Intermittent features of inertial particle distributions in turbulent premixed flame

Intermittent features of inertial particle distributions in turbulentpremixed flames

F. Battista,1 F. Picano,1 G. Troiani,2 and C. M. Casciola1,a)

1Dip. di Ingegneria Meccanica e Aerospaziale, Sapienza University, via Eudossiana 18, 00184 Rome, Italy2ENEA C.R. Casaccia, via Anguillarese 301, 00123 Rome, Italy

(Received 16 June 2011; accepted 16 November 2011; published online 30 December 2011)

Clustering is widely observed in many turbulent flows, where it results from the inability of finiteinertia particles to comply with the different time scales, which characterize a turbulent field.Depending on their inertia, particles are found to be instantaneously organized in clusters, whosesize depends on the Kolmogorov-Stokes number and which presumably form as a consequence ofparticle ejection from persistent vortical structures. In reacting flows, the abrupt acceleration of thefluid across the thin flame front due to combustion adds new and unexpected features. The particlesfollow such acceleration with a certain time lag which, coupled with the flame front fluctuations,gives rise to an entirely different mechanism of cluster formation. As suggested in previous studies,a possible indicator of this preferential localization is the so-called clustering index, quantifyingthe departure of the actual particle arrangement from the Poissonian distribution. Most of the clus-tering is found in the flame brush region, where it cannot be explained by the standard argumentsused in cold flows. Actually, the effect is significant also for very light particles, where the simplemodel we propose, based on the Bray-Moss-Libby formalism, is able to account for most of thedeviation from the Poissonian. When the particle inertia increases, the effect becomes larger and itis found to persist well within the region of the burned gases. The observed clustering is confirmedby a more precise analysis in terms of a generalization of the radial distribution function to inhomo-geneous, anisotropic flows. The results taken from a direct numerical simulation with single stepkinetics favorably compare with experiments on a premixed Bunsen turbulent flame. The presentfindings are expected to be of some relevance for the plenty of applications dealing with particlesin presence of combustion, e.g., liquid droplet swarms for combustion temperature control, sootdynamics, or combustion-oriented particle image velocimetry. VC 2011 American Institute ofPhysics. [doi:10.1063/1.3671734]

I. INTRODUCTION

Reacting flows, laden with particles of different inertiaand dimensions, are common in many fields of engineeringand physics. Dispersed aluminum particles are found in solidfueled rocket engines.1 They are embedded in the fuel matrixand, after ablation, released in the flow where they areoxidized, thus increasing the propulsion performances of thesystem. Undesirable particles are also found in reciprocatingengines where carbon soot may also form due to pyrolysis ofthe fuel droplet and its incomplete combustion.2,3 They alsofind application as transducers in particle image velocimetry(PIV), where the fluid velocity field is measured by thedisplacement of small seeding particles.4,5

The particle behavior in several flows, i.e., homogene-ous and isotropic flows, shear, pipe, and channel flows, is thetopic of many papers, which aim at understanding localiza-tion phenomena such as turbophoresis in wall bounded flowsor preferential accumulation and clustering in the smallscales of turbulence, see Refs. 6–12 among others.

Particle clustering has a strong influence on dropletsgrowth in clouds via coalescence,13 on particle settling14

and on inter-particle collisions.15 The spatial distribution of

droplets in jets is a topic of several works16–20 since it mayinfluence combustion processes and temperature distribu-tions in flames.21

In presence of spatial temperature variations, the tem-perature gradient may influence the particle spatial distribu-tion, as discussed in Ref. 22. Due to thermal diffusion, themean particle distribution is not uniform, with increasedlocalization near the minimum of the mean temperature. Theauthors demonstrate that the accumulation induced by thethermal gradient is stronger than the turbulence-inducedaccumulation found in isothermal conditions.

Purpose of the present paper is to discuss recent numeri-cal data23 concerning the effect of the abrupt accelerationinduced by the turbulent thin flame front on the statisticalproperties of inertial particle spatial distributions. The issuemay be relevant for evaluating combined inertia/combustioneffects on particles collisions and coalescence. Beyond beingof interest in itself, the analysis could be useful in estimatingthe characteristic dimension of soot particles, which grow bycoalescence in the burned gas region2 and in providing indi-cations on the behavior of liquid droplet swarms that canfound application in flame temperature control systems.24

In addition to the more familiar effects observed in coldflows, e.g., ejection from persistent vortical structures andconcentration in large strain regions, inertial particles presenta)Electronic mail: [email protected].

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PHYSICS OF FLUIDS 23, 123304 (2011)

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further interesting peculiarities in turbulent premixed com-bustion. Some of these are more or less known, or expected,e.g., thermophoresis as unbalanced thermal drift in thepresence of extreme temperature gradients, or abrupt meanconcentration decrease, induced by the strong thermal expan-sion across the flame front. Others are new, or at least muchless discussed in the scientific literature, and associated withthe sudden acceleration across the thin fluctuating turbulentfront. In order to address these combined effects of inertiaand combustion on particle motion and preferential concen-tration, a direct numerical simulation (DNS) of a turbulentpremixed Bunsen flame at Reynolds number ReD¼ 6000 hasbeen performed.23 The flow is endowed with four differentparticle populations evolved by a Lagrangian trackingmethod. The system consists of the low-Mach numberformulation of the Navier-Stokes equations equipped with anArrhenius one-step reaction model, which reproduces a leanCH4/Air premixed flame.25–27 In order to isolate the effectsof inertia from concurrent phenomena, thermophoresis andgravitational forces are neglected. At any rate, in the condi-tions addressed here, they are easily shown to be substan-tially irrelevant. The configuration simulated by DNS hasbeen also reproduced experimentally as a turbulent premixedmethane/air Bunsen flame seeded with alumina particles.Light scattering from the particles is used to obtain theinstantaneous particle distribution for comparison with thenumerical simulation.

A suitable Stokes number Stfl, called the flamelet Stokesnumber, expressed in terms of the characteristic time scale ofthe flame front, is found to properly parametrize the particledynamics.23 On top of the expected average density reductionacross the front, we find that the front fluctuation leads tostrong intermittency of the particle population. Two observ-ables, namely, the clustering index13 and a suitable general-ization of the radial distribution function9 proposed here, areused to quantitatively address the particles trend to get con-centrated in clusters of variable dimension depending on theirinertia. As main conclusion, we find that the particles mani-fest a strong tendency towards anomalous clustering in theregion spanned by the instantaneous turbulent flame front, asconfirmed by both numerical and experimental data. Theeffect is found to peak at order one flamelet Stokes number.

After introducing the basic modeling background inSec. II, the numerical method is briefly illustrated in Sec. III.The analysis in terms of the clustering index is described inSec. IV, which provides the relevant numerical and experi-mental observations completed with a short description ofthe experimental set-up.28 Section V is devoted to the radialdistribution function and finally a general discussion of themain findings is provided in Sec. VI.

II. THEORETICAL BACKGROUND

In the following, small spherical particles are consid-ered, neglecting inter-particle collisions and force feedbackon the fluid as appropriate for dilute suspensions.9 Underthese assumptions, with particle mass density much largerthan the fluid, qp=qf " 1, the equations describing the dis-persed particle dynamics read29

dxp

dt¼ v

dv

dt¼

uðx; tÞjx¼xp%v

sp

8>><

>>:; (1)

where xp and v are the particle position and velocity, respec-tively, while u x; tð Þjx¼xp

is the fluid velocity at the particleposition.

sp ¼d2

pqp

18!qf; (2)

is the relaxation time representing the particle response delayto fluid velocity fluctuations (here, dp is the particle diameter,qp and qf the particle and fluid density, respectively, and !the fluid kinematic viscosity).

