Rich Spray-Flame Propagating through a 2D-Lattice of Alkane Droplets in Air

Transcription

1 Rich Spray-Flame Propagating throgh a D-attice of Alkane Droplets in Air Colette Nicoli, Brno Denet, Pierre Haldenwang o cite this version: Colette Nicoli, Brno Denet, Pierre Haldenwang. Rich Spray-Flame Propagating throgh a D- attice of Alkane Droplets in Air. Combstion and Flame, Elsevier, 0, (1), pp.-. <./j.combstflame >. <hal-0> HA Id: hal-0 Sbmitted on Jan 0 HA is a mlti-disciplinary open access archive for the deposit and dissemination of scientific research docments, whether they are pblished or not. he docments may come from teaching and research instittions in France or abroad, or from pblic or private research centers. archive overte plridisciplinaire HA, est destinée a dépôt et à la diffsion de docments scientifiqes de nivea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o privés.

2 Manscript Click here to view linked References Abstract Rich Spray-Flame Propagating throgh a D-attice of Alkane Droplets in Air C. Nicoli 1, B. Denet, P. Haldenwang 1 1 MP, Aix-Marseille Université/ CNRS/ Centrale Marseille, UMR, 1 Marseille France, IRPHE, Aix-Marseille Université/ CNRS/ Centrale Marseille, UMR, Marseille France In a recent nmerical paper (Nicoli et al. Combst. Sci. echnol. vol. 1, pp. -; 0) [1], a model of isobaric flame propagation in lean sprays has been proposed. he initial state of the monodisperse mists was schematized by a system of individal alkane droplets initially located at the nodes of a facecentered D-lattice, srronded by a satrated mixtre of alkane and air. In the present stdy, the previos model is complemented with an original chemical scheme that allows s to stdy the combstion of rich alkane/air mixtres. he main parameters of this configration are s, the lattice spacing (in reactive-diffsive length nits),, the liqid loading (or eqivalence ratio relative to the fel nder liqid phase), and (with 0. ), the gaseos eqivalence ratio (i.e. that corresponding to the satrated vapor pressre in the fresh spray). We presently focs on sprays, the overall eqivalence ratio of which is within the range 1 ( ) 1.. For a large set of parameters, we retrieve a featre often observed on the rich side in the experiments: flame propagation in the presence of droplets can be faster than the pre premixed flames with the same overall eqivalence ratio. his is mainly observed when the lattice spacing is sfficiently large. However, the stdy nderlines the role played by the velocities of two particlar single-phase premixed flames: the vapor that only brns (if any) the mixtre de to the satrated vapor and the flame that propagates (if any) in a mixtre where all fel is vaporized and mixed. When the vapor is too slow (i.e. a feeble spray Peclet nmber), the spray-flame speed reslts from the competition between two mechanisms: a speed chemically enhanced de to some enrichment coming from vaporization (possibly and a slowing down in flame velocity becase the vaporization time scale sets the pace on combstion. On the other hand, for large spray Peclet nmber, the pper flammability limit is fond to be strongly enlarged, and the spray-flame propagates with the veloci. Moreover, the flame strctre deeply depends on lattice spacing: for a large lattice, the combstion stage mainly corresponds to a triple flame, with the diffsion flame that develops arond the oxygen pocket located behind the lean wing of the flame front (i.e. far from the droplets). On the other hand, as s decreases, this diffsion flame tends to be more and more incorporated into the flame front. Keywords : spray-flame; two-phase combstion; heterogeneos combstion; stratified combstion; droplet array combstion, flamelet model.

3 Nomenclatre D th thermal diffsivity F heat of reaction depending on nbrnt mixtre eqivalence ratio I W marker of diffsion flame (i.e. the negative part of the indexed reaction rate) latent heat of fel vaporisation e ewis nmber of species i in the mixtre P p F i Pe s Q R s A b pressre partial pressre of fel spray Peclet nmber effective heat of reaction at stoichiometry droplet radis lattice spacing activation temperatre adiabatic flame temperatre for stoichiometric gaseos mixtre temperatre of the fresh gaseos mixtre U ( ) adiabatic flame speed for the single-phase premixtre of eqivalence ratio U U ( 1) adiabatic flame speed for the stoichiometric single-phase premixtre W Y i Z F Z O Z e ( ) i reaction rate mass fraction of species i in the mixtre fel mixtre fraction (that follows the fel atoms) oxygen mixtre fraction (that follows the oxidizing atoms) Zeldovich nmber for stoichiometric gaseos mixtre adiabatic flame thickness for the stoichiometric gas mixtre eqivalence ratio local estimate of the nbrnt mixtre eqivalence ratio liqid eqivalence ratio of the fresh spray (or liqid loading) (initial) gaseos eqivalence ratio of the fresh spray (a fnction of the satrated vapor pressre) overall eqivalence ratio of the fresh spray thermal condctivity (here, a fnction of temperatre) redced mass fraction of specie i density redced temperatre

