Fábio Henrique

Eugênio Ribeiro

Radouan Boukharfane

Arnaud Mura
email: arnaud.mura@ensma.fr

Highly-resolved large-eddy simulations of combustion stabilization in a scramjet engine model with cavity flameholder

Keywords: High-speed flows, Large-eddy simulation (LES), Mesh reliability, Jet in supersonic crossflow (JISCF), Scramjet, Immersed boundary method (IBM)

Scramjet engines are high-speed airbreathing propulsion systems that do not require rotating elements to compress the air inlet stream. It is compressed dynamically through a supersonic intake system that is integrated in the forebody, thus leading to the required temperature and pressure levels for combustion to proceed within the combustor length. In such engines, the combustion chamber is crossed by a supersonic flow, which limits the time available to inject fuel, to mix it with oxidizer, to ignite the resulting mixture, and to reach complete combustion. Residence times can be increased thanks to cavities, which have the potential to stabilize combustion without excessive total pressure loss and are therefore used as flameholders in supersonic combustors. In the present study, we perform high-fidelity large-eddy simulations (LES) of a hydrogen jet in a supersonic crossflow (JISCF) of vitiated air, which is located upstream of a wall-mounted squared cavity. The performance of such high-fidelity LES does not only require the use of high-precision numerical schemes and reliable subgrid-scale models relevant to the so-called direct numerical simulation (DNS) limit, it is also strongly dependent on the mesh quality. Therefore, the present study places special emphasis on computational grid assessment through the introduction of a detailed numerical procedure, which aims at analysing mesh reliability. The corresponding procedure combines several verification subsets including (i) the inspection of distributions of the dimensions of the computational cells present at the wall location, (ii) the analysis of normalized velocity profiles and viscosity ratio in boundary layers, and (iii) the check of fields of some mesh quality indexes and associated distributions. For the geometry under consideration, it appears that the level of resolution imposed by a correct description of boundary layers leads to a mesh quality that is close to the one associated to DNS requirements. Combustion stabilization is then studied for two distinct values of the inlet vitiated airstream temperature. Two stabilization modes are recovered from the numerical simulations: cavity-stabilized and jet-wake stabilized regimes.

Introduction

Scramjet engine is a very promising propulsion strategy for hypersonic vehicles. Several fields of application, such as civil transport and launchers, are concerned. For access-to-space applications in particular, the use of airbreathing hypersonic propulsion systems could reduce significantly the launch costs. Preliminary studies showed remarkable and promising results in the use of this technology as a second stage of a three-stage-to-orbit launcher [START_REF] Smart | Orbital delivery of small payloads using hypersonic airbreathing propulsion[END_REF]. Thus, there is a special interest in the hypersonic vehicle as the promise of an airbreathing reusable system able to lift payloads into Earth orbit.

One of the main advantages of scramjets over concurrent technologies is related to the fact that it does not carry oxidizer, which is scooped directly from the atmosphere, thus providing a considerable weight reduction and allowing for higher payloads and specific impulses compared to standard rocket engines [START_REF] Heiser | Hypersonic airbreathing propulsion[END_REF][START_REF] Smart | Scramjets. Technical report[END_REF][START_REF] Segal | The scramjet engine: processes and characteristics[END_REF]. This also means reduced costs, reduced complexity, and also lighter ground support. However, high-speed flights in the atmosphere raise several specific issues related, among others, to aerodynamic heating and vehicle drag, which impose the cooling of both the vehicle and the engine for prolonged operations. The corresponding high temperature levels also contribute to the dissociation of molecules, e.g. oxygen and nitrogen molecules decompose and recombine, and they may even ionize at higher speeds [START_REF] Anderson | Hypersonic and high temperature temperature gas dynamics[END_REF][START_REF] Segal | The scramjet engine: processes and characteristics[END_REF]. Since it produces no thrust at zero flight speed, such an engine must be ignited after it has been accelerated to its takeover speed thanks to a secondary propulsion system. The issues associated to ignition and propulsion efficiency during this takeover stage are thus important in the scramjet development but they lie outside the scope of the present study, which is concerned with combustion stabilization, once significant flight Mach number values are reached.