In order to sharply focus on the effect of particle inertiain turbulent premixed flames we neglect the thermophoreticforce which may be shown to be negligible unless the par-ticles are extremely tiny.28 Concerning gravity, its relativeimportance with respect to the Stokes drag can be estimatedby the ratio (1% qf /qp)gsp /vREL, where g is acceleration ofgravity and vREL is the estimate for the relative particle/fluidvelocity. Gravity will be shown to be of minor importancecompared with the Stokes drag for the case of presentinterest.

The proper parameter controlling particle dynamics is asuitably defined Stokes number, ratio of the particle relaxa-tion time to the relevant time scale of the flow. In classicalincompressible turbulence, particles are able to follow quiteaccurately the velocity fluctuations when sp is smaller than

the Kolmogorov time scale, sK ¼ffiffiffiffiffiffiffiffiffiffiffiffiqf !="

q, with " the energy

dissipation rate, corresponding to the fastest frequency of thesystem,

StK ¼ sp=sK < 1: (3)

In turbulent premixed combustion the strong fluid accelera-tion due to the abrupt thermal expansion across the flamefront introduces an additional time scale. When the thermo-chemical conditions are such to have an instantaneous flame

front comparable with the Kolmogorov scale, g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiqf !3="4

q,

of the incoming turbulence (flamelet regime), the fastesttime scale is defined in terms of the laminar characteristicsof the front, i.e.,

sfl ¼dfl

Dufl; (4)

where dfl is the thermal thickness of the laminar flame andDufl the velocity jump across the front. The thermal thicknesscan be estimated as dfl¼ (Tb%Tu)/j!Tjsup, where Tb and Tu

are the temperature of burned and unburned mixtures,respectively, with j!Tjsup the maximum temperature gradi-ent modulus. Note that the thickness of the instantaneousflame where the actual reaction occurs is typically largerthan dfl. Across the flame thermal expansion drives a velocityjump Dufl¼ Sfl(Tb/Tu% 1), with Sfl the laminar flame speed.

123304-2 Battista et al. Phys. Fluids 23, 123304 (2011)


In these conditions, the relevant Stokes number controllingparticle dynamics can be taken as23

Stfl ¼sp Sfl ðTb=Tu % 1Þ

dfl: (5)

We stress that the flamelet Stokes number defined abovedepends on the particle relaxation time sp and on thermo-chemical properties of the flame, namely temperature ratioTb/Tu and flamelet thickness dfl.

We are now in the position to quantify the relevance ofgravity for our system. The relative particle/fluid velocity isof the order of the fluid velocity jump across the flame front,vREL&Dufl. Hence, for a density ratio, qp/qf¼ 4' 103, wehave

1%qf

qp

!gsp

vREL& gsp

Dufl¼ gsp

Sfl ðTb=Tu % 1Þ:

With the parameters used in the present simulation, see Sec.III, Sfl(Tb/Tu% 1)& 1, which, for our largest particles, corre-sponding to alumina particles of 20 lm, yieldsgsp/vREL& 6' 10%2, suggesting that in the worst case gravityis at least one order of magnitude smaller than Stokes drag.Clearly, the influence of gravity scales with particle diametersquared, hence for 5 lm particles the figure decreases toabout 3' 10%3.

III. METHODOLOGY

Particle dynamics in a turbulent premixed flame is stud-ied by means of an Eulerian DNS of a turbulent Bunsenburner coupled with a Lagrangian solver for particleevolution.

The Eulerian algorithm discretizes the low-Mach num-ber formulation of the Navier-Stokes equations30 in a cylin-drical domain, which describes a low Mach number flowwith arbitrary flame-induced density variations neglectingacoustics effects,

@q@tþ $ ) ðquÞ ¼ 0; (6)

@qu

@tþ $ ) ðquuÞ ¼ % $pþ 1

Re$ ) l $uþ $uT

" #$ %

þ 1

Fr2qf ; (7)

@qYR

@tþ $ ) ðqYRuÞ ¼ 1

ReScR$ ) ðl$YRÞ % xR; (8)

$ ) u ¼ 1

P1

RePr$ ) ðl$TÞ þ c% 1

cCe xR

& '; (9)

T ¼ Pq; (10)

with q, u, and p the density, the velocity, and the dynamicpressure, respectively,%T, P YR, and xR are the temperature,the thermodynamic pressure, the reactant concentration andits global reaction rate, respectively. l is the dynamic viscos-ity of the mixture, Re¼ q0U0L0/l0 is the Reynolds number,

and Pr¼ l0/(q0a) is the Prandtl number giving the ratiobetween thermal diffusivity and viscosity of the mixture(here, Pr¼ 0.6). The ratio of the heat capacity coefficientsis c¼ 1.33 and the ratio of momentum to reactant masstransport coefficients is given by the Smidth numberScR¼ l0/(q0Da) (here ScR¼ 0.6). The chemical kinetics isgiven by a simple one-step irreversible reaction transformingpremixed fresh mixture R into combustion productsP, with an Arrhenius model for the reaction rate: xR

¼ Da qYRð Þe%Ta=T with Da the nominal Damkohler number,Ta the activation temperature, and Ce the dimensionlessheat release. The model is tuned to reproduce a lean pre-mixed methane/air Bunsen flame with equivalence ratio/& 0.7 (Tb/Tu¼ 5.3, Ta/Tu¼ 30, Da¼ 45 000, Ce¼ 18.5 andtemperature dependent viscosity, l / T1=2). The ratioof laminar flame speed to fluid bulk velocity at the jetinlet is Sfl=U0 ’ 0:05, with laminar flame-thicknessdfl ’ 0:019 D.

The spatial discretization is based on central secondorder finite differences in conservative form on a staggeredgrid. The convective terms of the relevant reaction-advec-tion-diffusion equations for the scalars are discretized by abounded central difference scheme designed to avoid spuri-ous oscillations.31 This non-linear scheme is tuned to enforceboundedness with no significant extra-dissipation. The tem-poral evolution is performed by a low-storage third orderRunge-Kutta scheme.32

Velocity is prescribed (Dirichlet condition) at the inflowsection, by using a cross-sectional plane of a periodic turbu-lent pipe flow, provided by a companion time-evolvingDNS. A convective outflow condition is adopted togetherwith a traction-free condition at the side boundary whichmimics the open environment surrounding the flame.

The computational domain, [hmax'Rmax'Zmax]¼ [2p' 6.2D' 7D], is discretized by Nh'Nr'Nz¼ 128' 201' 560 nodes with radial mesh stretching (Dr¼ 0.00625D) for accurate resolution of the turbulent shearlayer (Kolmogorov lengths g ’ 0:005 D and g ’ 0:05 D inthe unburned and burned gas regions, respectively) and ofthe instantaneous flame front (4* 5 points within the actualinstantaneous flame thickness.23)

The parameters of the simulation correspond to a lean pre-mixed methane/air Bunsen flame (equivalence ratio /& 0.7)with diameter-based Reynolds number ReD¼U0D/!1¼ 6000,with U0 the bulk velocity, see Ref. 23 for comparisons withcorresponding experimental data.

Particles evolve by a Lagrangian tracking method,which integrates Eq. (1) by means of the same Runge-Kuttascheme used for the fluid phase. Interpolation of fluidvelocity at particle positions is done by using second orderLagrange polynomials, see Refs. 11, 18, and 23 for moredetails. Particles do not counteract on the flow (one-waycoupling) and inter-particle collisions are neglected.

