη 0Re?3/4,where Nη>1is a scaling parameter that needs to be defined.

The stirring frequency is determined using Kolmogorov scaling[33]as:

λ=54

νRe

Cλ 03

[( 0/ηe)5/3?1]

[1?(ηe/ 0)4/3](4)

Here,ν,the kinematic viscosity is typically taken to be ahead of the flame(i.e.,the reactant side)in the past work[29,33].The stirring time for a domain of size,L is then computed asτstir,L=1/(λL).

There are two coefficients in this model,Cλand Nηboth of which are related to inertial range turbulent scaling(and not to the reaction-diffusion processes solved in Eq.1).Once these coefficients are defined,there are no more adjustable parameters.Nηis typically in the range5-20[33]and does not affect turbulent diffusivity[33].Earlier,Cλ=1/0.067～15 was determined using dimensional arguments for high Re turbulence[6].The sensitivity of predictions in the flamelet regime to the choice of Cλand Nηwas also investigated [33].Since we are considering flames in new regimes and for conditions similar to those in the experiments,we revisit this issue by conducting a systematic study using the available conditions for the cases defined in Fig.1b.

In the current study,we revisit the choice ofν(computed from the fresh gas conditions) since in the TRZ and D/BRZ regimes preheating occurs and the choice of a reference tem-perature is not clear.Therefore,an alternative approach to computingλlocally using the localν(T)and then calculatingηe is considered.The spatially averaged value ofλis then used for the entire domain.Theλcomputed as above is,in general,smaller than the one computed based on the conditions from upstream of the flame due to the smaller local Re in the flame and the largerηe.The quantities,u and 0are assumed constant and equal to the values upstream of the flame due to lack of data at present.

A major difference between earlier studies using1DLEM model and the current one is in the details of the chemical kinetics.Here,to study flame structure in highly intense tur-bulence as in the D/BRZ regime we employ a more detailed12step,16species mechanism [35].This mechanism is known to capture extinction/re-ignition physics and includes key radicals that have very different transport properties than the reactants and major products. Therefore,by using a more detailed finite-rate kinetics,and realistic temperature dependent curve-fits for the binary diffusion coefficients and the thermal conductivity we include key chemical time scales.Thermally perfect gas assumptions are used to compute the thermal properties,and the mixture averaged formulation is used for computing the mixture aver-aged diffusivity.Dufour and Soret effects are neglected for this study and constant pressure (atmospheric)combustion is considered.

3Setup

The suite of test cases chosen for this study span the CF,the TRZ and the BRZ regimes as shown in Figs.1a and b.The laminar flame properties for all test cases are given in Table1, all of which correspond to a reactant temperature of300K and pressure of1atm.The flame thickness,δ00.5is computed from:

δ00.5=

T b?T u

dT/dx|c=0.5

(5)

and is considered to be the thickness of the reaction zone[40].Here,the subscripts,“b”and “u”refer to burnt and unburnt states and the progress variable is c=(T?T u)/(T b?T u). Similarly,the pre-heat zone thicknessδ0.3is computed at c=0.3.Note that these definitions are not unique in literature as other definitions based on scalar profiles have also been used to demarcate regions in the premixed flame as either pre-heat or reaction zones[8,10]. Here,we use the expressions shown above to be consistent with experimental data[40]. Other definitions are also considered in the discussion of results.

In Table1,the1DLEM predictions for the laminar flame thicknesses using the reduced mechanism are shown.The values are confirmed to agree in a separate analysis using the 1D CHEMKIN-PREMIX package[13]with both the reduced mechanism and the detailed GRIMECH3.0mechanism.The naming convention for all cases investigated in this work and tabulated in Table2is as follows.The first letter,“E”,“N”or“T”refers to whether the conditions of the flame are adapted from actual experimental configurations(“E”)or otherwise(“N”and“T”).Both“N”and“T”refer to numerical flames with two different initializations.In the former,the standard1D premixed flame profile is used with the region of reactants and products situated at either side of the flame-center.The schematic in Fig. 3a shows this initial condition.In the latter,the premixed flame profile is mirrored about the domain-center so that an island of products is surrounded by cold reactants on either side.

Table1Premixed unstretched

Laminar flame properties.All

test cases involve a methane/air

premixed mixture at300K and1

atm.Re=u 0/ν,Damkohler

number,Da=( 0/u )/(δ0L/s0L),

δ0 L =ν/s0L,η～ 0Re?3/4

Case s0Lδ0Lδ0L,maxδ00.5δ00.3

(cm/s)(mm)(mm)(mm)(mm)

EK1.40.70200.0750.630.640.79

EK1 1.00380.0370.440.420.48

EK40.70200.1070.630.640.79

EK370.60100.18 1.00 1.2 1.5

EK0.980.70200.1070.630.640.79

EK2.30.70200.1070.630.640.79

EK4.30.70200.1070.630.640.79

EK7.30.70200.1070.630.640.79

EK9.40.70200.1070.630.640.79

EK12.50.70200.1070.630.640.79

NK2060.60100.18 1.00 1.2 1.5

NK11930.60100.18 1.00 1.2 1.5

TK370.60100.18 1.00 1.2 1.5

TK2060.60100.18 1.00 1.2 1.5

TK11930.60100.18 1.00 1.2 1.5

Table2Geometry and turbulence parameters

Case u /s0L 0ηRe t Ka Da Ref.

(mm)(μm)

EK1.4 3.415.0358145 1.412.2Plessing et al.[24] EK1 3.2 1.675398 1.09.4Yuen and G¨u lder[40] EK4 6.55 1.6452100 4.0 2.4Yuen and G¨u lder[40] EK3724.13 1.7929238370.4Yuen and G¨u lder[40] EK0.98 3.5 3.0721450.98 1.4de Goey et al.[10] EK2.3 6.0 3.046258 2.290.83de Goey et al.[10] EK4.39.2 3.034393 4.310.54de Goey et al.[10] EK7.313.1 3.0265577.320.38de Goey et al.[10] EK9.415.4 3.0236599.360.32de Goey et al.[10] EK12.518.7 3.02080012.50.27de Goey et al.[10] NK20675.0 1.79137452060.13this work

NK1193241.3 1.795238011930.04this work

TK3724.13 1.7929238370.4this work

TK20675.0 1.79137452060.13this work

TK1193241.3 1.795238011930.04this work

Figure3b represents this configuration.The numerical value after the letter,“K”represents the Karlovitz number.

