Turbulent consumption speed via local dilatation rate measurements in a premixed bunsen jet

Combustion and Flame 160 (2013) 2029–2037

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Combustion and Flame

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Turbulent consumption speed via local dilatation rate measurementsin a premixed bunsen jet

0010-2180/$ - see front matter � 2013 The Combustion Institute. Published by Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.combustflame.2013.04.013

⇑ Corresponding author.E-mail address: [email protected] (G. Troiani).

Guido Troiani a,⇑, Francesco Battista b, Francesco Picano c

a Sustainable Combustion Laboratory, ENEA C.R. Casaccia, via Anguillarese 301, 00123 Rome, Italyb Department of Mechanics and Aerospace Engineering, University ‘‘La Sapienza’’, via Eudossiana 18, 00184 Rome, Italyc Linné Flow Centre, KTH Mechanics, Osquars Backe 18, SE-100 44 Stockholm, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 May 2012Received in revised form 13 November 2012Accepted 23 April 2013Available online 3 June 2013

Keywords:Premixed combustionTurbulent combustionFlameletPIVLIFFlame stretch

The mean local reaction rate related to the average expansion across the front and computed from themean velocity divergence is evaluated in this work. Measurements are carried out in a air/methane pre-mixed jet flame by combined PIV/LIF acquisitions. The procedure serves the purpose of obtaining valuesof a turbulent flame speed, namely the local turbulent consumption speed SLC, as a function of the posi-tion along the bunsen flame. With the further position that the flamelet assumption provides a propor-tionality between turbulent burning speed normalized with the laminar unstretched one and theturbulent to average flame surface ratio, the proportionality constant, i.e., the stretching factor becomesavailable. The results achieved so far show the existence of a wide region along which the bunsen flamefront has a constant stretching factor which apparently depends only on the ratio between turbulent fluc-tuations and laminar flame speed and on the jet Reynolds number.

� 2013 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

A common assumption in turbulent premixed combustionmodeling is that of flamelet regime, where the thin reactive flamefront is conveyed by the flow field. This front, being chemicallyreactive, propagates normal to itself towards the fresh reactantsof a premixed mixture at a velocity usually referred to as laminarcombustion velocity SL. As a consequence, the combustion rate _mcan be thought as proportional to the product of the laminarcombustion velocity with the reactive surface area. At a basic levelof complexity, with SL considered constant, the combustion rateincreases linearly with the flamelet surface. Actually, experimentalmeasurements in turbulent flames [1] suggest a non-linear growthof combustion rates at increasing turbulence levels, implying anon-trivial behavior of the laminar combustion velocity.

First attempts to model dependencies for SL began after Darri-eus–Landau [2,3] studies on laminar flame front instabilities. Intheir original formulation they considered an inviscid initially flatsurface of discontinuity separating two zones at different but con-stant densities mimicking a thin laminar flame front dividing reac-tants from combustion products. Boundary conditions at theinterface are such to impose a normal burning velocity equal tolaminar flame speed SLo = SL (function of thermo-chemical

parameters only), a constant mass flux and no jump of tangentialvelocity across the flame. The results is a linear and unstablegrowth of disturbances at all wave-lengths k of the kind ofa = Aexp(Xt ± i(kyy + kzz)), where a is the front position, X / SLokthe inverse of the time constant of the harmonic perturbation ap-plied to the front, y and z the transverse directions and x that nor-mal to the unperturbed front. Since the system is consideredinviscid and the only characteristic length is the flame front thick-ness, a simple dimensional prediction can directly lead to theabove expression for X. This unconditional instability has beenconfuted by experiments that undoubtedly can reproduce stableflames, as in cellular flames, see e.g., [4]. In particular, shorterwavelengths are thought to initiate transport mechanisms insidethe flame, influencing the flame structure and the normal burningvelocity such that SLo – SL. To take into account these phenomena,Markstein [5] prescribed different boundary conditions at theinterface, introducing a dependence of front velocity SL from itscurvature (1/R)

SL � SLo ¼ SLoðL=RÞ; ð1Þ

where L is the Markstein length, function of diffusive properties ofthe reactive mixture and of the order of the flame front thickness.This new boundary condition adds to the dispersion relation of Xa quadratic term (k2) which stabilizes the small wavelengths in sucha way that only a limited range of wavenumbers close to zero re-mains unstable.

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In a further step, Markstein [6] refined his previous positionassuming that the relevant quantity controlling the flame velocityis the curvature of the front with respect to the curvature of theflow measured by the divergence of the velocity flow field,

SL � SLo ¼ SLoL1Rþ $ � u

juj

� �� �; ð2Þ

where � denotes the perturbation from the plane solution.This modeling is also consistent with Karlovitz concept of flame

stretch [7], that is the relevant quantity controlling local flamespeed SL and flame quenching too. A further step towards the def-inition of suitable dispersion relation of X is that of consideringalso the effects of flow inhomogeneities upstream the flame front[8,9]. The result now is that flame velocity depends on the flamestretch k = 1/r(dr/dt) caused either by curvature effects ortangential velocity gradients at the interface,

SL

SLo� 1

� �¼ � L

SLo

1r

drdt

� �; ð3Þ

with r the elementary area defined on the flame front. Each pointbelonging to this area moves with a tangential velocity equal to thatof the flow ahead the flame surface.

