Numerical Investigation of the Stochastic Behavior of Light-Round in Annular Non-Premixed Combustors

ABSTRACT The ignition behavior of a non-premixed multiple-burner annular combustion chamber was investigated numerically, focusing on the stochasticity and average speed of the light-round mechanism ensuring flame propagation from burner to burner that have been observed experimentally. During the propagation sequence, the flame expansion process is tracked by a previously developed stochastic low-order ignition model adapted to full combustor ignition. A stochastic model based on the probability that a flame fragment coming from an ignited burner leads to successful ignition of the next un-ignited one is developed in order to explain and quantify the global ignition behavior of the combustor. The stochastic behavior of the rig, highlighted through the experimentally observed variability of the burner-to-burner propagation times during the ignition sequence, was clarified and quantified. The lean light-round ignition limiting conditions and the mean light-round speed measured experimentally are explained and reasonably accurately predicted, demonstrating the validity of the use of the probabilistic model together with the low-order ignition model for the combustor considered. The results presented in this paper can be used to predict the ignition envelope of annular gas turbines combustors at the design stage.


Introduction
Ignition of gas turbine combustors and rocket engines is important from a practical perspective and involves complex phenomena (Lefebvre, 1999;Mastorakos, 2009). The ignition process in gas turbines occurs in four phases Mastorakos (2009); Lefebvre (1999). The first phase is the generation of a kernel consisting of the initiation of a flame through a spark in a flammable mixture (Glassman and Yetter, 2008;Lefebvre, 1999;Lewis and von Elbe, 1987;Spalding, 1979). The second phase is flame propagation from the kernel. The third phase is the overall burner ignition characterized by the stabilization of the flame on a single injector. Stabilization is based on the physical mechanism of initiating combustion by bringing hot gases in contact with the reactants (Dunn-Rankin, 2008). The fourth phase is burner-toburner flame propagation called light-round, which has not been yet extensively studied in the laboratory. This latter phase is the topic of this article, and in particular the probabilistic behavior of the underlying processes for annular non-premixed systems.
In the context of single combustors, ignition has been studied experimentally (Lang et al., 2010;Mosbach et al., 2010;Read et al., 2010). Spark ignition has also been studied experimentally for non-premixed flames in various geometries (Ahmed et al., 2007a(Ahmed et al., , 2007bAhmed and Mastorakos, 2006, 2016 and through Large Eddy Simulation (LES) (Bulat et al., 2013;Jones and Prasad, 2011;Jones et al., 2012;Lacaze et al., 2009;Subramanian et al., 2010;Triantafyllidis et al., 2009). Moreover, low-order ignition models providing lower computational cost have been developed in order to provide easier prediction of the ignition behavior in non-premixed and spray single burners (Eyssartier et al., 2013;Neophytou et al., 2012). In addition, the ignition behavior of single premixed burners have been investigated with the model developed by Neophytou et al. (2012) and Sitte et al. (2016).
Recently, in non-premixed configuration, light-round has been investigated experimentally and numerically by Machover and Mastorakos (2016) with a linear combustor comprising five non-premixed swirl bluff-body burners. Results have suggested that, in the interburner region, bridges of positive albeit small flammability factor-defined as the local probability of finding a mixture within the flammability limits [see Mastorakos (2009) for a review of this concept]-allow flame fragments to travel from the recirculation zone of the ignited burner to the recirculation zone of the adjacent one, leading to burner-to-burner propagation. Furthermore, fuel supply appeared to be of major importance to successful propagation. Indeed, very low values of flammability factor measured in the bridges allowed a flame fragment to travel from burner to burner only seldom.
Experimentally, burner-to-burner flame propagation has been further investigated by Machover and Mastorakos (2016) in an annular non-premixed combustion chamber adapted from the premixed combustor used by Dawson (2013a, 2013b) consisting of a variable number of equally spaced swirling burners equipped with a central bluff-body and axial fuel injection. The global light-round behavior of the annular combustor comprising N b = 12, 15, or 18 burners was studied by investigating changes in spacing between burners (S), overall equivalence ratio (), and bulk velocity (U b ). Side visualization showed that burner-to-burner propagation occurring in the stratified inter-burner region consisted of a successful flame propagation event following a succession of failed events in which a flame fragment coming from the ignited burner failed to penetrate fully in the recirculation zone of the adjacent unignited burner. Turbulent dispersion process was considered dominant given that the time taken for successful burner-to-burner flame propagation was much larger than in premixed systems investigated in Bourgouin et al. (2013); Philip et al. (2015); Barré et al. (2014);Machover (2016); and Machover and Mastorakos (2017). The number of failed events appeared to vary significantly from burner to burner suggesting a stochastic global behavior. The variability was found to increase with augmenting S, diminishing , and increasing U b . Furthermore, the average speed of light-round augmented with increasing number of burners or with increasing overall equivalence ratio. However, it remained rather constant with augmenting bulk velocity.
Following that experimental work, the present article aims to explain, through modelling, the above trends. First, the numerical set-up is presented. Then, a probabilistic model is developed in order to explain the stochastic behavior observed in the experiment. The experimental results obtained are then re-analyzed and explained with emphasis on the stochasticity quantified through the model. Finally, the article concludes on the performance of the ignition and probabilistic models together for the prediction of ignition behavior in non-premixed configuration.