Cases EK1.4,EK1,EK4and EK37correspond to conditions in the experiments,specif-ically flame F1from Ref.[24],and flames M1,M9and M15from Refs.[40,41], respectively.Cases EK0.98,EK2.3,EK4.3,EK7.3,EK12.5correspond to conditions in Ref.

[10].The high Karlovitz number flames-NK206,NK1193,TK206and TK1193-retain all chemistry and turbulence parameters of the Case EK37,except u /s0L,which is increased to shift the operational regime from TRZ to D/BRZ.It must be noted that these are numerical flames that do not exist experimentally.

The flame EK1.4[24],is comparable to EK4but has an order of magnitude larger 0. The effect of the geometry-related 0on flame-propagation characteristics is assessed by comparing their heat-release signatures.The effect of Ka on flame-structure and broadening is studied by keeping chemistry and geometry(i.e., 0)unchanged for Cases EK37,NK206, and NK1193.Finally,the configuration of flames with the prefix“T”is used to study the impact of product dilution by unburnt fuel at high Ka.Here,the same three configurations, EK37,NK206and NK1193are again revisited with the alternative initialization,but keeping all other parameters unchanged.

3.1Numerical setup

Figures3a and b show schematically,the two type of premixed flame propagation prob-lems that have been simulated.The size of the LEM domain is influenced by geometry, which,in this study,is related to 0.In all simulations,detailed in Table3,the length of domain,X L>25 0.This choice was motivated by the requirement that flame-propagation be unconstrained by the imposed boundary conditions at the reactant(Dirichlet)or the prod-uct side(Neumann),and that the turbulent stirring action not perturb the domain close to the inflow.After preliminary simulations with various domain lengths,the above estimate

Table3Simulation parameters. Note that the“c”and the“d”setups are different in the manner the stirring frequency,λis computed(see text for details)Case X L LEMηe CλNη(cm)(μm)(μm)

EK1.415.016.0717 3.5 2.0 EK1a 5.017.9106 3.5 2.0 EK1b 5.020.81530 3.510.0 EK1c 5.015.026515.0 5.0 EK1d 5.015.026515.0 5.0 EK4a 4.817.3104 3.5 2.0 EK4b 4.820.31520 3.510.0 EK4c 4.821.726015.0 5.0 EK4d 4.821.726015.0 5.0 EK37a 5.029.558 3.5 2.0 EK37b 5.020.81290 3.510.0 EK37c 5.015.6214515.0 5.0 EK37d 5.015.6214515.0 5.0 EK0.98 5.017.035815.0 5.0 EK2.3 5.017.023315.0 5.0 EK4.3 5.017.016915.0 5.0 EK7.3 5.017.013015.0 5.0 EK9.4 5.017.011515.0 5.0 EK12.5 5.017.09915.0 5.0 NK206c 5.0 5.006515.0 5.0 NK206d 5.0 5.006515.0 5.0 NK1193 5.0 5.002515.0 5.0 TK37 5.015.6214515.0 5.0 TK206 5.0 5.006515.0 5.0 TK1193 5.0 5.002515.0 5.0

was found to be a conservative choice for simulations with both,large 0and,lean flames with large preheat zone thicknesses.

As the premixed flame propagates towards the reactants,a recentering procedure is per-formed at regular intervals,chosen here to be the time-period spanning a net displacement of12ηe.The recentering of the scalar field is performed using the spatial location of c=0.5 as the flame center.This recentering strategy is also retained when time-averaging the scalar profiles from the instantaneous snapshots for data comparisons.The procedure is,however, not without its ambiguities.For example,when multiple flame-crossings with respect to the chosen c(=0.5)exist,the choice of the spatial location for recentering is arbitrary.Here,this

issue is addressed partially by choosing the location with the largest|˙ωCH

4|from among the

multiple flame-crossing locations.Regardless of its limitations,the c?based re-centering method is used to be consistent with the experimental analyses of Yuen and G¨u lder[40].

The details of the numerical algorithm are given in Smith and Menon[33]and the model is used as before.Briefly,the reaction-diffusion equation,Eq.1and the temperature equa-tion Eq.2are solved using a finite-difference scheme on a uniformly discretized1D line. Backward Euler time-integration and second-order accurate central-difference for the spa-tial derivatives are used here.Fractional operator splitting method[29]is used such that the

scalar equations are first solved for a time-step(reaction,diffusion and stirring)without the source term to get the thermo-chemical state after mixing.Then the chemical mechanism is integrated over the time-step using a stiff ODE solver,DVODE[4]to get the final state.

To determine grid resolution,all cases are first run using1DLEM in the laminar premixed mode(i.e.,by disabling the stirring terms F k,stir and F T,stir)to confirm that both the flame-speed and the flame thickness are reproduced correctly(see Table1).For all cases,the resolution is chosen to ensure that the reaction rate gradient is resolved with at least fifteen points based on the analysis of the EK1flame with =1.0.A conservative estimate equal toδ0L,max/30where,

δ0L,max=

T b?T u

dT/dx max

(6)

is chosen as the grid-size to resolve the reaction rate profile adequately.

For large Re,the Kolmogorov length scale,ηe is comparable to or smaller than the lam-inar flame thickness.Therefore,to capture the eddy-reaction zone interactions accurately,

ηe is resolved with at least six grid points.In the context of the triplet mapping proce-dure for stirring,this resolution estimate permits the simulation of the smallest eddy of size

equal toηe.As mentioned in Section2,the model constant,Nηreduces the range of avail-able eddies without compromising the turbulent diffusivity.This is so because,for a given

value of the coefficient,Cλ,the turbulent diffusivity,D t can be derived to be of the form D t=(u 0)/Cλ[33],and is therefore unaffected by the choices made for Nη.However, the value of Nηshould still be chosen judiciously to ensure that eddy interactions on the order of the relevant flame-thickness definitions discussed in this section are possible in the

TRZ and D/BRZ regimes.Since we are spanning the CF-TRZ-D/BRZ regimes,parametric studies with different values of Nηand Cλare repeated here and results compared against measured data in order to assess the sensitivity of the prediction to these coefficients.