A number of experiments [1] mostly based on Eq. (3) and aimedat the evaluation of SLo and L have been performed in the past. Themost common configuration adopted for this kind of measure-ments is that provided by laminar spherical flames expanding ina quiescent ambient. Reasons for this choice are the relative simpletracking of the the flame radius r(t) of the traveling front and ananalytic expression for flame stretch k = 2/r(dr/dt).

Coming back to turbulent combustion modeling issues, thecombustion rate _m can be associated by means of the continuityequation not only to the laminar velocity SL and to a fluctuatingflame front surface AT, but also to a reference (usually mean) frontposition (of extension Ao) and to an equivalent velocity, i.e., the tur-bulent consumption speed Sc [1]:

_m ¼ quSLAT ¼ quScAo; ð4Þ

with qu the unburned mixture density. Consequently,

Sc

SL¼ AT

Ao; ð5Þ

from which, by means of Eq. (3), it follows

Sc

SLo¼ Io

AT

Ao; ð6Þ

with the stretching factor Io grouping the dependencies of the ratioSL/SLo [1].

As a matter of fact, estimates of Io are typically performed by2D–3D numerical simulations, where combustion rate and surfaceevolution are instantaneously available in the whole computa-tional domain [10,11]. Concerning experimental measurements,results can be obtained with reasonable effort only for simpleand highly symmetrical configurations, i.e. spherical or flat flames[12]. When geometries are only slightly more complicated, as forbunsen flames, the task becomes a challenge, since the measure-ment of the local turbulent combustion rate is extremely difficult.This is the case of the present work aimed at estimating the timeaveraged local combustion rate in a turbulent premixed jet flamefed with a methane–air mixture by combined PIV/LIF measure-ments. These measurements are instrumental in the evaluationof local and global turbulent consumption speeds. Comparisonwith global turbulent burning velocity data found in literature[13–15] provides confirmations that assumptions made for thepresent flames and discussed in details in next sections areappropriate.

Another key point addressed in this work is the investigation ofthe variability of the stretching factor Io along the turbulent flamebrush. In general, the degree of universality of the stretching factorIo is not obvious and the geometry dependency/independency isstill debated [1]. It is found that downstream a transitional zoneat the exit nozzle, whose extension seems to depend on Reynoldsnumber, local stretching factor evaluated by means of Eq. (6) as-sumes constant values larger than unity. Such values may varywith experiments differing to each other from Reynolds numberand equivalence ratio, that are global observables easily measur-able and predictable. This could be of importance in numericalmodeling where one of the major concerns is the definition of astretched laminar flame speed.

2. Governing equations

When a flamelet description of the problem is taken into ac-count, the dynamics of propagating flame fronts at Low-Mach con-ditions and Lewis number unity can be defined in terms of aprogress variable c, e.g. a normalized temperature c = (T � Tu)/(Tb -� Tu) with subscript u/b indicating unburned/burned mixture [16].Given the density of the mixture q and a molecular diffusion D anadvection/diffusion equation ruling the conservation of c reads

@ðqcÞ@tþr � ðqucÞ ¼ _xþr � ðqD$cÞ: ð7Þ

Its integration over a control volume V, embedding the wholeturbulent flame brush, leads to the determination of the averagemass burning rate introduced in Eq. (4), [13]

_m ¼ZV

_xdV ¼ quScAo: ð8Þ

From the experimental point of view, two problems arise whendealing with this definition. First, the reaction rate ( _x) cannot bemeasured directly; second, the consumption speed Sc dependsfrom the choice of the reference area Ao. Attempts to measure _xindirectly through the determination of the mass of reactants flow-ing by a control volume [17], have been performed in the past. An-other method consists in computing the divergence of theunconditioned average velocity field, as a measure of the dilatationeffects of temperature increase. In particular, the latter techniquehas been applied to a flame flowing towards a stagnation plate,so to have an almost statistical flat flame front [18].