Burner description
The combustor modeled in the present work is represented by drawings and a photograph in Figure 1 and Figure 2, respectively, and is described in detail in Machover and Mastorakos (2016). This annular non-premixed combustion chamber consists of a variable number of equally spaced bluff-body stabilised turbulent non-premixed burners placed in an annular configuration. The air is exhausted through a plenum connected to 12, 15, or 18 identical 150 mm long circular tubes with an inner diameter of D = 18.9 mm. Each tube is fitted with a centrally located duct that consists in a 5-mm-diameter tube of wall thickness 1.0 mm. To the end of the duct, at the exit of the burner, a conical bluffbody of diameter D a = 13 mm is attached, giving a blockage ratio of 50%. Each duct is connected to the methane supply. The pipe/bluff-body assemblies are arranged around a circle of diameter 170 mm and fixed between upper and lower plates. A six vane, α = 60°, counter-clockwise swirler (as viewed from the top of the combustion chamber) is fitted 10 mm upstream of each of the bluff-body giving a geometrical swirl number of 1.22 Worth and Dawson (2013a). The flames are confined within an annular enclosure that consisted of inner and outer tubes of diameter D in =127 mm and D out = 212 mm. Three sets of plates were manufactured with the same circumference enabling simulations with 12-, 15-, and 18-burner configurations to be performed. These correspond to inter-burner spacings of S 12 ¼ 2:33D (S ¼ 44:0 mm), S 15 ¼ 1:87D (S ¼ 35:3 mm), and S 18 ¼ 1:56D (S ¼ 29:5 mm), where S denotes the arc distance between the bluff-body centers. In the study, flame characterized by inter-burner spacing S, overall equivalence ratio Φ, and bulk velocity U b is denoted flame S U b .

Flow conditions
Flow conditions are the same as those studied experimentally in Machover and Mastorakos (2016). The process of light-round is investigated in ambient conditions of temperature (T ¼ 293 K) and pressure (1 bar) with air/methane mixtures at three interburner spacings S ¼ S 12 , S ¼ S 15 , and S ¼ S 18 ; at three overall equivalence ratios ¼ 0:30, ¼ 0:35, and ¼ 0:40; and at bulk velocities of air and methane reported in Table 1. Although these bulk velocities are lower than that encountered in realistic gas turbines, flow is turbulent in the combustion chamber (Reynolds numbers based on the bulk velocity at each burner's annular inlet and D range from Re ¼ 3950 to Re ¼ 7100). The flammability limits of the air-CH 4 mixture are lean ¼ 0:028 and rich ¼ 0:089; st ¼ 0:055 is the stoichiometric mixture fraction (Turns, 2000).

