J L Ruan

L Bouheraoua

P Domingo

Guillaume Ribert
email: guillaume.ribert@insa-rouen.fr

Simulation of a Scramjet Combustor: A Priori Study of Thermochemistry Tabulation Techniques

Keywords: Supersonic combustion, Large-eddy simulation, Cavity-based scramjet, Premixed Flamelet Tabulation

Large eddy simulations (LES) of a scramjet combustor are reported in this paper. The case under study is a cavity-based combustion chamber that is experimentally studied at the U.S Air Force Research Laboratory. The chamber is fed by eleven injectors. The computational domains are either simplified including only one or two injectors or complete with the 11 injectors. A good agreement is found between experimental data (velocities measured by PIV) and results from the LES if the kinetic used is chosen with care. A high temperature is found inside the cavity promoting a reactive zone located in the mixing layer where the flow velocity is high. At this location, the combustion occurs first in a diffusion dominated regime followed by the efficient burning of a well mixed mixture (rich then lean). A significant diffusion dominated burning is also found inside the cavity, mostly at the interface between the two recirculation zones. The simulation of the complete geometry revealed a transverse phenomenon on the temperature and mixing fields, but which had nevertheless little effect on the comparison with the available experimental data. A tabulation of the chemistry based on a premixed flamelet library without compressibility effects has been tested a priori on the results of the simulation with one injector. Good results on temperature and H 2 O fields are found. Significant localized discrepancies appeared on CO and CO 2 fields due to the complexity of the combustion regimes, while compressibility effects were found to be weak for the configuration studied.

Introduction

Despite the increase in high-performance computing, it remains challenging to simulate complete combustion chambers while ensuring accurate numerical methods, reliable description of the thermochemistry and efficient models for the phenomena unresolved by the mesh. In particular, it should be noted that few turbulent combustion models are specifically developed for the supersonic flow regime and the classical subsonic modeling techniques are very often transposed to the supersonic case [START_REF] Bouheraoua | Large-eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base[END_REF]. This is the case for tabulation techniques, that allow for drastically reducing the number of transported scalars [START_REF] Van Oijen | Stateof-the-art in premixed combustion modeling using flamelet generated manifolds[END_REF][START_REF] Pope | Small scales, many species and the manifold challenges of turbulent combustion[END_REF]. Indeed, a look-up table contains all information about the chemical system from a well-chosen collection of canonical flames or processes. Thermodynamical variables are then tabulated as functions of a small set of control variables that are transported with the flow [START_REF] Domingo | Large-eddy simulation of a lifted methane jet flame in a vitiated coflow[END_REF]. However, dealing with tabulated thermochemistry in a highly compressible mixture constitutes a challenge [START_REF] Saghafian | Large eddy simulations of the hifire scramjet using a compressible flamelet/progress variable approach[END_REF][START_REF] Saghafian | An an efficient flamelet-based combustion model for compressible flows[END_REF], as the compressibility effects are not generally accounted for in table constructions. For instance, let's consider the supersonic burner of Cheng [START_REF] Cheng | Raman measurement of mixing and finite-rate chemistry in a supersonic hydrogen-air diffusion flame[END_REF] in which a sonic hydrogen fuel jet at 545 K interacts with an annular jet of hot (1250 K) and vitiated air at Mach 2. Both flows exit into a quiescent atmosphere creating shocks as shown in Fig. 1 through the pressure field and the scatter plot of temperature, before combustion starts. The combination of hydrogen, oxidizer and hot temperature causes autoignition phenomena that occur at different locations in the jet and result in different combustion regimes as well as the birth of detonations [START_REF] Bouheraoua | Large-eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base[END_REF]. Building a look-up table thus becomes legitimate. Perfectly stirred reactor (PSR) have been computed and all thermochemistry infirmations (temperature, species mass fractions, etc.) are stored function of the progress variable for various values of the mixing and pressure levels between 20 and 220 KPa. However, using the inlet temperature of the experiment as an input data of the table (T mix 0 ), i.e. using Table 1 in Fig. 2, leads to incorrect predictions. Indeed, compressibility effects must be consid-Fig. 2 Compressibility effects on look-up table (TABLE 1 vs. TABLE 2). Comparison of temperature versus pressure scatter plots for the reactive case (left) and at two downstream locations (right) along the jet axis: x/D = 10.8 [A-A] and x/D = 21.5 [B-B], with D = 2.36 mm. Z 1 describes the mixing between hydrogen and vitiated air, and Z 2 the mixing between pure air and vitiated air. The value of temperature to enter the look-up table is T 0 . The correction (T corr 0 ) applied to TABLE 2 to account for compressibility effects is simply defined as the difference of the grey and green curves of Fig. 1 ered when building up the look-up table. In Cheng's burner, these effects come from the shock arrangements which appear before the combustion zone and which modify the values of thermodynamical variables such as temperature. This temperature modification can be deduced from Fig. 1 by the difference between the temperature from the green dash lines and the temperature from the grey dash lines (T corr 0 ). Using T shock 0 (= T mix 0 + T corr 0 ) for the creation of the look-up table (Table 2) instead of T mix 0 leads to a better description of the temperature field as shown either on the scatter plot of temperature (left) or on a cut of temperature at two different locations downstream the burner exit (right), in Fig. 2. In the case of hydrogen fuel, the gain in CPU time is small because the CPU cost of the chemical kinetic is not prohibitive. However, it becomes significant for hydrocarbons. Note that another advantage of chemistry tabulation consists in the easier way to introduce combustion subgrid modeling through presumed probability density functions. Similar questions are now addressed: is the use of a tabulation method for thermochemistry feasible in the case of a cavity-based scramjet configuration? Will the compressibility effects play also an important role in that case? In such a device, air flows at a supersonic velocity over a cavity that traps hot products coming from the decomposition of fuel and air by means of a primary vortex positioned inside the cavity. The reaction zone is located between the air flow and the cavity, and a secondary vortex acts as a fuel pool mixed with hot burned products in a dead zone at the front of the cavity [START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF]. Again, the effects of compressibility can play a major role in the creation of a look-up table. This difficulty is exacerbated by the wide variety of flame regimes that can be encountered: premixed, non-premixed, partially-premixed, or near (or in) the distributed-reaction-like regime [START_REF] Gonzalez-Juez | Advances and challenges in modeling high-speed turbulent combustion in propulsion systems[END_REF], i.e. when high-speed reacting flows have also high levels of turbulent fluctuations. To tackle this issue, a high-fidelity simulation of a scramjet combustor has been engaged. In the facility from AFRL [START_REF] Tuttle | Particle image velocimetry in an isothermal and exothermic high-speed cavity[END_REF], the air flow is accelerated through a nozzle to reach supersonic speed and enters the isolator at nominal Mach 2, with a total temperature and pressure of 589 K and 483 kPa, respectively. The air flow then meets the fuel (ethylene) inside the cavity where combustion has been shown to take place. The back of the cavity has an angle to limit streamwise pressure oscillations. An array of 11 fuel injectors are evenly distributed on the aft cavity wall. This experiment has already been studied by LES (Large-Eddy Simulations) or RANS (Reynolds Average Navier-Stokes) [START_REF] Baurle | Hybrid reynolds-averaged / large eddy simulation of a scramjet cavity flameholder[END_REF][START_REF] Hassan | Dynamic hybrid reynolds-averaged navier-stokes/large-eddy simulation of a supersonic cavity: Chemistry effects[END_REF][START_REF] Hassan | Dynamic hybrid reynolds-averaged navier stokes/large-eddy simulation of a supersonic cavity[END_REF][START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF]. Particle image velocimetry are available inside the cavity [START_REF] Tuttle | Particle image velocimetry in an isothermal and exothermic high-speed cavity[END_REF].

In the present study, a high-fidelity simulation of the AFRL test-bench is performed with a focus on the impact of the number of injectors. Indeed, simulating a single injector with periodic conditions on the sides can be enough to highlight the turbulence-flame interaction acting inside the cavity. However, to really encompass this interaction in detail, that is to say with transverse phenomena, it is necessary to simulate the complete configuration, which will necessarily entail a huge CPU cost. The choice of a combustion modeling will then be crucial and it is proposed, in this study, to validate a priori the use of a tabulated approach.

LES of scramjet combustor

Experimental and numerical set-up

Experiments conducted at the US Air Force Research Laboratory (AFRL) by Tuttle et al. [START_REF] Tuttle | Particle image velocimetry in an isothermal and exothermic high-speed cavity[END_REF] will then support the goal described above. Figure 3 is a summary of the experimental configuration and more details can be found in [START_REF] Tuttle | Particle image velocimetry in an isothermal and exothermic high-speed cavity[END_REF] or in numerical articles [START_REF] Baurle | Hybrid reynolds-averaged / large eddy simulation of a scramjet cavity flameholder[END_REF][START_REF] Hassan | Dynamic hybrid reynolds-averaged navier-stokes/large-eddy simulation of a supersonic cavity: Chemistry effects[END_REF][START_REF] Hassan | Dynamic hybrid reynolds-averaged navier stokes/large-eddy simulation of a supersonic cavity[END_REF][START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF]. The case under study has a fuel flow Actually, the distance between the first/last injector and the wall is 1.5 times higher than the distance between two injectors: 19.05 mm instead of 12.7 mm. The centerplane of the whole geometry is the centerplane of the 6-th injector which is located at z = -6.35 mm. The side walls are modeled with adiabatic non-slipping wall conditions. The mesh of the complete domain has 542 millions of cells and requires about 3 000 000 hours CPU time for 4 ms of statistics. The airflow is injected with homogeneous isotropic turbulence with an intensity of 10% of the local speed. Each injector has the same mass flow rate, which is 1/11 × 99 SLPM, leading to a velocity of 115 m/s per injector. Two mesh resolution have been used for the simulations with one injector, a coarse one with a resolution around 200 µm (RCA1) and a more refined one with a resolution around 100 µm (RFA1) in the reactive zones inside and above the cavity. For the simulations with two or eleven injectors, only the coarse resolution has been used.

