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Combustion and Flame 137 (2004) 129–147www.elsevier.com/locate/jnlabr/cn

Unsteady effects of strain rate and curvature on turbulpremixed flames in an inflow–outflow configuration

Nilanjan Chakraborty and Stewart Cant∗

CFD Laboratory, Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK

Received 2 February 2003; received in revised form 20 July 2003; accepted 30 January 2004

Abstract

Three-dimensional direct numerical simulation (DNS) studies of premixed turbulent flames have beenout using an inflow–outflow configuration at moderate Reynolds number(Re) and with single-step Arrheniuchemistry in the thin reaction zones regime. The compressible Navier–Stokes equations are solved togea transport equation for a reaction progress variable(c). Results are obtained for several quantities of interincluding the gradient magnitude of the progress variable(|∇c|) and the displacement speed(Sd) of each progresvariable isosurface. The probability density function (pdf) of displacement speed and the implications ofshape are discussed in terms of the relative magnitude of reaction rate and molecular diffusion effects. TSd itself, as well as the pdfs of its different componentsSr, Sn, andSt, in the present three-dimensional simulatiois found to be consistent with previous results based on two-dimensional DNS with detailed chemistry. Theof the assumption(ρSd)s ≈ ρ0SL is assessed on averagingρSd over the isosurfaces ofc across the flame brushThe unsteady effects of tangential strain rate(aT) and curvature(κm) on flame propagation are also considerCurvature and displacement speed are found to be negatively correlated, while the conditional pdf of tastrain rate and displacement speed at zero curvature locations also shows a negative correlation, againwith previous two-dimensional detailed-chemistry DNS. 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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1. Introduction

Premixed combustion is widely used in a rangeengineering devices such as gas turbines and spanition (SI) engines. Recent requirements for pollutcontrol have increased the importance of premicombustion, most notably in gas turbine applicatioand have led to the development of lean premiprevaporized (LPP) combustion technology. In mpractical applications, premixed combustion taplace in a turbulent environment. Recent increain computer capacity have made it possible todirect numerical simulation (DNS) to simulate a tubulent flame without modeling and to extract use

* Corresponding author.E-mail address:[email protected] (S. Cant).

0010-2180/$ – see front matter 2004 The Combustion Institutdoi:10.1016/j.combustflame.2004.01.007

information which can be used to support modelfor both Reynolds averaged Navier–Stokes (RANand large eddy simulation (LES). The high comptational cost limits the applicability of DNS to smaor moderate Reynolds numbers(Re), and DNS re-mains too expensive to simulate a three-dimensioturbulent flame taking full chemistry into accounTwo-dimensional turbulence is physically differefrom three-dimensional turbulence, and hencepresent study is focused on three-dimensional Dfor a premixed flame with simple chemistry. Maprevious premixed combustion DNS studies hbeen carried out either on a periodic domain oring nonreflecting outlet boundary conditions (e.g.,2]). In reality, many combustion devices have at leone inlet and one outlet. Premixed combustion Dstudies using an inflow–outflow configuration habeen carried out by Haworth and Poinsot [3] and

e. Published by Elsevier Inc. All rights reserved.

http://www.elsevier.com/locate/jnlabr/cnf

130 N. Chakraborty, S. Cant / Combustion and Flame 137 (2004) 129–147

Nomenclature

aT Flame tangential strain ratea0 Reference acoustic speedc Reaction progress variablec∗ Progress variable value defining flame

surfaceCP0 Reference specific heat at constant

pressureCV0 Reference specific heat at constant

volumeD Progress variable diffusivityD0 Reference progress variable diffusivityDa Damköhler numberE Nondimensional internal energyEa Activation energyG Field equation variableH Heat of the reaction per unit mass of

mixturek Turbulent kinetic energyk0 Initial turbulent kinetic energyKa Karlovitz numberl0 Reference length scaleL Length of the computational domainLe Lewis numberMa Mach numberN Local flame normal vectorP Nondimensional pressureP0 Reference-dimensional pressurePr Prandtl numberq Heat fluxR Specific gas constantR0 Universal gas constantRe Reynolds numberRet Turbulent Reynolds numberSc Schmidt numberSd Local flame displacement speedSL Laminar flame speedSn Normal diffusion component of

displacement speedSr Reaction component of displacement

speedSt Tangential diffusion component of

displacement speedST Turbulent flame speedt Nondimensional timetF Chemical time scalet0 Reference time scaleT Nondimensional temperatureTad Adiabatic flame temperatureT0 Reactant temperatureT Instantaneous-dimensional temperatureu Nondimensional fluid velocity vector

ui ith component of nondimensional fluidvelocity

u0 Reference velocity scaleu′ Nondimensional initial root mean square

fluctuation velocityu∗ Nondimensional fluctuating velocity

vector at inlet planeu∗i ith component of nondimensional

fluctuating velocity at inlet planeU Velocity vector at inlet planeUb Nondimensional mean inflow velocityUi ith component of nondimensional fluid

velocity at inlet planeνη Kolmogorov velocity scalew Chemical reaction ratex, y, z Cartesian coordinatesxi ith Cartesian coordinateYp Product mass fractionYp0 Product mass fraction in fresh gasesYP∞ Product mass fraction in fully burned

gases

Greek letters

α Heat release factorβ Zel’dovich numberγ Ratio of specific heatsδL Laminar flame thicknessε Dissipation rate of turbulent kinetic

energyη Kolmogorov length scaleκm Mean curvature of the flameλ Thermal conductivityλ0 Reference thermal conductivityµ Dynamic viscosityµ0 Reference dynamic viscosityν Kinematic viscosityρ Densityρ0 Reactant density, reference densityσ Surface density function (SDF)Σ Flame surface density (FSD)Σgen Generalized flame surface densityΣ ′ Fine-grained flame surface densityτ Heat release parameterτij Viscous stressτη Kolmogorov eddy turnover time

Averaging and filtering

(· · ·) Reynolds averaging or LES filteringoperation

(· · ·)s LES surface averaging operation

N. Chakraborty, S. Cant / Combustion and Flame 137 (2004) 129–147 131

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Trouvé and Poinsot [4], but a turbulent inlet flow fiewas not specified. Zhang and Rutland [5] carriedpremixed flame DNS using an inflow–outflow cofiguration with a turbulent inlet in an incompressibflow field. Even with very small Mach numbers(Ma),the reacting gas flow is affected by significant dilation as a result of heat release, and compressibeffects allow for coupled interaction of the flame, tubulence, and acoustic waves. In the present studyinflow–outflow configuration for compressible susonic combusting flows is implemented with a vieto studying the full physics of the problem.