The list of cases for the sensitivity study along with the model coefficients,Nηand Cλ

are given in Table3.Note that these parameters are identical for Case K xc and K xd.The only difference between these sets of cases is that a spatially averaged value of the stirring frequency across the flame is used in Case K xd whereas for Case K xc(and all the other cases in the table)the value ahead of the flame is used[29,33].

Since this is an unsteady simulation,the computational cost is significant especially since fine grid resolution and relatively detailed kinetics are employed.Small time step on the order of O(10?8)?O(10?7)s is needed and for statistical convergence,simula-tion time on the order of102milliseconds is required.Hence,1DLEM model has been parallelized using the Message Passing Interface(MPI)paradigm.The communications are implemented using a hybrid strategy involving both point-to-point(used typically in finite-volume or finite-element based CFD codes)and gather-scatter approach.The former strategy is used for parallelizing the discretized reaction-diffusion equations,where only the scalar field information in the cells across the processor-boundary is communicated. In contrast,the triplet mapping involves communication of segments of scalar fields possi-bly across multiple processors depending on the size of the stirring eddy.A communicator, dynamically created based on the eddy size,is used for performing the communications selectively across the processor space.This is in contrast to the alternative of a global gather-scatter approach that would be computationally prohibitive owing to the large volume of communicated data.

It is noted that the code cannot scale ideally due to latency imposed by stirring-related communications.This is so because,unlike conventionally point-to-point communications, the volume of stirring related communications remains unchanged(as the stirring statistics

are independent of domain decomposition)or increases if a large number of processors are used.The scalability on 200cores is roughly 60%for Case NK1193,which happens to be the most computationally intensive case in this study.For the extreme cases,e.g.,EK1and NK1193around 2000and 48,000single CPU hours (based on a Cray XE6system with a 2.3GHz AMD Opteron core)are needed,respectively.Note that our simulations span about 100-500turnover times to compute flame statistics and this is consistent with the long simulation times for large Ka flames mentioned by Aspden and Bell [2].4Results and Discussion 4.1Model parameter sensitivity

We begin by revisiting the sensitivity of the predicted scalar profiles and other flame statis-tics to the choice of the model parameters C λand N η.Table 3lists the values of the constants,C λand N ηused in the current study.These parameters were investigated ear-lier in the CF regime using 1-step kinetics [33]for configurations that involved an order of magnitude larger 0compared to the current cases.In the current cases we employ the more detailed kinetics noted above.

The time averaged profiles in Fig.4show a progressive broadening of the temperature profiles from Cases EK1to NK206relative to the reference laminar profile.In the cases K xa and K xb the temperature profiles are broader than for the cases K xc and K xd due to smaller C λ.Furthermore,for Case EK1and EK4the pairs K xa /K xb and K xc /K xd show great similarity between themselves due to identical values of C λ.Since C λaffects turbulent

05001000150020002500x (mm)

T (K )

EK1a EK1b EK1c EK1d Laminar (a) EK1x

048

0500100015002000x (mm)

T (K )

EK4a EK4b EK4c EK4d Laminar

(b) EK4x

?12

?8?4

048120500100015002000x (mm)

T (K )

EK37a EK37b EK37c EK37d Laminar

?12

?8?4

04812

0500100015002000x (mm)

T (K )

NK206c NK206d Laminar

(d) NK206x

Fig.4Time averaged temperature profiles centered at c =0.5

diffusivity there is similarity in flame-broadening among the cases with identical Cλbut different Nη.With increase in Ka the size of the smallest eddy determined by Nηalso becomes important and subtle variations between all four LEM distributions in Fig.4c can be seen for Case EK37.With further increase in Ka,the differences between NK206c and NK206d are even more obvious in Fig.4d.Although the parameters for Cases K xc and K xd are identical,the stirring frequency computation procedure is not.Since the Cases K xd consider a stirring frequency averaged across the flame,these flames are less broad compared to the Cases K xc owing to a smaller value ofλ.The extent to which they deviate depends on Ka;The lowerφ=0.6(i.e.,broader laminar unstrained flame profile)and the smallerηe of Cases EK37and NK206cause the initially laminar flame to respond more effectively to the stochastic eddies than the smaller Ka flames of EK1and EK4,although the stirring frequency for all cases changes by roughly the same factor.

The differences between the sub-cases of the three flames based on Ref.[40,41]are quantified in Table4where the various flame-thicknesses described in Section2are tabu-lated.These quantities are normalized with respect to the laminar flame value and they are essentially the non-dimensional temperature gradient since the term T b?T u appears in the numerator of the definitions of these thicknesses.The effect of Cλon the computed thick-

nesses is assessed by defining a stirring time-scale,τstir,η

E =η2E/D t,whereηE has been

used as the length-scale representing the smallest eddy that can be represented by the triplet map and D t=u 0/Cλ.For the purposes of analysis,the choice ofηE as a length-scale is reasonable because(a)ηE is the most probable eddy-size during the stirring events,given the form of Eq.3,(b)it is assumed that given the size range 0?ηE,it is the eddies on the order of size,ηE that contribute significantly to D t,and(c)ηE is on the order of the grid-size,and,therefore represents the local turbulent diffusion time-scale.Note that the time-scale,τstir,L,is a global,geometry-dependent timescale for the entire domain and not

a local time-scale.Thus,τstir,η

E is more suitable for comparing configurations with differ-

ent domain lengths.The value ofτstir,η

E decreases from1.8×10?5s for Case EK1a,down

to8.7×10?7s for Case NK1193.

