Using LES to predict ignition sequences and ignition probability of turbulent two-phase flames

Combustion and Flame 160 (2013) 1191–1207

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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e

Using LES to predict ignition sequences and ignition probability of turbulent two-phase flames Alexandre Eyssartier ⇑, Benedicte Cuenot, Laurent Y.M. Gicquel, Thierry Poinsot CERFACS, 42 Av. G. Coriolis 31057 Toulouse Cedex 01, France

a r t i c l e

i n f o

Article history: Received 10 June 2011 Received in revised form 3 April 2012 Accepted 13 January 2013 Available online 25 March 2013 Keywords: Ignition Two-phase flow Turbulent combustion LES

a b s t r a c t Ignition and altitude reignition are critical issues for aeronautical combustion chambers. The success of the ignition phase of a combustor depends on multiple factors, from the characteristics of the ignitor to the spray droplet size or the level of turbulence at the ignition site. The optimal location of the ignitor or the potential of ignition success of a given energy source are therefore parameters of primary importance in the design and the certification of combustion chambers. To study ignition, series of experiments are usually performed but they are costly especially when multiple spark locations must be tested. For the same reason, current unsteady simulations are useful but do not give reliable results, and require a lot of simulations if different locations are to be tested, which brings the CPU cost to unreasonable values. Alternatives are hence needed and are the objective of this contribution. It is proposed here to derive a local ignition criterion, giving the probability of ignition from the knowledge of instantaneous non-reacting two-phase (air and fuel) flow data. This model is based on criteria for the phases of a successful ignition process, from the first kernel formation to the early flame propagation towards the injector. Comparison with experimental data on an aeronautical chamber shows qualitative agreement, indicating that the proposed ignition criterion, coupled to a Large Eddy Simulation of the stationary evaporating two-phase non-reacting flow, can be used to optimize the ignitor location and power. Ó 2013 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction Ignition and high altitude re-ignition, are critical processes in aeronautical engines. Efficient and reliable ignition is a crucial point for the certification of engines, and innovative systems such as laser ignition are actively developed. Ignition is however a complex transient phenomenon, not yet fully understood and controlled. Experiments show that ignition by a single spark may be successful or fail in the same operating conditions so that an ignition probability must be used for engine design [1–3]. This probability is the result of the fluctuations of the fluid state at the ignitor location, and is conditioned by local turbulent mixing and velocity fields [4]. Previous numerical studies have been devoted to ignition in premixed gaseous flows [5,6] and in particular demonstrate the concept of minimum energy for successful ignition [7–9]. More recently, Quintilla [10] and later Ouarti [11] or Garcia-Rosa [12] used zero and one-dimensional sub-models to describe the very first moments of ignition. Less work has been devoted to the ignition of a two-phase mixture. Ballal and Lefebvre [13,14] proposed

⇑ Corresponding author. E-mail address: [email protected] (A. Eyssartier).

an extension of their analytical study on gaseous ignition to twophase flow, leading also to the determination of a minimum energy. A comparison with experimental data confirms that global tendencies are captured by their analysis and that ignition is strongly influenced by the droplet size. This study is however limited to a simple ideal case and misses other processes occurring in a real engine. The complexity of ignition in a real combustion chamber is illustrated by Boileau et al. [15] who simulated the ignition of a full annular combustor. They show that the ignition process develops in several steps, from the first flame kernel near the ignitor, to the stabilization of a flame on one injector and its propagation towards the neighboring burning sectors. Such simulations were performed using the Large Eddy Simulation (LES) technique, which has proven its ability to compute combustion chambers [16–21]. There are two ways to use LES to study ignition in turbulent flows:  Perform LES of ignition sequences [22,15,23–26]. This is a very expensive procedure to determine ignition probabilities [3]. Indeed, one full computation of an ignition sequence must be produced for multiple initial fields to ensure proper statistical sampling and given one spark position. This procedure needs then to be repeated for all spark positions which clearly prevents constructing probability maps numerically.

0010-2180/$ - see front matter Ó 2013 The Combustion Institute. Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.combustflame.2013.01.017

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 Build a model based only on a cold flow LES. This requires additional assumptions but is much faster because it needs only one two-phase flow LES but without combustion or the need to simulate directly the transient flow evolution (which can be long). Building a model based on cold flow data is the objective of the present paper. Several attempts already address such objectives either by use of the Reynolds Averaged Navier–Stokes (RANS) [4,27] or Large Eddy Simulation (LES) of cold flow predictions [28] with and without multi-phase flow models. Very few strategies however directly target the prediction of the ignition process as encountered in real gas turbine engines. Instead specific steps of the full ignition phase are studied by use of simpler experimental setups or flow configurations [29,2,30,3,1,23,4,28,27]. Prior to the presentation of an a priori strategy relying on two-phase evaporating LES predictions to evaluate the ignition probability, we will also use LES results of full ignition sequences to illustrate the statistical nature of ignition and to identify the phenomena which must be included in the model. For the a priori model called I-CRIT-LES, an ignition scenario is built, leading to a series of analytical criteria, which are all necessary for successful ignition. These criteria are exclusively based on the non-reacting flow (i.e. before ignition) local conditions. Applying these criteria to a set of instantaneous non-reacting solutions, used as a statistical sample of the flow, a probability of ignition may be calculated at each point in the combustion chamber. One objective of such a strategy is to provide a priori identification of the controlling processes of the early phase of any ignition sequence in real applications: i.e. up to the early instant where the flame front propagates away from the energy deposit location. Based on this diagnostic, engineers can access the potential limiting stages of such sequences based on the ignition scenario retained and eventually improve the design. The paper is organized as follows. First (Section 2), LES of full ignition sequences in a swirled kerosene/air flow are presented to illustrate the phenomena which the I-CRIT-LES model will have to take into account. Taking one flow realization, Section 3 establishes the ignition scenarii and criteria. Then Section 3.3 shows how a set of instantaneous flow solutions can be used together with the criteria of Section 3 to compute an ignition probability. Finally, the model for ignition prediction is applied to a real combustion chamber and compared to experimental results (Section 4).

2. LES of ignition sequences LES of ignition sequences have been performed for the MERCATO configuration of ONERA [31–35] in order to illustrate the leading physical parameters potentially of importance in the determination of the ignition probability in such a configuration [32,34]. For this numerical investigation only the early phase following the localized energy deposit is considered: i.e. the transient phase including the stabilization of the flame in front of the injector in not covered. In parallel, experimental ignition campaigns have been produced using either spark or laser ignitors to produce experimental estimates of the ignition probability at specific points. For the latter, several positions of the spark ignitor were tested: i.e. two axial distances (z = 26 mm and z = 56 mm) in the neighborhood of the combustor wall (Fig. 1a). Note that throughout the document, the axial coordinate attached to the configuration is chosen as follows: z-axis covering the length of the domain (i.e. main entrance, plenum, swirler and chamber); the x, y-axes refer respectively to the cross-stream and vertical directions of the configuration, Fig. 1a. The origin is located on the symmetry axis of the configuration at the exit section of the swirler which also coincides with the main entry plane of the combustion cham-