In fully premixed flames, the composition of threacting mixture may be described using a singleaction progress variablec defined as zero in fresreactants and unity in fully burned products. Withthe flame it is possible to identify a given isosurfaceprogress variable defined byc= c∗. This is not a ma-terial surface; it moves with a displacement speedSddetermined by the reactive–diffusive characterisof the flow. The displacement speed is an importquantity in models based on the flame surface den(FSD) [6] andG equation [7] approaches. It can bseen from the unclosed transport equation of the figrained FSD(Σ ′) [8], as well as from modeled equations for the FSD [6,8], thatSd appears in the flamcurvature and propagation terms. FurthermoreSd is astrong function of curvature, so the interdependebetweenSd and the propagation and curvature terplays an important role in FSD transport. Usually,Sdis modeled based on the assumption(ρSd)s ≈ ρ0SLwhere the laminar flame speedSL is either taken tobe constant or else has a prescribed strain and cuture dependence [7,9].

The approximate relation betweenSd and SLshown above is a corollary of a one-dimensional mbalance made under the flamelet assumption whenflame is in a quasi-steady state. In unsteady flaproblems the reacting flow field can be predicmore accurately if the transient behavior of the flais known in detail. Based on an order of magnituanalysis of theG equation, Peters [7] has shown thin the corrugated flamelet regime the unstrained lanar flame speed is greater than the Kolmogorov veity νη and hence quasi-laminar propagation effectsdominant, while in the case of the thin reaction zoregime curvature effects are more important. For lainar flames subjected to small stretch rates, curvaaffects the propagation speed through the Marksnumber [7], and more generally in unsteady flamthe stretch directly affectsSd. In turn, the stretchrate is dependent on strain, curvature, andSd, andit is known [8] that stretch is of key importancethe evolution of flame surface area. The link betwestretch andSd has been investigated by a numberauthors using two-dimensional DNS with compl

chemistry. Chen and Im [10] presented the effectflame stretch on the displacement speedSd based on astudy of CH4/air flames, while Echekki and Chen [have shown curvature dependence and secondarfects of strain rate onSd. Peters et al. [11] carried oua study for stoichiometric and lean CH4/air flames inthe thin reaction zones regime in which the effectscurvature and strain rate on the reaction compon(Sr), normal diffusion component(Sn), and tangen-tial diffusion component(St) of displacement speeSd were investigated.

In the present work, three-dimensional premixturbulent flame DNS with simple chemistry is usedextend the previous results. An initially planar ondimensional laminar flame is subjected to a turbulflow field, and under the effect of turbulence the flabecomes stretched and wrinkled. The principal aof the study are to establish the validity in threemensions of the conclusions that have been reacusing previous two-dimensional DNS, and to obtnew information on the behavior ofSd and its compo-nents in the thin reaction zone regime. The study aserves to introduce the present inflow–outflow conuration, to explain its advantages and potential forfuture, and to outline the issues concerned withimplementation of the turbulent subsonic compreible inflow condition. Results have been obtainedall major quantities of interest including strain raflame curvature, and displacement speed, and agment with previous work is good. In particular, thcorrelations betweenSd and κm are found to agreewith those of Echekki and Chen [1], while the proability density function (pdf) ofSd and its differentcomponents are found to be similar to those obtaiby Hilka [12] and by Peters et al. [11].

The remainder of the paper is organized aslows. Section 2 describes the mathematical baground and the formulation for the models of intest. The numerical implementation and the inflooutflow configuration are described in Section 3. Rsults and subsequent discussion are presented intion 4.

2. Mathematical background

To allow for three-dimensional turbulent flamsimulations within the limitations of available computer capacity, a single-step irreversible chemicalaction is assumed:

(1)reactants→ products.

A reaction progress variable is defined based onmass fraction of the product species,

(2)c= Yp − Yp0

Y − Y,

P∝ p0


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n-tingbe

d in

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ith

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ace

and the Arrhenius reaction rate for one-step chemitakes the following form in terms of the progress vaable:

(3)w = Bρ(1− c)exp

[−Ea

R0T

].

HereB is the preexponential factor andEa is activa-tion energy. The diffusion velocity appearing in tprogress variable balance equation can be assumobey Fick’s law of diffusion,

(4)ρVckc= −ρD ∂c

∂xk,

whereD is the mass diffusion coefficient, whichgeneral is a function of thermochemical variables.

With these assumptions the full set of nondimesional governing equations for compressible reacflow with single-step Arrhenius chemistry maystated in Cartesian tensor notation asmass continuity,

(5a)∂ρ

∂t+ ∂(ρuk)

∂xk= 0;

momentum in theith direction,

(5b)∂(ρui )

∂t+ ∂(ρukui)

∂xk= − ∂P

∂xk+ 1

Re

∂(τki )

∂xk;

energy equation,

∂(ρE)

∂t+ ∂(ρukE)

∂xk

= −(γ − 1)Ma2 ∂(Puk)

∂xk

+ 1

Re(γ − 1)Ma2 ∂(τkiui )

∂xk

+ τ

RePr

∂

∂xk

[λ∂T

∂xk

]

(5c)− τ

ReSc

∂

∂xk

[ρD

∂c

∂xk

];

balance equation for reaction progress variable,

(5d)

∂(ρc)

∂t+ ∂(ρukc)

∂xk= w+ 1

ReSc

∂

∂xk

[ρD

∂c

∂xk

].

The nondimensional numbers and parameters usethe above equations are

Re= ρ0u0l0

µ0, Pr = µ0CP0

λ0,

(6)Sc= µ0

ρ0D0, Ma = u0

a0,

whereRe, Pr, Sc, andMa are the Reynolds numbePrandtl number, Schmidt number, and Mach numrespectively, anda0 = √

γRT0 is the acoustic speed

The ratio of specific heatsγ , the heat release parmetersτ and α, the Zel’dovich numberβ, and thepreexponential factorB∗ are

γ = CP0

CV0

, τ = α

(1− α)= (Tad− T0)

T0,

(7)β = Ea(Tad− T0)

R0T 2ad

, B∗ = B

ρ0u0exp

[−β

α

].

The nondimensional temperatureT is given by

(8)T = T − T0

Tad− T0,

whereT denotes the instantaneous-dimensional vaT0 is the initial temperature andTad is adiabatic flametemperature, given byTad= T0 +H/CP0. This formmakes nondimensional temperature lie betweenand unity for adiabatic heating. The viscous strtensorτij and reaction ratew are

(9)τij = µ

[∂ui

∂xj+ ∂uj

∂xi

]− 2

3δij

[∂uk

∂xk

],

(10)w= B∗ρ(1− c)exp

[− β(1− T )

1− α(1− T )

],

while the nondimensional equations of state are giby

(11)P = 1

γMa2ρ(1+ τT ),

(12)

E = 1

γ(1+ τT )+ 1

2(γ − 1)Ma2ukuk + τ(1− c).

The equations are nondimensionalized using a sestandard values of the principal variables, namely,u0,l0, t0 = l0/u0, ρ0, andT0, whereu0 is taken to beequal to the unstrained laminar planar flame speedSL ,andl0 is taken to be equal to the domain sizeL. In ad-dition, standard and constant values of the transcoefficientsµ0, λ0, ρ0D0, andCV0 are chosen. Thenormalizing pressureP0 is chosen to be representtive of dynamic effects using the relationP0 = ρ0u

20,

while the internal energy is nondimensionalized wrespect toCP0T0.