In Table4,the effect of model parameters onδmax is more obvious for Flames EK37x

than the other cases.The value ofτstir,η

E ranges from2.7×10?6s to2.9×10?4s for

Cases EK37a-EK37d respectively,and,τstir,η

～1.8×10?5s-2.0×10?3s for Cases

Table4Model parameter

sensitivity Caseδmax/δ0L,maxδ0.5/δ00.5,Lδ0.3/δ00.3,L cδmax(c0L,δ

max

)

EK1a 1.122 2.071 3.9800.49(0.47)

EK1b0.804 1.081 2.0610.50(0.47)

EK1c 1.044 1.422 2.0440.49(0.47)

EK1d 1.046 1.369 1.8160.48(0.47)

EK4a0.975 2.920 3.3290.53(0.57)

EK4b 1.077 1.74811.7730.58(0.57)

EK4c 1.013 1.326 1.9650.56(0.57)

EK4d 1.020 1.399 1.5510.56(0.57)

EK37a0.63212.06017.8750.50(0.65)

EK37b0.777 3.428 6.2020.55(0.65)

EK37c0.883 2.091 4.5630.56(0.65)

EK37d0.944 1.620 2.0870.59(0.65)

EK1a-EK1d respectively.Similarly,the ratioδ0L/ηE～7?14for Cases EK1a-EK1d while it ranges from0.3?1.6for Cases EK37a-EK37d.The combination ofτstir,η

E andδ0L/ηE reproduces the influence of Ka on the premixed flame structure.Specifi-

cally,with smallerηE the probability of an eddy to perturb the flame-structure near the high-gradient region of the reaction zone increases.Given that the stirring procedure increases scalar gradients in the flame,repetitive stirring progressively increases the

maximum temperature(and other scalar)gradients proportional toτstir,η

E before the

diffusion processes can dissipate these gradients.The flame thickness,δmax therefore shows more sensitivity in the Cases EK37x with Ka=37than the Case EK1x with Ka=1.

In Table4,the thicknesses,δmax on one hand,and,δ0.5(the reaction zone thickness) andδ0.3(the preheat zone thickness)on the other,show an opposite trend,in general,

withτstir,η

E .This is related to their definitions.The flame thickness,δmax,computed

over0

least two orders of magnitude smaller thanτstir,η

E for the flames considered here.Consis-

tent with these arguments,there is a nominal increasing trend ofδ0.5andδ0.3withτstir,η

E in Table4.

Earlier in the discussions,the flame thickness,δmax was considered to be related to the reaction zone thickness for a laminar premixed flame.Instantaneously,in1DLEM,the max-imum gradient of temperature can occur anywhere in the pre-heat and reaction zones due to the stochastic stirring process.Thus,the magnitude of gradients is suggestive of the stirring activity,and the frequency of stirring,if large enough to overcome the attenuating effects of molecular diffusion,results in steeper gradients and smallerδmax values.The values of c corresponding toδmax are also given in Table4where its deviation from the values cor-responding to that of the laminar unstretched flame is significant for the EK37x flames relative to the EK1x and EK4x flames.Furthermore,within EK37x,the mean c location at

whichδmax occurs,shifts towards lower c values with decreasingτstir,η

E .These changes to

the flame structure imply that in EK37x,flame-broadening is active over a wider c-space through the creation of the temperature gradients via enhanced stirring.In contrast,in EK1x and EK4x,the maximum gradient coincides with that for the baseline laminar flame.This seems to suggest that the vicinity of the reaction zone is,in the average sense,relatively unaffected by the action of eddies.This contrasting behavior of the EK1x and EK4x flames in one hand and the EK37x flames on the other is consistent with the location of these flames in the regime diagram,Fig.1b.

To summarize the sensitivity-analysis study,the model parameters Cλand Nη

affect scalar profile distributions and flame-statistics via the stirring timescale,τstir,η

E .

The value ofνused in calculatingλaffects the predictions of thickness in the EK1x and EK4x to a lesser extent than it does for the EK37x series of cases.For the

experimental cases,different parameter sets performed well either in the pre-heat or the reaction zones while over/under predicting gradients in the other region,but the parameters

Cλ=15and Nη=5produced the best global agreement for all cases[34].Hence,for all cases that follow,the LEM parameter-set of K xd has been used and the suffix“a/b/c/d”

is dropped and the parameter set Cλ=15and Nη=5is used,similar to previous LEM investigations[5,29,33].

4.2Flame characterization

The1DLEM flames based on experiments,EK1.4,EK1,EK4and EK37are first discussed along with comparisons with data followed by the discussion of the numerical flames NK206,NK1193,TK37,TK206and NK1193.

4.2.1Temperature and reaction rate distributions

Fifteen snapshots of instantaneous fields of the temperature and the methane reaction rate,

˙ωCH

4are shown in Figs.5a-f.The figures are constructed by centering the images at the

spatial location corresponding to c=0.5.The snapshots in Fig.5a and c clearly highlight the dependence of the geometric(integral)length-scale on the unsteadiness of the profiles. For example,Cases EK1.4and EK4with similar Ka produce markedly different ensemble of instantaneous distributions.This is related to the almost O(1)difference in 0=15mm (Case EK1.4)relative to 0=1.64mm(Case EK1).Since bothηE and 0corresponding to

EK1.4are larger than those of EK4,larger segments of the EK1.4scalar fields are stirred

compared with that of Case EK4.Additionally,for Case EK1.4,theτstir,η

E is an order of

magnitude smaller than that of EK4.This causes multiple flames to be created,as seen

via the reaction rate profiles of5a.Figures5a and c show similarity in that˙ωCH

4is of

comparable magnitude.This is consistent with the behavior of flames in the TRZ where only the pre-heat zone is affected[22].

The time-averaged temperature profiles are compared with the baseline laminar profiles to assess the impact of turbulence-chemistry interaction.Figures5a,b and c show insignif-icant deviations in the pre-heat and reaction zones.Figure5d shows visible departure from the laminar distribution in the pre-heat zone,and these departures are much more evident in the D/BRZ cases of NK206and NK1193.However,the appearance of instants of“broad-ened”reaction rate profiles seen in Fig.5f,need not necessarily indicate a permanent shift in the heat-release signature.This aspect will be investigated when the heat-release distribution will be examined further below.