As emphasized above, there are several possible applications of scramjet engines, including civil high-speed aircrafts, space exploration as reusable stages for access to Earth orbit, and military hypersonic weapons [START_REF] Urzay | Supersonic combustion in air-breathing propulsion systems for hypersonic flight[END_REF]. Despite many impressive achievements obtained in the field since the early sixties, still today there are no scramjet engines being used in practical devices: many developments including experimental flights and ground-based testings are currently in progress. In such engines, the combustion chamber is crossed by a supersonic flow, which limits considerably the time available to inject fuel, mix it with the oxidizer, ignite the resulting mixture, and stabilize combustion. Combustion ignition and stabilization indeed appears as one of the most challenging issues [START_REF] Cai | Ignition processes and modes excited by laser-induced plasma in a cavity-based supersonic combustor[END_REF]. One promising solution is the wall-mounted cavity, which has been shown to be quite efficient in stabilizing combustion without excessive total pressure loss [START_REF] Ben-Yakar | Cavity flame-holders for ignition and flame stabilization in scramjets: an overview[END_REF][START_REF] Mathur | Supersonic combustion experiments with a cavity-based fuel injector[END_REF][START_REF] Gruber | Mixing and combustion studies using cavity-based flameholders in a supersonic flow[END_REF]. The recirculation zones that develop inside the cavity increase the residence time of intermediate combustion products that can continuously initiate chemical reactions.

However, the understanding of the interaction of the fuel jet with the wall-mounted cavity is far from being an easy task [START_REF] Cai | Large-eddy simulation of the fuel transport and mixing process in a scramjet combustor with rearwall-expansion cavity[END_REF][START_REF] Gong | Large-eddy simulation of hydrogen combustion in supersonic flows using an eulerian stochastic fields method[END_REF] and it has motivated a large amount of experimental studies in both reactive and non-reactive conditions. For instance, [START_REF] Ben-Yakar | Cavity flame-holders for ignition and flame stabilization in scramjets: an overview[END_REF] investigated hydrogen normal injection in air cross-flow upstream of a cavity simulating Mach 10 flight conditions. High-speed Schlieren visualization and planar laser-induced fluorescence (OH-PLIF) were used to characterize the compressible flow topology and combustion development. The first OH fluorescence events are found to take place in the recirculation zone upstream of the underexpanded jet and the signal of fluorescence extends further downstream along the outer edge of the jet. [START_REF] Micka | Dual-mode combustion of a jet in cross-flow with cavity flameholder[END_REF] studied the combustion characteristics of a dual-mode scramjet combustor with normal fuel injection upstream of a cavity flame-holder. Depending on the value of the vitiated airstream stagnation temperature, combustion is either anchored at the cavity leading edge and spreads into the main flow at an approximately constant angle (low stagnation temperature levels) or stabilizes a short distance downstream of the hydrogen injection, in its wake, and features a curved leading edge (high stagnation temperature levels). The combustion stabilization is analysed on the basis of premixed flame propagation and the possibility that it flashes forward to a relatively stable location in the hydrogen jet-wake. [START_REF] Sun | Flame characteristics in supersonic combustor with hydrogen injection upstream of cavity flameholder[END_REF] also analysed combustion in a supersonic combustor with normal hydrogen injection upstream of cavity flame-holders. Their investigation is based on the combination of experimental observations using OH-PLIF and numerical simulations performed within a hybrid RANS/LES framework.

It is shown that an approximately steady flame may be maintained in the cavity shear layer.

Hot combustion products can be transported towards the jet inlet stream through a process of interaction between the counter-rotating vortices issued from the jet and the cavity shear layer.