For a single isosurface denoted byc = c∗, theprogress variable balance equation can be writtethe form [1]

(13)

[∂c

∂t+ uk

∂c

∂xk

]c=c∗

= Sd|∇c|c=c∗ ,

whereSd is the displacement speed of the isosurfat c∗ given by

(14a)Sd =w|c=c∗ + ∂

∂xk[ρD ∂c

∂xk]c=c∗

ρc=c∗√

∂c ∂c |c=c∗.

∂xk ∂xk


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It is possible to split the molecular diffusion term acording to

∂

∂xk

[ρD

∂c

∂xk

]c=c∗

=Ni∂

∂xi

[ρDNj

∂c

∂xj

]c=c∗

(14b)− ρD

√∂c

∂xk

∂c

∂xk

∣∣∣∣c=c∗

∂Ni

∂xi,

where the first and second terms on the right-hside correspond to normal and tangential diffusion,spectively, and theith component of the flame normN is given by

(15)Ni = −∂c∂xi

|c=c∗√∂c∂xk

∂c∂xk

|c=c∗.

It should be noted that in this convention the flanormal points towards fresh reactants. FollowEqs. (14a) and (14b), one can rewrite the displament speedSd as

(16a)Sd = Sr + Sn + St,

whereSr is the reaction rate component given by

(16b)Sr = w|c=c∗ρc=c∗

√∂c∂xk

∂c∂xk

|c=c∗,

Sn is the normal diffusion component

(16c)Sn =Nk

∂∂xk

[ρDNj ∂c∂xj

]c=c∗ρc=c∗

√∂c∂xk

∂c∂xk

|c=c∗,

andSt is the tangential diffusion component

(16d)St = −2Dκm,

whereκm is the local mean curvature of thec = c∗isosurface given by

(16e)κm = 1

2

∂Ni

∂xi

∣∣∣∣c=c∗

.

In this convention, positive curvature is convexwards the reactants. It is also important to note frEqs. (14a), (14b), and (16a)–(16d) thatSd is governedby the local balance between the reaction rate anddiffusion rate, which together will determine the manitude and direction of the displacement speed.

The transport equation for the gradient magnituof progress variable(|∇c|) at the isosurfacec= c∗ isgiven by

∂(|∇c|)∂t

+ ∂(uk |∇c|)∂xk

+ ∂(NkSd|∇c|)∂xk

(17)= (δij −NiNj )∂ui

∂x|∇c| + Sd

∂Ni

∂x|∇c|.

j i

It is useful to compare the above equation withunclosed transport equation for the fine-grained Ftransport equation. At an isosurface the above eqtion is identical to the fine-grained FSD transpequation [8]:

∂Σ ′∂t

+ ∂(ukΣ′)

∂xk+ ∂(NkSdΣ

′)∂xk

(18)= (δij −NiNj )∂ui

∂xjΣ ′ + Sd

∂Ni

∂xiΣ ′.

Boger et al. [13] introduced the concept of a geneized flame surface density,Σgen, that does not depenon a particular choice of isosurface for the flame.the context of LES filtering the generalized flame sface density is given by [13]

(19)Σgen= |∇c|.In the limit of zero filter size(2 → 0) the flow be-comes fully resolved and under this condition tgeneralized FSD becomes equal to the surfacesity function (SDF) as defined by Kollmann and Ch[14]. Thus, for the purposes of DNS, the SDF(σ =|∇c|) can be viewed as equivalent to the generaliflame surface density. For reasons of numerical cvenience the transport equation (17) is carried alwith the simulation to extract fully resolved timedependent statistical information relevant to modelof the SDF. It should be stressed that the extra eqtion is used purely to simplify postprocessing andnot coupled back to the simulation.

3. Numerical implementation

For the present series of simulations the flais initialized using a one-dimensional planar laminflame solution. The numerical resolution is governby the flame resolution, with about 10 grid points usto resolve the laminar flame thicknessδL , as definedby

(20)δL = 1

Max | ∂c∂x

| ,

wherex is the direction of propagation of the lamnar flame. The parameterB∗ is adjusted to producevalue ofSL equal to unity. An isotropic homogeneoincompressible divergence-free initial turbulent vlocity field is generated using a spectral method [and corresponds to the Batchelor–Townsend enspectrum [16] which is realizable in decaying grgenerated turbulence. The compressible DNS cSENGA was used in which all first- and seconorder spatial derivatives are discretized using a 1order explicit central difference scheme [2]. Bounary points are treated with explicit one-sided fin


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Fig. 1. Description of the computational domain (shoelongated in thex direction).

differences of decreasing order down to 4th ordethe boundary. Time stepping is carried out usingexplicit 3rd-order low-storage Runge–Kutta meth[17]. The computational domain (shown schemacally in Fig. 1) is a cube, with the left face in thx direction taken as the inlet, and the right facethe same direction taken as the outlet. The domataken to be periodic in they andz directions. Bound-ary conditions for the inlet and outlet faces are impmented using the Navier–Stokes characteristic bouary condition (NSCBC) approach [18]. A standanonreflecting outflow condition is used with adjument to match the specified outlet pressure.

At the inlet, the density and all three compnents of a turbulent velocity field are imposed usthe local one-dimensional invisid (LODI) formulatio[19] within the NSCBC framework. The charactertic wave amplitude variations at the inlet boundaryset according to the prescribed inlet values of the tderivatives of density, progress variable, and all thcomponents of velocity. In this formulation, acousenergy from inside the computational domain isflected back into the domain at the inlet boundary.reduce the level of acoustic activity the inlet velocfield is made acoustically inert (incompressible). Cis taken to ensure that the flame remains sufficiefar from the inlet boundary throughout the simulatiso that the inlet values of the progress variable anddensity can be assumed to be independent of time

The success of the turbulent inlet boundary contion depends on specifying a turbulent velocity fiewhich is evolving with time at the inlet in a reasonabrealistic manner. This ensures that the simulationbegin to generate a truly turbulent solution in as sha distance from the inlet as possible. Akselvoll aMoin [20] specified an inlet turbulent velocity fielby using a precursor simulation of a simpler cold floproblem, but this approach is computationally expsive. Random perturbation of the inflow velocitysometimes used for natural shear flows [21], but

leads to excessively rapid decay where natural sis absent. In the present case, a three-dimensirandom velocity field with periodic boundaries oall sides is generated in Fourier space, usingBatchelor–Townsend energy spectrum [16] andisfying the continuity constraint for incompressibflow. The result is then Fourier-transformed to phical space on a single plane and applied to the iface of the domain. The approach is similar to thatscribed for DNS by Lee et al. [22] and subsequenused for combustion LES [6,23,24]. The specifiedlocity field is assumed to be frozen in time, andfluctuating part of the inflow velocity is extracted bscanning a plane through the field as suggestedZhang and Rutland [5]. The scanning plane is dplaced by an amount given byUc2t in the negativexdirection, whereUc is an appropriate convection vlocity in the positivex direction and2t is the timeelapsed fromt = 0. The time derivatives used for thNSCBC inflow boundary condition are then given