Four particle populations are considered to mimic alaboratory Bunsen premixed flame seeded with alumina par-ticles (qp¼ 4000 kg/m3) with diameters dp¼ 1 lm, dp¼ 5lm, dp¼ 10 lm, and dp¼ 20 lm. In the four cases, theflamelet Stokes numbers are, respectively, Stfl¼ 0.022,Stfl¼ 0.54, Stfl¼ 2.16, and Stfl¼ 8.65.

123304-3 Intermittent features of inertial particle distributions Phys. Fluids 23, 123304 (2011)


Particles are introduced in the field at a fixed rate withhomogeneous distribution at the jet inlet section with veloc-ity matching the local fluid. About 6' 106 particles areevolved in the simulation.

The simulation run for 30D/U0 to achieve the statisticalsteady state before collecting 100 sample fields, separated by0.125D/U0, for statistical analysis.

Figure 1 provides a snapshot of the instantaneous flowfield together with the configuration of the Stfl¼ 0.022 parti-cle population. The fluid axial velocity is shown by contourswith the reactant iso-level at YR=Y0

R ¼ 0:5 plotted as a blacksolid line.

IV. CLUSTERING INDEX

A. Numerical analysis

Figure 2 provides the instantaneous configuration of thefour particle populations. Colors encode the fluid density atparticle positions to help detecting the flame front. The hugevariations in the particle concentration between fresh andexhaust gas regions are induced by the sudden expansion dueto heat release. According to their inertia, the particles candifferently comply with the abrupt fluid acceleration. Theeffect is not confined to the front proximity and, dependingon particle mass, it can influence the particle distributionwell into the burned gas region.

A distinguishing feature of the particle distributions isthe presence of particle clusters, Figure 2. Clusters takesplace at smaller and smaller scales as the Stokes number isreduced. In the burned region, they are particularly welldefined for intermediate particle mass, Stfl¼ 0.54 and 2.16,

with the typical length scale of the voids large enough to beclearly appreciated at first sight. The lightest particles aremore evenly distributed both in the burned and unburnedregions recovering the tracer-like behavior characterized bya sharp jump of the particle number density across the front.

Particle segregation can be measured by the deviation ofthe actual particle distribution with respect to randomly dis-tributed independent positions. These reference conditionsare expressed by the Poisson distribution which gives theprobability p(n) to find n particles in a certain domain X ofvolume DV with characteristic length ‘ ¼

ffiffiffiffiffiffiffiDV3p

. Once theaverage particle number !n is known, the probability densityfunction (pdf) reads

P n; !nð Þ ¼!nne%!n

n!: (11)

A well known feature of Poissonian processes is that the

variance r2n ¼ ðn% !nÞ2 ¼ ðdnÞ2 equals the mean value !n.

When the particles are distributed according to the Poisso-nian distribution, they do not segregate. Particle segregationcan then be quantified in terms of the departure from thePoissonian distribution measured by the clustering index13

K ¼ ðdnÞ2

!n% 1: (12)

From the definition, it is clear that a Poisson distributionimplies K¼ 0, consistently with the absence of any cluster-ing or preferential localization effect. The smaller the ratioof variance to mean value is, the more uniformly distributedthe particle are. This corresponds to a negative clusteringindex K< 0. On the contrary, if particles preferentially ag-gregate in clusters, the variance exceeds the mean value,yielding a positive clustering index, K> 0. Since the cluster-ing index depends on the length scale ‘, K(‘), via !nð‘Þ andðdnÞ2ð‘Þ, all the above features are scale dependent. Addi-tionally for inhomogeneous processes, the clustering indexbecomes a field since it depends on the position of the con-trol domain X(x), K¼K(x, ‘).

The clustering index field can be exploited to quantifythe amount of particle segregation induced by a fluctuatingturbulent flame front. The clustering index for each particlepopulation is shown in Figure 3. The region spanned by theflame front is called the flame brush. It can be defined interms of the average progress variable !c, where the instanta-neous progress variable c ¼ 1% YR=Y0

R ranges from 0 in thereactants to 1 in the products. The flame brush is then theregion where 0 < !c < 1, whose boundaries are shown asthick lines in Figure 3. The clustering index K, shown by col-ors in the same figure, is positive in the flame brush and inthe outer region of the jet. In the outer region, the increasedvalue of K is indeed induced by the intermittency of theboundary between the hot jet seeded with particles and theexternal, particle-free, cold environment. From our presentpoint of view, the most significant feature is that evenextremely small particles, Stfl¼ 0.022, which behave liketracers under every respects, show large values of K in theflame brush region. Actually while clustering of finite mass

FIG. 1. (Color online) Iso-levels of instantaneous axial fluid velocity onthree azimuthal planes. The solid line shows the reactant concentration iso-level at Yr=Y0

r ¼ 0:5. The dots mark the instantaneous positions of particlesat Stlf¼ 0.022. Only one particle out of 40 is displayed to avoid blurring theimage.



particles is more or less expected, this behavior of tracer-likeparticles needs being properly understood before the cluster-ing properties of finite mass particles are addressed.

To try and explain this effect of the flame we resort tothe Bray-Moss-Libby (BML) description of the flamelet33

where the field is decomposed in two nearby regions, per-taining to burned (c¼ 1) and unburned (c¼ 0) gases, respec-tively, separated by the thin instantaneous front. Though notstrictly necessary, for simplicity, we further assume that theprobability to find n particles in a domain of size ‘ condi-tioned to the burned and unburned states, p(njc¼ 0/1), is in-dependent of position. We highlight that for vanishingflamelet Stokes number such conditional pdf’s shouldapproach a Poissonian distribution with suitable mean value.This assumption, exact for tracers, is reasonably accuratealso for sufficiently small particles Stfl + 1

" #.

As anticipated, in the flamelet context, the state of themixture is either c¼ 0 (unburned) or c¼ 1 (burned), and theprobability distribution for the state is p(x, c)¼ [a(x)d(c)þb(x)d(1% c)], where a and b are the probability to find theunburned or burned gases at x. By evaluating !c from theabove pdf it follows bðxÞ ¼ 1% aðxÞ ¼ !cðxÞ, see Refs. 28and 33. Assuming that the scale ‘ is sufficiently smallerthan the characteristic scale of variation of the average pro-gress variable (the flame brush characteristic dimension)

b ¼ 1=jr!cj, the joint probability distribution of the particlenumber in V, n, and of the instantaneous progress variable, c,is p(x;c,n)¼ p(x, c)p(njc), which on the basis of our assump-tion follows as:

pðx; c; nÞ ¼ 1% !cðxÞ½ -dðcÞ þ !cðxÞdð1% cÞf g pðnjcÞ: (13)

Starting from Eq. (13), the first and second order moments ofthe particle number pdf read

!nðxÞ ¼ ½1% !cðxÞ-nu þ !cðxÞnb; (14)

n2ðxÞ ¼ ½1% !cðxÞ-n2u þ !cðxÞn2

b; (15)

where !nu=b ¼ hnjc ¼ 0=1i is the mean particle number in thevolume V corresponding to fully unburned/burned conditionsand analogously n2

u=b ¼ hn2jc ¼ 0=1i.For tracers, the average particle number ahead and

behind the front are not independent since they are related tothe respective fluid density. Hence, the ratio !nu=!nb equals thegas density ratio across the thin flame front which is esti-mated by the expansion ratio s¼Tb/Tu, !nu=!nb ¼ s. Consider-ing Eqs. (12), (14), and (15), the clustering index reads

K0ðx; ‘Þ ¼!nuð‘Þ

s!cðxÞ½1% !cðxÞ- s% 1ð Þ2

!cðxÞ þ s½1% !cðxÞ-; (16)

FIG. 2. (Color online) Instantaneous particle con-figuration in a thin slice of width R=20 through theaxis. Top-left: Stfl¼ 0.022; Top-right: Stfl¼ 0.54;Bottom-left Stfl¼ 2.16; Bottom-right Stfl¼ 8.65.Colors encode the fluid density at particle positions,see legend.



where the subscript 0 stands for tracer. Hence, according tothe flamelet model, the clustering index of purely Lagrangiantracers deviates from zero in the flame brush where0 < !c < 1, to vanish in the burned and unburned regionswhere the statistics comes back to Poissonian.