More specifically, continuity equation can be written in theform:

r � u ¼ q@ð1=qÞ@t

þ qu � r 1q: ð9Þ

Now, defining the heat release parameter as

s ¼ ðTb � TuÞ=Tu ¼ ðqu � qbÞ=qb; ð10Þ

it can be re-casted in terms of the progress variable c and unburneddensity in order to obtain [19–23] and [24].

r � u ¼ squ

@ðqcÞ@tþr � ðqucÞ

� �; ð11Þ

which is the LHS of Eq. (7) [25,26]. In principle, the coupling of Eqs.(7) and (11) could lead to the evaluation of the reaction rate _x pro-vided that the molecular diffusion term is properly defined. Fromthe experimental point of view this is not obvious when the flamefront is an interface dividing reactants from products and the pro-gress variable is known in its average distribution only. To over-come this difficulty it is considered the averaged form of Eq. (7)

@ð�q~cÞ@tþr � ð�q~u~c þ �q gu00c00 Þ ¼ r � ðqDrcÞ þ �_x; ð12Þ

Fig. 1. Sketch of control surface SðVÞ.

G. Troiani et al. / Combustion and Flame 160 (2013) 2029–2037 2031

with ~� Favre averaging. In non dimensional formulation the previousequation assumes the form (for the sake of brevity non dimensionalvariables are indicated with the same notation used for thosedimensional)

@ð�q~cÞ@tþr � ð�q~u~c þ �q gu00c00 Þ ¼ 1

RePrr � ðqDrcÞ þ _x; ð13Þ

with the Strouhal number St = 1, Re the Reynolds number and Pr thePrandtl number, i.e., viscous to thermal diffusivity. Note that withthe flamelet hypothesis of the unity of Lewis number, Prandtl andSchmidt numbers are equivalent and in a air–methane flame of or-der one. When Reynolds number is sufficiently high as in turbulentflames the average field described by Eq. (13) is independent frommolecular diffusivity, being this therm OðRe�1Þ in contrast to otherterms that are Oð1Þ. Under this hypothesis, Eqs. (13) and (11) (in itsaveraged form) can be recasted together to give

r � �u ¼ squð _xÞ: ð14Þ

Another way to highlight the independence of the averagedadvection/diffusion equation of c from molecular diffusivity is toconsider its integral formulation. The integration of Eqs. (12) and(11) (in its averaged form) over the same volume V used in Eq.(8), after Green–Gauss theorem application, results inZV

qu

sr � �udV ¼

ZV

�_xdV þZ

SðVÞðqDrcÞ � ndS; ð15Þ

being n ¼ �r�c=jr�cj. It is worth mentioning that the volumeV and itsrelative surface SðVÞ enclose the whole volume in which the meanprogress variable �c goes from 0 to 1. A suitable treatment of the termðqDrcÞ is of difficult solution in experiments unless some simplify-ing hypotheses are made. In particular, both density and moleculardiffusivity coefficient are supposed to be constant and equal to aneffective value between burned and unburned regions, qeff and Deff.The result is that density and diffusivity do not participate to theaverage operation, i.e., ðqeff DeffrcÞ ¼ ðqeff Deffr�cÞ.1 When turbulentflame brush is far from cold boundaries, the surface SðVÞ can be di-vided into subdomains of the kind of those exemplified in Fig. 1 andthe surface integral term in Eq. (15) evaluated on the poligon definedby A � B � C � D vertices. The direction defined by r�c is normal toisolines �c ¼ const. By construction the scalar products n2 � r�c andn4 � r�c are null while n1 � r�c and n3 � r�c depend from the value

1 Note that, in the limit of Bray–Moss–Libby model [16,27] the Favre averagedprogress variable eC can be recasted in terms of the mean progress variable C and theburned/unburned densities, such that the parallelism between reC and rC isguaranteed.

assumed by r�c that for �c ’ 0 and �c ’ 1 is zero. The result is that thesurface integral is identically zero along the whole flame brush.

Within these two point of views it is always possible to recallfrom volume integral of Eqs. (14) and (15)

qu

s

ZVr � �udV ¼

ZV

_xdV ¼ _m; ð16Þ

which allows, by means of Eq. (8), to define the global turbulentconsumption speed:

SGC ¼1

Aos

ZVr � �udV : ð17Þ

Note that it is used a different notation, SGC in place of Sc, tohighlight the assumptions made to neglect diffusivity. Integrationover the whole flame brush volume would lead to an overall turbu-lent consumption speed of the flame. In this case we are interestedin a local characterization of the burning speed. To this purpose,the elementary volume dV can be associated to iso-�c surfaces withan arbitrary mean curvature and decomposed into the product A(s)ds, where A(s)’s are elementary surfaces at �c ¼ constant and theintegration path s is normal to A(s). Figure 2 is the sketch of a con-trol volume typical of a bunsen flame, for example, where meanflame curvature is not null as for planar flame fronts. Integrationof Eq. (17) is performed along path shown in figure, and whoseorthogonality to iso-�c’s is given in terms of jr�cj. Coming back toreference area problems, we tried to remove any arbitrariness inthe characterization of the reference surface Ao by defining it as