ber. For LES ignition sequences, only the position z = 56 mm and y = 57 mm is tested. For all experimental tests, i.e. transient operating conditions following an ignition energy deposit or stationary cold and reacting conditions, a specific experimental set-up is devised. To ease diagnostics, the combustion chamber is a simple rectangular box with optical accesses on both lateral sides and through the exit end wall (Fig. 1a). The fuel injector is typical of aeronautical systems and is made of one swirler hosting a centered liquid atomizer forming a hollow cone spray (Fig. 1b). The present case corresponds to kerosene combustion at a mean overall (liquid + gas) equivalence ratio of 0.95. Air and fuel are both injected at 285 K and the mean pressure in the chamber is 1 atm. Mass flow rates of air and kerosene are respectively 35.5 g/s and 2.26 g/s yielding a flow Reynolds number of 85,000 (based on the exit swirler section mean axial velocity and outlet swirler diameter). LES of the non-reacting two-phase flows were previously performed and validated against measurements [37,38,32,35]. These simulations use the WALE [39] sub-grid scale model associated with no-slip walls. Characteristic boundary conditions NSCBC [40] are used for inlet and outlet sections (see Table 1 for a recapitulation of the boundary conditions imposed in these simulations). A monodisperse Eulerian model as well as monodisperse and polydisperse Lagrangian models have been applied to the dispersed phase. Although clear room for improvements exits with both approaches when compared to experiments, the locally monodisperse Euler-Euler model provides similar predictions if compared to the polydisperse or the monodisperse Lagrangian simulations [38]. Note that in all of theses cases, issues related to wall impact or film formation are not addressed and slip wall conditions are used for the liquid phase (zero normal liquid velocity at the wall). Likewise, the locally monodisperse Euler-Euler formalism imposes a specific treatment at the liquid injection plane (Fig. 4). Details about the modeling methodology used here are available in the description of the FIM-UR methodology [36]. The imposed liquid inflow conditions rely for LES on the injector characteristics (type and spray angle) along with a mean diameter of 60 lm (Table 1) as reported experimentally [32,35]. Details about the actual grid resolutions used to reproduce the cold evaporating flow and the ignition sequences are summarized in Table 2. For both cases, grids are fully tetrahedra based meshes (Fig. 2). Local refinement is enforced in the ignition simulations in order to reduce potential modeling issues linked to the energy deposit model [23]. Figure 3 displays the field of axial mean gas velocity in the median vertical plane (x = 0) as obtained with the Euler-Euler approach. The complexity of the flow topology is here clearly illustrated, the central and corner recirculation zones being apparent. These flow structures control the droplet distribution as shown by Fig. 4. This configuration has been extensively studied in previous works [37,38] which are at the basis of the current investigation. Starting from this statistically stationary evaporating two-phase flow, a series of ignition sequences have been performed using LES and the energy deposition model of [23] for the sparking phase. For these transient simulations, the thickened flame model and a two-step reduced chemical scheme for kerosene/air combustion [41] are used during the flame propagation phase [15]. The energy deposited by the numerical spark has an integrated value of 100 mJ (25% of the reported experimental setup) and lasts rt = 50 ls. The size of the modeled spherical deposit is rs = 8 mm which is to be compared to the 10 mm gap between the electrodes of the experimental setup. Eleven ignition sequences were simulated for at most 3 ms by triggering the spark at 11 different times (i.e. initial flow conditions at sparking time) but always at the same location, producing the 11 curves of Fig. 5. The results of the 11 tests are very different, from fast ignition to immediate failure, also showing

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Fig. 1. Geometry of the MERCATO burner [32]. (a) Computational domain retained for LES detailed in Section 2 (black cross: spark location for ignition sequences); (b) detailed view of the injection system considered for the simulation [36].

Table 1 Flow boundary conditions used in LES. Gas boundary conditions Air inlet (plenum) Fuel inlet Outlet Walls Liquid boundary conditions Air inlet (plenum) Fuel inlet Outlet Walls

_ air ¼ 35:5 g=s T ¼ 285 K; m Y O2 ¼ 0:23; Y N2 ¼ 0:67 T = 285 K, Ubulk = 1.3  103 m/s Y O2 ¼ 0:23; Y N2 ¼ 0:67 P = 101,250 Pa No-slip adiabatic T = 285 K, al = 108 dl = 5 lm, Ubulk = 22 m/s _ l ¼ 2:26 g=s T ¼ 285 K; m dl = 60 lm Convective No-slip adiabatic

Table 2 Numerical parameters of the Euler–Euler LES used to simulate the MERCATO burner.

Mesh resolution Number of Nodes Number of Cells Mini cell size (m)

Stationary non-reacting

Ignition sequences

1,237,854 6,798,047 1.5  103

1,876,654 6,765,231 1.5  103

a series of intermediate behaviors where ignition starts more or less slowly and leads to complete ignition or failure. The snapshots corresponding to 3 different ignition times t0 are shown on Fig. 6

after 1 ms of development. According to Fig. 6, the sizes of the hot gas kernel at this instant are very different. One has already begun to shrink (Fig. 6a), another one has spread rapidly (Fig. 6c), while Fig. 6b displays an intermediate behavior, with a flame front highly wrinkled by turbulence. Note that for all the sequences (Fig. 5), the early instants following the energy deposit (Dt < 0.5 ms; Dt standing for the time to the sparking time t0) are very similar indicating that all initial kernel follow the same behavior and that the local conditions are favorable for the establishment of this early part of the sequence. After, 0.5 < Dt < 3 ms clear differences appear although the established flame front has not propagated too far away from the initial sparking location. In fact most views indicate that at these latter instants the flame front encompasses the sparking location and it is only its wrinkling and extent that differ. Note that all of these comments strongly rely on the actual ability of the models and the corresponding LES to be representative of such transient phenomena. This of course still needs to be fully validated. These results are nonetheless used as a first indicator of such evolving from which strategies can be built for industrial design and applications. The results of Fig. 5 illustrate some of the basic ingredients used in the I-CRIT-LES model: (1) ignition in this turbulent flow is strongly dependent on the instantaneous flow when the spark is activated and (2) a model for ignition cannot be built on average flow fields: ignition criteria must be applied on multiple realizations of the same flow and ignition probabilities should be obtained by averaging criteria obtained for these realizations. This is the basis of the I-CRIT-LES model developed hereafter. These

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Fig. 4. Mean liquid volume fraction field in the x = 0 plane.

Fig. 2. Typical mesh resolution used to perform: (a) LES of the stationary evaporating flow and (b) the initial phase of the ignitions sequences. Note the spherical patch of refined mesh at the location of ignition.

Fig. 3. Mean axial velocity field in the x = 0 plane (white line deNotes 0 m/s).

simulations also confirm that although the flame propagation towards the main injector is critical in obtaining a stabilized and fully ignited burner (operable for other operating conditions), the initial flow state near the time of sparking is also crucial in determining the ignition success. 3. A priori evaluation of a burner ignition probability map based on non-reacting simulations The first ingredient of the I-CRIT-LES procedure is to build a set of successive deterministic criteria to be evaluated on a single nonreacting flow realization obtained by LES. In this work, targeted towards aeronautical applications, ignition is produced by a spark or

R _ dVÞ in the Fig. 5. Ignition tests: time evolution of the total consumption rate ð x whole combustor. t0 being the sparking time.