In the present study, it is attempted to perform high-fidelity numerical simulations of a scramjet model representative of experimental investigations previously conducted at the University of Michigan [START_REF] Micka | Combustion characteristics of a dual-mode scramjet combustor with cavity flameholder[END_REF][START_REF] Micka | Combustion stabilization, structure, and spreading in a laboratory dual-mode scramjet combustor[END_REF]. In the experiments under consideration, a sonic jet of hydrogen is injected into a supersonic crossflow of vitiated air upstream of a wall-mounted cavity. The computations are performed with the highperformance computational solver CREAMS [START_REF] Martínez-Ferrer | A detailed verification procedure for compressible reactive multicomponent Navier-Stokes solvers[END_REF], 2017a) that has been developed to perform the numerical simulation of compressible reactive multi-component flows on massively-parallel architectures. The solver makes use of high-order precision numerical schemes applied on structured meshes. The simplified computational geometry consists of a constant section channel, followed by the wall-mounted cavity section, and finally the diverging section. This geometry is handled thanks to the recent immersed boundary method (IBM) introduced by [START_REF] Boukharfane | A combined ghost-pointforcing/direct-forcing immersed boundary method (IBM) for compressible flow simulations[END_REF]. The corresponding set of computations is expected to be useful to analyse the reactive flow topology and structure as well as combustion regimes, which can be investigated on the basis of standard turbulent combustion diagrams.

Combustion stabilization is studied for two distinct values of the inlet vitiated air-stream temperature, which are denoted RFSC-LST and RFSC-HST, for low-and high-stagnation temperature levels, respectively. A preliminary inspection of numerical data confirms that two stabilization modes can be observed: cavity stabilization mode for case RFSC-LST and jet-wake stabilization mode for case RFSC-HST.

The present manuscript is organized as follows, in section 2, the large-eddy simulation model and numerical methods are presented. It is followed by a short section where the computational setup is set forth. This section includes the presentation of the computa-tional domain, its discretization, and associated boundary conditions. The next part of the manuscript, section 4, provides a detailed analysis of computational resolution issues. Finally, some results of both non-reactive and reactive flow simulations are discussed in section 5. The manuscript ends with a brief conclusion section 6, where some perspectives for future works are also presented.

Governing equations and numerical methods

The present study is conducted within the large-eddy simulation (LES) framework, i.e., the largest scales of the flow are explicitly computed, while the subgrid-scale (SGS) effects, resulting from the smallest ones, are modelled. Thus, the following set of filtered transport equations, written in a conservative form, is used to describe the multicomponent reactive flowfield

∂ρ ∂t + ∂ρ u i ∂x i = 0 (1) ∂ ρ u j ∂t + ∂ ρ u i u j ∂x i = - ∂ p ∂x j + ∂ τ ij ∂x i - ∂ ∂x i ρu i u j -ρ u i u j (2) ∂ ρ e t ∂t + ∂ ρ u i e t ∂x i = - ∂ p u i ∂x i + ∂ u i τ ij ∂x j - ∂ J i ∂x i - ∂ ∂x i (ρe t + p)u i -(ρ e t + p) u i (3) ∂ ρ Y α ∂t + ∂ ρ u i Y α ∂x i = - ∂ ρY α V αi ∂x i - ∂ ∂x i ρY α u i -ρ Y α u i + ρ ωα (4)
where t denotes time, x i is the Cartesian coordinate in direction i (with i = 1, . . . , 3), u i is the velocity component in the same direction, ρ is the density, p is the pressure, e t = e + u i u i /2 is the total specific energy which is obtained as the sum of the kinetic energy and internal specific energy e, and finally Y α is the mass fraction of chemical species α (with α = 1, . . . , N sp ). The integer N sp denotes the number of chemical species. In the above equations, Φ is the filtered value of any quantity Φ, while Φ denotes its density-weighted or Favre filtered counterpart: Φ = ρΦ/ρ. The thermodynamic variables are interrelated through the filtered pressure field p = ρR T / W with R the universal gas constant, T the temperature, and

W -1 = Nsp α=1 Y α /W α
the molar weight of the multicomponent mixture (W α being the molar weight of species α). In Eq. ( 4), the quantities V αi and ωα denote the diffusion velocity and chemical production rate of species α, respectively. The filtered stress tensor is evaluated from