Low-order modeling
The burner-to-burner propagation is investigated numerically by use of a low-order ignition model, i.e., based on a steady-state cold flow CFD solution without simulating transient flow, introduced first and fully described by Neophytou et al. (2012). Low-order models provide simplified and fast modeling of ignition compared to models with reacting flows. In the present model, flame propagation is modeled by the motion of virtual flame particles in a turbulent flow field, leaving the thermal environment apart. The required input for the code to simulate ignition events is a time-averaged cold flow solution.
The fluid domain is filled with rectangular grid cells of size Δx and the algorithm follows a cellular automaton model, i.e., the state of each cell changes over time according to some fixed rules. Cells can transit through two different states successively: cold state and burn state. The initial state is cold for all the cells. The spark is simulated by switching the cells in the spark volume from the cold to the burnt state. A burnt cell can not get back to its previous cold state. When a cell switches its state from cold to burnt, a single virtual flame particle is emitted. The dynamics of the flame particles is modeled by a simplified Langevin model described by the stochastic differential Equations (1) and (2) (Pope, 2000). Whenever a virtual flame particle enters a cold cell, the latter switches to the burnt state and emits one additional flame particle. In the present model, the ignition impact on the cold flow field is not considered and the mixture supply within the combustion chamber is not included.
The latter equation consists in a linear drift towards the local time (isothermal) averaged velocity and an isotropic diffusion term. ΔX p;i denotes the distance a particle p travels in the direction i during the time step Δt with velocity U p;i . C 0 is a constant equal to 2.0 and N p a random variable that follows a normal distribution of mean 0 and standard deviation 1. The cold flow time-averaged CFD solution provides the other parameters. u 0 denotes the turbulent velocity fluctuation estimated as ffiffi ffi k p , where k is the turbulent kinetic energy, L turb the integral length scale, e U i the local mean velocity, and ε the local turbulent dissipation rate. Moreover, an extinction criterion based on the local Karlovitz number (Ka) and originally introduced by Abdel- Gayed and Bradley (1985) is applied to decide whether a U air and U CH4 refer to the bulk velocity of the air at the end of the tubes and of the methane at the duct exit, respectively. Experimental conditions are tested for the three spacings S ¼ S 12 , S ¼ S 15 ; and S ¼ S 18 . flame particle remains active or not. The flame particle quenches if Ka p exceeds the critical value Ka crit ¼ 1:5 (Abdel- Gayed and Bradley, 1985). The local Karlovitz number is calculated for each flame particle at every time step according to Equation (3).
Here ν is the kinematic viscosity and S L the laminar flame speed. S L depends on the local mean mixture fraction and is calculated by the ignition model according to data provided by Vagelopoulos and Egolfopoulos (1998) for methane. , L turb , and u 0 are provided everywhere within the combustion chamber by the cold flow CFD solutions used as an input for the low-order model. Locally, the mixture fraction follows a β-function probability density function (PDF) with mean and root mean square (r.m.s.) given by the cold flow CFD solution. The flammability factor F is calculated assuming a β-function PDF for the mixture fraction. The local mixing is taken into account as the local Ka diverges when the local mixture is not flammable (S L is zero), resulting in virtual flame particle quenching given that the limit set by the extinction criterion is exceeded.
The model, as described above, has been validated for ignition of non-premixed and spray recirculating flames (Neophytou et al., 2012), bluff-body premixed flames (Sitte et al., 2016), and realistic gas turbine combustors (Neophytou et al., 2011;Soworka et al., 2013), and is illustrated in Figure 3, where virtual particle quenching following two burning particle emissions is sketched.
Criteria based on some quantities relative to the cold flow have been proposed by Neophytou et al. (2012) for the choice of the time step (Δt) and the grid size (Δx). First, the time-step Δt should be smaller than the large-eddy time-scale for all the turbulent time scales to be captured by the code. The time-step being fixed, grid spacing boundaries are determined from the Langevin model. "p; i, ΔX p;i should be smaller than the distance a particle is carried by an eddy during Δt. Finally, the grid spacing should be larger than the smaller turbulent scales. In the turbulence spectrum, the middle of the dissipation range in Figure 3. Illustration of a single flame fragment propagation with the low-order model. In this example, a virtual burning particle is emitted from an ignited cell at t ¼ t 0 . Following the trajectory given by the simplified Langevin model (represented by the arrow in the figure), it reaches a combustible cold cell at t ¼ t 0 þ Δt that switches its state from cold to burnt. Then, the burnt state emits a virtual burning particle that attains an incombustible cold cell at t ¼ t 0 þ 2Δt, resulting in virtual particle extinction.
the Kolmogorov scale is at approximately 18η K , where η K is the Kolmogorov length scale (Pope, 2000). Regarding those guidelines, the retained values are Δx ¼ 2:0 mm and Δt ¼ 0:5 ms.