a laser which deposits a certain amount of energy in a cold mixture of gas and droplets. In the following and in agreement with experimental diagnostics targeting ignition probability measurements, an ignition sequence is considered as complete and successful if a stabilized flame is installed at the end of the process. This definition imposes that success is not solely determined by the local ignition of the first flame kernel (like in minimum ignition energy studies in premixed flames [8,9]) but also depends on global stabilization mechanisms themselves determined by the flow configuration and geometry. Empirically, ignition can be divided into a series of successive processes or phases of finite and a priori unknown duration, each representative of a specific mechanism. Each phase is hence associated to a criterion that is to be fulfilled for the ignition process to proceed to the next phase. This chain of processes starts with the creation of a small kernel just after the spark or laser discharge. To proceed to the next step, the deposited energy must be sufficient to sustain a first flame kernel until combustion takes over and produces sufficient heat to increase the initially sparked hot volume. Note that during this transient of unknown duration, the kernel may have been convected to another location or subject to local flow fluctuations (equivalence ratio fluctuations or turbulent field fluctuations). After a certain time and if the previous criterion is fulfilled, the initial flame kernel should reach a sufficient size, start to interact with the surrounding flow and convert into a turbulent propagating flame. Ignition is completed if this turbulent flame propagates upwards to the fuel injection system and stabilizes there. In annular combustion chambers (typical of gas turbines), the last phase towards full ignition of the engine

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Fig. 6. Snapshots of three characteristic ignition sequences 1 ms after the beginning of the deposition t0. The light gray isosurface denotes the Tgas = 1500 K surface. : failed ignition; : sharp ignition.

relies on the propagation of combustion from one burner to all the others. This last mechanism is a large scale phenomenon which depends on the burner geometry and operating conditions which will not be analyzed here. It requires full annular simulations as in Boileau et al. [15]. The full ignition of one sector can be intuitively recast into the steps and states shown in Fig. 7. Each step is built to ease identification of the leading mechanisms and the development of specific criteria detailed in Section 3.2. The justification for this specific sequencing stems from previous observations [29,2,30,3,1,4, 28,27] of ignition. Key mechanisms and associated input parameters are as follows:  Droplet dispersion in the chamber (Initial State in Fig. 7): the first step prior to igniting a burner is to inject fuel and mix it with air to create a locally ignitable mixture. For I-CRIT-LES this data is obtained by extracting statistically independent snapshots of the LES predictions of a cold (possibly evaporating) two-phase flow, which produces instantaneous distributions of liquid and gaseous fuel in the chamber. As underlined in Section 2, the local mixture fraction and velocity fields at the moment of ignition are crucial and are the primary inputs of the I-CRIT-LES model to determine if the local mixture is flammable or not. This initial state is evaluated by applying the notion of Flammability factor, PF as discussed in [29,1].  Spark or laser discharge and first hot gas kernel (Phase 1 in Fig. 7): ignition is triggered by an energy deposition which leads to complex processes including non-equilibrium physics. If sufficient, the energy deposition leads to the creation of a small hot

gas kernel whose temperature should rise above the ignition temperature. The size and temperature of this first kernel are critical for its later growth and this initiation of a first flame kernel.  Growth and convection of the flame kernel (Phases 2–3 in Fig. 7) constitute the essential elements to determine Pker or the probability of initiating a sustainable flame kernel [29,1]. Typically, if the vaporization time is small compared to the heat diffusion time, the hot gas kernel yields a growing flame kernel whose size increases at a speed dictated by the local flame curvature and characteristic scales at play [14]. One can distinguish two sequences [22,25]: – The early expansion phase (Phase 2 of Fig. 7) is dominated by the laminar flame speed (function of the local equivalence ratio), the flame curvature and the hot to cold flow densities. . . – The ‘‘turbulent’’ phase (Phase 3 of Fig. 7) that follows, involves large/resolved flow scale interactions between the flame front and its surrounding. Note that within these two sequences, strong flow/combustion interactions are at play (convection of the initial kernel to other locations by the mean flow for example) which cannot be simply recovered by a priori evaluation of simple criteria at a given Eulerian locations as devised in I-CRIT-LES. Transition to a turbulent flame appears only if the flame front reaches sizes comparable to the smallest turbulent structures. If so, the flame kernel starts to interact with turbulence, leading to wrinkling and stretching of the flame front. In most situations, this will accelerate flame propagation. If turbulence is too strong

Fig. 7. Schematics of the different processes identified for the construction of the I-CRIT-LES procedure: dark zones correspond to ignited regions and light gray represents the spray region.

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however, local or global flame quenching may occur [42,30,26]. This last point is not considered in the present analysis as turbulent quenching is not expected to be a major mechanism in the target applications: i.e. the size and amount of energy deposited by typical aeronautical spark ignitors way outsize the turbulent length scales encountered in such devices preventing aerodynamic quenching of the initial turbulent flame kernel as confirmed by the LES of Section 2. However another quenching mechanism may occur in case of flame interaction with cold walls. Only this last occurrence is considered in the present analysis. These two phases and issues have been extensively investigated for homogeneous and heterogeneous mixtures and translate into the notion of minimum ignition energy [14,43].  Propagation of the turbulent flame towards the injection system (Phase 4 in Fig. 7): usually the ignition system is located downstream of the injection system, and the turbulent flame kernel is created away from the stationary flame position of the fully ignited burner. In order to stabilize near the injector, the turbulent flame must thus in some cases propagate upstream, i.e. against the mean local flow direction and irrespectively of the initial energy deposit location. This upstream propagation is hence only possible in low axial velocity zones, around jets and requires a turbulent flame speed larger than the local flow speed [44,45,28]. Note that such requirements are not needed in recirculation zones usually found in the near field of swirled liquid injectors since the mean flow direction is here oriented towards the injector. A critical aspect of the flame propagation process (also present in Phase 3) is that while the turbulent flame front locally evolves, convection by the mean flow of the burnt gases or flame front elements makes the hot gas pocket to be in contact with more favorable locations. Typically a weak flame kernel deposited in a jet flow can enter a region more favorable to its propagation or even ignite a new region yielding a successful ignition sequence. The mechanisms at play during this flame or hot spot migration are clearly not just local and are a very difficult processes to model a priori without simulating them directly. Although solutions in a Lagrangian framework can be proposed [27], a simple local model is retained here knowing that it will need to be greatly improved for reliable predictions of the ignition probability maps.  Stabilization of the flame: this is the end of the ignition process of a single burner. It is not a specific phase of the ignition sequence but the desired result that is fulfilled if all phases 1– 4 and associated criteria are met. For the full annular burner ignition at least one additional step should be added, corresponding to flame propagation from ignited burners to their neighbors [15]. This last step will not be considered here. Note also that in the specific context of the in-flight re-ignition of a single burner, hot product pockets are present in the flow and re-ignition probability predictions could be addressed by the proposed strategy if the initial extinguished LES is considered. Although possible, this specific case is not covered here since very difficult to validate. 3.1. Ignition criteria Following the previous phenomenology (Fig. 7), five criteria are listed below. The objective of each criterion is to be representative of the critical physics necessary for successful ignition. Each condition must be fulfilled for the successful completion of each step and the start of the next one: C1: (Initial State) – The fuel distribution must guarantee a flammable mixture.