Relation (16) highlights that for light particles, a posi-tive clustering index, K0> 0, is mainly due to the fluctuationof the thin flame front which locally induces an intermittentstate switching between unburned (higher particle concentra-tion) and burned (lower concentration) conditions. The inter-mittency results in local density fluctuations, whichcorrespond to K0> 0. It is worth emphasizing that K0> 0 ishere due to the alternation between two purely Poissonainstates, which show no clustering when taken separately. Aswe will see this effect occurs also for inertial particles wherethe phenomenology is much more rich due to flame front

fluctuations/corrugations interacting with the typicalresponse time of the particle.

The bimodal description for the flame brush can beeasily extended to model the intermittent region separatingthe burned gas and the external particle-free environment.To this purpose we shall assume that the instantaneous inter-face between hot gas and environment is sufficiently thin, asexpected of high Reynolds number flows. Borrowing thenotion from the classical description of cold jets,34 we willrefer to this thin interface as the instantaneous viscous super-layer and will call super-layer brush the region spanned bythe fluctuating super-layer. Across the instantaneous inter-face, the average particle number jumps from !nb to !ne ¼ 0.After considering the limit se ¼ !nb=!ne !1, the emptyexternal region is formally characterized by a limiting Pois-sonian process with vanishing average particle number.According to the model, the clustering index pertaining tothe super-layer brush is described by relation (16) in the limits !1 with !nu replaced by !nb, i.e., K0 ¼ nb!ceðxÞ. In fullanalogy with the flame brush, the instantaneous indicatorfunction ce(x) is a singular bi-modal distribution which takesthe value 0 in the fully burned region and 1 in the externalfluid. Its average !ce goes smoothly from zero to one and itstransition defines the super-layer brush. The huge value of K,seen in Figure 3 in the outer part of the jet, can actually beinterpreted in this framework, after noting that the relevantStokes number should be defined in terms of the local timescale of the super-layer, which is typically larger than theflamelet characteristic time.

Figure 4 concerns the comparison between the numeri-cal results for the smallest, almost Lagrangian particles(Stfl¼ 0.022) and the model prediction (16). Despite thecrudeness of the model, the agreement is excellent.

FIG. 3. (Color online) Clustering index contours K, Eq. (12). Top-leftStfl¼ 0.022, top-right Stfl¼ 0.54, bottom-left Stfl¼ 2.16 and bottom-rightStfl¼ 8.65. The clustering index is evaluated on control volumes of charac-teristic size ‘¼ 0.125R. The solid lines (YR=Y0

R ¼ 0:05 and YR=Y0R ¼ 0:95)

delimit the flame brush.

FIG. 4. (Color online) Clustering index K0 for almost Lagrangian particles:the model (16) for tracers, left panel, is compared with the numerical resultsfor particles at Stfl¼ 0.022, right panel. See caption of Figure 3 for controlvolume dimensions and definition of the flame brush.



When the particle inertia is significant, the particle inter-action with the flame introduces additional effects dependingon Stfl, see Figure 3. The clustering index is in general largerthan the tracers. The peak clustering occurs within the flamebrush, though a significant value is also found in the burnedgas region. Indeed, the maximum peak clustering intensitytakes place for particles with Stfl¼ 2.16. The observation thatpeak clustering is achieved at flamelet Stokes numbers orderone confirms that Stfl is the actual parameter controllingparticle localization effects in turbulent premixed flames.Furthermore, it shows that the increased localization is asso-ciated with the increased relaxation time of the particles.

The model for tracers, Eq. (16), can be extended toinertial particles by assuming that the fluctuating flame frontseparates two states, now both non Poissonian and character-ized by !nu;b and dn2

u;b (hence Ku;b ¼ dn2u;b=!nu;b % 1) for the

unburned and burned mixtures, respectively. From the flame-let model, the same procedure used for Eqs. (14) and (15)yields

dn2 ¼ ð1% !cÞðn2u % nu

2Þ þ !cðn2b % nb

2Þ % 2!cð1% !cÞnu nb

¼ ð1% !cÞdn2u þ !cdn2

b þ !cð1% !cÞðnu % nbÞ2

¼ ð1% !cÞnuKu þ !c nbKb þ !cð1% !cÞðnu % nbÞ2 þ !n:

Hence, the clustering index becomes

K ¼ K0 þKusþ ðKb % KusÞ!cð1% sÞ!cþ s

; (17)

where K0 is the clustering index for tracers with a maximumin the flame brush, while the second contribution accountsfor the effect in the flame brush of the clustering occurring inthe pure states (Ku, b). In the range 0 . !c . 1, the latter is amonotone function of !c.

In fact, the data, we have, available show that K%K0 doexhibit a maximum inside the flame brush, suggesting thatadditional effects should be present beside the fluctuationbetween two pure non-Poissonian states. To investigate theseeffects, it is convenient to assume the tracer behavior as thereference condition. By defining the deviation of mean andvariance of particle number from the local !c-dependent valuepertaining to the tracers, !n0 and dn2

0, respectively,

!n ¼ !n0 þ D!n; (18)

dn2 ¼ dn20 þ Ddn2;

to first order in the corrections, the clustering index K(‘)reads

Kð‘Þ ’ K0 þDdn2

!n0% ðK0 þ 1ÞD

!n

!n0: (19)

In this approximation, the effects of inertia are lumped inthe last two terms in right-hand-side of Eq. (19). The first term,Ddn2=!n0, takes care of the inertia-induced alteration of the var-iance of the particle number in the control volume. The secondterm, %ðK0 þ 1ÞD!n=!n0, accounts for the variation of the meanparticle number. The agreement between the approximate clus-

tering index reported in Fig. 5 and the exact one reportedin Figure 3 is excellent. Figure 6 reports Ddn2=!n0 andðK0 þ 1ÞD!n=!n0 for particles with Stfl¼ 0.54;2.16;8.65. Con-cerning the most segregating particles, Stfl¼ 2.16, Ddn2=!n0 islarge in the flame brush region, especially near the burned gasside. We infer that the interaction between corrugated flame

FIG. 5. (Color online) Clustering index field reconstructed from relation(19). From left to right, Stfl¼ 0.54, Stfl¼ 2.16 and Stfl¼ 8.65. See caption ofFigure 3 for control volume dimensions and definition of the flame brush.