Cold flow CFD solution
Time-averaged CFD cold flow solutions of the 27 different flames described in Table 1 were computed with LES with the commercial software ANSYS Fluent. The incompressible Navier-Stokes equation solutions were obtained with the Smagorinsky-Lilly subgrid-scale model with default model constants. Spatial discretization was set up with least square cell based gradient reconstruction, second order upwind momentum, and second order mean mixture fraction. Finally, a bounded second order implicit transient formulation was adopted together with the PISO scheme for the pressure-velocity coupling. The computational domain comprised one single burner sector of the rig consisting of 1/12, 1/15, or 1/18 of the annular combustor. The domain was meshed with an unstructured tetrahedral grid refined in the region of the recirculation zone, and in the air and the fuel inlets. The grid for the 12-, 15-, and 18-burner configuration comprised 7.7×10 5 , 7.3×10 5 , and 6.5×10 5 cells corresponding to 1.5×10 5 , 1.4×10 5 , and 1.3×10 5 nodes, respectively, with minimum size 0.5 mm and maximum size 1.8 mm. At the exit of the air inlet, the tangential component of the flow induced by the swirl was set equal to the axial component following the velocity field experiments conducted in a non-premixed linear combustion chamber presenting very similar flow characteristics (Machover, 2016). This single unit had periodic boundary conditions on the walls disposed at the middle of the arc distance between burners. For the use of the cold flow field as an input in the low-order model, each individual burner CFD solution was interpolated on a regular grid of 2.0 mm spacing. For each case, the CPU time required by the desktop machine used (8 CPUs, Intel(R) Core(TM) i7-4820K @3.70Ghz, 32 GB RAM) was approximately 40 h. The full combustor CFD solution was obtained by concatenating the 12, 15, or 18 interpolated units. The CPU time required to compute 500 ignition trials with the low-order ignition model SPINTHIR (Stochastic Particle INTegrator for High-altitude Relight) was about 5 h with the desktop PC.

Stochastic light-round modeling
It has been shown experimentally that burner-to-burner propagation occurring in the inter-burner region consisted of a successful flame propagation event following a succession of failed events in which a flame fragment coming from the ignited burner failed to penetrate fully in the recirculation zone of the adjacent un-ignited burner (Machover and Mastorakos, 2016). The main idea of the following probabilistic model is to model the burner-to-burner propagation by a succession of failed and successful events occurring with a certain probability. A successful event corresponds to a flame fragment emitted from an ignited burner that successfully ignites the next un-ignited one.
We denote by δt the average time a flame fragment would take to travel from any ignited burner to any un-ignited adjacent one. We now denote by P the probability that during δt, the flame fragment leads to successful ignition of the burner. P depends on various parameters such as S, , U b , and the inter-burner turbulent pattern that is time dependant. If δt>T turb ¼ L turb =U b , T turb being the large-eddy turnover time, a flame fragment experiences all the turbulent time scales during δt. Thus, all the flame fragments experience the same turbulent environment in the inter-burner region and P is reduced to a function of S, , U b only, assuring that the sequence of events is independent and identically distributed. This hypothesis is verified a posteriori (Table 2).
Suppose an ignited burner (index k À 1) and let us determine the average time taken to ignite the adjacent un-ignited one (index k). The probability P δt that successful burner-toburner propagation occurs during δt is given by: Burner-to-burner propagation is successful in kδt if a successful event follows k À 1 unsuccessful ones. The probability associated, P kδt is hence given by: The average propagation time that takes a flame fragment to travel from one burner to the adjacent one i.e. the expected value of the burner-to-burner propagation time, T, is given by: Thus, the problem is reduced to a geometric sequence with success probability P. The expected value of a geometrically distributed random variable being 1 P , the average propagation time from burner to burner is given by: T ¼ δt P Finally, considering that the average speed of light-round S LR is given by: where S is the arc distance between two burners and that the average speed at which flame propagates during an event, V LR , is given by: Table 2. The ratio δt=T turb for the various conditions simulated, which assists in assessing the applicability of the probabilistic model.
we obtain the final relation: The model is illustrated in Figure 4, where a succession of four consecutive burners ignition is sketched. P and V LR are determined with the low order ignition model by performing simulations of a large number of full combustor ignition events, N e , presently 500. The study focuses strictly on the burner-to-burner propagation mechanism. Thus, it is assumed that the first burner of the annular combustion chamber was always successfully ignited. Consequently, results presented herein do not take into account ignition of the first burner. For each ignition sequence, burner-to-burner propagation is initiated by a 10 mm diameter spark situated in the first burner, leading to its successful ignition. The probability P that flame travels successfully from one burner to the next one can be quantified since the code does not take into account refuelling. Indeed, since virtual particles that had extinguished cannot go back to their previous ignited state, one failed event results in the cessation of burner-toburner propagation. In other terms, the bridges of positive flammability factor can be used only once by the flame fragment. Therefore, the first burner being always successfully ignited, the i-th burner in succession to the sparked one (i 2 ½1; N b À 1) will have a probability P i to be ignited long-term, in steady-state. Hence, after having performed N e simulations, the number of successful ignition N i of a burner i leads to the determination of P ¼ ðN i =N e Þ 1=i . This probability does not take into account the probability to ignite the first burner. Furthermore, V LR is obtained by dividing the distance i:S by the average time taken by the flame to travel from the first burner to the i-th burner. In practice, i was chosen as close as possible to the burner number opposite to the first one with values of P i greater than zero. . Illustration of the burner-to-burner propagation model. In this example, the first burner has already been ignited. A successful event results in the ignition of the second one. Then, the succession of three failed events followed by a successful event leads to the ignition of the next burner. The last burner ignites after a failed followed by a successful event.