C2: (Phase 1) – The discharge energy must be sufficient to create a first hot gas kernel in the very early time following the energy deposit. C3: (Phase 2) – The local conditions (vaporization time versus diffusion time) must allow the flame kernel to increase. C4: (Phase 3) – The flame must not quench. C5: (Phase 4) – The flame has to be able to propagate upwards at least provided the local conditions. These conditions need now to be transposed into mathematical expressions to be evaluated on instantaneous flow fields. 3.2. Modeling All expressions proposed below result from ideal and isolated 1D phenomena taking information from the assumed near stationary flow field described in an Eulerian framework. Interactions and transport that are effectively at play in a real ignition sequence are here not considered unless a Lagrangian framework or a full simulation is performed [46,47]. To obtain a numerically effective procedure, such alternatives are not developed or attempted here and certain effects are thus discarded. Typically, the ignition success of an energy deposit at a location where conditions are not favorable because of its transport to more favorable locations [47] cannot be captured. These processes mainly affect Phases 3 and 4. With the proposed modeling and criteria developed below only local one point statistics are accessible, which prevent to accurately predict or address such effects on the determination of Pign the probability of ignition success. Despite such clear limits, local variability at the time of sparking, as evidenced in Section 2, are imbedded in the procedure since captured by the set of instantaneous LES snapshots. 3.2.1. Criterion 1: Flammability During a two-phase flow ignition, the available fuel is the fuel vapor (usually small, especially in high altitude cold conditions) and the liquid fuel which will evaporate at the spark location. As a consequence, flammability of the mixture is evaluated by the total (gas + liquid) equivalence ratio, that must be in the flammability limits of the considered fuel. Defining the equivalence ratio as:

U ¼ Ugas þ Uliq ¼ s

YF q al þs l YO qY O

ð1Þ

where s is the stoichiometric ratio, YF and YO respectively the fuel and the oxidizer mass fraction, q the gas phase density, ql the liquid phase density and al the liquid volume fraction. Criterion 1 writes:

Ulow 6 U 6 Uhigh

ð2Þ

where Ulow and Uhigh are respectively the low and high flammability limits, known for usual fuels. Note that in the context of sprays, flammability limits are known to be strongly affected by local spray properties and corrections may be introduced to consider such effects [48]. Such corrections have not been introduced in this work but can readily be implemented as recommended by [48,49].

3.2.2. Criterion 2: Energy discharge and first kernel Details about spark ignition generation are comprehensively given for various types of ignitors (laser and electrode discharges) in [50–52,6,53,54] and briefly recapitulated in Section 2. The actual energy transferred to the flow is known to strongly depend on multiple parameters yielding a high degree of uncertainty of this crucial phase. For the following these are considered as input parameters whose characteristics (time, size, temperature. . . ) are specified by the end user. In a spray, droplet evaporation is much slower than the energy transfer from the ignitor to the gas: it is

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assumed here that droplets do not interact directly with the ignitor and are heated only by their interaction with the hot gas (Fig. 8). In practical devices, the spray is ideally very dilute at the spark location so that the energy absorbed by the liquid is very small. This observation is of course limited to specific conditions and most of the time (high altitude ignition), the liquid fuel injected in the burner may yield very large filaments or drops invalidating the set of hypotheses introduced here. However within the proposed hypotheses, the minimum energy required to install a first kernel is determined by considering the heating, evaporation and ignition of a homogeneous mist of droplets in an initially fuelvapor free gaseous environment (Fig. 8). Numerical tests (A) show that if the liquid is too far from saturation conditions (pressure, composition and temperature) when combustion starts, evaporation is too slow to feed the flame so that a criterion can be defined by introducing an ignition temperature Tign which is the temperature at which the chain-branching reaction balances the terminating reaction [55]. This may be expressed in terms of a gas temperature, that must stay above the ignition temperature Tign until the liquid has reached its saturation or boiling temperature Tcc (Fig. 9). If tcc is the time at which the liquid reaches saturation, Criterion 2 is:

Tðtcc Þ P T ign

ð3Þ

Criterion 2 requires the knowledge of the gas temperature temporal evolution T(t), which is obtained from the evolution equations for the gas (T) and liquid (Tl) temperatures at the spark location (typical evolutions are shown in Fig. 9). Leading terms to take into account are the heat diffusion, the heat transfer between the two phases and the source term (the spark energy during the deposition phase, S ¼ e_ dep ) all expressed in a simplified form (see Appendix B for detailed developments):

@T T 1  T T l  T S ¼ þ þ @t sdiff scond qC p @T l Tl  T ¼ A @t scond

Fig. 9. Typical evolution of gas and liquid temperatures when ignition occurs (thermal runaway after tcc).

scribe heat diffusion in the gas and heat transfer between the phases, ndim being the number of space dimensions (2D vs. 3D) and Nu the Nusselt number evaluated as in [56]. In these expressions, D is the heat diffusivity, dk is the kernel size, and dl is the droplet diameter. Solving Eqs. (4) and (5) yields:

Tðtcc Þ ¼ T d 

scond ðT cc  T l;0 Þ As

ð6Þ

where Td = T1 + S/qCp is the gas temperature at the end of the energy deposit phase and 1/s = 1/sdiff + 1/scond. Finally Criterion 2 becomes:

S

qC p

P ðT ign  T 1 Þ þ

scond ðT cc  T l;0 Þ As

ð4Þ

or

ð5Þ

! 2 S 4ndim dl ql C p;l P ðT ign  T 1 Þ þ al þ ðT cc  T l;0 Þ qC p 3Nu d2k qC p

These equations are written for the sparking phase and the heating phase of the droplets, before they reach Tcc, therefore with negligible evaporated fuel and negligible combustion. A unit Lewis number is assumed. The initial temperatures for the gaseous and for the liquid phase are respectively noted T1 and Tl,0. The parameter A = qCp/alqlCp,l is the liquid to gas ratio of heat capacities. The two 2 2 characteristic times sdiff ¼ dk =8ndim D and scond ¼ dl =6Dal Nu de-

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ð7Þ

ð8Þ

This criterion expresses that the energy deposit must be sufficient to heat the gas from T1 to at least Tign and the liquid from Tl,0 to Tcc while the gas temperature does not cool down to a temperature lower than Tign. Note that dl = 0 (i.e. al = 0) provides the gaseous limit of the criterion, i.e. the minimum ignition energy density for a gaseous mixture. 3.2.3. Criterion 3: Kernel growth The next step is to sustain the kernel and ensure that it can grow (Phase 2 in Fig. 7). If combustion is not strong enough, the heat release is not sufficient to compensate for the cooling of the gas due to diffusion and the kernel quenches. This third criterion is derived following the methodology of Ballal and Lefebvre [14]. Considering a hot gas kernel surrounded by a flame, in a mixture of fresh gas and droplets (Fig. 10), characteristic times of evaporation svap, diffusion sdiff and combustion scomb are compared. Combustion is sustained if the droplets can evaporate and burn before heat diffuses away:

sv ap þ scomb 6 sdiff

ð9Þ

As combustion is fast compared to diffusion and evaporation, scomb  0. The vaporization time is determined from the evaporation mass transfer as expressed in Spalding’s theory [57]:

sv ap ¼ Fig. 8. Model for the temperature profile at the end of the energy deposition.

ql d2l 6U Sh q D lnð1 þ BM Þ

ð10Þ

where BM is the mass Spalding number. After rearrangement of Eq. (9), one finds Criterion 3:

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r Q P d > dQ

and U  n > 0

ð13Þ

 At last, if:

d > rQ

ð14Þ

then Criterion 4 is true. Other quenching events are of course possible in the early phase of the ignition process. Aerodynamic quenching is a reported issue for which a criterion could be developed and included at this stage. Such events can indeed affect the early evolution of the flame kernel in regions of high strain and shear especially if the initial size of the deposit approaches the turbulent flow scales. This can appear in region of stagnation points or at the edges of recirculation zones if low energy sparking devices or highly focused laser ignitors are used. Fig. 10. Expansion of the initial laminar kernel.