4 sperscript indicates the vale at stoichiometric (resp. satrated vapor) conditions (resp. sbscript indicates the vale for fresh (resp. brnt) mixtre sbscript indicates the vale for liqid (resp. gaseos) phase 1. Introdction Combstion spreading throgh a spray concerns a large nmber of applications, sch as Diesel engines or rocket engines, gas trbines or indstrial frnaces. In a recent stdy [1], it has been performed nmerical simlations of isobaric flames propagating throgh a face-centred D-lattice of droplets. he lattice was conceived as a schematization of a particlar initial state of the fresh spray. Since the prpose was the lean sprays, it has been observed that the reslts very weakly depended on the chemical scheme: the classical onestep irreversible chemistry with a global exothermic reaction was fond to be sfficient for exhibiting the main featres of the combstion spread in a lean mist. Contrarily to that previos work on lean sprays, we are now interested in rich spray combstion. In [], we have shown that the reslts of nmerical simlations in rich sprays can strongly depend on the selected chemical scheme. Althogh the overall spray composition is spposed to be rich, the local eqivalence ratio can be fond lean far from the droplets, if the satrated vapor pressre of fel is low. o stdy the combstion in sch a heterogeneos medim, we have implemented a chemical scheme, which is a composition-corrected one-step global reaction. o obtain satisfactory properties we have proposed to adapt the heat release to the eqivalence ratio of the fresh gaseos mixtre, that reslts from droplet vaporization and the sbseqent mixtre of fel with air. his chemical scheme introdces two progress variables that allow s to adapt heat release to fresh composition. his procedre gives satisfaction on both lean and rich sides []. he problem of flame speed enhancement by droplets has a long history; an interesting smmary of the early works has been carried ot by Myers and efebvre () []. et s particlarly qote the works by Cekalin (1) [] and by Miztani and Nakajima () [-a, -b], who added kerosene droplets to a propane air mixtre and saw an increase in propagation speed. We also have to mention the pioneering works of Hayashi and Kmagai () [] and Hayashi et al. () [], who sed a Wilson clod chamber to prodce a nearly monodisperse spray. For polydisperse kerosene sprays, Polymeropolos and Das () [] observed that brning velocity reaches a maximm for a certain domain of droplet size. he sitation concerning the velocity increase is, however, not completely clear. For ethanol and isooctane sprays, Hayashi and Kmagai () [] and Hayashi et al. () [] reported velocity enhancement for rich sprays, and for lean sprays with large droplets. Bt, Ballal and efebvre (1) [] for isooctane, and Myers and efebvre () [] with six different fels, did not observe the enhancement effect for lean sprays. Or recent nmerical stdy on flames propagating in a lean droplet lattice [1] tends to confirm the latter observations: no increase in spray-flame velocity were noticeable for lean alkane sprays. here are nevertheless other experiments reported in the literatre where a velocity increase occrs: for instance, in the lean spray case, a more recent experimental stdy by Nomra et al. (000) [] on ethanol sprays in microgravity indicates larger propagation speeds in a spray than in the eqivalent premixed flame, when the droplet size belongs to some interval.

5 Becase it focses on the droplets, or small scale DNS stdy does not contain an important effect observed in the experiments: the role of the flame front instabilities (see, for instance, the recent works by Bradley et al. (0) [] and by Nassori et al. (0) [1] in the case of the expanding spherical sprayflames). iven the importance of these effects in nmeros experiments, we now discss this qestion more deeply. Varios experiments have revealed that flame spreading in a rich spray has a propagation velocity larger than that of the eqivalent premixed flame. In a nmber of these experiments, the flame front was fond corrgated, with a large nmber of cells. herefore, spray-flame speed enhancement cold possibly be explained by instabilities of the front. As a matter of fact, the interplay between instabilities and droplets seems to have a pecliar importance for the spherical flames, a case that has extensively been stdied in the recent years [-]: de to gas expansion and droplet inertia, front acceleration modifies vaporization which in trn modifies combstion spreading. Moreover, even for the pre gaseos flame, we know that the spherical flame is sbject to highly non-linear effects, leading to the creation of a lot of cells and even to an acceleration of the flame speed (called self-acceleration), see the experiments by Bradley et al. (001) []. In this regard, the experiments with droplets lead more or less to the same effects, except that the inhomogeneity seen by the flame front is here cased by droplets, and not by trblence. On the theoretical side, similar conclsions can be drawn. he creation of many cells in spherical premixed flames (withot droplets) has long been modeled in the case of the hydrodynamic instability with the Sivashinsky eqation, namely in D by Karlin and Sivashinsky (00) [], Frsenko et al. (00) [], (see al (000) [] in the D case, which cold be compared to the experiments with droplets). his model eqation contains the two main effects, creation of many cells and self-acceleration. In the related case of the Sivashinsky eqation close to a parabolic shape, it has been shown in Denet and Jolin (0) [1] that, as the stretch is redced (as the front is less and less crved), soltions with a lot of cells appear, the same effect as the one observed in the spherical configration. Every theoretical approach stresses on the major role played by high-level noise, which can here be triggered by the droplets. A droplet lattice, as depicted by the left (nbrnt) parts of the varios fields drawn in Fig.1, is a manner of controlling the spray initial conditions. Other attempts exist in the literatre. For instance, Mikami et al. (00) [] measred the flame spread along an array of anchored n-decane droplets. In this microgravity experiment, droplet size and transverse interdroplet distance were fixed, only the interdroplet distance in the direction of spreading was changed. he prpose was to investigate different modes of droplet combstion, from individal to grop combstion. he domain of overall eqivalence ratio considered in [] is mch higher than the one stdied here. In what concerns the recent spherical spray-flames as stdied in [-1], we wold warn the Reader that sch experiments cmlate many physical effects, and conseqently are not easy to interpret. For instance, the self-acceleration of flame is observed in [], bt the effect of instabilities seems to be limited to a relatively mild increase of 0% in the reported flame velocity; it cold be more in other experiments, particlarly at high pressre. Or present DNS stdy being performed in a small domain to focs on dropletflame interplay, we hence do not pretend to reprodce the featres observed in the above-mentioned experiments, especially the front instabilities. et s finally note that the present DNS -as described in detail below- solves the vaporization and the combstion of individal droplets. his is in contrast to simlations that se point droplets, sch as all