Chemistry of ethylene

The reduced kinetic scheme used in this study has 21 reacting species plus nitrogen [START_REF] Luo | Chemical explosive mode analysis for a turbulent lifted ethylene jet flame in highly-heated coflow[END_REF] and is named S22. It has been reduced from a detailed mechanism based on the mechanism called USC-Mech II from the University of South California, composed of 75 species and 529 elementary reactions. S22 has been compared with success to its seminal detailed mechanism for auto-ignition, extinction in Perfectly-Stirred Reactor (PSR) and premixed laminar flames for a large range of initial conditions [START_REF] Luo | Chemical explosive mode analysis for a turbulent lifted ethylene jet flame in highly-heated coflow[END_REF]: pressure P ∈ [1 -50] bar, temperature T ∈ [1000 -1800] K and equivalence ratio φ ∈ [0.5 -1.5]. Autoignition computations for lean and rich mixtures have been performed with S22 at the pressure of the air inlet, P = 0.66 bar, but varying the inlet temperature. In Fig. 4 (left), the ignition delay decreases when increasing the inlet temperature but also when the equivalence ratio increases. As a consequence, a rich mixture would burn faster than a lean one. Kopp et al. [START_REF] Kopp | Oxidation of ethylene-air mixtures at elevated pressures, part 1: Experimental results[END_REF] have found that under high-pressure conditions (around 20 bar), the richer the mixture is, the faster it reacts. However, under low-pressure conditions (around 1 bar) the leaner the mixture is, the faster it ignites. Fig. 4 (right) shows a comparison between the numerical results from S22 and the experimental data from Kopp et al. [START_REF] Kopp | Oxidation of ethylene-air mixtures at elevated pressures, part 1: Experimental results[END_REF] at low pressure, P = 1.1 bar. The simulations and experiments show strong disparities for φ = 0.5 and φ = 1. Kopp et al. also propose a detailed mechanism Aramco 1.3 [START_REF] Metcalfe | A hierarchical and comparative kinetic modeling study of C 1 -C 2 hydrocarbon and oxygenated fuels[END_REF] which could overcome the previous defect of S22. But such mechanism is composed of 346 species and thus does not fit a 3D large eddy simulation. S22 will be kept with the knowledge of this deficiency.

Comparison between 1, 2 and 11 injectors

Averaged fields of temperature, heat release rate and fuel mass fraction are displayed at the centerplane of the injector in the transverse direction (y = -19.57 mm) in Fig. 5, Fig. 6 and Fig. 7 for RCA1, RCA2 and RCA11, respectively. The cases RCA1 and RCA2 are showing very similar results where each injector is performing the same way. The injection jet is parallel to the direction of the injection and the resulted flame has a V-shape with the same intensity. The case with eleven injectors RCA11 is performing differently, especially for the injection jets which are not parallel to each other. The combustion is also found more intense in the vicinity of the injectors located at the centerplane. The difference is most probably due to the side walls, which were prescribed as periodic in the cases RCA1 and RCA2, and which create here a transverse oscillatory mode which deflects the injection jets. In addition, the average temperature is also higher in RCA11 than in RCA1 or RCA2 by 100 K in the most reactive zones.

The simulations are then compared to the experimental data of Tuttle et al. [START_REF] Tuttle | Particle image velocimetry in an isothermal and exothermic high-speed cavity[END_REF] in the centerplane between two injectors (between 6-th and 7-th for RCA11). The averaged wall pressure is displayed in Fig. 8(a). RCA1 and RCA2 have very similar wall pressure, whereas the discrepancy between RCA1 and RCA11 is about 1 kPa. However, the experiment is fairly well predicted by all cases, except for the last point in the cavity ramp. The averaged streamwise and transverse velocities profiles are displayed in Fig. 9. The averaged streamwise velocity is still very well predicted by all the cases, only slight differences are observed between them. The simulations still over-predict the transverse velocity, regardless of the number of injectors involved. The difference between RCA1 and RCA2 is in average smaller than the one with RCA11, probably due to the periodic conditions prescribed at the side walls.

Comparison between injectors of the case RCA11

The eleven injectors case (RCA11) is showing some disparities on velocity, temperature and pressure profiles depending on the position of extraction.

Comparisons have been done on the centerplanes between injectors 4-5, 6-7, 9-10, and 10-11 for averaged wall pressure inside the cavity (Fig. 8(b)), averaged temperature profiles and averaged velocity profiles (Fig. 10). In the region prior to the cavity, the wall pressure is higher at the centerplane between injectors 10 and 11, which is closer to the side wall. The wall pressure stays relatively the same for each centerplane in the cavity, except in the vicinity of the reattachment shock system where a higher wall pressure is found for the centerplane between injectors 4 and 5 and the lowest one between injectors 10 and 11. The discrepancy in temperature is slight between centerplanes 4-5, 9-10 and 10-11. But the centerplane near the central injector behaves differently. This centerplane has a higher temperature at the front of the cavity and a lower temperature at the rear of the cavity. The temperature discrepancy can reach 250 K in some regions. The combustion zone is in fact shifted from the large recirculation area to the small recirculation area in the centerplanes near the central injector (6-th), resulting in a difference in temperature. The averaged streamwise velocity is still not modified and stays relatively the same at each centerplane. The averaged transverse velocity is however dependent of the location of the centerplane, especially at the vicinity of the shock system.

The isosurface of temperature (T = 2000 K) colored by the values of mixture fraction is displayed in Fig. 11. A lower temperature and a leaner mixture can be observed in the vicinity of the central injector. This indicates that the amount of cold airflow entering through the rear of the cavity is higher in the central injector region, thus decreasing the temperature and the mixture fraction. The origin of the airflow massively entering in the central injector region is difficult to capture and in the absence of more complete experimental results, it is too early to decide if this is a real feature of the experimental set-up or a numerical artefact. The differences between the simulation of the complete set-up and one or two injectors simulations can be considered unimportant if only stability issues or combustion regime analysis are considered. However to use simulation to fully characterize and design such a device, the full geometry must be included. Transporting a semi-detailed chemistry with its heavy load on memory and CPU time is then to be avoided. In the rest of the paper, the possibility of using a tabulated approach which allow to drastically decrease the CPU time [8-10, 20, 27] to describe the chemistry is explored.

A priori tabulated chemistry

As discussed in Ruan et al. [START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF], the dominant combustion regime found in this device is the premixed regime. A tabulation of the chemistry based on a premixed flamelet library [START_REF] Van Oijen | Stateof-the-art in premixed combustion modeling using flamelet generated manifolds[END_REF] without accounting for eventual compressibility effects has then be attempted and tested a priori on the results of the simulation with one injector presented in the previous section, but with a fine resolution (case RFA1) to minimize the impact of under resolution of the chemistry on the mesh.

One dimensional laminar premixed flames

One dimensional laminar premixed flame propagations were computed with the solver REGATH [START_REF] Candel | Progress in transcritical combustion: Experimentation, modeling and simulation[END_REF][START_REF] Pons | Mass transfer and combustion in transcritical non-premixed counterflows[END_REF][START_REF] Ribert | Counterflow diffusion flames of general fluids: oxygen/hydrogen mixtures[END_REF] at P = 66 kPa and T = 329 K in the flammability limits range of ethylene, which is φ ∈ [0.4, 5], and over a distance of

-0.1 0 -0.1 0 -0.1 0 -0.1 0 -0.1 0 -0.1 0 -0.1 0
Fig. 9 Averaged streamwise (top) and tranverse (bottom) velocity profiles inside the cavity. Experiment (pattern), RCA1 (red line), RCA2 (green line) and RCA11 (blue line, between 6-th and 7-th) at the centerplane between two injectors.

10 meters. The purpose of this study is not to reach equilibrium with 1D flame propagation which in some cases require a computational domain of hundreds of kilometers, but to build a laminar premixed flame table which contains all the possible solutions of the LES. Therefore, the last point of each flamelet is in general not the equilibrium state.

The temperature profiles of laminar premixed flames at different equivalence ratios are displayed in Fig. 12. Superadiabatic temperature (SAT) phenomenon can be observed for high equivalence ratios and is identified by a temperature peak in the flame front which is higher than the adiabatic temperature. This phenomenon is present in hydrogen flames [START_REF] Gicquel | Influence of differential diffusion on superequilibrium temperature in turbulent non-premixed hydrogen/air flames[END_REF] and it has been investigated in hydrocarbon flames at rich mixtures numerically and experimentally by Liu et al. [START_REF] Liu | Numerical study of the superadiabatic flame temperature phenomenon in hydrocarbon premixed flames[END_REF] for C 1 -C 3 hydrocarbons and numerically by Meeks et al. [START_REF] Meeks | Computational simulation of diamond chemical vapor deposition in premixed C 2 H 2 /O 2 /H 2 and CH 4 /O 2 strained flames[END_REF] for CH 4 and C 2 H 2 . Aside from the temperature peak, SAT is also associated with superequilibrium concentrations of some combustion products and with negative heat release rate (endothermic reactions). Meeks et al. [START_REF] Meeks | Computational simulation of diamond chemical vapor deposition in premixed C 2 H 2 /O 2 /H 2 and CH 4 /O 2 strained flames[END_REF] argue that SAT phenomenon is due to high gas velocities in the burning region at high equivalence ratios, while Babkin et al. [START_REF] Babkin | Superadiabatic temperature phenomenon in the combustion process due to a competition between chemical reactions[END_REF] pointed out that SAT generally occurs due to competition between molecular diffusion and heat transfer processes. The right hand side figure of Fig. 12 is a close view of the temperature profiles for equivalence ratio φ from 1.6 to 5. In the case of ethylene-air combustion, SAT can be observed from φ = 1.7 with a slight temperature peak until φ = 3.5.