The time averaged temperature profiles of the EK1,EK4and EK37flames are shown in Fig.6a-c respectively,and compared with data[40].The quantity,T±σis also plotted in red dashed lines in Figs.6a-c to quantify the mean fluctuations due to turbulent convection.The predictions show similarities as well as differences with data.In Fig.6a the temperature gradients near the pre-heat zone are overpredicted by the stand-alone 1DLEM model.In the reaction zone(c=0.5)and beyond,however,the agreement is good.In EK4,the pre-heat zone predictions are good whereas the reaction zone gradients predicted by LEM are larger.Finally,in EK37,the preheat zone thickness is slightly over-predicted.The standard deviation for the three flames is equal to zero at c=0.5due to the re-centering of the propagating flame at c=0.5.The upper estimate,T+σ,however,exceeds the product temperature in the zero-gradient post-flame region and drops below the reactant temperature level(300K)just before the beginning of the pre-heat zone.This is so because this estimate is computed by adding or subtracting σfrom the mean temperature,regardless of the position along the flame.For Cases EK1, EK4and EK37the data is within the bounds of T±σ.These differences from data are to be expected since1DLEM,does not model effects of mean flow anisotropy or mean flow imposed strain.

The normalized average temperature distributions for all cases are compared in Fig.6d. The trend of profile broadening with increasing Ka is apparent.Moreover,for the D/BRZ flames NK206and NK1193,reduction in the temperature gradient is also effected in the

?30?20?10

01020

?1000

?500

050010001500

20002500T (K ), 10× w d o t

C H

(k g /m 3

/s )x (mm)

(a) EK1.4

?8?6?4?202

?1000

?500

05001000150020002500T (K ), 5× w d o t C H 4

(k g /m 3

/s )

x (mm)

(b) EK1

?8?6?4?202

?1000

?500

05001000150020002500T (K ), 10× w d o t C H 4

(k g /m 3/s )

x (mm)

?8?6?4?202

?1000

?500

05001000150020002500T (K ), 40× w d o t C H 4

(k g /m 3

/s )

x (mm)

(d) EK37

?8?6?4?202

?1000

?500

05001000

1500

20002500T (K ), 40× w d o t

C H

(k g /m 3

/s )

x (mm)

(e) NK206?8?6?4?202

?1000

?500

050010001500

20002500T (K ), 40× w d o t

C H

(k g /m 3

/s )

x (mm)

(f) NK1193

Fig.5Instantaneous temperature and scaled reaction rate (˙ωCH 4)distributions.The laminar distribution is shown using red dashed lines.The first and last instantaneous snapshots are shown using blue and black lines respectively

c >0.5region of the flame when comparing to EK37with the same φbut at a much smaller Ka =37.

4.2.2Flame thickness

The flame thickness,δ0.5is plotted in Fig.7a and compared with data from Refs.[40,41].The plotted data also include 1DLEM predictions of flames from Ref.[10]for which δ0.5data is unavailable.The simulated flames from Ref.[10]have a mixture composition,φ=0.7whereas those from the Refs.[40,41]span an equivalence ratio range,φ～0.6?1.0.It

?8?6?4?2

024

500

1000150020002500T (K )

x (mm)

(a) EK1

?8?6?4?2

024

500

1000150020002500T (K )

x (mm)

(b) EK4

?8?6?4?2

500

1000150020002500T (K )

x (mm)

0246810

0.2

0.40.60.81c

x (mm)

EK1EK4EK37NK206NK1193

(d) Comparison of c-distribution

Fig.6Time averaged temperature distribution in (a),(b)and (c)and progress variable distribution in (d).Data[40]is plotted with symbols

is seen that for δ0.5,the agreement between the 1DLEM predictions and data is reasonable for Ka <5beyond which deviations become significant.Additionally,the 1DLEM results based on Ref.[10]deviate sharply from the Cases EK37,ND206and https://www.doczj.com/doc/151888608.html,pared

to these three cases,the estimated turbulent diffusivity (D T =u 0/C λ)in terms of u /s 0L

(D T =(u /s 0L )s 0L 0/C λ)is four times larger for the flames from Ref.[10]due to a twice as large s 0L and 0(see Table 2).As a result,the thickening of the flames based on Ref.

[10]at c =0.5is more rapid with u /s

.Note that the 1DLEM results based on Ref.[10]10

1010

u /s 0L

δ0.5/δ0

0.5,L

(a)δ0.510

0.4

0.60.81

u /s 0L

δT i /δ0T i ,L

(b)δT

Fig.7Comparison of 1DLEM predicted flame thicknesses with data.Red square symbols represent data

from Ref.[40]in (a)and Ref.[10]in (b).The blue triangles represent cases EK x corresponding to the Yuen and G¨u lder [40]experiment and the other NK x cases.The black circles represent the cases EK x corresponding to Ref.[10]

are in agreement with EK4and EK1(that follow the setup and conditions of Ref.[40])for Ka<4due to their location near the CF-TRZ border.In this region of the regime diagram, the effects of pre-heat zone thickening is theoretically expected to commence[22].The increase inδ0.5are consistent with the discussions in Sec.4.1.

On one hand,the behavior among the1DLEM predictions ofδ0.5corresponding to the two different experiments is consistent.On the other,the experimental data corresponding to Ref.[40]shows a weakly decreasing trend with u /s0L for Ka>5.It may be noted here that while the authors in that work note a slight increase in dimensional thickness with

u /s0

L ,it is predominantly due to leaner flames used in the cases with high u /https://www.doczj.com/doc/151888608.html,pared

to the laminar unstrained value,the flames are thinner when plotted in a normalized form. Reduced thickness of flames have also been observed by Dinkelacker et al.[8]in their swirl based burner where the mean strain due to large eddies was attributed to this trend.But contrary views can be found in the literature.For instance,Sankaran et al.[27]in a study involving a slot burner determined the contribution of small eddies for flame-thickening to be a significant factor regardless of the positive mean strains consistent with the experimen-tal results of Mansour et al.[16].In the1DLEM,only the interaction of isotropic eddies on the flame front is modelled whereas large scale features related to the geometry cannot be modelled.On one hand this methodology helps in studying the influence of the small eddies on the flame front in isolation to large scale effects.But on the other,the absence of the large scales impacts the interpretation of the deviations from data.