6 D nmerical stdies on spray-flame, as in Aggarwal and Sirignano () [0], Silverman et al. () [1], Sard, et al. (00) [] and Neophyto and Mastorakos (00) [], or in the D analysis by Reveillon and Vervisch (00) []. By contrast, the present DNS flly resolves the droplets as in the nmerical approach by Kikchi et al. (00) []; here, we additionally allow the droplets to be moved by the gas expansion. he paper is strctred as follows. First, the spray-flame modelling is presented, the chemical scheme being briefly described. Second, the flame spread throgh the lattice of droplets is nmerically stdied. he spray-flame speed is defined, and the flame strctre is discssed. Finally, the role played by the varios parameters is described.. Heterogeneos modelling of spray flames At low pressre, flame thickness in standard sprays appears large in comparison with droplet interspacing. In the recent years, this property allowed s to resort to a homogenization process for developing an appropriate nmerical modelling [, -]. In sch an approach which also neglects droplet inertia, liqid fel appears as an additional species only allowed to enter into the chemical scheme after a vaporization step. Several spreading regimes have been predicted [], in particlar an intrinsic oscillatory regime, which had been observed experimentally by Hanai et al. [] and by Atzler []. he existence of this regime, which occrs as a Hopf bifrcation, does not reqire the presence of differential diffsivity effects [-]. At moderate and high pressres, spray-flame thickness can no longer be large enogh -in comparison with droplet interspacing- to allow any process of homogenization. In sch a system where both phases are initially in eqilibrim, the spray-flame tends to be controlled by vaporization, the chemical heat release permitting the vaporization of the droplets one after another. herefore, the spray-flame propagates within a heterogeneos (or stratified) mixtre with droplets. he mist initial strctre is represented as a face-centred D-lattice of alkane droplets in a pre-mixtre of alkane-air (see Fig.1). hen, the droplets can move as the flame propagates. his work is hence devoted to spray with droplet inter-distance not small in comparison with the characteristic combstion scales (at least, of the same order). Or nmerical modelling starts from the sal set of conservation laws: mass, momenta, energy and species. he non-dimensional form of the conservation laws is performed with the se of the nits related the stoichiometric (gaseos) premixed flame, flame thickness, flame velocity, as already sed in [1]. A brief description of the overall approach is provided in Appendix A..1 eneral considerations Since the accrate chemical schemes for alkane are too complex for efficient nmerical simlations, a simplified chemical kinetics is generally recommended. It is long known that the classical one-step Arrhenis law (eqipped with constant heat release) largely overestimates the adiabatic flame temperatre on the rich side. o overcome the difficlty to assess the main rich flame characteristics (as speed and temperatre), we have considered [] a simple modification of the one-step chemical scheme: heat release becomes a fnction, denoted by F, of the fresh mixtre eqivalence ratio (see below the definition of ). In practice, this model is able to correctly mimic the premixed single-phase flame characteristics (adiabatic

7 flame temperatre and flame speed). As a matter of fact, this adaptation acconts for all the species in eqilibrim at the actal flame temperatre. his approach of flamelet type reqires to know for any point within the comptational domain- the actal composition of the corresponding fresh premixtre, already denoted. Section is mainly devoted to the description of this flamelet approach. et s first introdce the irreversible single-step reaction for alkane-air combstion n 1 n 1 CnH his single reaction allows s to characterize the local mixtre composition by, the field of eqivalence ratio, which reads N N n ( ) O N nco ( n 1) H O ( ) N (1) O O Y M Y M F O O O F F where i stands for the stoichiometric coefficient of species i, () M i, the molar mass of species i. In fresh air, N O. is the molar ratio of nitrogen to oxygen, while the mass ratio is N / O N M N OM O.. At stoichiometry, let s denote by accordance with Q, the combstion heat related to the flame temperatre (i.e. YQ F b CM p F F b ), in () We denote by U, the laminar (single-phase) flame speed, that characterizes the combstion of a premixed fresh mixtre of nity eqivalence ratio (i.e. 1 ). associated with gaseos fresh [resp. brnt] mixtre. hs, D th,b denotes the thermal diffsivity coefficient of the brnt gases in the stoichiometric (gaseos) flame. he non-dimensioning process described in Appendix A ses l D U and RD thb, D U RD thb, as length and time scales, respectively. Frthermore, we handle temperatre and species mass fractions nder the redced forms ( ) ; b Y Y (.a-b) (i=f for the alkane fel, i=o for oxygen and i=p for combstion prodcts) Appendix A recalls the system (A.-A.) of governing eqations that are derived from the general laws of conservation. Eqation (A.) deserves a particlar attention 1 V. div( Dth ) F( ) W(, i, ) () t where W, the reaction rate, is defined in (A.), and i i i, D th is a strongly non-linear fnction of temperatre, which prohibits the diffsion phenomena at low temperatre. Vector V stands for the velocity field that is governed by the Navier-Stokes eqations (A.-A.).