dk P dl

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ndim ql 3U Sh q lnð1 þ BM Þ

ð11Þ

which compares the kernel size dk to the droplet size dl. If the kernel size is too small compared to the droplet diameter, heat diffusion is too fast compared to evaporation and the flame cannot survive. 3.2.4. Criterion 4: Wall quenching Previous studies [58–61] have shown that a flame cannot exist below a critical wall distance called the quenching distance dQ. The quenching distance has been evaluated for various laminar flames interacting with walls, and is usually expressed as a Peclet number PeQ ¼ dQ =d0l where d0l is the local laminar flame thickness. This number approaches 3.0 for usual hydrocarbons (1.7 for H2/O2) when the flame propagates normally to the wall (Head On Quenching). A characteristic time of the interaction tQ can be defined to represent the time needed to quench the flame. This time expressed in a non-dimensional form reads sQ ¼ t Q =t0l where t0l ¼ d0l =S0l . For typical hydrocarbons sQ  2.1. Defining d as the distance of the ignition kernel to the wall, knowing PeQ and sQ, criterion 4 is expressed differently depending on the wall distance d (Fig. 11):  First, if the kernel is initiated within the quenching layer, i.e. if d 6 dQ, it has to leave this zone before t = tQ. Hence to survive:

d 6 dQ

and U  n >

dQ  d >0 tQ

ð12Þ

where U is the local flow velocity and n the wall normal.  Second, even if d > dQ, the flame may be convected within the quenching layer. It therefore has to be strong enough before entering this zone. It is assumed that the flame is strong enough when it has developed during at least sker = tcc + sdiff which represents the time to reach equilibrium (i.e. non-super adiabatic temperature). This leads to the definition of an ‘‘overlayer distance’’ rQ = (U  n)sker and the second case:

Fig. 11. Definition of wall distances for Criterion 4.

3.2.5. Criterion 5: Upstream flame propagation At this stage, a fully turbulent flame is supposed to be present in the neighborhood of the ignitor. To ignite the complete chamber, this turbulent flame kernel should at some point propagate upstream to anchor in the near field of the main fuel injector yielding a fully stabilized flame. As already discussed, this is a complex issue probably requiring full LES of ignition sequences (something we do not want to perform within the present model) or the use of Lagrangian simulations of kernel/flame trajectories (see for example [47]). Criterion 5 of I-CRIT-LES considers a simpler condition which is more restrictive and limited since based on an evaluation at the energy deposit location so that history and convective effects potentially bringing flame elements to favorable locations are not captured by this simple condition. Despite this clear shortcoming and the limitations of such a strategy in terms of predictability, the main objective is here a reduced algorithmic effort and a fully local tool that can solely rely on one point field descriptions. Consequently if one assumes that ignition can occur if the flame travels in the axial direction at a net velocity (i.e. with respect to the observer) that allows the front to go upstream or towards the injector: the turbulent flame speed ST must be larger than the mean flow velocity component aligned with the injector orientation axis and noted U:

U  ST 6 0

ð15Þ

where U is the local flow temporal mean value and ST  SL ðUÞ þ 2:5u0 is the turbulent flame speed [62–66,28] with u0 being the turbulent velocity fluctuation. The corrected laminar flame speed SL ð/Þ for two-phase flow flames is for lean combustion as in [14]:

0

11=2

1 C Bsv ap SL ð/Þ ¼ @ þ  2 A D S0L

;

ð16Þ

where, D is the gas thermal diffusivity. Note that other expressions are available in the literature for extended ranges of conditions [27,48,49]. The chosen expression is retained for its simplicity. However the limits introduced by the initial assumptions needed to ease derivation of Criterion 5 will not adequately cover the true physics possibly encountered during Phase 4. All five criteria are necessary for ignition within the context of the I-CRIT-LES procedure. The resulting ignition prediction is evaluated through the ignition index noted Iign = C1  C2  C3  C4  C5, where Ci takes the value 1 if Criterion i is met and is 0 otherwise. At this stage, Iign is the result of satisfied or failed observations issued by analytical expressions estimated locally from one flow snapshot. It is a deterministic quantity which can take only 0 or 1 local values depending on the spatial evolution of the various variables (liquid mass fraction, fuel mass fraction, gaseous and liquid temperatures. . . ) used in the analytical expressions.

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3.2.6. Multiphase flow sensitivity and contributions to the global ignition index The primary contribution of the above criteria is that it is applicable to multiphase reacting flows. A direct consequence is that most of the criteria require multiple inputs including characteristic quantities of the disperse phase. Two distinct components of the ignition probability are targeted by the proposed criteria: (a) the first three criteria aim at evaluating Pker when faced with multiphase flows and (b) the remaining criteria target the issue of turbulent multiphase flames. This last issue is addressed indirectly through the use of turbulent scales and more or less ad hoc correlations available in the literature. The first three criteria are more fundamental and it is of interest to illustrate their dependencies on local flow quantities. The following analysis concentrates on stagnant homogeneous flow problems where the ignition index simply reduces to Iign = C1  C2  C3. For illustration and analysis a quiescent two phase kerosene air mixture is considered as initial condition for an ignition test. For this study, the energy deposit is fixed at Edep = 50 mJ and the liquid

1199

volume fraction is set at al = 3  104 for an overall equivalence ratio of 0.95 and a liquid temperature of Tl = 285 K. Since the flow velocity is zero everywhere, criteria C4 and C5 do not apply. Evolutions of C1, C2, C3 and the resulting Iign as functions of droplet diameter, gas temperature and pressure are first presented on Fig. 12. With such initial conditions and no flow, ignition is more difficult at high pressure: almost all criteria become 0 when pressure is higher than 30 bars. For C1 (flammability limits), the critical pressure noted Pil and identified by the light gray iso-surface on Fig. 12a, indicates the limits above and below which flammability limits are not reached. As expected higher initial gas temperatures lead to easier ignition and higher values of Pil. For C2 (the kernel generation), shown on Fig. 12b, a dominant dependency on the gas temperature is also observed. For C3 qualifying the kernel growth and shown on Fig. 12c, the limiting pressure Pil depends mainly on the droplet diameter. The Pil surface obtained from the combination of Criterion 1 to 3 leads to the complex shape of Fig. 12d. This last result clearly indicates that ignition of real gas turbine engines at high pressure and cold conditions is difficult

Fig. 12. Isosurfaces of ignition criteria and the global index in a pressure, initial gas temperature and droplet size diagram (light part means the criterion is validated, i.e. =1). The spark energy is 50 mJ and the initial fuel volume fraction is al = 3  104.