(21)∂u∗

i

∂t≈ −Uc

∂u∗i

∂x,

where u∗ is the fluctuating velocity vector on thscanning plane. The fluctuating part of the inflow vlocity vector is given by

(22)U (y, z, t)−Umeane1 = u∗(L−Uc2t,y, z),

whereU(y, z, t) = u(0, y, z, t) andUmeanis the de-sired mean velocity in thex direction at the inletIt remains to establish a relationship betweenUmeanandUc, where it is expected thatUmean will be onthe order of the laminar flame propagation speed,is likely to be smaller than the rms fluctuation manitude u′

inlet of the inlet velocity. It is important tonote that Taylor’s hypothesis is unlikely to be vain these circumstances. Indeed, Lee et al. [22] uUc =Umeanand showed for a highu′

inlet/UmeanthatTaylor’s hypothesis does begin to fail. A pragmaapproach suggested by Wille [23] for combustiLES is to takeUc = u′

inlet, which for a given time in-tervalδt allows a notional turbulent eddy to move ba distanceUcδt while simultaneously turning over ba distanceu′

inletδt , and hence to mimic some featurof an evolving turbulent field. A simple extension [is to takeUc = Umean+ u′

inlet, which is able to re-vert to the original form of Taylor’s hypothesis fosmall values ofu′

inlet/Umean. Inlet turbulence generated using this approach is very much more accuthan random perturbations while avoiding the comtational expense of a full precursor simulation. Giva short distance within the computational domainwhich to develop further, the resulting turbulenceentirely satisfactory for the present purpose. Resdemonstrating the validity of the approach are psented below.


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4. Results and discussion

Before presenting the results of the simulatioit is useful to estimate the Damköhler and Karlovnumbers so that the combustion regime can be idefied. Here, DNS is carried out for a flame with modate heat release(τ = 3) in a domain of 96× 96× 96points. The values of the physical and numerical pameters for the simulations are given in Tables 1 anrespectively. In nondimensional units, the initial itegral length scale(l) is about 0.25 and the initiaturbulence intensity is given byu′/SL = 7.1907. TheReynolds number based on the laminar flame spand the domain size isReSL = SLL/ν = 25, whereL = 1 andSL = 1.0. Then the turbulence ReynoldnumberRet ≈ 45 andl/δL ≈ 2.4. The Kolmogorovtime scaletη is given by 5.182 × 10−3l0/u0 andthe flame time can be estimated bytF = D/S2

L. Forthe present case withLe = 1.0 and Sc= 0.7, tF ≈D/S2

L = 0.057l0/u0, and so the Karlovitz number igiven by

(23)Ka= tF

tη≈ 11,

and from the definition of the Damköhler number itfound that

(24)Da = (l/u′)tF

≈ 0.61.

Based on these values, it can be concluded [8]combustion is taking place in the thin reaction zon

Table 1Nondimensional parameters used for the simulation

Physical property Value

Heat release parameter,τ 3.0Zel’dovich number,β 6.0Reynolds number,Re 25.0Mach number,Ma 0.014159Prandtl number,Pr 0.7Schmidt number,Sc 0.7Lewis number,Le 1.0α = τ/(1+ τ )= (Tad− T0)/Tad 0.75

Table 2Numerical parameters used for the simulationa

Numerical parameter Value

Grid spacing,2x 1.053× 10−2

Time step,2t 5× 10−6

Kolmogorov scale,η 1.44× 10−2

Kolmogorov eddy turnover time,tη 5.182× 10−3

Initial turbulent intensity,u′/SL 7.1907

a All numerical parameters are nondimensionalized wrespect to appropriate reference quantities.

regime, and that for the present values ofu′/SL andl/δL the flamelet assumption remains valid [4,25].

In the present study an initially planar laminflame is allowed to interact with the turbulent enviroment for just over two initial integral eddy turnovtimes. It is recognized that this simulation time spremains fairly small, but previous DNS studieswhich the simulations were run for the same or esmaller times have provided useful informationturbulent combustion (e.g., [1,11]). Simulations werun on a PC having dual 2.4 GHz Athlon processand 2 Gb of memory, and took about 28 days of Ctime. After one integral eddy turnover time, the flamhas adjusted itself reasonably well to the turbulentvironment, and after 1.5 integral turnover times gloparameters such as turbulent kinetic energy andbulent flame speed are no longer changing rapiFig. 2a shows the temporal evolution of the turbulkinetic energyk which is found to be decaying wittime. The decay ofk with respect to its initial valuek0 is found to be consistent with the results of Let al. [22]. The corresponding spatial profile ofk rel-ative to the inlet valuekinlet is shown in Fig. 2b atime 2τf0 for the domain ahead of the flame. The stial decay ofk with distance from the inlet followspower law with an exponent of−1.35, in good agreement with classic experimental data [26]. The timevolution of the variance of vorticity(w′

iw′i ) and the

variance of dilatation(θ ′θ ′) are presented in Figs. 2and 2d, respectively. The time evolution ofw′

iw′i is

found to be qualitatively consistent with the findinof Lee et al. [22]. The decay ofw′

iw′i

in Fig. 2cis indicative of the temporal evolution of the turblent kinetic energy dissipation rate, and comparingmagnitudes ofw′

iw′i andθ ′θ ′, it can be seen that th

ratio θ ′θ ′/w′iw

′i remains small throughout the sim

lation. This indicates that the turbulence energy ininertial scales is much larger than that in the acouscales. These results provide valuable evidencethe present formulation for inlet turbulence is safactory.

Clearly a major advantage of the inflow–outfloconfiguration lies in the potential to achieve statical stationarity and, hence, to collect statistics baon long-time averages. In practice a limit is set bytendency of the flame to broaden and ultimatelycape from the domain. This can be overcome throthe use of larger domain sizes and by careful ctrol of the inlet mean velocity, and this will form thbasis of a future study. It remains a matter forbate whether it is possible to attain a genuinely steflame in this configuration, but it is certainly possibto attain a quasi-stationary state as shown by Zhand Rutland [5]. Certainly the present results (e


bulent

nce

(a) (b)

(c) (d)

Fig. 2. Evolution of turbulence quantities. (a) Temporal evolution of turbulent kinetic energy normalized by initial turkinetic energy(k/k0). (b) Spatial variation of turbulent kinetic energy normalized by inlet turbulent kinetic energy (k/kinlet).