τ ij = 2 µ( S ij -S kk δ ij /3) with S ij = ∂ u i /∂x j + ∂ u i /∂x j /2
denoting the resolved strain-rate tensor. It is noteworthy that the retained system of filtered equations is similar to the one previously considered by [START_REF] Ragab | An investigation of finite-difference methods for large-eddy simulation of a mixing layer[END_REF], [START_REF] Piomelli | Large-eddy simulation: achievements and challenges[END_REF], [START_REF] Kosovic | Subgrid-scale modeling for large-eddy simulations of compressible turbulence[END_REF][START_REF] Dubois | The subgrid-scale estimation model applied to large eddy simulations of compressible turbulence[END_REF].

For the present set of numerical simulations, standard modelling assumptions are retained. A mixture-average formulation is used to describe multicomponent molecular diffusion [START_REF] Hirschfelder | The theory of flame propagation[END_REF]. Moreover, it is assumed that the filtered molecular diffusion fluxes and filtered heat flux can be deduced from their instantaneous expressions, but applied to filtered quantities. The components of the filtered molecular diffusion flux of chemical species α are thus approximated from

ρY α V αi = -ρ D m α W α W ∂ X α ∂x i + ρ Y α V c i (5)
where X α denotes the molar fraction of species α and D m α is the matrix of the diffusion flux coefficients of the same chemical species. It is worth noting that the last term of Eq. ( 5) is a correction term used to enforce total mass conservation. Indeed, at each time step, each component of the correction velocity

V c i = Nsp β=1 D m β (W β / W)∂ X β /∂x i
is evaluated and added to the filtered velocity component u i so as to enforce compatibility between the discrete forms of total mass and species mass fractions conservation equations.

The i-component of the filtered molecular heat flux is expressed as follows:

J i = -λ ∂ T ∂x i + Nsp α=1 ρ Y α V αi h α ( 6
)
where λ is the thermal conductivity of the multicomponent mixture as evaluated from the filtered composition.

The subgrid-scale (SGS) stress tensor

T ij = ρu i u j -ρ u i u j is modelled within the Boussi- nesq's framework. Its deviatoric part is thus evaluated from T ij -T kk δ ij /3 = -2µ SGS ( S ij - S kk δ ij /3
), where µ SGS = ρ ν SGS is the SGS eddy viscosity and T kk denotes the isotropic contribution. The closure of the SGS mass fluxes relies on the standard turbulent diffusivity assumption, which may be expressed in the following general form:

T ϕ,i = ρϕu i -ρ ϕ u i = ρ ϕu i -ϕ u i = -ρD SGS ∂ ϕ ∂x i (7)
where ϕ denotes any scalar quantity, D SGS = ν SGS /Sc SGS is the turbulent diffusivity and Sc SGS corresponds to the SGS turbulent Schmidt number, the value of which is set to 0.7. Finally, the last term of the filtered energy equation, see Eq. (3), may be rewritten as follows:

(ρe t + p)u j -(ρ e t + p) u j = ρc p T u j -ρ c p T u j Q j + 1 2 ρu i u i u j -ρ u i u i u j ψ iij - 1 2 T ii u j ( 8
)
where Q j = -λ SGS ∂ T /∂x j is nothing but the SGS heat flux also modelled using the gradientdiffusion approximation, with λ SGS = µ SGS c p /Pr SGS the turbulent thermal conductivity and Pr SGS the SGS turbulent Prandtl number. The value of Pr SGS is set to 0.7. In Eq. ( 8), the quantity ψ ijk denotes the triple velocity correlation tensor, which is closed by retaining the [START_REF] Daly | Transport equations in turbulence[END_REF]. At this level, it must be precised that the SGS eddy viscosity µ SGS is evaluated from the wall-adapting local eddy (WALE) model of [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF]:

Daly and Harlow expression

ψ ijk = C c3 ν SGS ∂T ij /∂x k with C c3 = 0.08
µ SGS = ρ(C w ∆) 2 (S d ij S d ij ) 3/2 ( S ij S ij ) 5/2 + (S d ij S d ij ) 5/4 (9)
where C w = C s √ 10.6 is the WALE model constant, ∆ = (∆x 1 ∆x 2 ∆x 3 ) 1/3 is the characteristic mesh size, and S d ij is the traceless symmetric part of the square of the resolved velocity gradient tensor:

S d ij = 1 2 ∂ u i ∂x l ∂ u l ∂x j + ∂ u j ∂x l ∂ u l ∂x i - 1 3 ∂ u m ∂x l ∂ u l ∂x m δ ij (10)
The following set of computations have been performed with C s set to its standard value C s = 0.18. In this respect, it should be emphasized that the present closure of the filtered Navier-Stokes equations gathers the same modelling ingredients as those previously retained by [START_REF] Techer | Mixing of fuel jet in supersonic crossflow: estimation of subgrid-scale scalar fluctuations[END_REF]. Finally, the filtered chemical rates ωα that appear in the right-hand-side of the filtered species mass fraction transport equations are represented using either the perfectly stirred reactor (PSR) or the unsteady partially stirred reactor (U-PaSR) closure [START_REF] Berglund | Finite rate chemistry large-eddy simulation of self-ignition in supersonic combustion ramjet[END_REF]Moule et al., 2014a Computations are performed with the computational fluid dynamics (CFD) solver CREAMS.

The reader is referred to [START_REF] Martínez-Ferrer | A detailed verification procedure for compressible reactive multicomponent Navier-Stokes solvers[END_REF] and to [START_REF] Boukharfane | A combined ghost-pointforcing/direct-forcing immersed boundary method (IBM) for compressible flow simulations[END_REF] for an exhaustive presentation of this solver including its application to a large number of verification test cases. Its main features are as follows: the treatment of the inviscid component of the transport equations relies on a seventh-order accurate weighted essentially non-oscillatory (WENO7) reconstruction of the characteristic fluxes. In practice, the numerical solver uses a high-order accuracy finite difference scheme, and the application of the nonlinear weighting procedure of the WENO7 scheme is conditioned to a shock sensor that involves the local values of the normalized spatial variations of both pressure and density [START_REF] Buttay | Analysis of small-scale scalar mixing processes in highly under-expanded jets[END_REF]. The viscous and molecular diffusion fluxes are evaluated thanks to an eighth-order centered finite difference scheme. The temporal integration is performed using a third-order Runge-Kutta algorithm. Finally, the present set of computations makes use of a recent immersed boundary method (IBM) to describe the wall-mounted cavity. The corresponding IBM strategy combines direct-forcing (DF) and ghost-point-forcing (GPF) algorithms [START_REF] Boukharfane | A combined ghost-pointforcing/direct-forcing immersed boundary method (IBM) for compressible flow simulations[END_REF].

Computational setup

Figure 1 displays the supersonic combustion facility studied at the University of Michigan.

It consists of a two-dimensional Mach 2.2 nozzle, a constant area isolator that extends over 400.0 mm up to the leading edge of a rectangular cavity, which is followed by a 349.0 mm long and 4 degree diverging section dumping into a 152.0 mm diameter exhaust. The cavity spans the entire width of the test section, which is equal to 38.1 mm. It is 50.8 mm long and 12.7 mm high, thus featuring a rather small length to depth ratio (l 2 /h 2 ≤ 10.0) in such a manner that it can be considered as an open cavity [START_REF] Lawson | Review of numerical simulations for high-speed, turbulent cavity flows[END_REF]. Upstream of this wall-mounted cavity, a sonic hydrogen jet is injected into the vitiated air supersonic crossflow .

Figure 1: Schematics of the combustion facility with the computational domain highlighted

The computational domain, highlighted in Fig. 1, is restricted to the consideration of the transverse jet and cavity flowfield. It can be decomposed into three distinct parts: a constant section channel s 1 with length L x 1 ,s 1 = l 0 + l 1 = 94.5 mm and height L x 2 ,s 1 = h 1 = 39.4 mm, a section s 2 with length L x 1 ,s 2 = l 2 = 50.8 mm featuring the wall-mounted cavity with depth h 2 = 12.7 mm, and a four-degree diverging section s 3 with length L x 1 ,s 3 = l 3 = 76.2 mm. The total length of the computational domain is thus L x 1 = 221.5 mm and it is L x 3 = 38.1 mm in the spanwise direction. Section s 1 features the hydrogen injection port with diameter D 1 = 2.5 mm located along the combustor centerline at 44.5 mm upstream of the cavity leading edge.