In order check at high equivalence ratios what would be the length needed in the 1D laminar premixed flamelet computed with REGATH to reach equilibrium as determined by EQUIL, the computational domain for the 1D laminar premixed flame has been extended to 5 kilometers for φ = 4. It is then verified that the equilibrium values for temperature and concentrations are reached (see Fig. 13). For higher equivalence ratios, an even longer domain would be needed to reach equilibrium. As building a table with these very long domains would take for ever, a first table has been constructed with v for all equivalence ratios. However, it will be checked (section 3.3) that this is sufficient to contain all the compositions found in the LES, or if further refinements need to be done in some regions.

Expressions of progress variable

The definition of progress variable is a key point in tabulated chemistry and is not straightforward [START_REF] Najafi-Yazdi | Systematic definition of progress variables and intrinsically low-dimensional, flamelet generated manifolds for chemistry tabulation[END_REF][START_REF] Niu | An optimization-based approach to detailed chemistry tabulation: Automated progress variable definition[END_REF][START_REF] Ribert | Hybrid transported-tabulated strategy to downsize detailed chemistry for numerical simulation of premixed flames[END_REF]. It is defined as the ratio of a linear combination of species mass fractions on the same quantity at equilibrium:

C = Y C Y C,eq (1)
So the progress variable C takes value in [0, 1], with the extrema 0 and 1 corresponding to fresh gases and burnt gases at equilibrium, respectively. Y C can be defined either from the reactants plus N 2 or from some major combustion products chosen carefully. In this work, we will be only interested in building the progress variable from the main products of ethylene-air combustion. Therefore, Y C will be constructed with Y CO , Y CO2 and Y H2O . Two expressions of Y C are studied and compared:

Y C1 = Y CO + Y CO2 Y C2 = Y CO + Y CO2 + Y H2O (2)
The mass fractions at equilibrium could be taken from the 1D laminar premixed flame results if they are not far from the equilibrium, which is not the case in this study at high equivalence ratios. Thus, mass fractions from EQUIL [START_REF] Lutz | EQUIL : A CHEMKIN implementation of STANJAN for computing chemical equilibria[END_REF] are used to compute Y C,eq , in order to avoid discontinuities when prolongating the flame library outside the flammability limits. Verifications are shown in appendix A.

Construction of laminar premixed tables

Laminar premixed flame tables are generally constructed from 1D flames within the flammability limits and may be extended to all equivalence ratios by interpolation between fresh gases and equilibrium state. The advantage of using REGATH is that the solver can compute flames outside of the flammability limits, thus reducing the prolongation by interpolation which is not always appropriate. This method has already been investigated by Duboc et al. [START_REF] Duboc | Description of kerosene / air combustion with hybrid transported-tabulated chemistry[END_REF][START_REF] Duboc | Evaluation of chemistry models on methane/air edge flame simulation[END_REF][START_REF] Duboc | Hybrid transported-tabulated chemistry for partially premixed combustion[END_REF] for methane-air combustion by extending the table to φ = 5.5 (the flammability domain of methane-air is φ ∈ [0.6, 1.4]). The results were found to be more convincing than prolongation with equilibrium values. Here, the 1D flame simulations with REGATH have therefore been extended from φ max = 5 to 50 at P = 66 kPa and T = 329 K with a computational domain of 10 meters long. The flame speed and the thermal flame thickness are displayed in Fig. 14. The highest speed and the lowest thickness are found for φ = 1.17 and worth respectively s L,max = 88 cm/s and δ L,min = 0.38 mm at displayed with a color representing the intensity of heat release rate: blue for negative values and the shading from purple to yellow for low to high values. The region with negative heat release rate corresponds to the area with SAT (see Sec. 3.1), reactions are endothermic in this region in order to tend towards adiabatic temperature. The highest heat release rates are, as expected, found near the stoichiometric mixture (Z = 0.06366). The red curves in Fig. 15 represent the last point (X = 10 m) of each flamelet, as C is an injective function of X, C(X = 10 m) = C max , which means that all points below this curve are tabulated. There is a small region near Z = 0.1 where the progress variable is superior to 1. That phenomenon occurs more severely when the progress variable is not well defined, since one or several ma-jor species are missing to describe correctly this region, or when the number of points is not representative. The phenomenon of C > 1 can be due to diffusion. The expression C 1 is therefore not valid in this study since the number of points with C 1 > 1 is significant compared to C 2 where less than 0.003% of cells are concerned. The table based on C 1 is consequently discarded, and C 2 is renamed as C in the following.

Besides, the red curve starts to fall down for Z > 0.57 (φ > 19.5) and goes under the LES results for Z > 0.6, which means that the table does not contain those points. It needs to be extended to include higher progress variables. The prolongation can be done by interpolating between the last point of the flame and the equilibrium data, or by extending the computational domain in that region. The second technique is preferred in this work. The 1D computational domain has been extended to 1 km for Z > 0.57 until Z = 0.835, because a longer domain is required for higher mixture fraction to reach equilibrium. The new curve is represented in green color in Fig. 15.

At this stage, we would like to know whether the prolongation outside the flammability limits with non-propagating 1D flame (REGATH) gives more consistent results than the one with equilibrium prolongation. Three tables with different prolongations were built and compared to investigate that question. Their characteristics are shown in Tab. 2. As the temperature inside the cavity is not constant and is always inferior to 450 K (for the non-reactive simulations), a fourth table (TAB2b) was constructed at 450 K and 66 kPa to investigate the impact of the fresh gases temperature on the results. The flamelet libraries are displayed in Fig. 16.

A priori comparison with LES field

The a priori comparison is done on an instantaneous field for an adiabatic computation with one injector and a fine resolution (RFA1). The interpolation of the table data into the computational domain has been performed in the following way:

• First, the mixture fraction Z and the progress variable C are computed using the gas composition from the LES mass fractions (Y k ); • Then, using Z and C computed previously, the corresponding value of each tabulated quantity ϕ tab is extracted. In the few locations where C > 1, possibly due to diffusion not accounted for in the table, the values are taken at C = 1. The comparison has been done for the temperature and the main products mass fractions (Y CO , Y CO2 , Y H2O ).

The instantaneous mixture fraction and progress variable for the case RFA1 are displayed in Figs. 17 and 18, respectively. The mixture is found very rich at the front of the cavity (Z > 0.2), and lean to moderately rich in the large recirculation area (Z ∈ [0.03, 0.1]), except at the exit of the injector where the mixture fraction is expected to have very high values. On the other hand, the progress variable is globally high in the whole cavity with C > 0.7. Values of C higher than 0.9 are found in the large recirculation area, while the small recirculation area is having much smaller values of progress variable C ≈ 0.75. This result is actually expected, since the time required to reach equilibrium increases strongly with large values of mixture fraction.

The instantaneous temperature field is displayed in Fig. 19 for cuts in the spanwise direction (z) and in Fig. 20 for cuts in the streamwise (x) and the transverse (y) directions for LES results and laminar premixed table (TAB2) interpolations. The z-cuts are still taken at the injector centerplane and the centerplane between two injectors. The first y-cut is localized in the mixing layer, while the second corresponds to the middle line of the cavity. The x-cuts are from the interface between the recirculation zones (x = 20 mm from the cavity front) and the middle line of the large recirculation area (x = 39 mm). At the first glance, temperatures from LES and TAB2 appear very close to each other. The only noticeable difference lies in the small recirculation area where TAB2 underpredicts the temperature. In order to better quantify the discrepancies detected in the instantaneous temperature fields, values at eight positions inside the cavity with constant x are picked up. These positions are the same as those used for the velocity profiles which are: x = 2, 11, 20, 30, 39, 48, 57, 66 mm from the cavity front. The mixture fraction and the progress variable profiles are first displayed in Fig. 21. The mixing appears efficient and the mixture fraction presents smooth variations. The progress variable is more affected by the turbulence and then features strong spatial gradients. The temperature and the H 2 O mass fraction profiles for TAB1, TAB2 and TAB3 are displayed in Fig. 22 and compared to the LES values in the injector centerplane. Readers are reminded that the only difference between these three tables is the way the prolongation outside the flammability limits was done (see Tab. 2 of the previous section). TAB1 has some shortcomings in predicting the temperature or the H 2 O mass fraction outside the flammability limits. The two other tables do not have this issue and have very similar results. The only difference between TAB2 and TAB3 lies in a small region at the exit of the injector where TAB3 has more accurate results. This analysis shows that table prolongation with equilibrium hypothesis is not accurate in the current simulation and it is preferable to use 1D laminar premixed flames even outside the flammability limits. Compared to the LES, the temperature from the table is weaker at the front of the cavity by about 100 K, and in regions with strong variations of mixture fraction where the discrepancy can reach 200 K. The differences in H 2 O mass fraction can be related to the difference in temperature.