(c) Variance of vorticityw′iw

′i . (d) Variance of dilatationθ ′θ ′. Vorticity and dilatation are normalized by appropriate refere

quantities. Simulation time is normalized by the initial integral eddy turnover timeτf0 .

hing

leof

erseanme

sloc-theed

ayfor

msionpen-ture.-en-e

but0.8es

Fig. 2a) suggest that the turbulence may be reaca quasi-stationary state.

Fig. 3a shows the initial progress variable profiof the laminar flame as compared with the profilethe turbulent flame brush averaged in the transvdirections. The averaged turbulent profile showsincrease in the flame brush thickness due to flawrinkling. Fig. 3b is a snapshot of the flame in thex–z plane aty = 0.5, in which contours of the progresvariable are presented along with the projected veity vectors. From Figs. 3a and 3b it is evident thatinitially flat flame has become curved and stretch

under the influence of the turbulence. In Fig. 3b it mbe observed that the progress variable contoursc � 0.5 are not parallel to each other, which confirthat combustion is taking place in the thin reactzones regime where turbulent eddies are able toetrate the preheat zone and distort the flame struc

The turbulent flame speedST, evaluated by integration of the instantaneous reaction rate over thetire domain, is shown in Fig. 4a. The turbulent flamspeed increases from the initial laminar value,the rate of rise slows down markedly after abouteddy turnover time, and after 2 eddy turnover tim


ro-e.

asi-ent

meluesssainstnotom-

utade

S.

nt

anr-

tedof

sgThegerl-

me,

(a)

(b)

Fig. 3. (a) Comparison of reaction progress variable pfiles in the initial laminar and subsequent turbulent flam(b) Contours of reaction progress variable in thex–z planeaty = 0.5. Velocity vectors projected onto thex–z plane aresuperimposed.

there is little further change, suggesting that a qustationary state is at least possible. The displacemspeedSd and mass weighted displacement speedρSdvary from onec isosurface to another across the flabrush, and this can be seen in Fig. 4b, where vaof Sd/SL and ρSd/ρ0SL ensemble-averaged acroeach progress variable isosurface are plotted agprogress variable. It should be noted that this isa simple average over transverse planes as is cmonly done in support of RANS modeling [4], binstead the choice of averaging method has been min the context of sub-grid scale modeling for LEIn LES the surface average of a quantityQ is given

(a)

(b)

Fig. 4. (a) Temporal evolution of the normalized turbuleflame speedST/SL . (b) Variation ofSd/SL , (Sr + Sn)/SL ,andρSd/ρ0SL across the flame.

by (Q)s =Q|∇c|/|∇c|, where the overbar denotesLES filtering operation. It is a property of LES filteing that lim2→0 (Q)s =Q, where2 is the filter size.Following this argument, Fig. 4b can be interpreas showing the variation of the LES filtered valuesSd/SL andρSd/ρ0SL with filtered reaction progresvariable in the limit of zero filter size, with averaginin the transverse directions on each isosurface.technique is essentially similar to that used by Boet al. [13] to eliminate the scatter from their LES fitered data. It can be seen from Fig. 4b that whileSdincreases with progress variable through the flathe value ofρSd remains close toρ0SL throughout,


adlyeed.s--6b)ntityeed

el-ionead

ist al.-

enre-

heecthly

re-ss–

ce

is-sur-

ith

two,andrn-l-24].cu-forress

of

o-e–sis.algedtandctd as

t al.r-

the

and

as the density decrease due to heat release brocompensates for the increase in displacement spAlso plotted in Fig. 4b is the contribution to the diplacement speed(Sr + Sn)/SL arising from the reaction rate and normal diffusion components (Eqs. (1and (16c)), averaged on the same basis. This quaremains very close to the mean displacement spSd throughout the flame but differs fromSL towardsthe fresh gas side. This finding supports the moding assumptions of Peters [7] for the thin reactzones regime, in that the unsteady fluctuations ahof the reaction zone are increasing the value ofSr +SnaboveSL while remaining of the same order, andconsistent with the results obtained by Peters e[11] from two-dimensional DNS with complex chemistry. It follows from the close agreement betweSr + Sn andSd that the ensemble average of themaining contributionSt arising from tangential dif-fusion (Eq. (16d)) remains negligible throughout tflame brush. This term may be expected to affthe displacement speed locally if the flame is higcurved.

The pdf of the reaction progress variable is psented in Fig. 5a, which shows a classic Bray–MoLibby (BML)-type bimodal pdf [27]. According tothe BML formulation the density is given by

(25)ρ = ρ0

1+ τc.

Differentiating Eq. (25) with respect to time and spaand enforcing mass continuity yield the expression

(26a)∂uk

∂xk

∣∣∣∣c=c∗

= τSd|∇c|(1+ τc)

∣∣∣∣c=c∗

.

This may be rewritten using Eq. (25) as

(26b)ρ01

τ |∇c|∂uk

∂xk

∣∣∣∣c=c∗

= (ρSd)c=c∗ ,

which is an expression for the mass-weighted dplacement speed on a given progress variable isoface.

The variation of the quantity

1

τ |∇c|SL

∂uk

∂xk

∣∣∣∣c=c∗

is plotted across the flame brush in Fig. 5b along wthe variation ofρSd/ρ0SL taken from Fig. 4b. It isclear that there is good agreement between theindicating that the dilatation due to heat releasethe behavior of|∇c| across the flame together govethe variation ofρSd. This result is important both fundamentally and from the point of view of LES modeing based on the flame surface density approach [

The contributions of the reaction rate and molelar diffusion source terms in the balance equationreaction progress variable are plotted against prog

(a)

(b)

Fig. 5. (a) pdf of reaction progress variable. (b) Variation(1/(τ |∇c|SL))∇u compared withρSd/ρ0SL plotted acrossthe flame brush.

variable in Fig. 6a. The normal diffusion compnent, tangential diffusion component, and reactivdiffusive imbalance are also plotted on the same baIt is important to note that this is not a conventionbudget plot but, as before, the quantities are averaover each progress variable isosurface to undershow the contributions of reaction and diffusion affethe displacement speed. This plot can be regardesimilar to the LES budgets presented by Boger e[13] but taken in the limit of zero filter size. To undestand the behavior ofSr, Sn, St, andSd on differentcisosurfaces it is important to know the behavior ofterms presented in Fig. 6a on thec isosurfaces of in-terest. It can be inferred from Eqs. (14a) and (16a)


if-ns-

The

eedlaynd

romof

torg-

le-inas-ofum

iuslo-dif-e is

s,ter-Theo-u-ngev-onatterin

scalDF

uvéxtly,e

ith

eanfef-te

ce-of

erial6a))thatithsgat

er

is

iling-l.

u-ighheb-

(a)

(b)

Fig. 6. (a) Variation of reaction rate and molecular dfusion rate terms in the reaction progress variable traport equation. Reaction rate= w, molecular diffusionrate= ∇(ρD∇c), normal diffusion term= N∇(ρDN∇c),tangential diffusion term= −ρD∇N |∇c|. All the termsare ensemble averaged over the respectivec isosurfaces.(b) Variation of surface-averaged SDF across the flame.scatter of SDF is shown by the dots.