Ao ¼1

s1 � s0

Z s1

s0

AðnÞdn; ð18Þ

with s0 and s1 corresponding to values of the integration path atreactant (�c ¼ 0) and product (�c ¼ 1) positions, respectively. Theadvantage of this definition, is that now the reference area is equiv-alent to an average surfaces intrinsically defined by the mean flametopology. In addition, Eq. (16) gives the following definition of heatrelease

s ¼ qu

_m

ZVr � �udV ; ð19Þ

which, integrated over the whole volume occupied by the flamebrush, takes into account partially incomplete combustion effectsand more in general all the heat losses that are not participatingto thermal flow expansion [28]. Finally, Eq. (19) has been coupledwith Eqs. (17) and (18) to give the expression of the local turbulentconsumption speed,

SLC ¼_m

qu

RV r � �udV

s1 � s0R s1s0

AðnÞdn

Z s1

s0

r � �uAðnÞdn: ð20Þ

Fig. 2. Sketch of control volume dV.

2032 G. Troiani et al. / Combustion and Flame 160 (2013) 2029–2037

Useless to say that application of Eq. (20) deserve specialattention. First of all, in the region near the nozzle exit thermal gra-dients may be relevant and it may happen to have integrationpaths not wide enough to guarantee r�c ¼ 0 at the extrema. Thiscould lead to under-estimation of the average mass burning rate_m. This is the reason that led us to not consider data near the noz-

zle, see e.g. Figure 6, where values starts from y > 5 mm. Second,remnant integration paths have been extended well into reac-tant/product regions.

To conclude this section, it is worth stressing that there are sev-eral slightly different definitions of turbulent burning speed. To beconsistent with literature, the authors decided to adopt thenomenclature used in the review of Driscoll [1] that appears tobe currently well accepted by the combustion community, as alsoconfirmed by the recent paper[29]. Hence the velocities evaluated(via local dilatation rate measurements) by Eqs. (17) and (20) havebeen referred to as global and local consumption speed,respectively.

Fig. 3. Mean progress variable �c for Re = 15,000, U = 1. Gray-tones: black, �c ¼ 0(reactants); white, �c ¼ 1 (products). Every gray-tone corresponds to steps of �c ¼ 0:1.

Fig. 4. Labeled iso-lines: mean progress variable �c. Conditions for Re = 5000, U = 1.Filled circles: integration path, gray-scale intensity proportional to velocitydivergence as reported in the legend.

3. Experimental setup and measurement methodology

Turbulent air–methane jet flames, stabilized by a diffusive pilotflame as in [30], are realized for the measurement of the local tur-bulent consumption speed. Different conditions are achieved byvarying the bulk Reynolds number, Re (based on bulk velocityand nozzle diameter – 18 mm) and the equivalence ratio, U, in or-der to obtain different ratios of turbulent velocity fluctuations tolaminar burning speed URMS/SLo. Turbulent velocity fluctuationsare measured 5mm above the nozzle exit and on the symmetry axisof the bunsen burner.

Flame front detection is performed by the acquisition of fluores-cence signal emitted by OH radicals. To that end, a Nd:YAG laserbeam is delivered through a tunable dye laser coupled with a sec-ond-harmonic generator crystal in order to shift the laser wave-length from 532 nm down to 282.93 nm, corresponding to theQ1(6) absorption line of OH. A suitable cylindrical lens expandsthe beam into a 350 lm thick laser sheet. The resulting OH fluores-cence emission is around 309 nm and is then collected by a1024 � 1024 pixels ICCD (2 � 2 pixel binning) equipped with a78 mm Nikon quartz lens, resulting in a map of 512 � 512 equiva-lent pixels with a resolution of 160 lm for each equivalent pixel.Furthermore, a narrow pass-band filter, 10 nm wide and centeredat 310 nm, isolates the relevant spectral line. The fluorescence sig-nal from OH radical is proportional to its concentration and relativemeasurements are possible (absolute measurements of concentra-tions are instead prevented from non-radiative disexcitation chan-nels that are generally active together with detectable fluorescenceemission). These indicate that the fluorescence signal risesabruptly across the flame front, where the OH radicals are formed,and then decreases more smoothly, within the product region. Theexplanation resides in the fact that, within the products, OH radi-cals disappear at a much slower rate than that characterizing theirformation in the flame. This asymmetric behavior can be used todistinguish reactants from products, hence unveiling front posi-tion. The front is located by the isoline that better correlates withthe maximum gradient of the OH signal. A full description of theexperimental set-up and methodology to extract flame front posi-tion from fluorescence signal is described in [31]. With instanta-neous front information in hands, images are binarized by settingvalues 0,1 to those zones belonging to reactants or products andthe averaging of 400 of this kind of images gives the mean front po-sition, i.e. the mean progress variable �c, as reported in Fig. 3.