A comparison between TAB2 and TAB2b has been performed to investigate the impact of initial temperature of the laminar premixed flame tables. Again, temperature and H 2 O mass fraction are displayed in Fig. 23 for the injector centerplane. The temperature of the freestream in the case TAB2b is higher because the temperature of the fresh gases is 450 K instead 329 K. TAB2b is having slightly better performance than TAB2 inside the cavity, but the opposite occurs when it comes to the mixing layer. Actually, the average temperature field of the non-reactive case has shown that only the inner cavity is featuring temperatures around 450 K, the temperature in the mixing layer is lower and approaching the freestream temperature (329 K). An appropriate way would be building a table including compressibility effects. Besides, there is almost no difference for the mass fraction of H 2 O between TAB2 and TAB2b. The initial temperature of the laminar premixed flames has only a slight im- pact on the results suggesting that compressibility effect can be neglected in that device. TAB3 will be used for the next comparisons.

Profiles of mass fractions of CO and CO 2 for LES and TAB3 are displayed in Figs. [START_REF] Meeks | Computational simulation of diamond chemical vapor deposition in premixed C 2 H 2 /O 2 /H 2 and CH 4 /O 2 strained flames[END_REF] and 25 respectively in the injector centerplane and in the centerplane between two injectors. The slight discrepancies that were observed for the temperature in the injector centerplane are increased for the mass fractions of CO and CO 2 . The difference stays relatively low at the large recirculation area for the profiles in the centerplane between two injectors, except at the rear of the cavity (x = 66 mm). At the front of the cavity, TAB3 and the LES are totally different regardless of the centerplane. Actually, when Y CO is overpredicted, Y CO2 is underpredicted at the same area. The sum of both mass fractions Y CO + Y CO2 is displayed in Fig. 26. The fitting is nearly perfect in the large recirculation area, but still some discrepancies persist in the small recirculation area (x = 2, 11 mm) and in the interface between the recirculation zones (x = 20 mm). The reason for the differences found between LES and TAB3 can be due to several factors that are be discussed in the next section.

Origin of discrepancies between tabulated approach and LES

Dilution by burnt gases, nonpremixed flame regime or strong scalar dissipation are the most probable origin of the discrepancies observed between tabulation and transported chemistry. These three options are investigated in the following.

Dilution by burnt gases

The dilution rate is the ratio of the burnt gases in the mixture and is defined as:

D r = Y BG 1 -Y N2 (3)
where Y BG is the mass fraction of the burnt gases. A species is tagged as burnt gas when it is neither a hydrocarbon (C n H m ) or a species from air (O 2 or N 2 ):

Y BG = 1 -Y CnHm -Y O2 -Y N2 (4)
The instantaneous field of dilution rate is displayed in Fig. 27 is strongly diluted by burnt gases with D r > 0.5. The front of the cavity is roughly composed of half burnt gases and half fuel (D r ∈ [0.5, 0.7]). The dilution rate is the highest in the large recirculation area where only burnt gases are present (D r > 0.9). Finally, the cavity ramp is featuring dilution rate from 0.4 to 0.7 because of the air entering at this area. Strong dilution is found everywhere inside the cavity, and yet the comparison between LES and TAB3 was satisfying in the most parts of the cavity, therefore, dilution could only be the origin of some minor discrepancies between the LES and the tabulated chemistry. Dilution without heat losses has in fact no impact on premixed flamelet properties [START_REF] Wang | Self-similar behavior and chemistry tabulation of burnt-gas diluted premixed flamelets including heat-loss[END_REF].

Nonpremixed flame regime

The instantaneous heat release rate and the flame index are displayed in Fig. 28 at four cuts in the streamwise direction. The cuts are representing the area of the small recirculation zone (x = 2 mm), the interface between the recirculation zones (x = 20 mm), the large recirculation zone (x = 39 mm) and the rear of the cavity (x = 66 mm). No combustion is found in the small recirculation zone, so the burnt gases of this region come from the interface between the recirculation zones where the combustion regime is mostly non- premixed. Trying to predict a nonpremixed flame with a premixed flame table is expected to be improper. However, nonpremixed combustion is also found in the large recirculation area, and the premixed table is accurate in that region. Fiorina et al. [START_REF] Fiorina | Approximating the chemical structure of partially premixed and diffusion counterflow flames using FPI flamelet tabulation[END_REF] found that a laminar premixed flame tabulation (FPI solver) could be efficient in predicting nonpremixed flames if the mixture fraction is close to the stoichiometric value. Indeed, chemistry near stoichiometry is too fast for any diffusive processes to have an impact on it, and thus becomes independent of the flame configuration. The mixture fraction in the large recirculation area is between 0.05 and 0.10 which is close to the stoichiometric value (Z st = 0.06366), chemistry is well predicted by the premixed flame tabulation regardless of the combustion regime. On the other hand, the interface between the recirculation zones is featuring mixture fractions of 0.15 to 0.25, chemistry is slow in that region so diffusive effects can intervene and make the species concentration dependent of the flame configuration. Another reason would be that the small recirculation area is acting like a perfectly stirred reactor (PSR), and the time required to reach the values given by the premixed flame tabulation is too long to be achieved, since the residence time of this region is only a few milliseconds [START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF]. The mixture fraction in the small recirculation area being almost constant, a mixture composition at Z = 0.279 is then extracted from the LES and used as an initial condition for PSR computations with SENKIN [START_REF] Lutz | SENKIN -A Fortran program for predicting homogeneous gas phase chemical kinetics with sensitivity analysis[END_REF]. The mass fractions of CO and CO 2 are 0.078 and 0.081 respectively for the LES, and 0.158 and 0.013 respectively for TAB3. The mass fractions of CO and CO 2 after 100 s physical time computation are displayed in Fig. 29. The mass fraction of CO is increasing while the one of CO 2 is decreasing, then evolving towards the values of TAB3. However, even 100 s is not sufficient for the mass fractions to reach Y CO,T AB3 and Y CO2,T AB3 . A premixed flame tabulation could never predict the burnt gas composition in this region which is fed by the rich side of a diffusion burning flame structure. The discrepancies at the front of the cavity being clarified, we are now interested in those in the large recirculation zone, which are probably due to strong scalar dissipations.

Scalar dissipations

Three scalar dissipations are now investigated in this section. They are the mixture fraction dissipation χ Z , the progress variable dissipation χ Y C and the cross scalar dissipation χ ZC : where the coefficients D Z , D Y C and D ZC are:

     χ Z = D Z |∇Z| 2 χ Y C = D Y C |∇Y C | 2 χ ZC = D ZC |∇Z||∇Y C | (5)
             D Z = λ ρC p D Y C = D CO |∇Y CO | + D CO2 |∇Y CO2 | + D H2O |∇Y H2O | |∇Y C | D ZC = D Z D Y C (6)
Instantaneous fields of these scalar dissipations are displayed in Fig. 30 for cuts in the injector centerplane and the centerplane between two injectors, in Fig. 31 for cuts in the streamwise direction showing planes at the small recirculation zone (x = 2 mm), the interface between the recirculation zones (x = 20 mm), the large recirculation zone (x = 39 mm) and the rear of the cavity (x = 66 mm). Strong dissipation of the mixture fraction is found at the exit of the injector, following the path of the fuel injection. The mixing layer above the small recirculation zone and the interface between the recirculation zones are featuring scalar dissipations of moderate intensity. The difference between χ Z and χ Y C is that the dissipation of the progress variable is weak at the interface between the recirculation zones and moderately strong in the whole mixing layer. The scalar dissipations profiles in Fig. 32 show that the large discrepancies between LES and TAB3 for Y CO and Y CO2 (see Fig. 24 and 25) in the large recirculation zone occur in zones with strong dissipation of mixture fraction. The dissipation of progress variable in the mixing layer only leads to small differences between LES and TAB3.

In order to link the scalar dissipation to the combustion regime, the regions with high values of χ Z in the injector centerplane at the fourth and fifth cuts (x = 30 and 39 mm, respectively) have been investigated along with the flame index (Fig. 33). In the fourth cut, strong scalar dissipations are located in regions with rich premixed combustion regime (F I = 1). While in the fifth cut, the strongest dissipations are found at the exit of the injector (y ∈ [-20, -17] mm) with nonpremixed combustion regime (F I = 0). At the middle of cavity (y > -15 mm), rich premixed and nonpremixed combustion regimes are found in zones with moderately strong scalar dissipations. The values of χ Z are insignificant for areas with lean premixed combustion regime (F I = -1) in both cuts. The largest discrepancies between the LES and TAB3 are actually located in regions with moderate scalar dissipations and featuring premixed combustion regime (see temperature profiles in Fig. 22). Regions in the large recirculation area with nonpremixed combustion are well predicted for the temperature, even with strong scalar dissipations.