In Fig.7b,an alternative definition of the normalized reaction zone thickness,δT

i =

(T i?T u)/(dT/dx)max is considered to facilitate comparison with data from Ref.[10].Here, T i is the inner layer temperature at which maximum temperature gradient is attained.The 1DLEM predictions related to these flames[10]as well as the numerical flames based on Ref.[40]are in agreement with each other.The trend observed in the data of a general reduc-tion inδT i with increasing u /s0L is also qualitatively captured.However,there are deviations as well.The1DLEM data points below Ka<37show a5%decrease with respect to the laminar flame value whereas,at the higher end of Ka=1193,the value decreases to40%of the laminar value.The experimental data from Ref.[10],however,decreases to less than 60%of the laminar value within just Ka<15.As discussed above,the effect of mean strain can possibly explain the larger reduction of thickness in the reaction zone,and this feature is not present in the1DLEM model.

In Table4,Cases EK1d and EK4d predict a mean c T i=(T i?T b)/(T b?T u)within98% of the laminar value of c T i.The TRZ flame with a higher Ka,Case EK37d,is still only less than10%of the laminar value.Only at larger Karlovitz numbers,the D/BRZ Cases,NK206 and NK1193are respectively,associated with17%and27%smaller values as shown in Table6.The gradual change in the location of the reaction-zone progress variable,c T i and

by inference,δT

i is consistent with expectations of enhanced eddy interactions with the inner

layer at large Ka.

The progress variable gradient distribution is plotted in Fig.8.The data plotted from Ref.[10]withφ=0.7and Ref.[36](φ=0.73)correspond to3D corrected quantities whereas the experimental flame gradients from Ref.[40]are2D.It has been shown that the 3D progress variable gradients are larger than their2D counterparts[10,38].But as shown by de Goey et al.[10]the qualitative trends are expected to be the same.

In Fig.8a,the flame surface density data from Yuen and G¨u lder[40,41],computed as the gradient of c,is plotted for their M1(φ=1.0),M9(φ=0.7)and M15(φ=0.6) flames along with the1DLEM results corresponding to these.Here we emphasize that since the1DLEM line is normal to the flame surface,the gradients are inherently3D[14].It can be observed from Fig.8a that the data from Ref.[40]shows relatively little sensitivity

0.510

1234c

d c /d s (1/m m )

c=0.5

M1 [40]M9 [40]M15 [40]EK1EK4EK37

(a) Comparison with data from Ref. [40]

for φ =0.6 ? 1.0.0

0.51

1234c=0.5c

d c /d s (1/m m )

Laminar Flame [36]EK0.98EK4.3EK12.5EK4

M9 [40]

Ka=0.59 [36]

(b) Comparison with data from Refs. [10,36] for φ ～ 0.7

Fig.8The distribution of the time-average of ?c conditioned on c

to either φor Ka .The LEM predictions,however,show sensitivity to φvia the location of the maximum gradient,and,Ka ,via the global reduction of gradients with increase in Ka .The reduction in temperature gradients with increase in Ka has been explained above in terms of turbulent diffusion.The sensitivity to φis expected due to the larger flame thickness associated with lean laminar unstretched flames relative to richer flames,and the concomitantly enhanced interactions of small eddies with the flame front.

In Fig.8b,data from Sweeney et al.[36]is plotted along with M9data [40],EK4and three flames from Ref.[10].In this subfigure,all cases correspond to φ～0.7.The CF flame

of Ref.[36]has a Ka=0.59,φ=0.73, 0=2.0mm and u /s 0L

=1.33,and has a configura-tion similar to EK0.98-D8.9.The CF data for gradients from [36]deviates from that of for the TRZ flame M9[40]and is expected,given the differences in their Karlovitz numbers.The location of the maximum gradient is around c =0.5whereas the laminar flame and CF flame data from Ref.[36]are at approximately c =0.55.The LEM predictions are slightly higher and closer to 0.6.Given the trends of the LEM gradient predictions in the TRZ,and the distribution of the experimental CF data,it appears that the gradients are overpredicted.Overprediction of Case EK4with respect to M9is still expected since,as explained above,2D gradients are in general,smaller than their 3D counterparts [10,38].

The flame thickness for Cases EK37,NK206and NK1193(cases with identical 0and φ)is shown in Table 5.Since the effect of turbulence on the reaction zone structure is one of the main interests,a definition based on the peak value of ˙ωCH 4is also used in addition to the ones discussed in Section 4.1.The c values at which the different thicknesses are defined are shown in Table 6.As in Table 4,there is a direct correlation between τstir,ηE and δmax due to turbulent stirring in the 1DLEM model.Also,the increasing value of δ0.5from EK37to NK1193correlates with the time averaged temperature distributions shown in Fig.6d.The thickness,δCH 4is close to the laminar value for flame EK37but progressively increases by 10%and 44%for flames NK206and NK1193respectively.These values indicate that the region in the flame where peak value of |˙ωCH 4|occurs,has on an average,a smaller value of the temperature gradient (i.e.,large flame thickness).

The averaged value of c corresponding to the flame thicknesses discussed above,is shown in Table 6.The value c δ=δmax progressively shifts to smaller c values indicating that the probability of finding larger temperature gradients shifts towards low tempera-ture regions of the pre-heat zone.This trend is similar to that of Table 4where within a

Table5Turbulent flame thickness computed according to various progress variable definitions.All quantities are normalized by their corresponding unstretched laminar flame values denoted by subscript,“L”

Caseδmax/δ0L,maxδ0.5/δ0.5,LδCH

4/δCH

4,L

EK370.944 1.620 1.038 NK2060.877 2.576 1.108 NK11930.559 5.982 1.445

given flame-configuration(for example EK4a-EK4d)the value of cδ=δ

max decreases with

decrease inτstir,η

E .The value of cδ=δ

CH4,L

,however,remains close to that of the laminar

value(albeit with a small increase)indicating that while the scalar gradients have decreased

(based on the discussions ofδCH

4,discussed above),the temperature at the location of

maximum consumption of CH4itself remains nearly the same.