Results and discussion
In this section, burner-to-burner flame propagation is investigated numerically with the loworder ignition model together with the probabilistic model in order to quantify the stochasticity of the process. The stochastic behavior of the combustor, the lean light-round ignition limits and the mean light-round speed measured experimentally are examined.

Cold flow and flame propagation pattern
Cold flow CFD solutions obtained with LES are shown for flame 12_0.40_18 in Figure 5. The axial component of the velocity takes high values in the expanding annular flow coming out from the burner air inlet and in the central gas jet. A central recirculation zone is observed. Small side recirculation zones are also visible at the limit of the annular jet formed by the expanding flow. The flammability factor takes high values in the expanding annular air inlet region, small values in the central jet region, and bridges of small flammability factor in the inter-burner region are visible. Generally, the flow is very similar to that from a non-premixed configuration studied in detail by Machover and Mastorakos (2016), where a full flow description is provided.
Burner-to-burner propagation modeling is shown in Figure 5, where flame spread around the annular combustor is tracked with the low-order ignition model in the 12burner configuration for flame 12_0.40_18. One failed propagation from burner to burner (from the central burner to that on the left) and one successful burner-to-burner flame propagation (from the center burner to that on the right) are shown. In both cases, burning virtual particles spread into the region of the sparked burner (t ¼ t 0 ms) and are convected downstream into the adjacent burner region (t ¼ t 0 þ 5 ms). Either all burning particles extinguish along the bridge of small flammability factor in the inter-burner region (t ¼ t 0 þ 15 ms) and burner-to-burner propagation does not occur (failed propagation from the central burner to the left adjacent one), or burning virtual particles successfully reach the recirculation zone of the adjacent un-ignited burner (successful propagation from the central burner to the left one). These numerical results are in accordance with that experimentally observed in Machover and Mastorakos (2016), where a failed and a successful propagation sequence are shown.