In rich premixed regions with moderate scalar dissipations, solely the concentrations of CO and CO 2 are largely affected, the mass fractions of H 2 O between LES and TAB3 were found to be very close to each other. A study on the sensitivity of these species to the gradient of equivalence ratio has been performed for one dimensional laminar premixed flames. The concentrations difference CO and CO 2 for mixtures around the stoichiometry, and stays small for H 2 O. For rich mixtures, only the mass fraction of CO is significantly affected by the change in equivalence ratio. As the mixture fraction in the large recirculation area is around the stoichiometry and the progress variable in this region being higher than 0.9, the large discrepancies found between LES and TAB3 for CO and CO 2 could be explained by these species being more sensitive to mixture change. In presence of high χ Z , even for premixed regime, the effect of the local gradient of mixture fraction cannot be neglected with a tabulated approach context. New methods including gradient of mixture fraction in the library construction could help [START_REF] Nguyen | Multidimensional flameletgenerated manifolds for partially premixed combustion[END_REF][START_REF] Lodier | Composition-space premixed flamelet solution with differential diffusion for in situ flamelet-generated manifolds[END_REF]

Conclusions

The large-eddy simulation (LES) of combustion in a cavity-based scramjet has been performed with success with a dynamic approach for subgrid fluxes modeling and a reduced but accurate kinetics. The numerical results are in fair agreement with the experimental measurements available for the AFRL testbench whatever the number of injectors used in the simulation. However, only the simulation of the complete geometry revealed a transverse phenomenon, i.e. the apparition of a M structure of the flame when the exact dimensions and boundary conditions of the cavity are included. In this device, the combustion takes place in region where the flow is largely subsonic and preferentially in the mixing layer over the cavity. Diffusion layers are also present. Highly refined LES has proved to be a reliable tool for gathering information about the dynamics of the combustion in this complex configuration without any need to tune or adjust models. An a priori tabulated approach for the chemistry was tested based on a collection of one-dimensional premixed low-Mach flames. Good results on temperature and H 2 O fields are found but some discrepancies appeared on CO and CO 2 fields due to the variety of combustion regimes. As expected, shortcomings of the tabulation are present in zones where non-premixed regime is dominant and far from stoichiometry especially in the small recirculation. The impact of strong mixture fraction scalar dissipation in region where combustion is premixed has been evidenced implying that the gradient of mixture fraction should be included in the construction of the look-up table. Never-theless, as the pressure in the cavity is relatively uniform and the comparison between LES and TAB3 (built with constant pressure P = 66 kPa) led to satisfying results for a quantity such as temperature, there is no need to account for the compressibility effect in the tabulation for this scramjet. Tabulation constructed with premixed flamelets including the effects of Z gradients, nonpremixed flamelets, taking into account the dilution by burnt gases in the case of a non adiabatic system, could be then envisaged in the future to model the combustion.

Acknowledgments

Computations ressources were provided by CRIANN and GENCI (allocations 2017 and 2018-020152).

Compliance with Ethical Standards

A Progress variable

The construction of progress variable needs to satisfy two conditions: (I) the progress variable C must be a monotonically increasing function of the physical coordinate X; (II) all tabulated quantities must be injective functions of the progress variable C. The verification is done for φ ∈ [0.4, 5] in Fig. 35 for (I) and in Fig. 36 for (II).

Fig. 1

 1 Fig. 1 Scatter plot of temperature versus pressure for an instantaneous non reactive solution [4] of Cheng's burner. The compressible effects on the mixing temperature are highlighted by green dashed lines. In the insert on the right: pressure (left) and temperature (right) fields.

Fig. 3

 3 Fig. 3 Sketch of the AFRL supersonic wind tunnel.

Fig. 4

 4 Fig. 4 Comparaison of ignition delay for S22 at several initial conditions.

Fig. 5

 5 Fig. 5 Averaged temperature, heat release rate and fuel mass fraction at y = -19.57 mm (injector center) for RCA1.

Fig. 6

 6 Fig. 6 Averaged temperature, heat release rate and fuel mass fraction at y = -19.57 mm (injector center) for RCA2.

Fig. 7

 7 Fig. 7 Averaged temperature, heat release rate and fuel mass fraction at y = -19.57 mm (injector center) for RCA11. Coarse mesh.

  (a) Experiment (pattern), RCA1 (red line), RCA2 (green line) RCA11 (blue line, between 6-th and 7-th injector). (b) Experiment (pattern). RCA11: injectors 4-5 (red line), 6-7 (green line), 9-10 (blue line) and 10-11 (yellow line).

Fig. 8

 8 Fig. 8 Averaged cavity wall pressure. Comparison between the numerical and the experimental data at the centerplane between two injectors.

Fig. 10

 10 Fig.10Averaged temperature (top), streamwise (middle) and tranverse (bottom) velocity profiles inside the cavity, comparison between the numerical and the experimental data for coarse mesh at the centerplane between the two injectors for RCA11: experiment (pattern), injectors 4-5 (red line), 6-7 (green line), 9-10 (blue line) and 10-11 (magenta line).

Fig. 11 Fig. 12

 1112 Fig. 11 Instantaneous isosurface of temperature at 2000 K for RCA11, colored by the values of mixture fraction.

Fig. 13

 13 Fig. 13 Temperature (left) and main product mass fractions (right) along X for φ = 4. The dashed line represents the equilibrium value.

Fig. 14

 14 Fig. 14 Laminar premixed flame speed (S L ) and thickness (δ L ) within the flammability limits (left) and in the whole computed range of equivalence ratios (right).

Fig. 16

 16 Fig. 16 Laminar premixed flamelet libraries: (a) TAB1, (b) TAB2, (c) TAB2b, (d) TAB3. Tabulated (-), interpolated (---).

Fig. 17

 17 Fig. 17 Instantaneous field of mixture fraction inside the cavity: centerplane between two injectors (top) and injector centerplane (bottom). The stoichiometric line is displayed in white color.

Fig. 18

 18 Fig. 18 Instantaneous field of progress variable inside the cavity: centerplane between two injectors (top) and injector centerplane (bottom). The stoichiometric line is displayed in white color.

Fig. 19

 19 Fig. 19 Temperature comparison with LES results: LES (top), TAB2 (bottom).

Fig. 20

 20 Fig. 20 Temperature comparison with (a) constant y cuts at mixing layer (y = -7 mm) and middle line of the cavity (y = -14 mm): from top to bottom, LES, TAB2 at y = -7 mm, then LES, TAB2 at y = -14 mm; and (b) constant x cuts at the interface of the recirculation zones (x = 20 mm from the cavity front) and the middle of the large recirculation area (x = 39 mm): from left to right LES, TAB2 at x = 20 mm, then LES, TAB2 at x = 39 mm.

Fig. 21

 21 Fig. 21 Instantaneous mixture fraction and progress variable profiles inside the cavity in (top) the injector centerplane, and (bottom) the centerplane between two injectors.

Fig. 22

 22 Fig. 22 Instantaneous (top) temperature and (bottom) mass fraction of H 2 O profiles inside the cavity in the injector centerplane: comparison between LES and tables with different methods outside the flammability limits.

Fig. 23

 23 Fig. 23 Instantaneous (top) temperature and (bottom) mass fraction of H 2 O profiles inside the cavity in the injector centerplane: comparison between LES and tables with different initial temperatures.

Fig. 24

 24 Fig.[START_REF] Meeks | Computational simulation of diamond chemical vapor deposition in premixed C 2 H 2 /O 2 /H 2 and CH 4 /O 2 strained flames[END_REF] Instantaneous CO mass fraction profiles inside the cavity in the injector centerplane (top) and the centerplane between two injectors (bottom): comparison between LES and TAB3.

Fig. 25

 25 Fig. 25 Instantaneous CO 2 mass fraction profiles inside the cavity in the injector centerplane (top) and the centerplane between two injectors (bottom): comparison between LES and TAB3.

Fig. 26 Fig. 27

 2627 Fig. 26 Instantaneous Y CO + Y CO 2 profiles inside the cavity in the injector centerplane (top) and the centerplane between two injectors (bottom): comparison between LES and TAB3.

Fig. 28 Fig. 29

 2829 Fig. 28 Instantaneous heat release rate (top) and flame index (bottom) in the small recirculation zone (x = 2 mm), the interface between the recirculation zones (x = 20 mm), the large recirculation zone (x = 39 mm) and the rear of the cavity (x = 66 mm). The stoichiometric line is represented in white color. Case RFA1.

Fig. 30

 30 Fig. 30 Instantaneous scalar dissipations in the centerplane between two injectors and in the injector centerplane. The stoichiometric line is displayed in white color. Case RFA1.

Fig. 32 Fig. 33

 3233 Fig.32Instantaneous scalar dissipations profiles in the injector centerplane (top) and in the centerplane between two injectors (bottom). Case RFA1.

Fig. 35

 35 Fig. 35 Progress variables C 1 (left) and C 2 (right) in physical space at equivalence ratio φ ∈ [0.4, 5]: blue, green and red colors correspond respectively to lean, stoichiometric and rich mixtures.

Fig. 36

 36 Fig. 36 Tabulated temperature and major combustion product mass fractions against progress variables C 1 (left) and C 2 (right) at equivalence ratio φ ∈ [0.4, 5]: blue, green and red colors correspond respectively to lean, stoichiometric and rich mixtures.

Table 1

 1 Parameters of reactive cases with different number of injectors in mm. z 0 = 6.35 mm and i ∈ [-5, +5]. BC: boundary conditions. Coarse mesh.

	Case	Cavity width Injector center coordinate	Side BC
	RCA1	12.7	z = z 0	Periodic
	RCA2	25.4	z = ±z 0	Periodic
	RCA11	152.4	z i = z 0 + i × 12.7	Adiabatic non-slip wall

Table 2

 2

	Table	1D Flame for φ in Prolongation for φ in Temperature (K)
	TAB1	[0.4, 5]	[0, 0.4[ ∪ ]5, +∞[	329
	TAB2	[0.4, 19.5]	[0, 0.4[ ∪ ]19.5, +∞[	329
	TAB2b [0.4, 19.5]	[0, 0.4[ ∪ ]19.5, +∞[	450
	TAB3	[0.4, 74.5]	[0, 0.4[ ∪ ]74.5, +∞[	329

Look-up table features. P = 66 kPa.

  in the injector centerplane and in the centerplane between two injectors. The whole cavity

	x= 2 mm	11 mm	20 mm	30 mm	39 mm	48 mm	57 mm	66 mm
	T (K)
	T TAB2 (K)
	T TAB2b (K)
	y (mm)
	1000 2000	1000 2000	1000 2000	1000 2000	1000 2000	1000 2000	1000 2000	1000 2000
	x= 2 mm	11 mm	20 mm	30 mm	39 mm	48 mm	57 mm	66 mm
	Y H2O
	Y H2O,TAB2
	Y H2O,TAB2b
	y (mm)
	0	0.1	0	0.1	0	0.1	0	0.1	0	0.1	0	0.1	0	0.1	0	0.1

  . Sensitivity of CO, CO 2 and H 2 O to the equivalence ratio: ∆Y k vs. progress variable C for mixtures around the stoichiometry (left) and for rich mixtures (right).