8

9 gases to the brnt gases. In the present work, we select two mixtre variables which involve either fel and prodcts mass fractions, or oxidant and prodcts mass fractions, namely: YF YP YO Z F Y and P ZO M M M M (.a-b) F F P P O O P P Note that p is a negative coefficient in eqations (.a-b) and (A.) for i=p. Conseqently, nder the above assmptions of eqal diffsivity of the species, the Z O fields are obviosly governed by the nonreactive eqation Z F and Z 1 V. Z div ( D Z ) t which reslts from combinations of eqations (A.). Hence, at the flame characteristic length scale, which is spposed mch smaller that the droplet interspacing, Z and Z F O can be considered as conserved when crossing the flame front. Since Z F and Z O are conserved throgh the flame, their ratio Z / Z is also conserved. F O ZF / ZO YF ( FMF / PMP ) YP YO ( OMO / PMP ) YP OMO FMF () where YP 1YF YO YN and YN N MN FMF OMO NMN herefore, Now, in the fresh gases, F O F Z redces to Z Y M F F F F, whereas O Z to Z / Z is nothing bt, the eqivalence ratio of the fresh gases, since nbrnt gases of the flamelet to () Z Y M. F O O O O Z / Z redces in the Z F / Z O Y F Y O M M O O F F () hs, as given by eqation (), the ratio ZF( x, yt, )/ ZO( x, yt, ) allows s to derive an estimate of, the pstream mixtre eqivalence ratio, from any local composition within the flame (especially in the reaction zone). Heat release inside the flame thickness can then be adapted to (within some ncertainties nevertheless since the assmption of eqal diffsivities does not exactly hold). his approach will be sed throghot the paper, which conseqently considers the spray-flame as a set of local flamelets propagating in a heterogeneos medim. his concept reqires that the characteristic length scale of composition variations is mch larger than the flame thickness, which is the main assmption rling the paper. et s remark that in [-] is which allows the athors to adapt heat release and activation energy in pace with the composition variation; the reslting one-step kinetics is implemented and checked with respect to a conter-flow diffsion flame. From the qantitative point of view, it has been shown in [] that the following general form of the adjstable qantity F( ) gives reliable reslts for the single-phase premixtre in what concerns the alkane/air flame speed and temperatre O

10

11 end of a rapid stage in the spray-flame propagation process (the instant of the snapshots is indicated in Fig. by a vertical arrow). Spray-flame periodically consmes the droplets, which remain at rest ntil they start to vaporize as indicated in Fig.1.a, where the field of fel mass fraction is represented. It also indicates that a certain amont of fel is nbrnt behind the (rich) spray-flame. Fig.1.b presents the field of oxygen mass fraction, while Fig.1.c corresponds to the temperatre field. Fig.1.d shows the field of heat release. As for Fig.1.e, the negative part of the indexed reaction rate [] is plotted for displaying the diffsion flames (see below); different elementary flames are noticeable, and will be discssed below. he nmerical experiments aim at determining the flame speed when combstion propagates throgh an array of fel droplets positioned at the nodes of the face-centered lattice for a given rich spray. We are interested in the inflence of the spray composition (i.e., ) and s, the lattice spacing. Droplet radis is obviosly a fnction of liqid loading and lattice spacing. hree dimensionless lattice spacings are investigated: s, s and s 1 (in nits of stoichiometric premixed flame thickness, as described in Appendix A). Five different initial premixtres srronding the droplets are considered: 0, 0., 0., 0. and 0.. Evidently, when is fixed, the amont of fel initially nder liqid phase increases as increases. In the same manner, for fixed s and, droplet radis increases as increases. For initiating combstion in the lattice, we follow the same procedre than that sed in or previos works [1-]: close to the open (right) end of the lattice, the temperatre field is imposed with the profile of a singlephase flame. his allows s to vaporize and ignite the first droplet of the spray. o follow the combstion spread, we compte ( x), the mean temperatre averaged in the periodic (y-)direction (i.e. transversally to the y propagation). hen, we decide to define x, the spray-flame position, as the locs where sf y ( x sf ) Spray-flame speed

12 Figre : Front position vs. time for varios liqid loading ( 0, 0., 0., 0., 0., 1.0; 0. ; s and iso-octane/air spray; droplet radis corresponding to each is provided in nits of stoichiometric premixed flame. Vertical arrow indicates the instant of the snapshots of Fig.1) In this section, we present reslts on spray-flame speed at constant gaseos eqivalence ratio. In other words, this nmerical experiment considers a constant (satrated) vapor pressre; as the initial twophase mixtre is spposed in eqilibrim at a given temperatre, the initial temperatre is a logarithmic fnction of thanks to the Clasis-Clapeyron relation of the considered fel, as described in Appendix B. is then modified by changing, the initial liqid loading. Considering the intermediate lattice spacing s, Figs. and show the flame front position for 0. and 0., respectively. Note that a single-phase premixed flame can propagate in both initial premixtres, slowly at 0. and more rapidly at 0.. In Figre, we can observe the actal way the flame propagates in the lattice for 0.. he front position (defined as the transversally averaged isotherm ( x ) 0.) is plotted verss time for varios overall eqivalence ratios (i.e. for increasing liqid loading). he lower (marked) crve corresponds to the front position of a single-phase premixed flame with the (initial) gaseos eqivalence ratio (i.e. 0, R 0 ). he mean slope of each crve is here negative and its absolte vale determines the flame speed for a given. It can be seen that, for this lattice spacing (s=), the single-phase premixed flame with the overall (gaseos) eqivalence ratio of 0. propagates significantly slower than all the considered sprayflames. One also observes that combstion propagates throgh the lattice in an nsteady manner [1], since it reslts from sccessive stages of droplet vaporization, species mixing, and reaction. Fig. incldes a lean case with the overall eqivalence ratio 0. (i.e 0. ): this is the slowest spray-flame. It can be observed that the increase of the slope is not monotonos with the overall eqivalence ratio. he maximm in slope is more or less obtained for the stoichiometric spray (i.e. 1.0 ). When liqid loading frther increases the spray-flame speed decreases and seems to rapidly reach an asymptotic vale. y sf