The use of a synchronized PIV system, instead, probes the veloc-ity field on the same plane scanned by the LIF laser sheet. PIV sys-tem consists of a 54 mJ Nd:YAG laser equipped with a 60 mm focal

length camera working at a resolution of 1024 � 1280 pixels. Arearecorded by CCD is 88.76 � 110.94 mm2. Pulse to pulse delay is of70 ls which guarantees a maximum particle displacement lessthan one-quarter of the interrogation window size (32 � 32 pixels)which present 50% overlapping for velocity field estimation (fulldetails of the technique and measurement uncertainties in[31,32]). By moving normal to �c iso-lines, values of the mean veloc-ity divergence are extracted. Figure 4 reports the superimpositionof �c iso-lines and integration paths (only two for the clarity ofthe figure) with the corresponding values of the mean velocitydivergence. In the reactant (�c < 0:05) and also in the product(�c > 0:95) zones the flow can be considered at constant tempera-ture and hence incompressible with velocity divergence valuesclose to zero. The fact that there the velocity divergence is not ex-actly zero can be considered as the overall estimate of the PIV erroron the measurement of the velocity gradients, and the proceduredescribed in Appendix B can give the global error of the local tur-bulent consumption speed SLC.

Along each integration path, the mean unconditioned velocitydivergence is integrated according to Eq. (20) and two adjacent

Fig. 5. Average of turbulent velocity fluctuations measured along �c ¼ 0:5 isolinenormalized with those found at the inflow, i.e. URMS. Error bars denote standarddeviation.

G. Troiani et al. / Combustion and Flame 160 (2013) 2029–2037 2033

paths define the elements of area A(n) of Eq. (18), as shown withdotted lines in Fig. 4.

4. Results and discussion

In this section results for local turbulent consumption speed arepresented. A compilation of flames at different values of the ratioURMS/SLo are realized by varying methane-air mass flow rate _mand equivalence ratio U, see Table 1. An inspection of second andthird columns highlights a quite wide variation of the ratio URMS/SLo with values ranging between 2.42 and 6.36, while the turbulentReynolds number ReK (ReK = URMSK/m, with K the integral lengthscale) at most doubles its value (136–287). It is worth stressingthat the integral length scale K is obtained by integration of longi-tudinal velocity autocorrelation at the jet axis, 5 mm above thenozzle exit. As a consequence results discussed hereafter will bereported as a function of URMS/SLo only, neglecting any parameter-ization with ReK [33,34]. To corroborate this choice it can be ofinterest the analysis of deviation of turbulent velocity fluctuationswithin flame brush from those measured at the inflow. To this pur-pose, turbulent velocity fluctuations along the �c ¼ 0:5 isoline areaveraged and normalized with those found at the exit nozzle. Re-sults in Fig. 5 confirm that turbulent characteristics along the flamecan be controlled by inflow turbulent conditions.

A typical result of local turbulent consumption speed measure-ment by means of Eq. (20) is presented in Fig. 6. In the left panel,the local turbulent consumption speed normalized with the corre-sponding unstretched laminar velocity (SLC/SLo) is reported and itsamplitude is proportional to the circle radii depicted in the figure.SLo has been obtained by Chemkin software monodimensional lam-inar simulations fed with GRI-Mech 3.0 reaction mechanism. Eachcircle corresponds to an integration path (see Fig. 4) and its centerplaced where the mean progress variable interpolated along thepath is �c ¼ 0:5. In the right panel of the same figure, the non-dimensional local turbulent consumption speed is reported againstthe projection of the circle centers on the y-axis. Note that in orderto highlight the symmetry of reaction rate distribution, open andfilled symbols have been used to discriminate between the leftbranch of the flame section x < 0 and the right one x > 0. Both pan-els shows qualitatively and quantitatively, that the consumptionspeed increases noticeably from the exit nozzle (y = 0) to the flametip. In particular, just above the exit nozzle SLC tends to assume val-ues comparable or even lower than the laminar flame velocity SLo,but moving downstream the increase of combustion speed be-comes remarkable, even five times larger the laminar combustionone.

Table 1Experimental conditions: first column, bulk Reynolds number and equivalence ratio;second column, integral length scale measured by PIV velocity data along thesymmetry axis of the jet at 5 mm above the nozzle exit; third column, ratio betweenturbulent velocity fluctuations and laminar unstrained flame velocity; fourth column,turbulent Reynolds number ReK = URMSK/m; fifth, sixth and seventh columns:normalized local turbulent consumption speed, stretching factor and turbulent toaverage flame surface, respectively. Conditions labeled with w have been obtained bymoving upstream the position of a perforated plate along burner axis to reduceturbulent fluctuations.