		0.012	CO								0.008	CO
		CO2								CO2
		H2O								H2O
		0.01
											0.006
		0.008
	∆ Y k (-)	0.006								∆ Y k (-)	0.004
		0.004
											0.002
		0.002
		0									0
		0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	1	0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	1
						C (-)									C (-)
	Fig. 34

Funding: This study was funded by the French Department of Defense (DGA) (grant number 2015089).


Jiangheng L Ruan

Guillaume Ribert
email: guillaume.ribert@insa-rouen.fr

Pascale Domingo

Stabilization and extinction mechanisms of flames in cavity flameholder scramjets

Keywords: Supersonic combustion, Large-eddy simulation, Cavity-based scramjet

Stabilizing a flame in scramjet engines is a current technological challenge. Indeed, the residence time in the combustion chamber is very short, thus limiting the mixing efficiency and consequently, the combustion. This problem is presently addressed by simulating the scramjet combustor of the U.S Air Force Research Laboratory in which ethylene is injected from the back of a slanted cavity to react with the supersonic airflow. Large-Eddy Simulations (LES) are performed for different cases showing a dependence of the combustion regimes encountered in the scramjet cavity with the fuel flow rate. For a medium-high flow rate, a mix of lean and rich premixed reaction zones is found along with non-premixed flames. At medium flow rate, rich premixed flames disappear but the flame remains stable in the cavity. The mechanism of flame stabilization is detailed and the LES tool has proven its efficiency in such challenging configuration. Finally, a mechanism of extinction is proposed when the fuel flow rate is further reduced.

Introduction

Scramjet engines are good candidates for hypersonic flight applications [START_REF] Segal | The scramjet Engine[END_REF][START_REF] Urzay | Supersonic combustion in air-breathing propulsion systems for hypersonic flight[END_REF]. However, at such high speeds, the residence time of air and fuel in the combustor becomes so short that the mixing and burning processes become ineffective if no recirculation zone is encountered. The problem of flame-holding, along with the understanding of the mechanisms of flame stabilization and flame extinction, are thus key elements in the development of a future reliable scramjet engine. Among the different strategies to trap the fuel and air in recirculation zones [START_REF] Segal | The scramjet Engine[END_REF][START_REF] Urzay | Supersonic combustion in air-breathing propulsion systems for hypersonic flight[END_REF], the use of cavity has been viewed as a promising solution [START_REF] Ben-Yakar | Cavity flame-holders for ignition and flame stabilization in scramjets: An overview[END_REF] since it reduces drag, lowers total pressure losses and minimizes aerodynamic heating [START_REF] Barnes | Cavity-based flameholding for chemically-reacting supersonic flows[END_REF]. Resolving problems related to cavity scramjets requires the treatment of various problems such as the effect of cavity geometry and fuel injection parameters which ultimately condition the impact of heat release on the structure and the dynamics of the shear layer above the cavity, the recirculation zones, the flame regimes or the conditions of flame blowout. The result is a complex flow-field that greatly complicates the identification of the flame-holding mechanisms and the prediction of flame stability. Nevertheless, among the various observations made on such device, it may be noted that inside the cavity, a primary vortex with a strong circulation is responsible for the mass exchange with the free-stream zone while a secondary vortex is supposed to act as a fuel pooling in a dead zone. In addition, the effect of heat release on the local flow field reduces the growth rate of the shear layer and increases the average residence time of the recirculating fluid relative to a non-reacting case. Finally, cavity blowout limits can be extended if the residence time increases, by changing the geometry of the cavity, increasing the stagnation temperature and pressure of the free-stream, or changing the location of the injector and its fuel rate. It becomes then a formidable challenge to predict such behaviors and the use of numerical simulations appears as a necessary tool [START_REF] Gonzalez-Juez | Advances and challenges in modeling high-speed turbulent combustion in propulsion systems[END_REF]. In particular, the use of Large-Eddy Simulation (LES) for supersonic combustion is now becoming relevant [START_REF] Berglund | Les of supersonic combustion in a scramjet engine model[END_REF][START_REF] Saghafian | Large eddy simulations of the hifire scramjet using a compressible flamelet/progress variable approach[END_REF][START_REF] Bouheraoua | Large eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base[END_REF][START_REF] Vincent-Randonnier | An experimental and computational study of hydrogen-air combustion in the LAPCAT II supersonic combustor[END_REF][START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF] even if for highspeed combustion, the choice of turbulent-combustion models can be challenging [START_REF] Miller | Survey of turbulent combustion models for large-eddy simulations of propulsive flowfields[END_REF][START_REF] Ruan | Simulation of a scramjet combustor: A priori study of thermochemistry tabulation techniques[END_REF] since the majority of models in literature have been developed for low-speed combustion. Besides, combustion can occur near or in the distributed-reaction-like regime [START_REF] Gonzalez-Juez | Advances and challenges in modeling high-speed turbulent combustion in propulsion systems[END_REF], i.e. when high-speed reacting flows have also high levels of turbulent fluctuations, combustion regime that is difficult to model. In this study the LES tool is used to reproduce the processes arising in the AFRL (Air Force Research Lab) combustor [START_REF] Tuttle | Particle image velocimetry in a non-reacting and reacting high-speed cavity[END_REF]. Experimental data are used to validate the numerical approach and support an analyze on the flame behaviors and regimes in the case of a stable and of a vanishing flame.

Configuration and numerical set-up

Experimental cases

The experimental configuration under study is a supersonic cavity flameholder [START_REF] Tuttle | Particle image velocimetry in a non-reacting and reacting high-speed cavity[END_REF]. It consists in a converging-diverging nozzle followed by an isolator that is H = 50.8 mm high and L iso = 177.8 mm long. The bottom wall of the test section has a divergence angle of ↵ = 2.5 , which begins at the end of the isolator. The cavity is located 76.2 mm downstream of the end of the isolator and has a depth of D = 16.5 mm with a floor length of 46.1 mm (L = 66 mm length to the ramp midpoint). The width of the facility is constant with a value of In the isolator, air is at Mach 2 and the total temperature and pressure are 589 K and 483 kPa, respectively. Ethylene (C 2 H 4 ) is injected at 310 K at the rear of the cavity which makes the auto-ignition impossible and necessitates the use of spark plugs to force ignition. Three fuel loadings [START_REF] Tuttle | Particle image velocimetry in a non-reacting and reacting high-speed cavity[END_REF] have been simulated (see Table 1). For the cases MHF (Medium-High Fuel) and MF (Medium Fuel), experimental data for velocity and mean fuel concentrations (only MF) are available inside the cavity [START_REF] Tuttle | Particle image velocimetry in a non-reacting and reacting high-speed cavity[END_REF][START_REF] Do | Simultaneous gas density and fuel concentration measurements in a supersonic combustor using laser induced breakdown[END_REF]. The flame is found stable and located inside the cavity contrary to the case LF (Lean Fuel) for which the flame extinguishes. about the experimental configuration can be found in numerical studies [START_REF] Hassan | Dynamic hybrid reynolds-averaged navierstokes/large-eddy simulation of a supersonic cavity[END_REF][START_REF] Baurle | Hybrid reynolds-averaged / large eddy simulation of a scramjet cavity flameholder[END_REF][START_REF] Hassan | Dynamic hybrid reynolds-averaged navierstokes/large-eddy simulation of a supersonic cavity: Chemistry effects[END_REF].