Ignitability limits
For all the flames studied, the calculated probability of successful propagation is plotted as a function of the bulk velocity and overall equivalence ratio for different inter-burner spacings in Figure 6, together with the lean light-round propagation limit obtained experimentally by Machover and Mastorakos (2016) in Figure 7. The lean light-round propagation limit circumscribes the stable lightround region where successful propagation occurs in the long term and the region where the burner-to-burner propagation never occurs. Beyond the lean light-round ignition limit, any (U b ,) point at fixed S leads to successful burner-to-burner propagation, whereas below that limit, no light-round occurs. The probabilities P of successful light-round, as given by the model, are denoted by the color of the square in Figure 7. A few trends are clearly visible. First, decrease of the interburner spacing results generally in increase of P. Indeed, a decrease of the length of the bridges of small flammability factor between the burners results in a greater probability for a flame fragment not to extinguish during its propagation. Second, an increase in overall equivalence ratio leads to an increase of P, resulting from the higher values of flammability factor in the inter-burner area evidenced in Machover (2016) where detailed laser diagnostics have been performed in a linear non-premixed combustion chamber adapted from that of the present study. Third, a bulk velocity increase mostly results in a decrease of P, a (a) S = S12 (b) S = S15 (c) S = S18 Figure 6. Simulated probability of successful propagation as a function of the bulk velocity and different overall equivalence ratios for the flames studied. Each image shows a different inter-burner spacing: (a) 12 burners, (b) 15 burners, and (c) 18 burners.
trend explained by flame fragments more inclined to quench in an environment with higher strain rate. In general, the model provides a diminishing P as the experimentally observed lean ignition limit is approached, which suggests that the model has promising predictive capability. These results explain the light-round envelopes obtained experimentally. At fixed U b , with gradually diminishing , P decreases until it reaches zero preventing propagation from burner to burner. The result is illustrated for flames out of the stable light-round region or close to its limit at small overall equivalence ratio and high bulk velocity (S 0:30 18), as P drops to zero for the three inter-burner spacings considered, explaining that in these conditions, no burner-to-burner propagation was observed experimentally (Machover and Mastorakos, 2016). Furthermore, for every inter-burner spacing configuration, at fixed , P decreases with an increase in U b . This explains the positive slope of the lean light-round ignition limits evidenced experimentally given that with bulk velocity increase, higher overall equivalence ratio is necessary for successful burner-to-burner propagation. Finally, for every inter-burner spacing configuration, at fixed and U b , increase in P with an inter-burner spacing decrease explains the wider ignition envelopes obtained experimentally in Machover and Mastorakos (2016). Future work could consist in full ignition envelopes prediction by simulating S 0:20 U b and S 0:25 U b flames.

Variability
The measurements conducted on the annular non-premixed combustor showed that the time taken between two consecutive burner ignitions varied significantly during the burner-to-burner propagation sequence (Machover and Mastorakos, 2016). This variability was found to be higher by decreasing the number of burners, by diminishing the overall equivalence ratio and by increasing the bulk velocity (Machover and Mastorakos, 2016). The probability density function (PDF) of the experimentally measured propagation time between two consecutive burners and the PDF of the geometric distribution at each value in time using the simulated probability P represented in Figure 6 are plotted together in Figure 8 for four different flames. The geometric distribution fits reasonably well with the experimental values. This shows that the burner-to-burner propagation time follows actually a geometric distribution of success probability P that is very close to the simulated one. Moreover, the variability trends observed are explained by the probabilistic model. Indeed, it is shown in Figure 8 that the variability increases with an increase in U b at fixed S and (Figures 8a and 8b), with a decrease in at fixed S and U b (Figures 8a and  8c) and with a reduction of number of burners at fixed and U b (Figures 8b and 8d), all of which is explained by the drop of P highlighted above.