Re/U K (mm) URMS/SLo ReK SLC/SLo Io AT/AM

5000/1 2.45 2.42 218 1.82 1.10 1.655000/0.8 2.04 2.24 136 1.80 1.12 1.618000/1 1.52 2.67 179 2.32 1.28 1.8110,000/1 2.38 2.98 287 3.00 1.56 1.9310,000/0.8 2.10 3.02 189 3.46 1.76 1.9715,000/1 0.99 4.24 212 4.20 1.74 2.4115,000/0.8 1.06 6.36 222 5.15 1.94 2.6415,000/1w – 3.67 – 3.93 1.69 2.3215,000/0.8w – 4.83 – 4.09 1.88 2.59

As a first verification of the consistency of these results, the SLC

distribution along the flame brush has been averaged (obtainingagain the global turbulent consumption speed SGC) and resultscompared to those found in literature, usually obtained with con-sumption speed measurements from mass flow rate _m and averageflame surface, [13–15]. Averaging of local data has to take into ac-count the influence of the turbulent flame brush extension [35,36]to be consistent with those one could obtain by means of Eq. (17).Such an equivalence is demonstrated in Appendix A. Figure 7shows the result of this comparison with filled-diamond symbols(�) referring to experiments carried out in the present work. Globalturbulent consumption speed increases with the ratio URMS/SLo asalso literature data do, and the associated error bars are withinthe dispersion of other results (see also section Appendix B for er-ror analysis). As a further comment we observe a remarkablematching (not shown) between data presented in Fig. 7 and rear-ranged according to the scaling law reported in [37], i.e,SGC=SLo /

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðURMS=SLoÞðd=dtÞ

p(d is the bunsen diameter and dt the

thermal flame thickness).To better understand the effect of turbulence on the global con-

sumption speed a more thorough analysis should take into accountthe reciprocal variation of burning speed and the degree of wrin-kling of flame front. To this end it is quite straightforward to recastEq. (6) with the definition of R leading to

SLC

SLo¼ Io

AT

Ao¼ Io

RRAðnÞdn

Ao: ð21Þ

The mean flame surface density R is the ratio between turbu-lent flame surface area and its embedding volume and it is a rela-tively simple observable accessible from the experimental point ofview. A possible complication when evaluating R is that it is a var-iable defined in a three-dimensional space and it has to be ex-tracted from two-dimensional data, usually an image. To this endthe knowledge of the orientation of the flame normal relative tothe image plane is necessary. Images from two orthogonal planesmay be used to evaluate the mean crossing angle along the lineof intersection of the two planes. In the case of Bunsen burners asimplification may be of help and reduce the number of imageplanes to only one. The assumption is that Bunsen flames havemean flame orientation with respect to its axis that is axisymmet-ric. Literature on this concern reports value of the mean directionalcosine of 0.7 obtained in experiments and direct numericalsimulations as well, see e.g., [38–41]. In fact, by means of instanta-neous OH-LIF measurements, it is possible to compute the meanflame surface density R.

Figure 8 compares the variability of normalized global turbulentconsumption speed and turbulent area, see Eqs. (6) and (18) for

Fig. 6. Left panel, normalized local turbulent consumption speed SLC/SLo for a bunsen flame with Re = 5000 and U = 0.8. Speed amplitude proportional to the circle radiidepicted in the figure. Right panel, projection of circle centers on the y-axis. Open symbols, x < 0; filled symbols, x > 0.

Fig. 7. Collection of average values of SGC/SL. Symbols: (�) present results; (�)Shepherd et al. 2001 [13]; (h, M) Gülder et al. 2000 [14].

Fig. 8. Average values for normalized global consumption speed (circles) andturbulent area (diamonds) vs URMS/SLo, see Eqs. (6) and (18) for normalizationdetails.

Fig. 9. Surface averaged stretching factor Iovs URMS/SL.

2034 G. Troiani et al. / Combustion and Flame 160 (2013) 2029–2037

normalization details. They both have a non linear dependencefrom turbulent velocity fluctuations, but to a different degree. Thisis shown more clearly in Fig. 9 where their ratio, i.e., the stretchingfactor Io is reported. It assumes values closer to one as the turbu-lent velocity fluctuations tend to the laminar velocity SLo and ithas values increasingly larger than unity for stronger velocity fluc-tuations, as reported in DNS studies [42,43].

Global analysis discussed until now and summarized in Figs. 8and 9 highlights the strong dependence between laminar flamevelocity and turbulence fluctuations, but at the same time evi-dently masks the behavior of the stretching factor along the flamesurface of the bunsen flames studied, where the constant action ofturbulence is not obvious. To unveil the local character of Io a scat-ter plot of local turbulent consumption speeds and turbulent areasis reported in Fig. 10. A linear regression has been applied for tworepresentative cases, Re = 15,000 and Re = 8000 both at U = 1,giving slopes of 1.61 ± 0.025 and 0.97 ± 0.055 (shown in figure),respectively. It is evidenced that the stretching factor, i.e. the angu-lar coefficient, has nearly constant values for each of the experi-

Fig. 10. Scatter plot of SLC/SLovs AT/AM. Two best-fits represent values of Io = 0.97 andIo = 1.61. In the inset, compensated plot of (SLC/SLo)/Io vs axial coordinate y.