from the above discussion that the displacement spvariation across the flame is a result of the interpof reaction rate, molecular diffusion, and SDF, ahence the validity of the assumption(ρSd)s ≈ ρ0SLacross the flame is determined by these factors. FFig. 6a it is evident that, towards the leading edgethe flame, molecular diffusion rate is the only facthat determinesSd as the reaction rate here is ne

ligible. With increasing progress variable, the mocular diffusion rate attains a broad maximum withthe front half of the flame and then starts decreing until it changes sign just beyond the middlethe flame. The reaction rate term attains a maximtowards the burned gas side of the flame(c ≈ 0.8),as expected in the context of single-step Arrhenchemistry. The molecular diffusion term attains acal minimum towards the trailing edge and as thefusion rate here is negative while the reaction ratalways positive, the competition of these two termalong with the SDF variation across the flame, demines the magnitude of the displacement speed.magnitude of the mean tangential diffusion compnent is small compared with that of the normal diffsion term throughout the flame brush, and the chain sign of the mean molecular diffusion term is goerned entirely by the behavior of the normal diffusicomponent. The surface averaged SDF, and the scof SDF at differentc isosurfaces, is also presentedFig. 6b. It is clear that at a givenc isosurface there ia significant variation of SDF determined by the locurvature and strain rate on the isosurface. The Sprofile is found to be skewed as observed by Troand Poinsot [4] and Boger et al. [13] in the conteof FSD transport for RANS and LES, respectivewith a maximum atc ≈ 0.65. This plot can also bregarded as the variation of the generalized FSD wthe filtered value ofc in the limit of zero filter width.Referring back to Fig. 4b, the sharp peak in the mdisplacement speedSd towards the leading edge othe flame is now seen to be due to the combinedfect of a positive increasing molecular diffusion raand a small SDF magnitude.

Also in Fig. 4b it can be seen that the displament speed is sufficiently high that an isosurfaceprogress variable cannot be considered as a matsurface. From the expression (Eqs. (14a) and (1for Sd it can be seen that there is nothing to ensureSd is a positive semidefinite quantity. In flames wfinite thickness,Sd can have locally negative value[11,12] where thec isosurface concerned is movinin the direction opposite to the local normal at thpoint. The probability of finding negativeSd can beseen from the pdf ofSd as presented for a numbof c isosurfaces in Fig. 7a. Comparing the pdf ofSdfor different levels ofc it can be seen that therea sharp peak close toSd/SL = 1 for locations to-wards the unburned gas side(c ≈ 0.1) and that thepeak value drops and the pdf broadens as the traedge is approached (c≈ 0.9). These findings are consistent with the pdf ofSd presented by Peters et a[11] from two-dimensional complex chemistry simlations. Fig. 7a also shows that the probability of hpositiveSd is greater towards the trailing edge of tflame. Referring to Fig. 6a, it is evident that the pro


(a) (b)

(c) (d)

(e)

Fig. 7. Pdfs of displacement speed and its components on isosurfaces of progress variable. (a) Pdf of displacement speedSd.(b) Pdf of normal diffusion componentSn. (c) Pdf of tangential diffusion componentSt. (d) Pdf of reaction componentSr. Ineach plot the progress variable runs fromc= 0.1 to 0.9 in steps of 0.2. (e) Pdfs ofSd and combined reaction and normal diffusioncomponents(Sr + Sn) on the isosurface atc= 0.8, close to the location of maximum reaction rate.


tiontiveeg-is

te iste

edlityustnceere

Asy ofn-ted

u-isluethatheentgh-asin

or--

ofb-

ntis-

tentn-

if-Thethe

NSof

seere-

rdss theabil-eeur-

ighe-n inofal

e)).

isis

te

ability of finding high positiveSd is greater towardsthe burned gas side since the highly positive reacrate is greater in magnitude than the highly negamolecular diffusion rate. By the same token, a native Sd may result in this region if the oppositetrue. It is possible also to have negativeSd towardsthe leading edge of the flame where the reaction ranegligible, provided that the molecular diffusion rais locally negative on a particularc isosurface. Thiscan happen only if the isosurface is strongly curvdue to turbulent motion. Nevertheless the probabiof finding locally negative displacement speed mbe small since the mean reactive–diffusive imbalaw + ∇(ρD∇c) is strongly positive throughout thflame (Fig. 6a). This is confirmed in Fig. 7a, wheit is clear that the probability of finding negativeSd issmall.

The pdf of the normal diffusion componentSn ofthe displacement speed is presented in Fig. 7b.the burned gas side is approached the probabilitfinding negativeSn increases markedly, which is cosistent with the mean normal diffusion rate presenin Fig. 6a, and with the behavior ofSn as reportedby Peters et al. [11]. The pdf of the tangential diffsion componentSt is presented in Fig. 7c, and itclear that this pdf is sharply peaked close to a vaof zero. It can also be seen from Figs. 4b and 6athe mean ofSt remains close to zero throughout tflame brush. The pdf of the reaction rate componSr is presented in Fig. 7d, and shows that the hiest values ofSr are attained towards the burned gside, consistent with the reaction rate plot shownFig. 6a. The pdf of the combined reaction and nmal diffusion components(Sr +Sn) close to the location of maximum reaction rate(c = 0.8) is presentedin Fig. 7e together with the pdf ofSd. The pdf ofSr + Sn is similar in shape and magnitude to thatSd, but it is interesting to note that there is lower proability of finding negativeSr + Sn than negativeSd,which implies that the tangential diffusion componeis mainly responsible for the observed negative dplacement speed. All of these findings are consiswith previous simulations carried out in two dimesions with detailed chemistry [11].

pdfs of flame surface curvature are shown for dferent progress variable isosurfaces in Fig. 8a.pdfs are nearly symmetric, with sharp peaks nearzero curvature location, consistent with previous D[1,3,28–30]. This explains why the mean valuesSt and tangential diffusion rate are close to zero (Figs. 4b and 6a). The curvature pdfs towards theactant side of the flame are slightly skewed towanegative values. The skewness decreases towardburned gas side because of the increased probity of finding high positive displacement speed (sFig. 4b), which leads to rapid destruction of flame s

face area and, hence, to reduced probability of hnegative curvature. This effect is mirrored in the dcreasing skewness towards the positive side seeFig. 7c in the pdf ofSt towards the burned gas sidethe flame, since this quantity is directly proportionto the negative of curvature (Eqs. (16d) and (16The tangential strain rate on ac isosurface is given by

(27)aT = (δij −NiNj )∂ui

∂xj

∣∣∣∣c=c∗

.

The pdf of this quantity is plotted in Fig. 8b and itclear that the probability of finding positive values

(a)

(b)

Fig. 8. (a) Pdf of mean curvatureκm on progress variableisosurfaces atc= 0.1 to 0.9. (b) Pdf of tangential strain raaT on progress variable isosurfaces atc= 0.1 to 0.9.


acementlacementerence.