13 Figre : Front position verss time for varios liqid loadings 0, 0., 0., 0., 0.. ( 0. and lattice spacing s=, iso-octane/air spray) et s now consider Fig., where the vapor pressre is chosen higher (i.e. 0. ). U ( 0.), the speed of the single-phase flame that spreads in the initial satrated vapor is large. hen, nlike the previos case, the liqid loading is of little help in terms of propagation. he maximm spray-flame speed is again achieved for stoichiometric overall eqivalence ratio. If the liqid loading is still increased the spray-flame speed slightly decreases. his slowdown with increasing eqivalence ratio is, however, very weak in comparison with the single-phase premixed flame. his observation shows that fel nder liqid phase does not play the same role as the fel nder vapor. As we shall see, this role depends on the droplet size (i.e. mainly on the lattice spacing) and the initial vapor pressre. he fact that the droplet size plays a role in spray-flame propagation can already be seen for instance in Fig., where the flctations in flame front position increase as the liqid loading increases, i.e. as the droplet size increases. As stdied in the previos contribtion [1], those flctations in front position are de to the vaporization stage (if needed), which depends on the droplet radis sqared. We now investigate the role played by the satrated vapor (i.e. by ) for the same intermediate lattice spacing s. In Fig., the non-dimensional spray-flame speed is plotted with respect to, the overall eqivalence ratio of the spray, for varios eqivalence ratios related to the vapor. In other words, the spray-flame speed is plotted for varios (gaseos) eqivalence ratio. In Fig., the crves of spray-flame speed are again compared with the single-phase premixed flame velocity. 1

14 Figre : Spray-flame speed verss overall eqivalence ratio of the spray for different gaseos eqivalence ratios ( s, heavy fel oil/air spray; nmerical predictions are compared with the experiments by Ballal & efebvre for 1 ) For the sake of validating the nmerical approach, it can be checked that the experimental observations made in [,,1] are retrieved by or nmerical reslts: - for moderately lean or stoichiometric sprays, the flame velocity is higher withot droplets (the overall eqivalence ratio being given). In Fig., we have added five experimental points from Ballal & efebvre [] for 1 (Sater mean diameter in the experiment being in link with the lattice spacing sed in or nmerical predictions). he agreement between experiment and present modelling appears reasonably satisfactory. - for rich sprays (say 1. ), this sitation reverses for all vales of : rich spray-flame speed is always higher than single-phase flame speed. It is worth noting that for very low vapor pressre (i.e. 0 ) a spray-flame does exist, even thogh its velocity is qite low. In other words, combstion ensres vaporization at a pace that allows its own propagation! From Fig. (i.e. for s ), we additionally have to nderline another trend: for 0, 0., 0., i.e. when corresponds to a non-flammable single-phase mixtre (in other words, combstion needs droplets to propagate), the spray-flame speed is an increasing fnction of. Otherwise, for higher, the spray-flame speed exhibits a maximm in the vicinity of the (overall) stoichiometry. astly, we observe that, when the overall eqivalence ratio still increases, the spray-flame velocity does not

15 vary so mch. In other words, on the far rich side, it seems that a constant amont of liqid fel is involved in spray-flame propagation.. Role of the lattice spacing In the previos paragraphs, increasing the liqid loading led s to slightly increase the droplet radis (in D, as the sqare root of ). A more efficient increase in droplet radis is achieved when the lattice spacing is enhanced: for fixed, the droplet radis increases linearly with s. For three different vales of gaseos eqivalence ratio ( 0. ; 0. ; 0. ), the role played by the lattice spacing is now stdied. More precisely, for each, the spray-flame velocity is estimated as a fnction of the overall eqivalence ratio for three lattice spacings (s=, and 1). Figre (resp. and ) plots the spray-flame speeds for 0. (resp. for 0. and 0. ). On each figre, the dashed line recalls the 1-phase flame speed verss gaseos eqivalence ratio (since ), while the experimental data from Ballal & efebvre [] are reported for 1 [Note the Sater mean diameter stdied in the experiment corresponds to s and not to s 1 ]. he spray-flames characterized by 0. need to vaporize droplets, previosly to any propagation. Hence, Fig. indicates an important role played by lattice spacing on the spray-flame speed. For small lattice spacing (say, s ), droplet radis being small, the droplets qickly vaporize and contribte to achieve a rather rapid propagation. For larger lattice spacings ( s ; s 1 ), droplet radis being higher, the vaporization is slower and imposes its pace to the combstion. When the overall eqivalence ratio is high, Fig. indicates that propagation velocity of the spray-flame with small lattice spacing [i.e. s ] follows the same trend as the single-phase flame: spray-flame speed decreases as eqivalence ratio increases. his featre is de to the rapid vaporization and mixing of the fel, which then brns accordingly with the chemical scheme.

16 Figre : attice spacing effect on spray-flame velocity: spray-flame speed verss overall eqivalence ratio for s=, and 1 and 0. (comparison with the experiments by Ballal & efebvre for 1, heavy fel oil/air spray) Figre treats of a slightly different sitation since vapor pressre is high enogh ( 0. ) to obtain a flammable (initial) single-phase premixtre. he spray-flame now reqires a smaller amont of fel coming from the liqid loading to reach the near-stoichiometric conditions in the premixtre. herefore, for relatively small droplet size (cases s= or s=), the spray-flame presents a maximm of its velocity. he intensity of the maximm is nevertheless lower than the maximm single-phase flame speed, since non-zero vaporization time scale impacts on combstion. he latter impact is weaker for small lattice spacing: for s, the crve of spray-flame speed roghly adopts the same pattern than that of the single-phase flame. On the very rich side of the overall eqivalence ratio, spray-flame with small lattice spacing again experiences the rich chemistry, whereas the spray-flame with large lattice spacing, i.e. with a large radis, maintains a large part of the liqid fel non reacting. In other words, althogh vaporization of large droplets is a slow process, the rich spray-flame with large droplets propagates faster than the rich spray-flame with small droplets. Figre presents the spray-flame speed when the vapor pressre is large (i.e. 0. ). If the lattice spacing (or the droplet size) is small, the liqid loading rapidly participates to the spray-flame propagation. In this case (say, s ), the spray-flame shares the same trends as the single-phase flame: the maximm flame speed occrs at (overall) stoichiometry, and a flammability limit occrs on the rich side, too. Otherwise, for large droplets (say, s 1), the spray-flame propagation ignores the liqid loading and propagates at the same speed as wold propagate a single-phase flame in the satrated vapor [since U ( 0.) 0., as discssed below].