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and can be guaranteed only if proper atomization of the liquid fuel jet is ensured, along with a dispersion mechanism that can produce local conditions at the sparking location that are well within the flammability limits. In the former diagnostics, Edep is fixed and local conditions are varied. An alternative use of the criteria is proposed in Fig. 13: keeping the pressure and gas temperature constant to 101,300 Pa and 284 K, the dependency of Iign to the deposited energy to the gas can be investigated. As expected, C1 (Fig. 13a), produces a Edep independent threshold corresponding to the rich flammability limit of the model kerosene fuel under these conditions (Utot = 0.95). Likewise, and if the initial kernel is generated, its expansion is inversely proportional to the liquid phase local droplet diameter, dl, as evidenced by Eq. (11) and confirmed by Fig. 13c. Note that the actual dependency of C3 is relative to the energy deposit size (dk in Eq. (11)) which underlines the importance of the sparking device.

a 0 or 1 ignition index. However exact conditions (temperature and composition of the liquid and gas phase) at the location and moment of ignition fluctuate due to the unsteady nature of the flow. As confirmed by the LES’s of Section 2 these will affect the ignition success. A probabilistic model is clearly a better indicator and to meet this requirement, the I-CRIT-LES uses a single non-reacting LES to evaluate ignition probability without actually simulating ignition events. In this approach, each instantaneous LES solution is viewed as one flow realization, the finite set of instantaneous fields being used as a statistical ensemble of samples representative of the turbulent field. This requires that the solutions are taken after statistical convergence has been reached and that solutions are sampled at a sufficiently low frequency to guarantee that they are uncoupled. Applying criteria C1 to C5 to a series of N realizations (instantaneous solutions at t = tk) allows to build a probability for the local ignition index:

3.3. Ignition probability

pðIign Þ ¼

The ignition scenario and criteria described in Section 3 are developed in a fully deterministic framework: given initial conditions, the model predicts if ignition will be successful or failed with

This statistical estimate is a three-dimensional field giving the one point probability of completion of all ignition steps at each location in the burner. Using expression Eq. (17), one can also evaluate the

N 1X Iign ðtk Þ N k¼1

ð17Þ

Fig. 13. Isosurfaces of ignition criteria and the global index in a liquid volume fraction, droplet diameter and the energy deposit diagram for MERCATO conditions (light part means the criterion is validated, i.e. =1).

A. Eyssartier et al. / Combustion and Flame 160 (2013) 1191–1207

probability of completion of one of the criteria. Assuming statistical independence of the criteria C1 to C5, one can write Q pðIign Þ ¼ 5i¼1 pðC i Þ, where the probabilities p(Ci) are built as in Eq. (17). Tests (not presented here) show that these two definitions of p(Iign) do not provide the same results. Indeed, criteria C1 to C5 are not statistically independent, especially C2 and C3 (the amount of the energy deposit will directly influence the size of the kernel at the beginning of Phase 3, Fig. A.19). The present approach also allows to analyze the causes of failure when the ignition probability, p(Iign), is lower than 0.5, identifying the limiting criterion as the criterion with the lowest probability p(Ci). 3.3.1. Comparison of I-CRIT-LES against experimental data To fully apprehend the potential and the limitations of the current procedure, simple configurations for which ignition maps have been obtained experimentally are considered and compared to I-CRIT-LES estimates. To do so, two laboratory set-ups are used:

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(a) a purely gaseous jet flame configuration [29] (Fig. 14a and b) a two-phase swirled bluff-body evaporating flow [3] (Fig. 14b). Each case is decently simulated by use of gaseous or Euler/Euler evaporating non-reacting LES as detailed in Table 3. For case (a), the jet issues a homogeneous mixture of methane (70%) and air (30%) with a jet Reynolds number of Re = 5800 while case (b) injects liquid n-hepthane into air through a jet located at the center of the bluff-body wall resulting in a mean overall equivalence ratio of 0.9. For purely gaseous flames, all I-CRIT-LES criteria degenerate to the model of Lacaze et al. [23] except for the convection criterion (C5) which results into a strictly positive upstream propagation speed (no downstream propagation of the flame), while Lacaze et al. [23] use a threshold of 5 m/s because of quenching events reported experimentally. Although the two criteria are motivated by different physics, in the specific context of jet flames the aerodynamics quenching correlates with the flow velocity field so both criteria will coincide if the local difference between the axial

Fig. 14. Ignition probability maps for (a) & (c) the gaseous jet flame of [29] and (b) & (d) the two-phase recirculating burner (case S1) of [3]. Top line, experimentally measured maps and bottom line, the corresponding I-CRIT-LES results.

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Table 3 Numerical parameters of the cold non-reacting gaseous and Euler–Euler LES used in the I-CRIT-LES evaluation. Gaseous jet simulation

Buff-body two-phase simulation

Mesh resolution Number of nodes Number of cells Mini cell size (m)

2,080,000 12,180,000 2  106

1,876,654 2,853,000 2  104

Models SGS closure Liquid phase model

WALE [39] –

WALES [39] Locally mono-dispersed model [37,36]

component of the gas velocity and a 5 m/s turbulent flame guaranties an upstream propagation direction. Correcting Eq. (15) to take into account this experimental observation and considering that C5 correlates to local turbulent quenching as evidenced experimentally, one gets the map of Fig. 14c, in agreement with [23]. This comparison with experimental results shows that I-CRIT-LES can provide comparable maps to experiments provided an accurate evaluation of aerodynamic quenching events as reported experimentally which is not directly addressed in I-CRIT-LES. Differences underline the limits of the procedure that can hardly accurately represent stabilization processes through local evaluations of dedicated criteria. However, if the initial phases C1–C3 dominate Pign as evidenced in this jet flame ignition sequences good tendencies are recovered. Application of I-CRIT-LES to the bluff-body two-phase flow case is shown on Fig. 14d. Although unable to recover the locus of 100% of success of ignition, I-CRIT-LES reproduces the overall shape of the ignition probability map. These last results further highlight the difficulty of properly evaluating the probability of ignition in complex configurations involving recirculating flows. Indeed, in such flows, the initial kernel trajectory and flame trajectories are important: even if it is generated in a non-flammable region, a hot kernel can be transported by the flow into more favorable regions and lead to ignition [1]. Likewise flame elements can propagate in regions of favorable ignition while others can simply be convected away from the injector. Both phenomena relative to Phases 3 and 4 are not predictable accurately with the local estimates as proposed in I-crit-LES. This is clearly evidenced by the comparison with the predicted map and the experimental finding. Potential directions of improvements to account for these physics could be to account for such effects by introducing pseudoLagrangian tracking as proposed by [28,27]. 4. Application to an aeronautical chamber 4.1. Mercato experiments: Comparison of I-CRIT-LES with LES’s and experimental data Despite the clear limitations of the I-CRIT-LES model, the proposed procedure is applied to the MERCATO configuration described in Section 2. The first intent of the exercise is to check if I-CRIT-LES is able to match the success/failure of the individual deterministic ignition sequences presented in Section 2 and conducted by use of LES. This comparison provides first insights on the ability of the procedure to capture local condition effects at the time of sparking. For the test, the cold two-phase flow fields at the various t0 are used. The outcomes are fields of Iign locally taking binary values (0 or 1) for the entire computational domain indicating favorable or non-favorable conditions for ignition at this given instant. In this test, the analysis is limited to the location where the energy is actually deposited. Results for all 11 individual instantaneous fields of Section 2 are shown on Fig. 15. The methodology predicts no ignition for t0 = 459.75 ms