(a) (b)

(c) (d)

Fig. 9. Contours of joint pdfs of tangential strain rate, mean curvature, and displacement speed on the isosurface atc= 0.8, closeto the location of maximum reaction rate. (a) Joint pdf of tangential strain rate and mean curvature. (b) Joint pdf of displspeed and curvature. (c) Joint pdf of displacement speed and tangential strain rate. (d) Conditional joint pdf of dispspeed and tangential strain rate taken at zero curvature. Joint pdf magnitudes decrease from the center to the circumf

veious

a-ms arateNSnyva-tionthe

metingthet theRe-e-

Thee iserelyo-

int

much higher than the probability of finding negativalues across the flame brush, supporting prevfindings in two and three dimensions [1,3,28–30].

The joint pdf of tangential strain rate and curvture on thec = 0.8 isosurface close to the maximureaction rate location is shown in Fig. 9a. There inegative correlation between the tangential strainand curvature on this isosurface. Some previous Dstudies without heat release [29,30] did not find asignificant correlation between strain rate and curture. It has been suggested [3] that the correlabetween strain rate and curvature could be due to

higher turbulence levels present ahead of the flathan behind it, as in the present case. It is interesto note that the configuration of the domain andturbulence intensity in the present case are abousame as those used by Haworth and Poinsot [3].cently, Renou et al. [31] found similar correlations btween strain rate and curvature from experiments.joint pdf of displacement speed and mean curvaturpresented in Fig. 9b, where it can be seen that this a negative correlation. This result is qualitativeconsistent with the previous results [1,10,11] for twdimensional DNS with detailed chemistry. The jo


alan

(a) (b)

(c)

Fig. 10. Contours of joint pdfs of components of displacement speed and mean curvature on the isosurface atc = 0.8, closeto the location of maximum reaction rate. (a) Joint pdf of reaction componentSr and mean curvature. (b) Joint pdf of normdiffusion componentSn and mean curvature. (c) Joint pdf of tangential diffusion componentSt of displacement speed and mecurvature. Joint pdf magnitudes decrease from the center to the circumference.

rateor-

ce istheinenteroith

n-

re

re-woega-

sententim-al-entFornit-

pdf of displacement speed and tangential strainis presented in Fig. 9c, and shows a very weak crelation. Nevertheless some strain rate dependenpresent and can be elucidated [1] by looking atconditional joint pdf at zero curvature as shownFig. 9d. The tangential strain rate and displacemspeed are found to be negatively correlated at zcurvature locations, which again is consistent wprevious two-dimensional results [1].

It is interesting to examine how the different cotributions to displacement speedSd are affected bycurvature and strain rate. The joint pdf of curvatu

andSr, Sn, andSt are presented in Figs. 10a–10c,spectively. In Fig. 10a it is possible to discern that ttrends are present. For positive and moderately ntive values of curvature,Sr andκm are positively cor-related, but a weak negative correlation is also prefor larger negative values of curvature. In the prescase the positive correlation narrowly prevails. Silar behavior was observed by Peters et al. [11],though the negative correlation was more prominin that case, and some discussion is warranted.unity Lewis number flames, the reaction rate per umass on a givenc isosurface remains fairly con


ichenential

lace-

of

res--r

r-m-are

DF

ueshatos-re

adsson

e-glyhighen-s atis-ing

skly

iveandfer-nt

otviorbe-

ndla-

ectF

turee a

glyin

t is

6d)

ec-

thesing

ithan-atlyown

ctsnsef-be-

nal

mdnce

isof

ionlygsareif-

t

or-ainmes

vi-0,

u-an

e-ntoled

e

stant, so the curvature dependence ofSr must appearthrough changes in the SDF (see Eq. (16b)), whis affected indirectly through the correlation betwecurvature and strain rate. The dilatation on a givisosurface is the sum of the normal and tangenstrain rates, and can be expressed in terms of dispment speed and SDF following Eq. (26a), to yield

(28)aT + an = ∂uk

∂xk

∣∣∣∣c=c∗

= τSd|∇c|(1+ τc)

∣∣∣∣c=c∗

.

This equation indicates that there is a critical valuetangential strain rate(aT > τSd|∇c|c=c∗ /(1 + τc∗))above which the normal strain rate becomes compsive in nature. If it is assumed thatSd scales approximately with SL(1 + τc∗) then the critical value foaT becomes of the order ofτSL |∇c|c=c∗ , which isfrequently exceeded on ac isosurface as can be veified from Figs. 4b, 6b, and 8b. Under such a copressive normal strain rate the isoscalar surfacesmoved closer to each other, leading to a higher Sand, hence, to lower values ofSr. High values of tan-gential strain rate are correlated with negative valof mean curvature (Fig. 9a), and for curvatures tare becoming less negative and finally turning pitive the tangential strain rate falls, leading to moextensive normal strain rates via Eq. (28), which leto thickening of the flame, lowering the SDF andincreasingSr. Thus, the principal correlation betweeSr and κm turns out to be positive. In the small rgions of the flame where the curvature is stronnegative, the displacement speed can take veryvalues (Fig. 9b). Due to the higher turbulence intsity in the reactants, the negatively curved regionthe leading edge of the flame move with a higher dplacement speed, and this leads to a local thickeneffect which decreases the SDF and so increaseSr.This secondary effect is responsible for the weanegative correlation betweenSr andκm also seen inFig. 10a. Peters et al. [11] found that this negatcorrelation was dominant for negative curvatures,gave an explanation in terms of the enhanced difential diffusion of light radical species. In the preseformulation the effects of differential diffusion are nincluded, and it is interesting that the same behais observed. Similar considerations apply to thehavior of the normal diffusion componentSn shownin Fig. 10b. The normal diffusion rate (Eq. (14b) athe numerator of Eq. (16c)) may be shown to be retively insensitive to curvature, so that the main effof curvature onSn arises from changes in the SDvia the correlation between strain rate and curvaas described above. The net effect is to producpositive correlation between curvature andSn, with aweak negative correlation also apparent for stronnegative values of curvature. This result also isagreement with the findings of Peters et al. [11]. I

evident from Fig. 10c thatSt andκm are strongly andnegatively correlated, as expected from Eqs. (1and (27).

The correlations between tangential strain rateaTandSr, Sn, St are presented in Figs. 11a–11c, resptively. From Fig. 11a it is apparent thatSr and aTare negatively correlated, which is expected sinceSDF increases as the flame is subjected to increaextensive strain rates. In Fig. 11b,Sn andaT are foundto be negatively correlated which is consistent wthe increase in normal scalar gradient with higher tgential strain rates. It is evident from Fig. 11c thSt and aT are positively correlated, which is puredue to the strain rate–curvature correlation as shin Fig. 9a.