The histogram ofδ0.5is compared with data for flame EK4in Fig.9a where the peak-location is captured accurately as the value equal to the laminar value,δ00.5.The distribution to the left of the peak location also closely matches the measurements.Since these corre-spond to temperature gradients that are larger than the laminar flame values,the ability of the triplet mapping process to stochastically model the increase in scalar gradients due to isotropic eddies is demonstrated.The under-prediction to the right of the peak location is apparent from the temperature distribution shown in Fig.6b,where at c=0.5the temper-ature gradient is larger than the laminar baseline.The1DLEM model does not include the effect of large-scale and anisotropic strain on the flame structure,hence these deviations from the data are expected.In Figs.9b-d the probability density function(PDF)ofδ0.5has been compared against a log-normal distribution fitted from the predictions.The calculated distribution appears to converge to a log-normal curve-fit for Case NK1193.

The joint PDFs of the temperature gradient and c are shown in Fig.10for some cases. These figures graphically represent the trends of Tables4,5and6.For EK1and EK4the deviation from the laminar state is small as evident in the closeness of the mean and the small variance.However,the EK37flame shows significant variance through out the flame thereby capturing the Ka number effect wherein eddies on the order of flame-thickness are able to broaden the flame.The deviations from the laminar state are even more significant for the D/BRZ flames of NK206and NK1193,although the mean value at the locations close to c=0.85(where CH4is consumed)is not noticeably affected.

4.2.3Heat release

The instantaneous global heat-release rate(HRR)integrated over the1D domain is shown in Figs.11a-f for some cases.The total HRR is computed as H RR tot=

L 0(?

k=1

f,k

˙ωk)ds where, h0f,k is the heat of formation.As mentioned in the dis-

cussion of Fig.5,Cases EK1.4and EK4differ markedly in the instantaneous realizations

Table6Progress variable values corresponding to the flame thicknesses computed in Table5.The corresponding unstretched laminar flame values are given in parentheses Case cδ=δ0(cδ=δ0

)cδ=δ

CH4

(cδ=δ

CH4,L

) EK370.60(0.65)0.84(0.83)

NK2060.54(0.65)0.85(0.83)

NK11930.48(0.65)0.85(0.83)

九年级课文翻译完整版

九年级课文翻译 Document serial number【NL89WT-NY98YT-NC8CB-NNUUT-NUT108】

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九年级英语u n i t课文翻译 Document serial number【KK89K-LLS98YT-SS8CB-SSUT-SST108】

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155******** 一单元 SECTION A 1a 我通过制作抽认卡来学习。通过和朋友一起学习。通过听磁带。通过做抽认卡。通过向老师求助。通过读课本。通过制作单词本。 1c A：你怎么为考试而学习。B：我通过参加学习小组来学习。 2a 1、你是通过看英文录像学英语的吗？ 2、你曾和朋友们练习过对话吗？ 3、听磁带怎么样？ 4、大声朗读以练习发音怎么样？5、我曾经通过参加学习小组的方式学习过吗？ 2b A是的，我通过那种学习方式学到了很多。B、哦，是的，它提高了我说英语的能力。C、有时那样做。我觉得他有用。D、不。（通过看英语录像学习）太难了，无法理解录像中的人所说的话。 2c A你曾经通过参加学习小组来学习吗？B、是的，我参加赤字，通过那种方式我学到了很多。 Grammer Focus 你怎么为准备一场考试而学习？我靠听磁带。你怎样学习英语？我通过参加学习小组来学习。你通过大声朗读来学习英语吗？是的，我是。你曾和朋友们练习过对话吗？哦，是的，他提高了我说英语的能力。你曾经通过参加学习？小组来学习吗？是的，我参加过。通过那种方式我学习到了很多。 3a如何才能学得最好这星期我们询问了新星高中的同学关于学习更多英语的最佳方法的问题。许多同学说他们通过使用英语为学习它，一些还有很特别的建议。比如，李莉莲说学习新单词的最好的方法是阅读英语杂志。她说记忆浒音乐的歌词也有一些作用。当我们问及学习语法的问题时，她说：“我从不学习语法。它太枯燥了。” 魏明有不同的看法。他学习英语已经6年了，并且确实喜欢英语。他认为学习语法是学习一门语言的一种好方法。他还认为观看英语电影也不错，国灰他可以看到演员说话的情形。但是，有时候他发现看英语电影是件很头痛的事情，因为那些演员说话太快了。刘畅说加入学校英语俱乐部是提高英语最好的方法。学生有很多练习的机会并且他们也有很多乐趣。她补充说和朋友练习会话一点用处也没有。“我们会因为某件事变得很激动，最后用汉语来讲，”她说。 3b A：我正在作一个关于学习英语的调查。我能问你一些问题吗？B：当然。A：太好了！你叫什么名字？B：魏明。A：那么你是怎样学习英语的，魏明？B：…… 4 A：你列词汇表吗？B：噢，是的。我常那样做。SECTION B 1a我不会发其中一些单词的音。我不会拼写一些英语单词。我听不懂英语口语。我在语法上犯错误。我读得很慢。1b我不知道怎么使用逗号。2a 1、不能正确发音。 2、忘记很多生词。 3、人们和我说话时我不能每次都听懂。 4、不能理解杂志中的单词。 5、没有获得很多写作训练。 2B A、你可以一直将生词写在你的笔记本里，并在家学习它们。B、你应该找一个笔友。C、听力能起作用。D、为什么不加入一个英语俱乐部来练习说英语呢？ 2C A：我没有搭档来练习英语。B、也许我应该加入一个英语俱乐部。 3a我是怎样学习英语的去年英语课对我来说很难。首先，对我来说听懂老师说话很难。开始，她说的太快，我不能听懂每个单词，后来，我意识到如果你听不懂每个单词并没有关系。而且我害怕在班上说话，因为我认为同学们可能会嘲笑我。我也不是总能造出完整的句子。然后我开始看英文电视。那很有用。我认为做大量听力练习是成为一个好的语言学习者的秘决之一。另一件我觉得很难的事是英语语法。所以我决定在每节课上记大量语法要点。然后我开始用我正在学的语法自己写新句子。这样作用处之大令人惊奇。现在我很喜欢学英语并且这学期我得了个A。我的老师对我印象很深。作者觉学英语很难是因为……1、老师发音差。2、她说话时人们总是嘲笑她。3、她在造完整的句子方面有困难。4、英语语法很难。当她开始…她的英语提高了。5、和说英语的朋友一起出去。6、大量的听力练习。7、在自己组织的句子里使用语法。 3b 亲爱的，我知道学英语不容易，但我有一些想法可能有用。你说你不能理解说话太快的人。那么，你可以尽量听最重要的单词，而不是每个单词。 4 1、关于学英语什么不容易。2、就这一点你作了什么？ 3、你最喜欢的学习更多英语的方式是什么？韩文说如果人们语速太快听力有时就很难。 SELF CHECK 1 你应该在词汇表中写下新的英文单词。2、如果你不知道怎样拼写生词，就查词典。3、最好的提高你的英语（水平）的方法是加入英语俱乐部。4、另一件他觉得很困难的事是英语语法。5、这种纸摸上去非常柔软。 2 READING Section 2使用词典词典是有用的学习工具，但许多英语单词有不同的含义和用法。我们需要确定我们从词典中找到的含义与语境匹配。我们该怎样解决我们的烦恼？无论贫富、老少，我们都有烦恼。并且除非我们解决了问题，否则我们会轻易变得不开心。为我们的问题担忧会影响我们在学校的表现。它也会影响我们同家人相处的方式。所以我们该怎么解决我们的烦恼呢？有许多方法。它不是烦恼----它是挑战。享受面对它（的过程）。通过学会忘记