Time of complete ignition and flame speed
The effects of variations in U b , in , and in S on the average speed of successful propagation of a flame fragment V LR are shown in Figure 9. Again, some trends are clearly visible. First, at fixed and U b , a decrease in the inter-burner spacing results in a higher V LR . This can be explained by the lower path a flame fragment would travel in the bridges that follow the radially expanding annular air inlet, as shown in Machover (2016). Second, at fixed S and , increasing U b mostly results in the increase of V LR , a trend explained by the faster convection by the flow. Third, when increasing , with S and U b remaining fixed, there is tendency for higher speed of successful propagation, although a strong correlation was not obtained. Experimentally, for every configuration, the average speed of light-round S LR (defined by the average speed at which burner-to-burner propagation occurs) was determined by averaging a number of individual burner-to-burner propagation speeds Machover and Mastorakos (2016). These results are compared with the values of S LR ¼ P:V LR , with P and V LR obtained numerically with the low order ignition model, at fixed inter-burner spacing in Figure 10.
The results obtained showed that δt>T turb except for flame 18_0.40_18 as shown in Table 2. Consequently, the model is valid in regard to the applicability hypothesis for all the other flames.
At high inter-burner spacing and low overall equivalence ratio, the experimental data are correctly retrieved with accurate prediction of a number of trends and values by the numerical model. Indeed, as shown in Table 2, δt approaches T turb for high overall equivalence ratio and small inter-burner spacing, which can explain the better applicability of the model for very lean mixtures and high inter-burner spacing, as shown in Figure 10. These results show both some ability of the code to track the propagation of a flame from burner to burner, and the accuracy of the flame fragment propagation characterization that is the founding principle of the present probabilistic model. Furthermore, the experimental trends can be interpreted by the above expression of S LR . Indeed, when fixing and U b , a decrease of S results in the increase of P and V LR together, explaining the increase of S LR when increasing number of burners. Likewise, when fixing S and U b , an increase of results in the increase of P and V LR together, explaining the increase of S LR when increasing the overall equivalence ratio. However, at fixed and S, an increase in U b leads to decrease of P and increase of V LR at the same time, a result interpreting the unexplained rather constant evolution of speed of lightround when increasing the bulk velocity evidenced in Machover and Mastorakos (2016). Finally, S LR is proportional to the probability that a flame fragment travels successfully from burner to burner, which justifies the low light-round speeds found in these very lean stratified flows.

Conclusions
The ignition behavior of an annular non-premixed combustion chamber comprising 12, 15, or 18 individual bluff-body swirl non-premixed burners has been examined numerically focusing on the process of light-round, namely the burner-to-burner flame propagation. Previous experiments conducted with the same non-premixed combustion chamber have shown that the flame propagation has a stochastic nature; the probability that a flame fragment travels from one burner to the next controls the burner-to-burner propagation. A model based on the probability of a flame fragment to travel from one burner to the next has been developed in order to quantify the light-round process. Together with a stochastic low-order ignition model adapted to the present configuration, interpretations of experimental results obtained previously with the same combustor were made. First, the numerical results were in accordance with the lean burner-to-burner ignition limit general shape and the wider ignition envelopes obtained experimentally when diminishing the inter-burner spacing. Second, an approach of the stochastic global behavior was suggested by the statistical trends obtained with the probabilistic model. Thereby, the variability trends of the time taken from a burner to ignite the next one during the ignition sequence were interpreted by the numerical model. Finally, average speeds of light-round obtained numerically were in rather good agreement with those measured experimentally, especially at low overall equivalence ratio and large inter-burner spacing. Limitations of the model could explain some discrepancies between simulations and experiments at low spacing between burners and high overall equivalence ratios. The model allowed interpretation of the trends obtained when bulk velocity, overall equivalence ratio, and inter-burner spacing varied. The present work highlights the statistical nature of ignition in non-premixed annular systems.
In the present study, the lightround process has been investigated in inhomogeneous fuel/air mixtures. However, real gas turbines involve turbulent spray combustion and additional complexity arise with flame propagating across a droplet-laden air flow. Indeed, a series of interdependent physical phenomena occurs in two-phase flows: atomization and evaporation of the liquid fuel, interaction between liquid droplets, turbulence and mixing in the gas phase, and process of forced ignition itself. Moreover, the present work has been conducted with laboratory-scale multiple-burner combustion chambers open downstream to the atmosphere at ambient conditions. In practice, relight occurs at high altitude and more research is necessary in order to study burner-to-burner flame propagation at different conditions of temperature and pressure. The present low-order model has shown good predictive capability of ignition probability trends in single burners operating with sprays (Neophytou et al., 2012). Moreover, effects of pressure and temperature variation are indirectly taken into account by the model through the laminar flame speed and the flammability factor contours. Similar efforts in more complex configurations with liquid fuel spray flames at realistic conditions of temperature and pressure should be conducted in order to fully understand ignition in gas turbine annular combustors.
Nomenclature N b number of individual burners of the annular combustor D inner diameter of the individual burners δt flame fragment average propagation time P flame fragment successful propagation probability φ overall equivalence ratio S inter-burner spacing S LR average speed of light-round T average burner-to-burner propagation time U b bulk velocity V LR average speed of a flame fragment