G. Troiani et al. / Combustion and Flame 160 (2013) 2029–2037 2035

ments carried out for a wide range of turbulent area and that suchvalues are in a general agreement with those reported in Fig. 9.Reasons for the overestimation of global values of Io are to be foundin regions characterized by low surface wrinkling where the ratioAT/AM is closer to unity and deviations from linearity are present.

To give a physical meaning to this behavior it may be helpful tolook at the development of Io in the physical space. In the figure in-set turbulent consumption velocities appear compensated with thecorresponding linear fit underlining that low-wrinkled regions ofthe flame are close to the nozzle where turbulence is not yet fullydeveloped. Just downstream the nozzle exit the flame front is cor-rugated by large scale velocity fluctuations induced by shear layerinstabilities. Conversely, at flame tip turbulence developed and awider spectrum of velocity scales is established and able to wrinklethe flame front at even finer scales. This is evidenced by the prob-ability density function (pdf) of flame front curvatures where thosecomputed at flame tip have larger tails, impling a higher occur-rence of large curvatures. This behavior is displayed in Fig. 11 withthe negative values of curvature indicating fronts convex to reac-tant regions. This observation highlights the presence of two dis-tinct zones along the flame height, one near the nozzle where aclear scaling of consumption velocity with flame wrinkling doesnot hold and another in the upper part of the flame front wherethis scaling does exist. Comparison with curvature statistics is in-

Fig. 11. Curvature (normalized with thermal thickness, 0.34 mm for experiment atRe = 15000 U = 1. Continuous line, flame front curvature close to the nozzle, meancurvature k dth = �0.08. Dashed line, flame front curvature close to the flame tip,mean curvature kdth = �0.28.

tended in the present work at introducing different qualitativebehaviors of flame-turbulence interactions. Quantitative analysiswould necessitate a deeper insight into the fluid dynamic charac-teristics of the flow field at the flame front that is beyond theaim of this work. Moreover, it is evident that first and second mo-ments of curvature pdf are not sufficient to describe interplayswith stretching factor. The mean values of curvatures are too closeto zero and curvature fluctuations are not so different in the twocases reported in Fig. 11. To this purpose it is useful to remind thatan estimation of standard deviation in a gaussian-like distibutioncan be given by the value of abscissa corresponding to the half-width of the curve. In addition, it must be mentioned that wrin-kling at finer scales of flame surface is not the unique effect in-duced by turbulence; in fact, velocity strain rates and flamecurvatures can modify transport phenomena and affect the laminarcombustion velocity [44,45]. For the sake of completeness, themethodology followed to extract front curvatures is reported inthe following. Calculation of flame front curvature necessitatespreliminary manipulation of flame front image. First of all, flamefront image pixels were sorted to form a connected line. Second,two fourth order polynomials fx(s) and fy(s) parametrized withthe curvilinear abscissa s were fitted along a 15 pixels strip of flamefront. With a resolution of 0.125 mm/pix the flame strip lengthranges between 1.9 mm and 2.7 mm. With the two polinomialsavailable, flame front curvature follows from

k ¼ f 0xf 00y � f 00x f 0y� �

f 02x þ f 02y

� ��1:5; ð22Þ

where f0 = df/ds and f00 = d2f/ds2 [46,47].This is an interesting result from the point of view of modeling,

since it states that I0 could be evaluated from global values such asturbulent velocity fluctuations at the nozzle exit, but it remainssubstantially constant along the flame height. This corroboratesthe flamelet hypothesis contained in the model of Eq. (21).

5. Conclusions

In this work, the mean velocity divergence is integrated overthe whole flame brush of turbulent premixed jet flames. The inte-gration is made along paths normal to the mean progress variableisolines. This serves the purpose of measuring the mean reactionrate and consequently the local turbulent consumption speed SLC.By means of the definition of a suitable reference area based onthe sole topology of the flame brush the arbitrariness in the choiceof the reference area is removed. These measurements combinedwith the definition of the SLC given by Eq. (20) put in evidencethe variation of the consumption speed along the flame height.In particular, near the jet exit a velocity close to the unstrainedlaminar combustion velocity SLo is found. This is consistent withthe fact that the flame front is weakly wrinkled by turbulence.Moving far away from jet exit, SLC increases reaching its maximumat the flame tip. A collection of flames with interactions betweenturbulence and combustion at different degrees, has been exploredand the global consumption speed obtained. It is found that globalconsumption speed values increase with the ratio URMS/SLo and areconsistent with literature data. Moreover, flame wrinkling is com-pared to consumption speed variation and a non linear behavior ofthe stretching factor Io evidences an increasingly higher interactionbetween laminar flame and flame stretch. The work ends with thelocal analysis of turbulent consumption speed and flame wrinklingcharacteristics along the flame brush. Despite jet flames and free-jets in general have quite complex velocity field, results show thatcorrelation between normalized turbulent consumption speed SLC/SLo and turbulent flame area AT/Ao does exist and can be related tothe global stretching factor Io previously discussed.