FIG. 6. (Color online) Contributions to the clustering index from the two

last terms in right-hand-side of Eq. (19). Top line: Ddn2=!n0. Bottom line:ðK0 þ 1ÞD!n=!n0. From left to right, Stfl¼ 0.54, Stfl¼ 2.16 and Stfl¼ 8.65. Seecaption of Figure 3 for control volume dimensions and definition of theflame brush.



front and particle inertia is a source of particle clustering viathe increase of the particle number fluctuation. The small posi-tive value of ðK0 þ 1ÞD!n=!n0 tends to slightly mitigate theincrease of the clustering index which, for this kind of particles(Stfl¼ 2.16), is quite large. For particles with larger inertia, e.g.,Stfl¼ 8.65, both corrections increase to a comparable valueresulting in a less intense global clustering. Consistently, maxi-mum clustering is actually achieved at intermediate Stokesnumber.

The data discussed so far show that a wrinkled, thinflame front has a more complex impact on the particle distri-butions than the simple alternation between two simple, non-Poissonian states. Specifically, the effect of inertia is to sig-nificantly enhance the fluctuation of the particle numberfound in a given (small) volume, Ddn2, leading to theincrease of the clustering index over the value which pertainsto purely Lagrangian tracers.

B. Experimental analysis

The results discussed in the context of DNS can beextended to actual experimental data. To this purpose an air/methane stoichiometric mixture, seeded with aluminaqp ’ 4000Kg=m3" #

particles with diameter of 10 lm, hasbeen injected into a Bunsen device with nozzle diameterD¼ 18mm at Reynolds number ReD¼ 8000. The character-istic time scale of the corresponding laminar flame front is ofthe order of sfl ’ 0:16ms (Ref. 23) leading to a flameletStokes number Stfl¼ 4.4.

Snapshots of the particle distributions are obtainedadopting a commercial PIV system based on a 532 nm,54 mJNd:YAG laser with a pulse-to-pulse delay of 70 ls. As usualthe laser beam goes through a cylindrical lens to generate alight sheet of thickness 0.45 mm. The camera is equippedwith a 60 mm focal length lens and with a 1280' 1024 pixelCCD corresponding to a field-of-view of 110.7 mm' 88.52mm. More details on the experimental setup can be found inRefs. 23 and 28.

The left panel of Figure 7 shows the raw data corre-sponding to a Mie scattering image. An image analysis algo-rithm is used to extract the particle positions and to count theparticle numbers in control volumes of dimension of1.26' 1.26' 0.45 mm (0.14' 0.14' 0.05 R) centered atdifferent positions within the field of view of the system.The box counting algorithm allows to determine the cluster-ing index field K by evaluating for each box mean particlenumber and variance. The clustering index field is reportedin the right panel of Figure 7. An estimate of the flame brushis obtained by looking at the iso-levels of mean particle con-centration. Its boundary is reported as thick black solid linesshown superimposed on the clustering index field. The typi-cal particle density in the experiments is nearly half the valueused in the DNS. The correspondence between numerics andexperiments is qualitatively excellent, see Figures 3 and 7,respectively. The quantitative discrepancy is fully explainedby considering that the clustering index depends strongly onthe local particle density through the average particle numberin the control volume. This effect is particularly apparent inEq. (16) for tracers. Accounting for the factor two in the

particle density between numerics and experiments allows toconclude that, also in quantitative terms, the agreement isreasonable.

As a note, the significant clustering found in the numeri-cal simulation in correspondence of the boundary betweenhot fluid and external environment (super-layer brush) doesnot show up in the experiment. The reason is that innumerics we are actually dealing with an unbound jet withno particles in the external region. On the contrary, in theexperiment, the Bunsen flame is placed inside a cylindricalvessel, whose lateral boundary is far enough from the jetaxis to prevent the alteration of the flame dynamics. Never-theless, the particles get trapped in the vessel so as to samplethe field also in the external part. As a consequence, no sub-stantial particle density contrast occurs between the hot jetand its environment.

V. THE RADIAL DISTRIBUTION FUNCTION

In alternative to the description in terms of deviationfrom the reference Poissonian distribution, particle clusteringis also conveniently described in terms of the radial distribu-tion functions g(‘), see Refs. 7 and 12. It provides the proba-bility to find a couple of particles with distance in theinterval [‘, ‘þ d‘] normalized with the probability thatwould result from a purely random spatially independentarrangement of particles. In presence of small scale cluster-ing, the probability to have a second particle in the vicinityof a previous one is much larger than expected on the basisof the random distribution, hence, the increase above one ofg(‘) at the given scale. Classically, the radial distributionfunction is introduced7,35 in the context of homogeneousflows as

gð‘Þ ¼ 1

4p‘2

dN ðlÞd‘

VT

N T ; (20)

where N T ¼ 05NðN % 1Þ is the total number of pairs in thesystem of volume VT and N ð‘Þ is the number of pairs in thesystem with distance less or equal to ‘. As shown in the

FIG. 7. (Color online) Experimental determination of the clustering indexin a Bunsen turbulent stoichiometric methane flame at ReD¼ 8000 endowedwith alumina particles of diameter dp¼ 10 lm, Stfl¼ 4.4. Mie scatteringimage, left panel, and clustering index, right panel. The solid lines demar-cate the flame brush region.



Appendix in full detail, this definition can be generalized toinhomogeneous flows as

gðx; ‘Þ ¼ dhncðx; ‘Þid‘

d‘

dhncðx; ‘Þi0; (21)

where nc x; ‘ð Þ, Eq. (A25), is the number of pairs found onaverage in a ball of radius ‘ centered at x and nc x; ‘ð Þ0 is thecorresponding number of pairs for a distribution with thesame particle density and vanishing pair correlation, seeAppendix.

It is well known7 that for homogeneous turbulent flowsg(‘) displays a scaling law at small separation, g(‘)& ‘%f.The scaling exponent f is known to be a function of theStokes number and quantifies the intensity of small scaleclustering. Actually, the exponent f is related to importantgeometrical features of the spatial distribution. Specifically,D2 ¼ 3% f is the so-called correlation dimension of themulti-fractal measure associated with the particle density.36

All these considerations extend to the present case where theradial distribution function depends on position, although thegeometrical multi-fractal characterization becomes morecomplex due to the strong inhomogeneity.

There is a direct connection between radial distributionfunction and clustering index. In homogeneous flows giventhe pair correlation function g(‘)¼ g(‘)% 1 one readily finds,see Appendix

K ¼ 4p!n

V

ð‘

0

gð~‘Þ~‘ 2d~‘: (22)

where the control volume is taken as a ball of radius ‘.For tracers, the flamelet model (16) predicts that K0=!n is

independent of the scale ‘, implying from Eq. (22) that thetracer radial distribution function g0 reads

g0 ¼ 1þ K0ð‘Þ!nð‘Þ

: (23)

We expect that prediction (23) holds only at sufficientlysmall scales since the model relies on control volumes ofscale ‘ much smaller than the flame brush thickness b.

Figure 8 shows g0 % 1 ¼ K0ð‘Þ=!nð‘Þ vs ‘/R for spherescentered on the axis of the jet at z¼ 7R. The squares areDNS data for almost Lagrangian particles with Stfl¼ 0.022.As apparent from the figure, g0% 1 is nearly constant atsmall scales as expected from Eq. (23). The circles are theestimate obtained from relation (16)

g0 % 1 ¼ K0

!n¼ 1

s!cðxÞ½1% !cðxÞ- s% 1ð Þ2

ð!cðxÞ þ s½1% !cðxÞ-Þ2; (24)

where !c is taken from the DNS. The behavior at large scale,where the inhomogeneity of the flow is dominant, is irrele-vant to our purposes. The small departure from the constantvalue at small scale evidenced by the fit (0.02R<‘< 0.07R,solid line) is related to numerical inaccuracies in the interpo-lation of !c from the cylindrical grid to the small spheres cen-tered on the axis of the jet. Nonetheless, as apparent from thefigure, the agreement is quite reasonable in the range of

scales where we expect the theory to work properly. Westress that the kind of behavior just described is limited topurely Lagrangian tracers. For inertial particles instead weshould expect g ‘ð Þ / ‘%f, hence K ‘ð Þ / ‘3%f.