It is important to recognize that curvature effeare implicitly present in the strain rate correlatioshown in Fig. 11. To isolate only the strain ratefects, it is necessary to look at the correlationtween tangential strain rateaT andSr, Sn, St in a verynarrow band around zero curvature. The conditiocorrelations betweenaT andSr, Sn are presented inFigs. 12a and 12b, respectively. It is evident froFig. 12a thatSr andaT are negatively correlated, anthis is due to the tangential strain rate dependeof the SDF. Fig. 12b suggests thatSn is negativelycorrelated with tangential strain rate, which alsoa result of the tangential strain rate dependencethe normal scalar gradient. The tangential diffuscomponentSt is not shown since it goes identicalto zero in the limit of zero curvature. These findinindicate that the effects of tangential strain rateimplicitly embedded in the reactive and normal dfusive components ofSd, which is consistent withthe modeling assumptions forSr + Sn in the thin re-action zones regime [7]. The reactive componenSrand normal diffusive componentSn are both foundto be weakly correlated with curvature, also in accdance with the modeling assumption [7] that the mcurvature dependence of displacement speed cothrough the tangential diffusion componentSt. Theseresults are in good qualitative agreement with preous two-dimensional detailed-chemistry DNS [1,111].

5. Conclusion

Three-dimensional compressible DNS of turblent premixed flames has been carried out ininflow–outflow configuration using one-step Arrhnius chemistry. Incompressible turbulence is fed ithe computational domain through the inlet. Detaistatistics of the displacement speedSd have beenobtained and suggest that the assumption(ρSd)s ≈ρ0SL holds good in most of the parts of the flam


al.

(a) (b)

(c)

Fig. 11. Contours of joint pdfs of components of displacement speed and tangential strain rate on the isosurface atc= 0.8, closeto the location of maximum reaction rate. (a) Joint pdf of reaction componentSr and tangential strain rate. (b) Joint pdf of normdiffusion componentSn and tangential strain rate. (c) Joint pdf of tangential diffusion componentSt and tangential strain rateJoint pdf magnitudes decrease from the center to the circumference.

ive

hanmethin[7].nF

ameTheity

ofo-

ntush.-ndthetentn-ndnd

brush. The cumulative contribution of the reactcomponent(Sr) and normal diffusion component(Sn)

of the displacement speed is found to be larger tthe laminar flame speed while remaining of the saorder, due to unsteady effects experienced in thereaction zones regime, as pointed out by PetersThe behavior ofρSd across the flame brush is showto be linked to the variation of dilatation and SDacross the flame. From the pdf ofSd it is clear thatthe most probable displacement speed is of the sorder as the corresponding laminar flame speed.pdf of Sd suggests that there is a finite probabil

of finding negative displacement speed. The pdfSr + Sn suggests that the tangential diffusion compnent of displacement speedSt is primarily responsiblefor locally high negative values of the displacemespeed seen at isolated points through the flame brThe pdfs ofSd and Sr, Sn, St are found to be consistent with the results reported by Hilka [12] aPeters et al. [11]. The pdf of flame curvature andpdf of tangential strain rate are found to be consiswith previous findings in both two and three dimesions [1,3,27–29]. The tangential strain rate is fouto be negatively correlated with curvature as fou


ofzerofonnt

the

1].ivelyenal

entis

ted.

is-ativeofnd

eednd

neen

ain

tersur-erises

n-try,tionoftryofr.

ortduegecalhis

6)

int-on

ic,

2)

1.95)

nt,

iv.

cal79,

(a)

(b)

Fig. 12. Contours of conditional joint pdfs of componentsdisplacement speed and tangential strain rate taken atcurvature on the isosurface atc= 0.8, close to the location omaximum reaction rate. (a) Conditional joint pdf of reacticomponentSr and tangential strain rate. (b) Conditional joipdf of normal diffusive componentSn and tangential strainrate. Joint pdf magnitudes decrease from the center tocircumference.

by Haworth and Poinsot [3] and Renou et al. [3The displacement speed and curvature are negatcorrelated as found by Echekki and Chen [1], Chand Im [10], and Peters et al. [11] in two-dimensionDNS with detailed chemistry. Based on the presthree-dimensional DNS with simple chemistry, itfound that the instantaneous displacement speed(Sd)

and the tangential strain rate are weakly correla

The conditional pdf of tangential strain rate and dplacement speed at zero curvature shows a negcorrelation, which is consistent with the findingsEchekki and Chen [1]. The reactive component anormal diffusive components of displacement spare found to be weakly correlated with curvature, athe trend ofSr and Sn variation with curvature isfound to be qualitatively similar to that of the leaflame data presented by Peters et al. [11]. It has bfound from the conditional correlations ofSr andSnwith tangential strain rate that the influence of strrate is implicitly present in the behavior ofSr + Sn,which supports the modeling assumptions of Pe[7] for the thin reaction zones regime. The weak cvature dependence ofSr and Sn suggests that thcurvature dependence of displacement speed athrough the tangential diffusion rate contributionSt,also in accordance with Peters [7].

In the present study with a realistic three-dimesional turbulent flow field and single-step chemismost aspects of the flame structure and propagaare well captured; but for a fuller understandingthe flame behavior the effects of detailed chemismust be taken into account along with the effectsdifferential diffusion due to nonunity Lewis numbeThese issues will be the subject of future work.

Acknowledgments

The authors are grateful for the financial suppof the Gates Cambridge Trust. Thanks are alsoto Karl Jenkins and Stephen Tullis of the CambridCFD Laboratory for useful discussions and practihelp. Special thanks are due to Evatt Hawkes foractive help and valuable technical input.

References

[1] T. Echekki, J.H. Chen, Combust. Flame 106 (199184–202.

[2] K.W. Jenkins, R.S. Cant, Flame kernel interactionturbulent environment, in: C. Liu, L. Sakell, T. Beauner (Eds.), Proceedings Third AFOSR ConferenceDNS and LES, Rutgers University/Kluwer AcademDordrecht/Norwell, MA, 1999, pp. 605–612.

[3] D.C. Haworth, T.J. Poinsot, J. Fluid Mech. 244 (199405–436.

[4] A. Trouvé, T. Poinsot, J. Fluid Mech. 278 (1994) 1–3[5] S. Zhang, C.J. Rutland, Combust. Flame 102 (19

447–461.[6] E.R. Hawkes, PhD thesis, Engineering Departme

Cambridge University, Cambridge, 2000.[7] N. Peters, Turbulent Combustion, Cambridge Un

Press, Cambridge, 2000, pp. 66–169.[8] S.B. Pope, Int. J. Eng. Sci. 26 (1988) 445–469.[9] T. Poinsot, D. Veynante, Theoretical and Numeri

Combustion, Edwards, Philadelphia, 2001, pp. 27–171–268.


8)

roc.

é,

27

ousch

on

ntort,

4–

–

:ge

&18.2)

r-nd

01)

28

6)

t.

93)

57

i-er

tan-

é,

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unsteady effects of strain rate and curvature on turbulent premixed flames in an inflow–outflow...

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