2036 G. Troiani et al. / Combustion and Flame 160 (2013) 2029–2037

Acknowledgments

The authors gratefully acknowledge Prof. C.M. Casciola for hisuseful suggestions during the preparation of this manuscript.

Appendix A. Averaging of local turbulent consumption speed

Averaging of local turbulent consumption speed obtained by Eq.(20) has to be carried out with the due care to take into account thelocal flame brush area. Recalling Eq. (20) with terms outside theintegral grouped together in the symbol C we have

SLCi¼ C

RVir � �udV

Ai; ð23Þ

where SLCiis the local turbulent consumption speed referred to the

ith volume Vi which the whole volume of the flame brush has beendivided into. Now, to obtain an averaged value of the local turbulentconsumption speed that can be representative of the flame as awhole, the average of values given by Eq. (23) has to consider theweight of local flame brush areas Ai, i.e,

SGC ¼ hSLCii ¼

PiSLCi

Ai

Atot: ð24Þ

Plugging (23) into (24) we have

SGC ¼P

iCR

Vir � �udV

Atot¼CR

Vtotr � �u

Atotð25Þ

that is formally equivalent to Eq. (17), when Vtot and Atot are consid-ered as the whole volume and area of the flame brush, respectively.

Appendix B. Measurement uncertainties

The divergence errors for the computation of the consumptionspeed SLC by Eq. (17) could be mainly attributed to uncertaintiesassociated with measurements of the velocity field, the spatial res-olution and the algorithm used for the derivatives. A more feasibleand direct method consists in evaluating the extent to which veloc-ity field deviates from null divergence in the unburned regions,where incompressibility is well supposed to hold. Figure 12 reportsin its top panel the behavior of mean velocity divergence along theburner axis (continuous line). It is interesting to note that in theunburned gas region (x < 20 mm), where incompressibility hypoth-

Fig. 12. Top panel, example of decomposition of the kind of Eq. (26): continuousline f, dash-dotted line ~f . Bottom panel, difference f 0 ¼ f � ~f . In this case f ¼ r � �u.for Re = 8000 and U = 1.

esis should hold, the velocity divergence fluctuates around the zeroand has the same tendency approaching the burned region(x > 90 mm). Moreover a local regression method [48] (dash-dottedline) serves the purpose of extracting the fluctuations (shown inthe lower panel of the same figure) from the local averaged signal,which seems to be of the same kind and amplitude along the wholepath crossing the flame brush. This observation leads to the conclu-sion that such fluctuation can be taken into account for the assess-ment of the divergence error amplitude.

To this end, we consider an integral function F and adecomposition

F ¼ eF þ F 0 ¼Z

xð~f þ f 0Þdx ð26Þ

with f ¼ r � u;~f and f0 the average and the fluctuation, respectively.The corresponding root mean square value is

F 02 ¼Z

xf 0ðxÞdx

Zy

f 0ðyÞdy ¼Z

x

Zy

f 0ðxÞf 0ðyÞdxdy: ð27Þ

Considering now a variable change of the kind of y = x + dx and thatthe correlation coefficient RðdxÞ isZ

xf 0ðxÞf 0ðxþ dxÞ ¼ LRðdxÞf 02; ð28Þ

with L the entire length of the integration path s, Eq. (27) becomes

F 02 ¼Z

dxLRðdxÞf 02dðdxÞ: ð29Þ

Integration of Eq. (29) let the correlation length lc to be put into theprevious equation

F 02 ¼ Llcf 02: ð30Þ

Hence, the error made in the integration of the velocity divergenceis

rr ¼ffiffiffiffiffiffiF 02

p¼

ffiffiffiffiffiffiffiffiffiffiffilcLf 02

q; ð31Þ

and its propagation in the consumption speed SLC follows as

rSLC ¼rrs; ð32Þ

used for the evaluation of error-bar amplitude of Fig. 7. A similar ap-proach can be also followed for turbulent area errors, rAT .

At this point, evaluation of Io errors (rIo ) from its definition in Eq.(21) can be obtained by a standard propagation error technique [49].

Given a polinomial function of the kind of

Y ¼ bXp11 Xp2

2 ; ð33Þ

where b is a constant and measurement uncertainties rX1 and rX2

propagate and affect rY according to

rY ¼ Yp1

X1

� �2

r2X1þ p2

X2

� �2

r2X2

" #1=2

: ð34Þ

In the case of Eq. (21) b = AM/SLo, X1 = SLC, X2 = AT, p1 = 1 and p2 = �1,and uncertainties on stretching factor Io are

rIo ¼SLC

SLo

AM

AT

1SLC

� �2

r2SLCþ �1

AT

� �2

r2AT

" #1=2

; ð35Þ

which are reported by errorbars of Fig. 9.

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