The corresponding computational grid features approximately 42, 000, 000 points and it is refined at some locations including the vicinity of walls, the hydrogen jet exit, and the shear layer that develops above the cavity. The characteristic dimensions of the resulting computational grid elements remain bounded between 15 µm and 275 µm, see Table 1.

∆x1 (m) ∆x2 (m) ∆x3 (m)
Min value 5.0 • 10 -5 1.5 • 10 -5 5.0 A normalized mesh concentration parameter ∆x * i is introduced to follow the spatial discretization evolution with respect to its smallest and largest characteristic dimensions along each of the three directions. It is evaluated from ∆x

* i = (∆x i,max -∆x i )/(∆x i,max -∆x i,min ),
where ∆x * i is the mesh concentration parameter, ∆x i the current (i.e., local) computational grid size, ∆x i,min its smallest value, and ∆x i,max its largest value. Thus, the value of ∆x * i ranges between zero, for small concentration of computational nodes (i.e., large characteristic mesh dimension), and unity, for high concentration of computational nodes (i.e., small characteristic mesh dimension). Figure 2 reports the normalized mesh concentration ∆x * 1 , ∆x * 2 , and ∆x * 3 along the three spatial directions. It is noteworthy that the mesh origin has been set at the center of the fuel injection exit diameter. The bottom wall, including the wall-mounted cavity geometry, is modelled using the IBM. Except for the fuel inlet boundary condition (BC), which is set as a supersonic inlet BC, the corresponding boundary condition is processed as a solid (i.e., impermeable) adiabatic wall with the energy and scalar zero-flux Neumann boundary conditions imposed using the GPF framework, and the zero-velocity Dirichlet boundary condition (i.e., no-slip wall) imposed through the combined GPF-DF framework, see [START_REF] Boukharfane | A combined ghost-pointforcing/direct-forcing immersed boundary method (IBM) for compressible flow simulations[END_REF] for further details. On the left side of the computational domain, the vitiated air inlet boundary condition is set as a supersonic inlet BC. The lateral and top boundaries are processed as slip walls. Finally, on the right side of the computational domain, the supersonic outlet is processed using extrapolation together with an additional sponge region combining both grid coarsening and explicit filtering, thus following the strategy previously retained by [START_REF] Buttay | Analysis of small-scale scalar mixing processes in highly under-expanded jets[END_REF][START_REF] Buttay | Highly resolved numerical simulation of combustion downstream of a rocket engine igniter[END_REF]. Further details about the fuel and vitiated air inlet conditions are provided below. With the objective of imposing vitiated air inlet temperature and velocity profiles as realistic as possible, preliminary simulations of the airflow in the isolator have been conducted

for both conditions considering M 0 = 2.2 and T 0 = 1100 K for case RFSC-LST, and M 0 = 2.2 and T 0 = 1400 K for case RFSC-HST. The pressure at the inlet of the computational domain is determined from an isentropic evolution, with the value of the heat capacity ratio γ being set to 1.313 and 1.287 for cases RFSC-LST and RFSC-HST, respectively, thus resulting in pressure values equal to 55.410 kPa and 55.426 kPa.