常用翻译技巧和方法

常用翻译方法和技巧 1. 四种翻译方法 1.1直译和意译所谓直译，就是在译文语言条件许可时，在译文中既保持原文的内容，又保持原文的形式——特别指保持原文的比喻、形象和民族、地方色彩等。每一个民族语言都有它自己的词汇、句法结构和表达方法。当原文的思想内容与译文的表达形式有矛盾不宜采用直译法处理时，就应采用意译法。意译要求译文能正确表达原文的内容，但可以不拘泥与原文的形式。（张培基）应当指出，在再能确切的表达原作思想内容和不违背译文语言规范的条件下，直译有其可取之处，一方面有助于保存原著的格调，另一方面可以进新鲜的表达方法。 Literal translation refers to an adequate representation of the original. When the original coincides or almost tallies with the Chinese language in the sequence of vocabulary, in grammatical structure and rhetorical device, literal translation must be used. Free translation is also called liberal translation, which does not adhere strictly to the form or word order of the original.（郭著章）直译法是指在不违背英语文化的前提下，在英译文中完全保留汉语词语的指称意义，求得内容和形式相符的方法。意译是指译者在受到译语社会文化差异的局限时，不得不舍弃原文的字面意义，以求疑问与原文的内容相符和主要语言功能的相似。（陈宏薇）简单地说，直译指在译文中采用原作的的表达方法，句子结构与原句相似，但也不排除在短语层次进行某些调整。意译指在译文中舍弃原作的表达方法，另觅同意等效的表达方法，或指对原作的句子结构进行较大的变化或调整。（杨莉藜） Literal translation may be defined as having the following characteristics: 1, literal translation takes sentences as its basic units and the whole text into consideration at the same time in the course of translating. 2, literal translation strives to reproduce both the ideological content and style of the entire literary work and retain as much as possible the figures of speech and such main sentence structures or patterns as SV,SVO, SVC, SVA, SVOO, SVOC, SVOA formulated by Randolph Quirk, one of the authors of the book A Comprehensive Grammar of the English Language. Free translation may be defined as a supplementary means to mainly convey the meaning and spirit of the original without trying to reproduce its sentence patterns or figures of speech. And it is adopted only when and where it is really impossible for translators to do literal translation. (Liu Zhongde). 练习： 1. He walked at the head of the funeral procession, and every now and then wiped his crocodile tears with a big handkerchief. 他走在送葬队伍的前头，还不时用一条大手绢擦一擦他的鳄鱼泪。 2. It’s an ill wind that blows nobody good. 对有些人有害的事情可能对另一些人有利。 3. Every dog has his day. 人人皆有得意日。

九年级英语课文翻译

年度九年级英语课文翻译

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第一课 One day in ancient Greece, King Hiero asked a crown maker to make him a golden crown. At first, he was very happy with it. 古希腊的一天,海尔罗国王请一位制作皇冠的人给他制作一顶金皇冠.起初,他对皇冠感到非常高兴. “It’s a nice crown, isn’t it?”he asked his men. Later, however, he began to doubt that it was a real golden crown. “Is it made completely of gold?”he wondered. He sent it to Archimedes and asked him to find out the truth. “这是一个很漂亮的皇冠,不是吗?” 他问仆人.然而后来他开始怀疑皇冠

是否是一顶真的金皇冠.“这真是纯金的吗?”他纳闷着.他把皇冠送到了阿基米德那里,叫他搞清楚真相. “This problem seems difficult to solve. What should I do?”thought Archimedes. “这个问题似乎很难解决.我该怎么办?”阿基米德想. Archimedes was still thinking about this problem as he filled his bath with water. When he got into the bath, some water ran over. 当阿基米德往浴池里倒水的时候他仍然思考着这个难题.当他进入浴池时,一些水溢了出来. “That’s it!”shouted Archimedes.