Numerical configuration

The simulations have been performed with the CORIA in-house code, SiTCom-B [START_REF] Bouheraoua | Large eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base[END_REF][START_REF] Guven | Large-eddy simulation of supersonic H 2 /O 2 combustion: application to a rocket-like igniter[END_REF][START_REF] Duboc | Hybrid transported-tabulated chemistry for partially premixed combustion[END_REF][START_REF] Duboc | Evaluation of chemistry models on methane/air edge flame simulation[END_REF]. The conservation equations of species mass fractions, with complex molecular transport properties, momentum and total sensible energy are solved in their fully compressible form over the structured mesh in a finite volume formulation resorting to a skew-symmetric-like scheme for the convective fluxes [START_REF] Ducros | Largeeddy simulation of the shock/turbulence interaction[END_REF] and to a fourth-order centred scheme for the viscous and diffusive fluxes. An addition of second and fourth-order artificial dissipation terms [START_REF] Swanson | On central-difference and upwind schemes[END_REF][START_REF] Tatsumi | Flux-limited schemes for the compressible Navier-Stokes equations[END_REF][START_REF] Swanson | On some numerical dissipation schemes[END_REF] is used to overcome spurious oscillations [START_REF] Petit | Large-eddy simulation of supercritical fluid injection[END_REF]. Time is advanced with a fourth order Runge-Kutta scheme. Navier-Stokes characteristic boundary conditions are applied to describe the air and fuel inflows while the outflow uses a zero-gradient condition. An immersed boundary method (IBM) [START_REF] Merlin | Immersed boundaries in large eddy simulation of compressible flows[END_REF][START_REF] Bouheraoua | Large eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base[END_REF] is used to model the solid surface of the wind tunel. Unsolved subgrid-scale fluxes are modeled by the Germano closure [START_REF] Germano | A dynamic subgrid-scale eddy viscosity model[END_REF]. Sub-grid Prandtl and Schmidt numbers are set to 0.9. Mixture diffusion coefficients for each species [START_REF] Bird | Transport Phenomena[END_REF] are considered based on binary diffusion coefficients [START_REF] Hirschfelder | Molecular Theory of Gases and Liquids[END_REF]. The upper and lower walls of the scramjet are modeled with an adiabatic non-slip condition and side sections use periodic conditions since only one injector is considered. A detailed chemistry containing N s = 22 species and 206 reactions [START_REF] Luo | Chemical explosive mode analysis for a turbulent lifted ethylene jet flame in highly-heated coflow[END_REF] is used to describe the combustion of ethylene with air. Note that it was not possible [START_REF] Ruan | Large eddy simulation of supersonic combustion in cavity-based scramjets[END_REF] to obtain a stable flame using a kinetic scheme with fewer species such as the ten reactive species mechanism from Jachimowski [START_REF] Jachimowski | An analytical study of the hydrogen-air reaction mechanism with application to scramjet combustion[END_REF]. A no-model assumption is used to compute the chemical source terms as in [START_REF] Bouheraoua | Large eddy simulation of a supersonic lifted jet flame: Analysis of the turbulent flame base[END_REF][START_REF] Guven | Large-eddy simulation of supersonic H 2 /O 2 combustion: application to a rocket-like igniter[END_REF]. It has been verified in [START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF][START_REF] Ruan | Simulation of a scramjet combustor: A priori study of thermochemistry tabulation techniques[END_REF][START_REF] Ruan | Large eddy simulation of supersonic combustion in cavity-based scramjets[END_REF]] that these assumptions do not modify the results and scenarios described hereafter. The use of the quasi-laminar-chemistry approach has been found acceptable in [START_REF] Gonzalez-Juez | Advances and challenges in modeling high-speed turbulent combustion in propulsion systems[END_REF] for small scramjet-like problems.

As in [START_REF] Hassan | Dynamic hybrid reynolds-averaged navierstokes/large-eddy simulation of a supersonic cavity[END_REF][START_REF] Baurle | Hybrid reynolds-averaged / large eddy simulation of a scramjet cavity flameholder[END_REF][START_REF] Hassan | Dynamic hybrid reynolds-averaged navierstokes/large-eddy simulation of a supersonic cavity: Chemistry effects[END_REF], the nozzle is not computed but the 3D simulations start at the beginning of the isolator. The computational domain is extended 81 mm after the cavity. At inlet boundary, the three components of velocity profile are given (in m/s) by U = 1.22U 0 (y/H) 1/7 (1 y/H) 1/7 with U 0 set to 727 m/s and corresponding to a Mach number of 2, V = W = 0. Turbulence in injected at the inlet to trigger the boundary layer development with an intensity of 10% of the local axial velocity and distributed evenly on the three components of the velocity. The mesh resolution uses 45 millions of cells, which are constant in x-direction ( x = 200 µm) but are refined near the wall and the mixing layer in y-direction ( y 2 [150 200] µm and z 2 [160 300] µm). Near the injector the mesh refinement in y-and z-direction leads to 10 the number of cells to describe the diameter of the injector. A more refined mesh using 300 millions of cells has been used to validate the procedure and analysis [START_REF] Ruan | Large eddy simulation of supersonic combustion in cavity-based scramjets[END_REF]. A wall-law model based on zero-equation mixing length model and using the van Driest damping function [START_REF] Cabot | Large-eddy simulations with wall models[END_REF] is applied at the upper and lower wall of the simulation domain since the dimensionless wall distance y + ⇡ 10. In the cavity, y + ⇡ 1. Statistics have been gathered over 6 ms while the empirically estimated cavity residence time [START_REF] Davis | Stirred reactor analysis of cavity flameholders for scramjets[END_REF] is 0.9 ms. at the beginning of the cavity and at the exit of the injector for the case MF, but mainly above the ramp of the cavity and at the exit of the injector for the case MHF. Inside the cavity, an average gas temperature is found around 1800 K for both cases. A species stratification is observed. In particular, oxygen is absent at the front of the cavity, filled with ethylene and combustion products, but is present near the injector. Averaged streamlines colored by axial velocity are shown in Fig. 3. Both primary and secondary recirculation zones can be clearly identified. The fresh air feeds the cavity with the large recirculation zone, while the smallest one located upstream traps hot products mixed with unburnt ethylene. Comparisons between the numerical solutions and the experiment are displayed in Fig. 4 for the velocity profiles taken at x = 2, 11, 20, 30, 39, 48, 57, 66 mm from the cavity front corner. The comparison of velocity profiles shows only a slight difference between the experiment and the numerical data for streamwise velocity. The discrepancies are located in first planes (x = 2 and 11 mm), in the mixing layer. The mixing layer in the experiment is thicker just behind the sudden expansion area, but the correct thickness is recovered further downstream. A better description of the mixing layer above the cavity would certainly have required a specific simulation of the nozzle [START_REF] Fiévet | Numerical investigation of shock-train response to inflow boundary-layer variations[END_REF][START_REF] Fiévet | Effect of vibrational nonequilibrium on isolator shock structure[END_REF] out of scope of this study. The transverse velocity (Fig. 4) shows significant discrepancies between LES predictions and experimental measurements inside the cavity. One may remark that in the work from Hassan et al. [START_REF] Hassan | Dynamic hybrid reynolds-averaged navierstokes/large-eddy simulation of a supersonic cavity: Chemistry effects[END_REF], the transverse velocity has a similar behavior as in the present paper. The question is open on whether those discrepancies are related to modeling issues or to experimental uncertainties due to the difficulties encountered by PIV in a cavity. The equivalent fuel concentration (X C2H4,eq ) in Fig. 5 is defined as X C2H4,eq = P Ns i K i X i , with X i is the molar fraction of species i, which is defined as C n(i) H p(i) O q(i) with C, H and O the atoms of carbon, hydrogen and oxygen respectively, and where K i = p(i)/p(C 2 H 4 ). Fairly good results are observed for the centerplane of the injector at x = 8.9 mm and x = 44.5 mm. Overall, the trend is recovered showing a penetration of the fuel inside the cavity up to its front where the fuel is trapped. All these results are consistent with other literature articles for which RANS (Reynolds Average Navier-Stokes) [START_REF] Baurle | Hybrid reynolds-averaged / large eddy simulation of a scramjet cavity flameholder[END_REF] or different RANS-LES hybrid approaches [START_REF] Hassan | Dynamic hybrid reynolds-averaged navierstokes/large-eddy simulation of a supersonic cavity: Chemistry effects[END_REF] have been tested. Further comparisons were made on the velocity and wall pressure profiles for the MHF case and a good agreement was found by comparison with the experimental data [START_REF] Ruan | Analysis of combustion modes in a cavity based scramjet[END_REF][START_REF] Ruan | Simulation of a scramjet combustor: A priori study of thermochemistry tabulation techniques[END_REF].

Validation

Flow analysis

Stabilisation mechanisms and combustion regimes

The goal of the cavity is to increase the residence time and favor combustion, this effect being also present in the cavity employed in a subsonic combustor, as discussed in a recent review paper from Zhao et al. [START_REF] Zhao | A review of cavity-based trapped vortex, ultracompact, high-g, inter-turbine combustors[END_REF]. Note that it would be legitimate to wonder how the impact of the cavity is modified if the air inlet becomes subsonic (out of scope of the present study). The mechanism might not be that different that the one described below, except that the mixing layer would be thicker and the mixing would be more efficient. In our study, a stable flame is found experimentally for both fuel loadings of 99 and 56 SLPM. This feature has been recovered by the present numerical simulations as shown in Fig. 2. For both cases, the whole cavity is at subsonic speed and two recirculation zones are created. For the case MHF (Fig. 6a), zone B features a large recirculation zone with high velocities (up to 350 m/s) and high temperature (up to 2500 K) [START_REF] Ruan | Large eddy simulation of supersonic combustion in cavity-based scramjets[END_REF]. The temperature of the flow near the cavity aft wall can reach 2000 K allowing the incoming airflow to be heated before encountering ethylene present in the cavity and thus support combustion. In zone A, i.e. at the front of the cavity, the flow velocity is one order of magnitude less than in zone B with a temperature below 1500 K. The heat release rate snapshots (Fig. 6) show that the flame fronts are located at the exit of the injector, the interface between the two recirculation zones and the mixing layer between the cavity and the freestream. The isoline Mach = 1 indicates that combustion occurs preferentially at subsonic speeds, i.e. inside cavity. The middle of the cavity and the interface between recirculation areas are driven by moderately strong positive and weak negative heat release rate respectively. Negative values of the heat release rate occur when the local mixture is in a state of over-equilibrium. As shown in Fig. 5 and 2(c), the fuel contained in the recirculation zone at the front of the cavity does not react because there is no more oxygen left. However at the top of zone A, hot ethylene (around 1500 K) starts to react with the oxygen coming from the airflow in the mixing layer developing above zone B. The strongest combustion activity is in zone B, with the effect of heating the airflow coming from the back of the cavity. This hot air can then react with the fuel leaving the injector and the resulting combustion products will then feed the zones A and B with hot gases. This mechanism of recirculating hot and reactive gases allows to maintain a stable flame in the cavity.

For the case MF (Fig. 6b), the quantity of fuel going into zone A is smaller. The amount of fuel flowing from zone A to the mixing layer is also reduced compared to MHF. The fuel shortage in the mixing layer above the large recirculation area and near the cavity ramp is responsible of the low heat release rate present in this area. This results in high temperatures at the front of the cavity and the only "cold" area of the cavity is located at near the ramp as observed by Baurle [START_REF] Baurle | Hybrid reynolds-averaged / large eddy simulation of a scramjet cavity flameholder[END_REF] with RANS simulations.