17 Figre : attice spacing effect on spray-flame velocity: spray-flame speed verss overall eqivalence ratio for s=, and 1, and for 0. (comparison with the experiments by Ballal & efebvre for 1, heavy fel oil/air spray) he previos comments on Figs. - invite s to stress on the following featres, that have already been observed in nmeros experimental and nmerical stdies [, 1, 0, 1,, ] : - spray with small droplets (i.e. short vaporization time scale) tends to brn with the same trends as the single-phase premixtre. - when the overall eqivalence ratio is large enogh (say, 1. ) spray-flame speed is always higher than the single-phase flame which tends to extingish according to the single-step chemical scheme. he difference between spray-flame and single-phase flame becomes more prononced as droplet size increases. In other words, the presence of large droplets cases some amont of fel (i.e. nder liqid phase) to be maintained ot of the combstion. his point will be clarified below. Frthermore, the vaporization time scale of the droplets ndobtedly plays a role in the spray-flame propagation. Since droplet radis depends on both lattice spacing and liqid loading, the point deserves to be stdied specifically. his is the object of the next paragraph.

18 Figre : attice spacing effect on the spray flame velocity: velocity verss overall eqivalence ratio for s=, and 1 and 0. (comparison with the experiments by Ballal & efebvre for 1, heavy fel oil/air spray). Detailed role of droplet radis We now consider a spray with fixed temperatre and given chemical composition:, its overall eqivalence ratio, is hence determined, as well as, the eqivalence ratio related to its satrated vapor. Different lattice spacings are considered. We present a series of figres (i.e. Figs. -) for sprays with a fixed overall eqivalence ratio. In each figre, a certain amont of fel is spposed in vapor form and the rest nder liqid state; the size of the corresponding fel droplet is a parameter still free. Or nmerical stdy then consists in compting the spray-flame speed dependence on the radis of these fel droplets. Figre concerns a slightly rich spray ( 1. ). It presents the spray-flame speed verss droplet radis for varios gaseos eqivalence ratios (or varios vapor pressres). For this case of overall eqivalence ratio, the reslts show that the single-phase flame speed (i.e. U ( 1.) 0. ) is always larger than any twophase flame of the same overall composition. et s observe additionally that three crves (i.e. for 0., 0., 0. ) correspond to a non-flammable gaseos srronding of the droplets. It is noticeable that these crves are close to each other. In other words, the spray-flame needs to resort to vaporization of some liqid fel. In that case, vaporization rles propagation, and the spray-flame speed behaves in inverse ratio to droplet radis (as already stdied in lean spray, see Ref. [1]). Frthermore, let s recall that the initial vapor (in fig.) is only flammable for 0. and 0., for which the corresponding single-phase flame speeds are U ( 0.) 0. and U ( 0.) 0.,

19 respectively. herefore, from Fig., it is clear that droplet vaporization enhances flame speed, whatever the droplet size and the considered vapor pressre. Bt, this enhancement reaches the single-phase flame speed of the same overall composition, only for vanishing droplet size. o sm p, Fig. is characteristic of the sitation for U ( ) U ( ), where U ( ) represents the laminar flame speed of the single-phase premixtre obtained for (which is also the spray-flame speed for a spray with infinitely small droplets) Figre : Spray-flame speed verss droplet radis for varios (and 1. ). Figre now concerns a moderately rich spray ( 1. ). Spray-flame speed is again plotted verss droplet radis for varios vapor pressres in this figre, which exhibits two different behaviors. he first case corresponds to the reslts obtained for 0., for which we have U ( 0.) 0. and U ( 1.) 0.. his sitation [i.e. for U ( ) U ( ) ] exhibits a flame speed enhancement with droplet radis. his behavior is explained thanks to the following rationale: for very small droplets R 0, vaporization and mixing are immediate so that we start from the rich single-phase case with the speed U ( 1.) 0.. On the other hand, for large droplets we retrieve the single-phase with the speed U ( 0.) 0.. In other words, the droplets are large enogh to have no time to flly vaporize dring the spray-flame propagation. his is the sense of the pper crve of Fig. (i.e. for 0. ), where spray-flame speed increases with droplet radis: the larger the droplets, the lesser liqid fel is involved in combstion. 1

20 Figre : Spray-flame speed verss droplet radis for varios (and 1. ). In fig., the second case considers the other for crves ( 0., 0., 0., 0. ), for which U ( ) U ( ). hese crves are non-monotonic as droplet size increases: if droplet radis slightly increases, less liqid fel is involved in combstion, and the spray-flame speed slightly increases; now, if the droplet radis is too large, the spray-flame speed asymptotically tends to U ( ), since the vaporization time tends to infinity; therefore, each crve reaches a maximm for an intermediate droplet radis, and afterwards decreases to the asymptotic flame speed U ( ) [which can be a zero vale]. Figre : Spray-flame speed verss droplet radis for varios (and 1. ).