The radial distribution function g(x, ‘) at three differentaxial positions along the jet axis is presented in Fig. 9 for thefour particle populations. The three positions are selected tocheck the particle system in the reactants z¼ 3R, in the flamebrush z¼ 7R and in the product region z¼ 12R, respectively.The three positions are best visualized in Figure 3 where thethick lines denote the boundary of the flame brush. The gen-eralized definition of radial distribution function complieswith the requirement that it should approach one for large‘/R. In the reactants (top panel of Figure 9, z¼ 3R) g(‘/R)slightly exceeds one at small scales ‘=R+ 1. The slope ofthe logarithmic plot increases with the Stokes number. At thisstation, in the reactant region, the behavior is entirely consist-ent with the picture of particle clustering in cold flows.7,12

The middle panel of Figure 9 more or less correspondsto a position well inside to the flame brush, z¼ 7R. Notewor-thy g(x, ‘/R) gives evidence of substantial intermittency forall the particle populations, also for the lightest onesStfl¼ 0.022* 0.54. At small scales, the value considerablyexceeds one with a significantly increased slope. As for theclustering index, the basic effect already occurs for tracers(compare, e.g., the open squares in the reactants and in theflame brush, top and middle panel of Figure 9, respectively),it is enhanced by inertia and peaks for order one flameletStokes number. Actually, the radial distribution functionconfirms that particles with Stfl¼ 2.16 present maximumclustering consistently with the conclusion based on the anal-ysis of K, Figure 3.

Based on the radial distribution function, further down-stream, at z¼ 12R the lightest particles recover a more regu-lar spatial arrangement. The visual inspection of theinstantaneous configurations in Figure 2 confirms this trendshowing a reduced occurrence of particle clusters. The radialdistribution function of the heaviest particles instead still

FIG. 8. (Color online) Normalized clustering index, Kð‘Þ=!nð‘Þ, vs scale ‘=Rfor quasi-Lagrangian particles, Stfl¼ 0.022, at axial station z¼ 7R. Squares(red), DNS data; circles (blue), estimate from Eq. (16); solid line, fit of datafrom Eq. (16).



show intense clustering effects, see the instantaneous field ofFigure 2 with intense aggregates separated by definite voids.

We conclude that combustion has a strong influence onparticle cluster formation also for tiny particles. The effect

of inertia is significant and, as understood by the fact thatpeak clustering takes place at order one flamelet Stokes num-ber, it is related to the typical time scale of the flame front,which is considerably shorter than the turbulent time scale.

VI. REMARKS AND DISCUSSION

The turbulent fluctuations of thin flame fronts, like thoseoccurring in turbulent premixed flames, have been hereshown to strongly affect the instantaneous spatial distributionof the particles, inducing intermittent effect which cannot bepredicted in terms of mean fields. Specifically, data of adirect numerical simulation of a Bunsen premixed flameendowed with a particulate phase present strong particleclustering in the region spanned by the instantaneous flame(flame brush). With respect to standard localization effects incold flows,7–9 a peculiarity of this system is that evenLagrangian tracers which are purely advected by the flowmanifest intermittent distributions which lead to an overallnon-Poissonian statistics in the flame brush. To isolate thiseffect from that of particle inertia a simple model of thetracer behavior in turbulent premixed flames has been pro-posed to explain the observations. The model shows that thefluctuations of the instantaneous turbulent flame front inducethe alternation between two states characterized by two dif-ferent tracer densities which result in a positive clusteringindex. Inertia introduces additional effects which lead tolarge intensification of the clustering especially in the flamebrush region. The relevant parameter controlling clusteringis the flamelet Stokes number, ratio of the particle relaxationtime and the characteristic time scale of the flame front.

Table I provides a comparison between the flameletStokes number of the particle populations we have addressedin comparison with the corresponding Stokes numbers basedon the Kolmogorov time scale of the turbulence both in thefresh and in the burned gases. In reading, the data one shouldbe aware of the decrease of the Kolmogorov time scale asso-ciated with the change of density, viscosity, and dissipationinduced by the flame. In the fresh gases, the observed clus-tering is consistent with the classical picture7,37 for coldflows where Kolmogorov like turbulent fluctuations lead tosmall scale particle clustering for order one Stg, where theKolmogorov-Stokes number is based on the characteristictime associated with the Kolmogorov scale. Due to tempera-ture increase, Stg decreases substantially in the flame brushclearly indicating that the relevant time scale associated withthe clustering process in that region is significantly shorterthan the Kolmogorov time scale. Actually, Table I shows

FIG. 9. (Color online) Radial distribution function g(‘=R) for the differentparticles populations at different axial stations. From top to bottom: z¼ 3R,z¼ 7R and z¼ 12R.

TABLE I. Different possible definitions of the Stokes number for the fourparticle populations. From top to bottom: flamelet Stokes number, Eq. (5)and Kolmogorov Stokes number, Eq. (3), based on unburned and burned

conditions, respectively. The relevant Kolmogorov time scale is evaluated incorrespondence of the jet axis in the fully unburned and burned regions,

respectively.

Stfl 0.022 0.54 2.16 8.65

Stug 0.01 0.25 1. 4.

Stbg 0.00125 0.03125 0.125 0.5



that the appropriate time scale is here provided by the flamefront characteristic time, leading to order one flamelet Stokesnumber for the most segregating particles. The clear conclu-sion is that the effect is associated with the strong expansionof the flow across the flame. The phenomenology is, how-ever, substantially more complex than expected on the basisof the behavior of tracers, as understood from the fact thatthe spatial variation of the net segregation cannot bedescribed by the simple random alternation between twopure elementary states. We expect an interaction between theflame corrugations and the particles dynamics, a conjecturethat we are presently unable to check with the data availableto us which pertains to a single turbulent flame.

We are convinced that the phenomenology we describeis rather robust and, as an overall check, we have been ableto produce the experimental data taken from a premixed tur-bulent Bunsen flame seeded with 10 lm diameter aluminaparticles, which confirm the numerical results. Actually, thequalitative correspondence between numerics and experi-ments can be made quantitatively acceptable by consideringthe difference in average particle density between experi-ment and numerics. A more refined data analysis in terms ofradial distribution function confirms the global picture andallows to account for the scale dependence of the clusteringprocess.

To conclude, we have shown that flows with strong den-sity gradients, such as flames or shock waves, give rise tostrong intermittent features in inertial particle distributionswhich are essentially different from the classical segregationphenomena found in cold turbulence interacting with inertialparticles.

We stress that the present findings are expected to have acertain relevance for applications such as PIV for turbulentflames23 and combustion control through fuel/water droplets.Concerning PIV, cross-correlation algorithms for velocitymeasurements require the presence of a minimum number ofparticles in both the interrogation windows which images aredivided into. Particle voids with dimensions comparable to thewindow would drastically reduce the particle number,increase noise and uncertainty and make the estimate ofhigher order moments, e.g., velocity rms, extremely difficult.Under this respect, it should be noted that the operating condi-tions may be very usually close to the most critical situation.

ACKNOWLEDGMENTS

We want to thank the COST Action MP0806 Particles inTurbulence for supporting the present work and CASPUR com-puting Consortium where the simulations were performed.