The unsteady viscous flows inside the isolator have been simulated with the same LES solver (CREAMS). As emphasized above, the objective of these preliminary computations is only to impose realistic mean profiles for both the temperature and velocity at the inlet of the main simulations. Taking advantage of the results issued from these preliminary computations of the isolator, the filtered temperature and velocity components are settled from the computed profiles of the averaged temperature T and velocity components { u i } obtained at the outlet of the isolator. However, it should be acknowledged that, in comparison with some previous studies, e.g., [START_REF] Kawai | Large-eddy simulation of jet mixing in supersonic crossflows[END_REF], no special effort has been spent to represent the unsteady (i.e., turbulent) features of the supersonic boundary layer at the inlet of the main simulation. In [START_REF] Kawai | Large-eddy simulation of jet mixing in supersonic crossflows[END_REF], a concurrent supersonic turbulent boundary layer (STBL) simulation has been conducted by making use of rescaling and reintroduction procedures. The inflow conditions for the main JISCF simulation were extracted from a plane of the concurrent STBL simulation. Other strategies based on the use of synthetic inlet turbulence generators could have been also retained to model the present fluctuating inflow conditions but, previous analyses, see for instance [START_REF] Vedovoto | Influence of synthetic inlet turbulence on the prediction of low mach number flows[END_REF], showed that, even with properly set mean velocity and turbulence kinetic energy, the resulting flowfields still remain very sensitive to the arbitrary choice of the synthetic turbulence model. Therefore, for the purpose of the present study, which is focused on other specific issues, a simpler solution has been chosen with no fluctuations imposed at the inlet boundary.

The fuel injection boundary layer is taken into account by setting the following velocity

profile u(r, t) = f (t) • [u w -(U 1 -u w ) erf ((r -r 1 )/(2.0 α r 1
))] with α = 0.05, where U 1 denotes the hydrogen inlet bulk flow velocity, u w is the flow velocity in the direct vicinity of the wall, r is the distance to the center of the fuel injection, and r 1 = D 1 /2.0 = 1.25 mm is the radius of the fuel injection port. The function f (t) = 1.0exp (-5.0 t/t 1 ) has been used to avoid numerical instabilities that may arise when the mass flow rate is applied directly. In this expression, t 1 = 0.2 µs is the characteristic time required for the injection to be fully established.

Time is made non-dimensional by considering its product with the ratio of the fuel injection port diameter D 1 to the fuel inlet bulk flow velocity U 1 . The simulations are started at

t * = U 1 t/D 1 = 0.

Mesh reliability analysis

The retained numerical schemes together with the choice of SGS modelling closures, see section 2, are well-known to be essential ingredients of any LES computations [START_REF] Garnier | Large eddy simulation for compressible flows[END_REF]. They do not, however, constitute the sole issues that must be properly addressed: it is indeed also worth recalling that the computational resolution is itself of the utmost importance. For instance, in wall-bounded flows such as those considered herein, the integral scale, away from the walls, is proportional to the turbulent boundary-layer thickness and the resolution requirement is determined by the range of scales contributing to the desired statistics. Mostly, the turbulent kinetic energy production results from high-speed streaks, which remain confined within a very small region in the direct vicinity of the wall [START_REF] Kline | The structure of turbulent boundary layers[END_REF]. The corresponding flow structures are extremely small when compared to the overall flow dimensions. These small structures, however, play an essential role in the turbulent boundary layer dynamics and therefore need to be well resolved. The design of the computational mesh and associated levels of resolution thus result from a compromise between turbulent dynamics reproduction and CPU costs. Based on the non-reactive computed flowfields, the present section will provide a detailed mesh reliability analysis. In this respect, it must be acknowledged that obtaining a strict proof of computational resolution adequacy remains quite challenging and instead, as documented in the present section, one can gather various clues that all bring support to the mesh quality assessment analysis. The conclusion drawn from this mesh reliability analysis will be confirmed by the subsequent inspection of the computed reactive flows.