To characterize the flame regimes, i.e. to distinguish premixed from non-premixed combustion, the flame index [START_REF] Yamashita | A numerical study on flame stability at the transition point of jet diffusion flames[END_REF] is often used: F.I. = rY F .rY O2 . A ponderation was further introduced by Lock et al. [START_REF] Lock | Liftoff characteristics of partially premixed flames under normal and microgravity conditions[END_REF] to distinguish between lean and rich combustion:

F.I. = 1 2 Z Z st |Z Z st | ⇥ Ç 1 + rY F .rY O2 |rY F .rY O2 | å . (1)
In Eq. ( 1), Z is the mixture fraction from the Bilger et al.'s expression [START_REF] Bilger | On reduced mechanisms for methane-air combustion in nonpremixed flames[END_REF] and Z st the stoichiometric mixture fraction. Y O2 is the oxygen mass fraction while Y F is defined presently as 1) is then used to identify the regimes of combustion: F.I. = 0 for non-premixed flames, +1 for rich premixed flames and 1 for lean premixed flames. The flame index shown in figures 7 and 8 is only computed in zones where the heat release is greater than 1% of its maximum value. For the case MHF (Fig. 7), non-premixed flames are observed in the mixing layer above zone A, between zone A and B and near the middle of the ramp where an excess of carbon species react with an excess of hot air. Rich premixed flames are found in the mixing layer above zone B and in the region close to the middle of the cavity. Lean premixed flames are located near the cavity ramp. The regions with negative heat release rate (see Fig. 6) located at the interface between the recirculation zones or at the exit of the injector are only featuring rich premixed combustion regime. Neither lean premixed nor nonpremixed flames can be found with negative heat release rate. Furthermore, not all non-premixed regions are crossed by the stoichiometric line, the cavity is provided with rich non-premixed combustion near its center and lean non- premixed combustion at its rear.

Y F = Y C2H4 + Y C2H2 + Y CH3 + Y CH4 + Y C2H6 + Y C3H6 . Eq. (
The flame index has also been computed and displayed in Fig. 8 for the medium fuel loading case (MF). The cavity is found to be mainly driven by non-premixed flames. All the regions with significant heat release rate (see Fig. 6) are featuring flame index values of 0, corresponding to non-premixed combustion. Rich premixed regime is only observed at the exit of the injector and barely at the mixing layer above the small recirculation area. The mixing layer above the large recirculation area and the cavity ramp are still controlled by premixed combustion, but only lean mixtures can be found in these zones. A lower injection velocity creates less turbulence in the cavity which makes the mixing more difficult. Fig. 9 shows the conditional mean of heat release rate by flame index < !E | F.I. > adimensionalized by the overall mean heat release < !E > (blue patterns) and the contribution of each flame regime as characterised by the flame index to the heat release, C F I (red histogram) defined as follow:

R F I = < !E | F.I. > < !E > and C F I = R F I N F I N T (2)
with N T = P F I N F I . N F I is the number of points in the interval of considered flame index and N T the total number of points. The conditional mean heat release rate is similar for each flame index value: from R F I = 0.8 in the nonpremixed flames to R F I = 1.1 in the premixed flames regardless of the mixture. None of the regimes is only marked by either strong or weak combustion. The values taken by C F I indicate that, for the case MHF, most of the contributions to the overall heat release rate come from premixed flames, over 75%, where the lean premixed regime contribution is approximately 60%. For the case MF, the premixed regime corresponds to 40% almost entirely dominated by the lean side.

The decrease in fuel loading has then a significant impact on the combustion regime which in turns impacts the choice of a combustion modeling if either a tabulated approach of the chemistry or a very coarse resolution implying a subgrid model for the filtered source terms were to be retained. Despite this change in combustion regime, a stable flame is still observed.

Extinction mechanisms

Starting with the stable case MF (t = 0), the fuel injection velocity is decreased up to 36 SPLM linearly within 0.5 ms in order to capture the extinction mechanism of the combustor. Once 36 SPLM have been reached, the simulation is left to evolve to track the extinction mechanism. In the procedure followed, it is not guaranteed that extinction occurs strictly at 36 SPLM but it is verified that, as in the experiment, this mass flow rate cannot sustain the stabilization of the flame.

The snapshots of heat release rate and C 2 H 4 mass fraction are displayed in Fig. 10 for the injector centerplane. Instantaneous fields of HRR and Y C2H4 at the centerplane between two injectors are in appendix A. A transitional regime, where the successive velocity damping leads to combustion inside the injector, is observed up to t = 2 ms. Contrary to the cases MHF and MF, the fuel is not able to reach the middle of the cavity and ethylene is immediately consumed at the exit of the injector. Combustion at the mixing layer is only fed by the initial huge stock of fuel at the front of the cavity which is starting to be cleared out. The lack of ethylene in the large recirculation zone causes a fuel shortage at the rear of the cavity, where the air entering is not mixed anymore with burnt products leading to a cooling of the cavity. At the next instants t = 4 ms, ethylene is still burning at the exit of the injector, and the spare fuel at the cavity front has almost totally been consumed. The mixing layer between the freestream and the cavity is not anymore fed by the fuel from the small recirculation zone, so combustion vanishes in this region and the airflow entering the cavity cannot be heated. The reaction zone observed at the exit of the injector begins to decrease because the air coming from the main airflow is now cold and the reaction zone cannot resist anymore to the strong stretching.

Conclusion

The Large-Eddy Simulation of the AFRL supersonic cavity revealed the difference in combustion regimes when the incoming fuel flow was changed. At medium-high flow rate, the three combustion regimes, i.e. lean or rich premixed then non-premixed, are present in the recirculation zone which maintains the flame in the cavity. At medium flow rate, lean premixed and non-premixed flames are clearly observed in the main recirculation zone. For these two cases, the scenario of flame stabilization is the same: unburned fuel supplies a small recirculation zone where hot products are trapped; this stock of fuel then reacts with the incoming air which is more or less preheated depending on the flow of fuel injected. However, if the fuel flow decreases further, up to 36 SLPM, the fuel stock located in the small recirculation zone begins to decrease, limiting the auto-ignition of the mixture layer. Consequently, the energy released by combustion is less, leading to a drop in temperature in the main recirculation zone. There is therefore no longer enough energy to maintain combustion and the flame disappears.

Figure 1 .

 1 Figure 1.: Sketch of AFRL supersonic wind tunnel. 152.4 mm (see Fig. 1 for details). An array of 11 fuel injection ports of diameter d = 1.6 mm are evenly distributed on the aft cavity wall, at y = 19.57 mm.In the isolator, air is at Mach 2 and the total temperature and pressure are 589 K and 483 kPa, respectively. Ethylene (C 2 H 4 ) is injected at 310 K at the rear of the cavity which makes the auto-ignition impossible and necessitates the use of spark plugs to force ignition. Three fuel loadings[START_REF] Tuttle | Particle image velocimetry in a non-reacting and reacting high-speed cavity[END_REF] have been simulated (see Table1). For the cases MHF (Medium-High Fuel) and MF (Medium Fuel), experimental data for velocity and mean fuel concentrations (only MF) are available inside the cavity[START_REF] Tuttle | Particle image velocimetry in a non-reacting and reacting high-speed cavity[END_REF][START_REF] Do | Simultaneous gas density and fuel concentration measurements in a supersonic combustor using laser induced breakdown[END_REF]. The flame is found stable and located inside the cavity

Figure 2

 2 Figure2shows the average fields of C 2 H 4 and O 2 mass fractions, heat release rate (HRR) and temperature at the injector centerplane for a medium fuel (MF) and medium high fuel (MHF) loading. HRR fields indicates that a strong combustion occurs in the mixing layer developing

Figure 3 .

 3 Figure 3.: Averaged streamlines, colored by averaged axial velocity (case MHF).

Figure 4 .

 4 Figure 4.: Comparison between the numerical (lines) and the experimental (symbols [13, 14]) data inside the cavity. (a,b) Centerplane between two injectors. Case MF.

Figure 5 .

 5 Figure 5.: Plots of X C2H4,eq at two positions (case MF, injector centerplane). Bullets: experiment. Lines: LES.

Figure 6 .

 6 Figure 6.: Instantaneous fields of heat release rate taken at the injector centerplane. At the top, case MHF (a); at the bottom, case MF (b). White line: isoline Mach = 1.

Figure 7 .

 7 Figure 7.: Instantaneous flame index in the centerplane between two injectors, in the injector centerplane and in four planes with constant x in the cavity. -1: lean premixed, 0: nonpremixed, 1: rich premixed. The isoline Z = Z st is displayed with a white line. Case MHF.

Figure 8 .Figure 9 .

 89 Figure 8.: Instantaneous flame index in the centerplane between two injectors, in the injector centerplane and in four planes with constant x in the cavity. -1: lean premixed, 0: nonpremixed, 1: rich premixed. The isoline Z = Z st is displayed with a white line. Case MF.

Figure 10 .

 10 Figure 10.: Snapshots of heat release rate (left) and C 2 H 4 mass fraction (right) at injector centerplane for a lean fuel loading, presented every ms from t = 0 to 4 ms.

Table 1 .

 1 : Fuel loading in Standard Liters Per Minute (SLPM).

	Additional information

Acknowledgments

The present work has been supported by the French Department of Defence (DGA). Computations ressources were provided by CRIANN and GENCI (allocation 2018-020152).