APPENDIX: CLUSTERING INDEX AND RADIALDISTRIBUTION FUNCTION IN INHOMOGENEOUSCONDITIONS

Taken N particles in a domain D with volume VD, theprobability to find the particles in the configuration x1,…, xN is

pðx1;…; xN; xÞ ¼ hd½x1 % bx1ðxÞ-…d½xN % bxNðxÞ-i; (A1)

where x is the stochastic variable belonging to a space S,which determines the instantaneous configuration of the par-ticles. The characteristic function of a certain domain, a ballB‘ of radius ‘, say,

HðxÞ ¼1 x 2 B‘

0 x 62 B‘

8<

: ; (A2)

allows to count the instantaneous number of particles in thedomain,

nðB‘;xÞ ¼XN

p¼1

H½bxpðxÞ-: (A3)

The rth statistical momentum of the particle number in thedomain is

hnri ¼ð

S

XN

p¼1

H½bxpðxÞ-

!r

pðxÞdx

¼ð

DN

XN

p¼1

HðxpÞ

!r

pðx1;…; xNÞdx1…dxN; (A4)

where DN ¼ D'…'Dzfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{N

is the N-fold Cartesian product ofD, which specifies the phase space of the particle system.From Eq. (A4), the mean particle number in the ball B‘ is

hni ¼ð

DN

XN

p¼1

HðxpÞ

!

pðx1;…; xNÞdx1…dxN

¼ N

ð

DNHðx1Þpðx1; …; xNÞdx1…dxN

¼ N

ð

DNHðx1Þpðx1Þdx1

¼ N

ð

B‘pðx1Þdx1; (A5)

where the second line follows from the identity of all the par-ticles. The second moment of the number of particles in B‘ is

hn2i ¼ð

DN

XN

p¼1

HðxpÞ

" #XN

q¼1

HðxqÞ

" #

pðx1;…; xNÞdx1…dxN

¼XN

p;q¼1

ð

DNHðxpÞHðxqÞpðx1;…; xNÞdx1…dxN

¼ N

ð

DNHðx1Þ2pðx1; …; xNÞdx1…dxN

þ NðN % 1Þð

DNHðx1ÞHðx2Þpðx1; x2; …; xNÞdx1…dxN

¼ N

ð

B‘pðx1Þdx1 þ ðN2 % NÞ

ð

B‘'B‘pðx1; x2Þdx1dx2:

(A6)

Since p(x1, x2)¼ p(x1)p(x2)[1 þg (x1, x2)], with g(x1, x2) thepair correlation function, Eq. (A6) reads



hn2i¼N

ð

B‘pðx1Þdx1þN2

ð

B‘pðx1Þdx1

& '2

%N

ð

B‘pðx1Þdx1

& '2

þN2

ð

B‘'B‘pðx1Þpðx2Þgðx1;x2Þdx1dx2

%N

ð

B‘'B‘pðx1Þpðx2Þgðx1;x2Þdx1dx2¼hniþhni2

%hni2

NþNðN%1Þ

ð

B‘'B‘pðx1Þpðx2Þgðx1;x2Þdx1dx2:

(A7)

Considering the definition of clustering index, Eq. (12), andexpression (A5), the change of variables x1¼ x andx2¼ xþ ‘, yields

K¼%hniNþNðN%1Þ

hni

ð

B‘'B‘pðxÞpðxþ ‘Þgðx;‘Þdxd‘: (A8)

For homogeneous systems p(x)¼ p(xþ ‘)¼ p0. Hence,Eq. (A8) reads

K ¼ % hniNþ NðN % 1Þ

hni

ð

B‘'B‘p2

0gð‘Þdxd‘

¼ % hniNþ NðN % 1Þ

hnihni

NVB‘

* +2

VB‘

ð

B‘gð‘Þd‘; (A9)

where we used Eq. (A5) to express p0. When the volume VB‘of B‘ is much smaller than VD, the average number ofparticles in B‘ is much smaller than the total number of par-ticles, n+ N which is assumed to be a significantly largenumber, N " 1. Hence, the relation between clusteringindex and pair correlation function g(‘) becomes

K ¼ hniVB‘

ð

B‘gð‘Þd‘: (A10)

Under the same assumptions concerning the volume VB‘ andthe total number of particles, the general expression (A8) forthe clustering index for non-homogeneous statistics reduces to

K ¼ N2

hni

ð

B‘'B‘pðxÞpðxþ ‘Þgðx; ‘Þdxd‘: (A11)

In order to connect the definition of clustering index withother relevant statistics of the particle system, let us intro-duce the particle distribution associated with a single particleconfiguration, which, for conciseness, will be referred to asthe instantaneous particle distribution,

cpðxÞ ¼XN

k¼1

d xkðxÞ % x½ -: (A12)

Its average is the particle density

CðxÞ ¼ hXN

k¼1

d xkðxÞ % x½ -i

¼ Nhd x1ðxÞ % x½ -i ¼ Np1ðxÞ; (A13)

whose volume integral provides the average number of par-ticles in a given domain, (A5). The instantaneous particlenumber inside a domain B‘ðxÞ centered around x is

n x;B‘ðxÞ½ - ¼ð

B‘ðxÞcpð~xÞd~x: (A14)

Under the assumption that the two domains B‘ðxÞ and B‘0ðyÞare disjoint ðB‘ðxÞ \ B‘0ðyÞ ¼ ;Þ the number of particlepairs, nc, with one particle in B‘ðxÞ and the other in B‘0ðyÞ is

nc x;B‘ðxÞ;B‘0ðyÞ½ - ¼ð

B‘ðxÞ'B‘0 ðyÞcpð~xÞcpð~yÞd~xd~y: (A15)

In general, the two sets may have non empty intersection. Inthis case, we may partition each of them in the form

B‘ðxÞ ¼ B‘ðxÞnB‘0ðyÞ½ - [ B‘ðxÞ \ B‘0ðyÞ½ -;

and

B‘0ðyÞ ¼ B‘0ðyÞnB‘ðxÞ½ - [ B‘ðxÞ \ B‘0ðyÞ½ -;

respectively. The Cartesian product of the two sets can thenbe rearranged as

B‘ðxÞ ' B‘0ðyÞ ¼ B‘ðxÞ' B‘0ðyÞn B‘ðxÞ \ B‘0ðyÞ½ - ' B‘ðxÞ \ B‘0ðyÞ½ -f g[B‘ðxÞ \ B‘0ðyÞ½ - ' B‘ðxÞ \ B‘0ðyÞ½ -f g¼ B‘ðxÞ ' B‘0ðyÞ½ -DJ[ B‘ðxÞ ' B‘0ðyÞ½ -J ;

where the last line decomposes the Cartesian product in thejoint and disjoint parts.