and 460 ms (Fig. 15c and d respectively), almost no ignition but favorable zones for ignition very close to the spark for t0 = 459.25 ms and 460.5 ms (Fig. 15b and e respectively), and ignition for the other cases. These observations are in good agreement with the results of full LES presented in Section 2. Like in Fig. 5, failure of ignition started for t0 = 459.75 ms and t0 = 460 ms are predicted, as well as the sharp ignition obtained for t0 = 464 ms. I-CRIT-LES however does not predict late ignition (t0 = 459.25 ms and 460.5 ms of Fig. 5 to be compared to Fig. 15b and e), as this involves convection of the kernel, not considered here. To further test the model, experiments have been conducted to qualify the burner ability to ignite in various conditions [31,34,35]. Several positions of the spark ignitor as well as a laser ignitor are tested (at the axial locations z = 26 mm and z = 56 mm). For each case, ignition probability maps are constructed from the recording of success/failure of experimental ignition sequences. A direct comparison with the present model is thus possible. For this comparison, the ignition index is built from 200 independent instantaneous LES solutions over a total time of 20 ms, approximately corresponding to one flow-through time of the combustion chamber and 10 revolution times of the swirling flow.1 The I-CRIT-LES output is shown in Fig. 16. Ignition is most probable in recirculation zones, almost impossible close to the fuel injection and along the top and bottom chamber walls as well as in high speed flow zones. The probability is intermediate in the region surrounding the recirculation zones. This result is in good agreement with experimental findings obtained from the laser ignition campaign, indicated with the black and white arrows in Fig. 16. Note that differences do exist around the fuel injection axis where the predicted probability is high (i.e. white) while no ignition could be obtained experimentally. This is explained by the fact that the laser ignitor used in the experiment could not be focused at the center of the domain due to the presence of the spray and its diffraction effect on the laser beam. Likewise, levels of Iign in favorable regions are higher than reported probabilities observed experimentally. Despite these differences, the procedure is able to properly discriminate locations of potential ignition success from regions where ignition is statically difficult to obtain. In order to understand the cause of ignition failure as well as the interest of I-CRIT-LES in comparison to more conventional ignition criteria, additional information is provided on Figs. 17 and 18. Fig. 17 presents the probability of satisfaction of (a) C1 (flammability limits) and (b) C5 (propagation towards the injector) issued by I-CRIT-LES from the 200 instantaneous LES fields. The flammability factor, Fig. 17a, confirms that aside from the CRZ and near the fuel injection, all locations are good candidates to establish local combustion. This is however not sufficient to guarantee ignition as evidenced by experimental findings and the view of Fig. 17b which disqualifies regions of high axial gas velocity as targeted by C5. Note that the combination of both indices in I-CRIT-LES fully coincides with the index proposed by [23] developed for gaseous flows. Differences appear in the extension to two-phase flows (through the new expression of C1) and in the local threshold propagation speed given in Eq. (16) which in I-CRIT-LES targets locally the ability for a flame to propagate upstream and in [23] targets quenching events observed experimentally. For the operating conditions considered, these two criteria (C1 and C5) seem sufficient to discriminate high from low probability of successful ignition regions although more developments and investigations are clearly needed. To better apprehend the different contributions of all five criteria to Iign, the local limiting criterion is displayed in Fig. 18. The values 1–5 correspond to the criterion with the lowest probability (over all realizations), 6 indicating that all criteria are validated 1 The evaporation time scale is for this burner evaluated as twice one flow-through time indicating that in stationary conditions flow characteristics are dimensioned by the mixing of the dispersed phase.

A. Eyssartier et al. / Combustion and Flame 160 (2013) 1191–1207

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Fig. 15. I-CRIT-LES applied to single snapshots for the same times as in Section 2. White: I-CRIT-LES = 1 (ignition); Black: I-CRIT-LES = 0 (no ignition). Reminder of Fig. 5: circle denotes spark location. : failed ignition; ⁄: late ignition; ⁄⁄: successful ignition; ⁄⁄⁄: sharp ignition.

Fig. 16. Global ignition index in the x = 0 plane given by I-CRIT-LES for Edep = 100 mJ. Dark zones correspond to low ignition probability and the white iso-line coincides with Iign = 0.66 reported experimentally for the location of sparking tested in Section 2. Arrows report experimental tests: white indicates that ignitions have been observed (with a high degree of probability, i.e. above 0.66), black indicates that no ignition has been observed at these locations.

and ignition is thus highly probable (more than 0.5). The first dark zone, limited by the flammability condition C1, is at the fuel injector nozzle, where droplets accumulate because of the recirculation zone just downstream, yielding a mixture that is locally too rich.

The second dark zone corresponds roughly to a low speed area of the lateral recirculation zone, where the droplets also accumulate. The last dark region (next to the top and bottom walls) corresponds to the impact of the main flow on the wall. C2 is always fulfilled, meaning that the energy deposit is always sufficient for this case. In gray, C3 indicates that evaporation is too slow compared to diffusion to sustain the flame kernel. C4 applies only in a thin zone near the wall and does not appear to be a limiting criterion for this configuration. Finally C5 in light gray is limiting in zones of strong axial velocity, in the shear zone around the central recirculation where the turbulent flame speed is not sufficient to propagate the flame in the upstream direction. An alternative use of the ignition index is to determine the Minimum Ignition Energy (MIE) to be provided for successful ignition: this point is reached when C2 becomes the most restrictive. Note also that since the notion of MIE relates to an energy density that involves the actual diameter dk over which the deposit is produced, it will also affect C3. For the operating point under study, it was found that a minimal energy of about 20 mJ is necessary to guarantee the growth of an initial flame kernel over a relatively large area (about few dk large) that encompasses non-zero probability of the follow on criteria to ignite the chamber. Experimental tests were done with an energy range from 60 to 212 mJ, their purposes were however not to evaluate MIE.

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Fig. 17. Probability in the x = 0 plane of: (a) C1 and (b) C5 given by I-CRIT-LES for Edep = 100 mJ. Dark zones correspond to low probability of satisfaction of the studied criterion.

droplets) that is ignited by a 1D spark for which the local energy deposited by unit volume is [23]: 2 1 Edep 1 xx0 e2ð rx Þ e2 e_ dep ðx; tÞ ¼ pffiffiffiffiffiffiffi 2 ð 2pÞ rx rt

Fig. 18. Limiting criterion of I-CRIT-LES in the x = 0 plane. Black cross: spark location studied experimentally and by LES using Edep = 100 mJ. Numbering corresponds to the criterion number limiting the ignition success (from 1 to 5 for C1 to C5 respectively), 6 indicates ignition success with Iign P 0.5.

5. Conclusions Ignition sequences using LES have been performed for the MERCATO configuration. These simulations underline the large variability of the ignition process and its different phases. An analytical model, I-CRIT-LES, to predict ignition probability of liquid–gas mixtures from non-reacting fields in any configuration is then presented. This approach is based on the identification of successive phases including a propagation mechanism that need to be validated for the burner ignition to be fulfilled. The probabilistic nature of ignition is recovered with this new approach by applying the different ignition sub-criteria to a series of independent LES snapshots. Comparisons of the I-CRIT-LES against experiments which probability maps have been measured, confirm the potential and limitations of the model. Results are however encouraging and emphasize the need for such tools to improve predictability of igniting real configurations. Application of I-CRIT-LES to a real combustor confirms the potential of the approach and allows to identify the best ignitor locations as well as the potential causes of failure in the other regions. Appendix A. Validation of criterion 2 using 1D laminar numerical simulation To take into account the size of the energy deposit, one-dimensional simulations are performed to study the evolution of the twophase mixture (Fig. A.19) and to test the criterion associated (Fig. A.21). Here a simple 1D DNS is produced for a two-phase mixture of air and kerosene (only liquid initially and monodisperse