21

22 Analysis of spray-flame strctre.1 arge lattice spacing In the case of lean sprays, it has been fond [1] that the flame propagated from one droplet to the next in different ways, either with a very lean premixed flame, or with constant re-ignition of the next droplet in the case of very small eqivalence ratio relative to initial vapor. Once the next droplet was ignited in one way or another, a triple flame propagated arond the droplets, leaving behind the diffsion flame arond the droplet. Sch a propagation was fond to correspond to the third propagation mode observed in []. In the rich case, however, Roland Borghi sggested -see the sketches in the book by Borghi and Champion [] and also in Demolin and Borghi [] - that the diffsion flame shold be arond the oxidant; this was called pocket combstion. et s see what we obtain in the present simlations. In Fig., we show different fields characteristic of the spray-flame at a particlar instant for a the large lattice spacing s 1: fel and oxygen mass fractions, temperatre, heat release. he fifth field reqires a particlar comment: we have plotted I W, the negative part of the indexed reaction rate, which is defined as follows [] IW0 if I0 with I F O (1) IW WI I if I <0 where I is the so-called akeno flame index []. I is expected negative when the species diffse towards each other, as it occrs in a diffsion flame. In other words, plotting the negative part of W I helps s to localize the diffsion flames. he chosen time in Fig. corresponds to the late instants of the triple flame stage (i.e. jst before both triple flames meet); we do not actally show the complete comptational domain. As in the lean case [1], for a large vale of s, we observe triple flames, the wings of which are connected here to lean and rich premixed flames. Fig..e clearly exhibits the location of the diffsion flames. he first diffsion flame accompanies the triple flame, while the second one brns oxygen in excess between the previos vaporized droplets. As a reslt, the lean premixed flame does not brn all the oxidant available locally, and is followed by a diffsion flame srronding the oxidant (and not the fel as in the lean case), as sggested by Borghi. We are here at a late stage of droplet vaporization, it appears that a significant premixtre exists close to the droplet (we recall that the diffsion coefficient is larger for the oxidant than for the fel, i.e. a ewis nmber of 0. for the oxidant and 1. for the fel). As a matter of fact, the flame arond the droplet is a (slow) rich premixed flame. his rich premixed flame does not brn all the fel, since a significant amont of fel is fond at the right of the figre where the flame extingishes. In the same manner, the premixed flame does not brn all oxygen: in Fig..c, a feeble amont of oxygen is present at the right of the domain, and is slowly consmed thanks to the diffsion flame located at the right of the domain. 1

23

24

25

26 optimal radis being on the order of 0. times the stoichiometric flame thickness, and the maximm sprayflame speed being fond close to 0.. As for the comparison with the experimental literatre, it mst be condcted in terms of the fndamental laminar spray-flame speed, becase or DNS carried ot at the scale of the droplets is not able to simlate large scale instabilities, or spherical expansion instabilities as those occrring in the Wilson chambers. As mentioned in Introdction, those instabilities are very sensitive to pertrbations: a change in geometry, pressre, can trigger those instabilities, and increase the measred propagation velocity. his is a possible reason for the conflicting reslts fond in the experimental literatre. Whatever the scattering in literatre, or reslts corroborate an often observed featre on spray-flames: the rich flammability of sprayflames can be enhanced by a large extent, provided that droplet radis belongs to the appropriate range. Acknowledgements /CNES nder the contract CNES/0. Appendix A his addendm is devoted to derive a non-dimensional form of the governing eqations from the basis conservation laws, which read for energy and species ( C) p div(vc) p div( ) Qw(,Y,) (A.1) i t Yi div(vy i) div( DiY i) imw( i,y,) i (A.) t where w(,y,) YY exp- / (A.) i F O A is the reaction rate, and A the activation temperatre. Non-dimensioning is carried ot with the se of the characteristics of a flame of reference, which has been chosen as the stoichiometric single-phase premixed flame. Namely, these characteristics are: (a) the stoichiometric mass fractions denoted by Y for species i [ i, i F,O and p, for fel, oxygen and prodcts, respectively]; (b) the adiabatic flame temperatre ; (c) the corresponding thermal diffsivity in the brnt gases D ( )/( C ); (d) the laminar velocity of the stoichiometric flame U. th,b b b p Within the framework of a single step chemical scheme, note that, the adiabatic flame temperatre, allows s to define Q, the heat of reaction associated to expression (A.1) at stoichiometry, as follows Q C M Y (A.) p b F F F b b

27 As a matter of fact, the non-dimensioning we se resorts to the following theoretical expression of the laminar velocity, as derived by Jolin and Mitani [] (see also arcia-ybarra et al. []). b b b F F O, A b Ze bc p ( U ) M Y exp( / ) ith gaseos fresh [resp. brnt] mixtre. As classically, D th,b and define two scales for time and length, the so-called transit time and flame thickness, as follows D /(U ) RD th,b Hence, for the sake of simplifying the notations, l D /U RD th,b t, x, V,, D th,d,f i stand hereinafter for (A.) U allow s to (A..a-b) t, x l, V U,, D D,D D,Q Q (A..a-f) RD RD b th th,b i th,b Frthermore, temperatre and species mass fractions are made dimensionless as follows b ( ) ; Y Y (i=f for the alkane fel and i=o for oxygen) (A..a-b) i i i, We are now able to write Eq. (A.1-) in non-dimensional form as 1 V. div( D th ) F( ) W(, i, ) t where i 1 V. i div( D i i) iw(, i, ) t Ze 1 W(, i, ) FO exp Ze b b 1 ( 1) with the redced coefficients and the classical parameters, i i i F F (A.) (A.) (A.) M M (A.1) Ze ( ) and A b b ( ) (A..a-b) b b It is worth noting that F( ), the redced heat of reaction, is a qantity that is tneable in fnction of the local composition of the fresh mixtre, which is characterized by the field F( ) [see section ]. his takes accont of the heterogeneos composition of the mixtre. Accordingly with the present nondimensioning, we have F( 1) 1. Finally, we assme that all ewis nmbers [ e D D ] are constant (i.e. the ratio of thermal i th i diffsivity to moleclar diffsivity is spposed constant for each species I ; they are taken eqal to e 1. for the fel and e 0. for oxygen. he same hypothesis is set on the Prandtl nmber [ F O Pr vd th is spposed constant]. So that, we obtain the following system of governing eqations, that reads copled with the Navier-Stokes eqations 1 V. div( Dth ) F( ) W(, i, ) t i 1 1 V. i div( Dth i ) iw(, i, ) t e i (A.) (A.)