To avoid counting the same couple twice when dealingwith the intersection of the two sets, the expression for theinstantaneous number of couples should include a factor 1/2in front of the integral over the joint part.

nc x;B‘ðxÞ;B‘0ðyÞ½ - ¼ð

B‘ðxÞ'B‘0 ðyÞ½ -DJ

cpð~xÞcpð~yÞd~xd~y

1

2

ð

B‘ðxÞ'B‘0 ðyÞ½ -Jcpð~xÞcpð~yÞd~xd~y: (A16)

The correlation of the particle distribution is defined as

hcpðxÞcpðyÞi¼ hX

k;r¼1

Nr 6¼kd xkðxÞ % x½ -d xrðxÞ % y½ -i

¼ NðN% 1Þhd x1ðxÞ % x½ -d x2ðxÞ % y½ -i¼ NðN% 1Þp12ðx;yÞ

¼ ðN% 1ÞN

CðxÞCðyÞ 1þ gðx;yÞ½ -

¼ CðxÞCðyÞ 1þ gðx; yÞ½ -; (A17)

where we assume N to be large enough to neglect 1/N withrespect to 1. Hence, the average number of pairs is

hnc B‘ðxÞ;B‘0ðyÞ½ -i¼ð

B‘ðxÞ'B‘0 ðyÞ½ -DJ

Cð~xÞCð~yÞ 1þgð~x;~yÞ½ -d~xd~y

þ1

2

ð

B‘ðxÞ'B‘0 ðyÞ½ -JCð~xÞCð~yÞ 1þgð~x;~yÞ½ -d~xd~y:

(A18)



This allows to introduce the density of pairs with respect tothe volume of B‘ðxÞ,

h.ci ¼ limVB‘!0

hnciVB‘

¼ limVB‘!0

Ð

B‘ðxÞ'B‘0 ðyÞf ð~x; ~yÞCð~xÞCð~yÞ 1þ gð~x; ~yÞ½ -d~xd~y

Ð

B‘ðxÞdex

¼ð

B‘0 ðyÞf ðx; ~yÞCðxÞCð~yÞ 1þ gðx; ~yÞ½ -dey; ðA19Þ

where f(x, y)¼ 1 when (x, y) belong to the disjoined part ofthe integration domain and f(x, y)¼ 1/2 in the complemen-tary set. In fact f(x, y)¼ 1 when x 62 B‘0 and f(x, y)¼ 1/2when x 2 B

/‘0 .

With ~y ¼ yþ e‘ relation Eq. (A19) reads

h.ciðx; y; ‘0Þ ¼ CðxÞð

4pd‘

ð‘0

0

f ðx; yþ e‘ÞCðyþ e‘Þ

1þ gðx; yþ e‘Þh i

~‘2d~‘: (A20)

We are interested in having x¼ y, i.e., coincident centers ofthe two balls B‘ and B‘0 and f¼ 1/2,

h.ci ¼ CðxÞð

4pd‘

ð‘0

0

1

2Cðxþ e‘Þ 1þ gðx; e‘Þ

h i~‘2d~‘; (A21)

since we intend to consider the extension, g(x, ‘0), of the

classical concept of radial distribution function7,35 to inho-mogeneous conditions,

gðx; ‘0Þ ¼ dh.cid‘0

d‘0

dh.ci0¼ dh.ci

dh.ci0; (A22)

where h.ci0 corresponds to a particle distribution with thesame concentration C and vanishing pair correlation g (inde-pendently distributed particles in space, characterized by noclustering K¼ 0, (A11)). Inserting Eq. (A21) in Eq. (A22),we get

gðx; ‘0Þ ¼

Ð

4pCðxþ e‘Þ 1þ gðx; e‘Þ

h id‘

Ð

4pCðxþ e‘Þdb‘

: (A23)

We note, in passing, that by localizing Eq. (A23) in a specific

direction ‘ we may retain information on the angular depend-ence by addressing the angular distribution function (see alsoRef. 12 for this concept in anisotropic homogeneous flows),

gðx; ‘0; ‘Þ ¼ 1þ gðx; ‘0; ‘Þ: (A24)

The present definition of radial distribution function can berecast in a form better suited for a direct comparison with theclassical one used in homogeneous isotropic flows. Let usconsider the number of particle pairs contained in a sphere ofradius ‘. This amounts to considering again (A18) by takingy¼ x and ‘

0¼ ‘, i.e., B‘0ðyÞ 0 B‘ðxÞ,

hnci ¼ ð4pÞ2ð‘

0

ð‘

0

‘2x‘

2y

1

2Cðxþ ‘xÞCðxþ ‘yÞ

1þ gðxþ ‘x; xþ ‘yÞ$ %

d‘xd‘y; (A25)

where the factor is now f¼ 1/2. The derivative of nc withrespect to the radius of the sphere ‘ reads

dhncid‘¼ ð4pÞ2‘2Cðxþ ‘Þð‘

0

~‘2Cðxþ ~‘Þ 1þ gðxþ ‘; xþ ~‘Þ$ %

d~‘: (A26)

The corresponding derivative of the number of pairs onewould have with same particle density C and vanishing paircorrelation function is

dhnci0d‘

¼ ð4pÞ2‘2Cðxþ ‘Þð‘

0

~‘2Cðxþ ~‘Þd~‘: (A27)

Taking the ratio of the two expressions above, by compari-son with Eq. (A23), we obtain

gðx; ‘Þ ¼ dhncid‘

d‘

dhnci0¼ dhnci

dhnci0: (A28)

For homogeneous systems, the derivative of the number ofpairs for zero pair correlation function reduces to

dhnci0d‘

¼ 4p‘2C20

4p‘3

3¼ 4p‘2 hni

2

VB‘; (A29)

where we used that C0 ¼ hni=VB‘ . In conclusion, for homo-geneous systems, the radial distribution function g(‘) followsas:

gð‘Þ ¼ 1

4p‘2

dhncid‘

VB‘hni2

: (A30)

Let us show that Eq. (A30) exactly coincides with the defini-tion of radial distribution function provided, e.g., in Ref. 35for homogeneous systems. There a system of N particles isconsidered. Given the position xk of the kth particle, wedenote by nk(‘) the number of particles within a ball of radius‘ centered at xk excluded the kth particle itself. We callN ðlÞ ¼ 1=2

PNk¼1 nkð‘Þ the number of pairs in the whole

system, whose relative distance is less or equal to ‘. Clearly

N ðlÞ ¼ 1

2N

1

N

XN

k¼1

nkð‘Þ

!

¼ 1

2Nhnð‘Þi:

The number of pairs within a shell of radius ‘ and thicknessd‘ is

DN ðlÞ ¼ 1

2N

1

N

XN

k¼1

Dnkð‘Þ

!

¼ 1

2NhDnð‘Þi:

The number of pairs in the shell is normalized by the total num-ber of pairs in the system,N T ¼ 1=2NðN % 1Þ ’ 1=2N2,



DN ðlÞN T ¼ hDnð‘Þi

N:

Let us multiply both members of the equality by VT/DV,

DN ðlÞDV

VT

N T ¼hDnð‘Þi

DV

VT

N:

Since N=VT ¼ hnðlÞi=Vð‘Þ ¼ C0, we have

DN ðlÞDV

VT

N T ¼hDnð‘Þi

DV

Vð‘Þhnð‘Þi

:

Let us take the limit for vanishingly small thickness of theshell,

dN ðlÞdV

VT

N T ¼1

4p‘2

dhnð‘Þid‘

Vð‘Þhnð‘Þi

¼ hnð‘Þi4p‘2

dhnð‘Þid‘

Vð‘Þhnð‘Þi2

:

Considering that n ‘ð Þdn ‘ð Þ ¼ dnc, with nc the average numberof particles pairs within the ball of radius ‘, we finally get

1

4p‘2

dN ðlÞd‘

VT

N T ¼dN ðlÞ

dV

VT

N T ¼1

4p‘2

dhncid‘

Vð‘Þhnð‘Þi2

:

The left hand side of the above expression is the definition ofradial distribution function according to, e.g., Ref. 35, whilethe right hand side, Eq. (A30), is the form our generalizeddefinition of radial distribution function, Eq. (A28), takesunder the assumption of homogeneous particle distribution.

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Intermittent features of inertial particle distributions in turbulent premixed flame

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