tt 2 0

rt

ðA1Þ

R R In Eq. (A1), Edep ¼ e_ dep dt dV ¼ edep dV is the total amount of energy transferred by the spark to the gas, rx and rt are parameters that control the size and duration of the source term, x0 and t0 are the space and time coordinates of the deposit. Initial conditions are P = 1 atm and a temperature of 300 K for both phases. The droplets size is 60 lm and the liquid mass fraction 104. The spark characteristics are: rx = 0.5 mm and rt = 8.2 ls. The liquid fuel is kerosene (surrogate and chemical kinetic scheme from Franzelli and Riber (2010) [41]). The temporal evolution of the droplet (dashed line) and gas (solid line) temperatures are shown in Fig. A.19. In a first phase (t 6 td), the gas temperature increases sharply, under the effect of the energy deposition. Then in a second phase (td 6 t 6 tcc), heating of the droplets starts, while the gas cools down. After some time, the droplets reach the fuel’s saturation temperature Tcc, while the combustion reactions, activated by the hot gas temperature, start to release heat, resulting in a fast increase of the gas temperature and a second temperature peak at 0.4 ms. This last step corresponds to the creation of a first flame kernel. If the spark does not increase the gas temperature enough, the gas may be cooled too fast by heat diffusion, and fuel evaporation is too small so that reactions cannot start and ignition fails (dashed-dotted lines in Fig. A.20). A series of 1D planar spark ignition simulations are performed for different droplet sizes dl and spark energies Edep, keeping all other parameters unchanged. The resulting ignition map is compared to C2 in Fig. A.21, showing that C2 correctly predicts ignition limits for the presented test cases. Appendix B. Criterion 2 developments The evolution equations for the gas (T) and liquid (Tl) temperatures at the spark location, taking into account heat diffusion, heat transfer between the two phases and a source term S (the spark energy during the deposition phase, see phase 1 in Figs. 9 and A.19) are:

@T k U S ¼ r2 T þ þ @t qC p qC p qC p @T l U ¼ @t al ql C pl

ðB:1Þ ðB:2Þ

where k is the thermal conductivity, q the gas density, ql the liquid density and Cp the thermal capacity.

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Fig. A.19. Ignition of a homogeneous droplet cloud (Edep = 4.3 mJ).

Fig. A.20. Comparison of the temporal evolution of maximum gas temperature between successful (1) and failed (2) ignition.

is assumed to follow a parabolic profile (Fig. 8) with a maximum value T at r = 0 and a temperature T1 at r = dk/2, the Laplacian of T can be approximated by:

DT ¼

8ndim ðT 1  TÞ

ðB:4Þ

2

dk

Assuming Le = 1 (i.e.

k

qC p

¼ D), Eq. (B.1) becomes:

@T 8ndim DðT 1  TÞ 6 S ¼ þ al DNu 2 ðT l  TÞ þ 2 @t qC p dk dl Then, introducing: to:

2

The heat conduction flux U between the two phases is defined as:

UðT f Þ ¼ al kNu

2

d l scond ¼ 6adl DNu and sdiff ¼ 8n k D, Eq. (B.5) simplifies dim

@T 1 1 S ¼ ðT 1  TÞ þ ðT  TÞ þ @t sdiff scond l qC p Fig. A.21. Minimal energy as a function of droplet diameter dl for a stoichiometric mixture. 1 and 2 refer to Fig. A.20.

ðB:5Þ

ðB:6Þ

Similarly, Eq. (B.2):

@T l 1 ðT  TÞ ¼ A @t scond l

ðB:7Þ

qC

where A ¼ a q Cp

l l pl

6 2

dl

ðT f  TÞ

ðB:3Þ

where Tf is the droplet surface temperature and is equal to Tl as droplets are assumed to have uniform temperature. The exact resolution of Eqs. (B.1) and (B.2) is beyond the scope of this paper but a simple approximation of the gas temperature space shape within the deposit can be obtained by replacing the DT term in Eq. (B.1) by a simplified expression: if the temperature

These equations are valid for the energy deposition (Phase 1 in Fig. 9) and the heating phase of the droplets (Phase 2 in Fig. 9), before they reach Tcc.  Phase 1: energy deposition (0 < t 6 td):

S ¼ e_ dep ðx; tÞ

ðB:8Þ

Supposing in this phase that the liquid temperature is constant, equal to Tl,0, Eq. (B.6) writes:

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e_ dep ðx; tÞ @T 1 T1 T l;0 þ T¼ þ þ @t s sdiff scond qC p

ðB:9Þ

where s is defined as:

1

s

¼

1

þ

sdiff

edep P qC p ðT ign  T 1 Þ þ qC p

1

ðB:10Þ

scond

The solution of Eq. (B.9) is:



Z t e_ dep ð0; t0 Þ ts0 0 st T1 T l;0 e dt e TðtÞ ¼ T 1 þ þ þ sdiff scond qC p 0

Tðtd Þ ¼ T d ¼ T 1 þ

edep

R where edep ¼ e_ dep dt.  Phase 2: droplet pre-heating (td < t 6 tcc): The second phase lasts from the end of the deposition, td, to the time tcc when the temperature of the droplets reaches the saturation value Tcc (Fig. 9). In this phase, the energy source from spark is null (S = 0) and the one from combustion is still negligible. Hence the temperature evolution of the gas phase is:

Tðt Þ ¼ T d þ

Z

t



T1

þ

sdiff

0

T l;0

scond



t0 t 0 e s dt e s

ðB:13Þ

⁄

where t = t  td Then:



Tðtcc Þ ¼ T d þ

Z

tcc t d



T1

sdiff

0

þ

T l;0

scond



tcc td t0 0 e s dt e s

tcc t d

¼ T  þ ðT d  T  Þe 

h

T l;0

T1

where T ¼ s s þ s diff cond

ðB:14Þ

s

i

Now that T(tcc) is a unction of tcc, this last time has to be determined. Solving Eq. (B.7) for td 6 t 6 tcc, we obtain:

Z T l ðt  Þ ¼ T l;0 þ

t

0

A

scond

A t0  A t 0 Tðt 0 Þescond dt e scond

ðB:15Þ

which leads to:



Z

T cc ¼ T l ðtcc Þ ¼ T l;0 þ

tcc td

0

A

scond

A

t0

0



A ðt t Þ cc d cond

s

Tðt0 Þescond dt e

ðB:16Þ

We suppose in this phase that the gas temperature varies slowly, and in order to integrate Eq. (B.16), use T(t)  Td. Hence:

h  A i ðt t Þ  A ðt t Þ T cc ¼ T l;0 þ T d escond cc d  1 e scond cc d

ðB:17Þ

tcc  t d ¼ 

scond A

Ln

T cc  T d T l;0  T d

scond T cc  T l;0 A

T d  T l;0

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

[36] [37]

ðB:19Þ

[45] [46] [47]

Including Eqs. (B.12), (B.13), (B.14) and (B.19) in the criterion definition, Eq. (3); and supposing that Tl,0 and T1 are close:

Tðtcc Þ P T ign () T 1 þ

  edep scond T cc  T l;0 P T ign exp  qC p As T d  T l;0

! 2 4ndim dl ql C p;l ðT cc  T l;0 Þ 3Nu d2k

ðB:22Þ

References

ðB:18Þ



As Tcc  Tl,0  Td  Tl,0, a 1st order Taylor expansion gives:

tcc  t d ¼

Finally, one obtains:

[38] [39] [40] [41] [42] [43] [44]

After some rearrangements, one obtains:



ðB:21Þ

ðB:11Þ

ðB:12Þ

qC p

scond ðT cc  T l;0 Þ As

edep P qC p ðT ign  T 1 Þ þ al þ

A quantitative analysis of the magnitude of the different terms shows that e_ dep (0,t0 ) is several orders greater than the other two terms which can be neglected. Consequently, at the end of the sparking time (i.e. at t = td):



leads after another Taylor expansion of the exponential term and some rearrangements to:

ðB:20Þ

[48] [49] [50] [51]

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