28 where P stands for pressre. div( V ) 0 t V VV. Pr 1 div D ( ) 1 th V V P t (A.) (A.) System (A.-) has been solved nmerically with a mixed nmerical method that takes accont of the y-periodicity, for which Forier spectral methods is sed. As for the x-direction, the nmerical approach resorts to finite differences. A particlar treatment for compting pressre has been implemented [] and continosly improved [0], since the present isobaric combstion is characterized by low Mach nmber flows. As srface tension is neglected, the nmerical approach can consider a single-flid flow with very high variations of density, from the volmetric mass of a hot gas to that of a liqid fel. In sch a way, inertia of the dense phase is easily acconted for. o confine the liqid fel into the droplets, diffsivity is frozen at low temperatre [1]. ypical discretization is nodes in the (propagation) x-direction and Forier modes in the (transverse) y-direction. More precisely, abot nodes are at least present along a droplet radis. o check the nmerical precision, this discretization has been dobled; no change in the observed featres has been noticed. he limitation of the nmerical approach is that no immediate copling with sond can be envisaged, since acostic filtering is associated to low Mach nmber flow models, as well as highly compressible flow (or high speed combstion) cannot be considered. Another limitation is related to the absence of srface tension: if the length scale of the flow shears is smaller than the droplet size, freezing the diffsivity will be insfficient to confine the fel inside the droplets. Appendix B his addendm is devoted to answer a practical qestion: is it possible to change withot drastically affecting the characteristic qantities sed for the non-dimensional stage that has been described in Appendix A? his point is an important isse in terms of interpreting or reslts with respect to laboratory experiments. Or nmerical experiments have considered spray-flames, which are determined by the set of parameters that characterizes the initial mixtre: pressre P, temperatre, chemical composition (i.e. and given) and droplet geometry (i.e. s given). Evidently, sch a spray-flame cannot generally exist from the experimental point of view, since P, and are not independent. hese qantities are linked by the Clasis-Clapeyron formla on satrated vapor.

29 Frthermore, as flame of reference in the non-dimensional process, we have sed the stoichiometric single-phase flame defined by the set P,, 1. Obviosly, the latter only exists if the pair P, belongs to a certain domain, which is delimited by the conditions of satrated vapor. Hence, instead of considering the spray-flame determined by, s, for any P,, the experimentalist has to perform the experiment for P,, satisfying the conditions of satrated vapor. For example, we can investigate two opposite cases: (a) we fix and consider consider 0. P, or (b) we fix P and. Both strategies are stdied below, for the range 0 1 1, discarding the limit case. (a) we first consider that all spray-flames have the same initial temperatre. hen, p F, the satrated vapor pressre of fel, is given. herefore, is experimentally controlled by the amont of air added to the mixtre that srronds the droplets. We hence obtain the pressre of the experiment as o fix the idea, P reads for isooctane ( 1 ) 1 ( ) P p p p F O N O F O N F (B.1) P p 0 0p (B.) F F Expressions (B.1-) recall s that -for heavy alkanes- the total pressre is essentially determined by the amont of air. If we consider the range of variation 0 1 1, expression (B.) indicates that total. pressre will vary by a factor. Althogh flame speed is known to vary mildly with pressre, this manner of 1 monitoring is not satisfactory, since flame thickness varies as P. herefore, sing P, a certain pressre vale independent of, instead of trn to another manner of controlling (b) we now qestion the role of P, introdces a hge error in assessing the length scale. We mst., the fresh mixtre temperatre, which allows the experimentalists to monitor the fel satrated vapor pressre for a given total pressre (i.e. a given amont of air, essentially). In combstion, it is well-known that the most sensitive qantity to temperatre is the reaction rate. More precisely, it is known as an exponential fnction of the brnt gas temperatre, which depends linearly on [as recalled in Eq.()]. In other words, the isse concerns the error committed by the fact of sing, some constant vale of the eqilibrim temperatre, instead of. o fix the ideas abot the effects of sch a modification in reaction rate, let s estimate the error committed on the single-phase flame speed fact to se instead of link. U by the. Eq. (A.) recalls that reaction rate and flame speed sqared are in direct From the Clasis-Clapeyron formla, we relate and as follows.

30 where p F, and C pf 1 ln( ) ln( ) R C pf (B.) R are the satrated vapor pressre of fel, the latent heat of fel vaporization and the gas constant, respectively. vale of, say 0.. Now, from Eq. (A.-) and Eq. (B.), we can derive the following relation between U U and as U U ( ) Ze exp exp ln / / b where Ze R N C VR N RV N RV Note that NVR is a nmber that takes accont of vaporization and reaction. For the isooctane flame nder standard conditions, N is as small as 0.0. In conseqence, a ratio of is damped into VR U 1 0 U. by expression (B.). his corresponds to a relative error on the order of %. In other words, interpreting or reslts as obtained with the same initial temperatre introdces an error on the reaction rate largely within the error bars of the experiments, as well as within those of the nmerical methods. b (B.) (B.)