DNS wall-resolved LES ∆x + 10 -15 50 -150 ∆y + 1 < 2 ∆z + 5 10 -40
Table 3: Recommended computational mesh size in wall units [START_REF] Piomelli | Large-eddy simulations: theory and applications[END_REF][START_REF] Pope | Turbulent flows[END_REF][START_REF] Piomelli | Wall-layer models for large-eddy simulations[END_REF] There exist many criteria to asses the mesh resolution quality. Some of them are based on mathematical estimates, while others rely on rather heuristic arguments. For instance, Table 3 presents recommended orders of magnitude of computational mesh cell sizes for both DNS and wall-resolved LES computations. In this table, the values of ∆x + , ∆y + , and ∆z + denote the dimensions of the computational cells present at the wall location expressed in wall units, i.e., ∆x + = ∆x 1 /l w , ∆y + = ∆x 2 /l w , and ∆z + = ∆x 3 /l w . We recall that the wall characteristic length scale l w is given by l w = µ w / √ ρ w τ w , where τ w = µ w ∂ u 1 /∂x 2 w denotes the stress at the wall with u 1 the statistically-averaged value of the longitudinal velocity component [START_REF] Lesieur | Large-eddy simulations of turbulence[END_REF]. It is noteworthy that this value is presently estimated from the average of the filtered velocity component. The mesh resolution quality in the vicinity of the walls is then assessed through a more detailed analysis of the averaged boundary layer flow. To this purpose, we consider again normalized quantities. In this respect, the longitudinal component of the non-dimensional mean velocity is defined by u 1 /u τw with u τw = τ w /ρ w the wall friction velocity. Figure 5 displays the profiles, in wall units, of the normalized mean velocity u + together with the logarithm of the viscosity ratio, i.e., log µ SGS / µ , plotted versus the distance to the wall at several locations x 1 /D 1 . Here, the mean values have been obtained from an average performed over instantaneous snapshots of the median plane (i.e., x 3 /D 1 = 0.0). Despite the visible shift of the velocity in the logarithmic zone, these profiles confirm that the WALE model satisfactorily changes its behavior in the buffer layer that separates the logarithmic zone (y + > 30.0) from the viscous sublayer (y + < 5.0). Moreover, once plotted in logarithmic coordinates, the viscosity ratio increases almost linearly, which is consistent with its expected behavior [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF][START_REF] Techer | Mixing of fuel jet in supersonic crossflow: estimation of subgrid-scale scalar fluctuations[END_REF].

The assessment of the computational resolution is completed through a detailed mesh quality analysis. In this purpose, two quality indexes are considered. The corresponding quantities vary from zero (i.e., poor mesh resolution) to unity (i.e., good mesh resolution):

the higher the value of the index, the better the resolution. First, we consider the modified quality index of [START_REF] Pope | Ten questions concerning the large-eddy simulation of turbulent flows[END_REF], which is defined as follows: 0.069 [START_REF] Yoshizawa | Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling[END_REF]. According to [START_REF] Pope | Ten questions concerning the large-eddy simulation of turbulent flows[END_REF], the mesh resolution could be considered as sufficient, provided that the value IQ k exceeds 0.80, which means that at least eighty percent of the turbulent kinetic energy is captured at the resolved scale. However, this quality index may display some sensitivity to the SGS modelling and, therefore, it seems worthwhile to complete the analysis with other estimates. To this purpose, the quality index of Celik et al. ( 2005) is also considered. It is based on a comparison between the computational grid characteristic size and the Kolmogorov length scale, as given by IQ η = 1 + α η (∆/ η ) m -1 , with α η , m = (0.05, 0.5) and η evaluated from the following scaling rule η = ν 3 /ε 1/4 with ε being the turbulent kinetic energy dissipation rate. In this respect, it seems worth recalling that a standard criterion for DNS computation is k max • η = 3/2 in such a manner that, approximating k max from π/∆ [START_REF] Pope | Turbulent flows[END_REF], it leads to a value of ∆/ η approximately equal to 2.0. Once introduced in the expression of the quality index IQ η , this gives IQ η = 0.93.

IQ k = k/(k + k SGS ),
This means that, in regions characterized by IQ η ≥ 0.93, the computational resolution can be considered as equivalent to the fulfillment of a standard DNS resolution criterion.

F (x) = x 0 PDF (IQ) dIQ (11)
and its complement to unity [START_REF] Celik | Index of resolution quality for large eddy simulations[END_REF] Figure 7: Value of 1-F(IQ), i.e., value of the probability to have a quality index larger than IQ. The probability to reach a level of the quality index larger than 0.80 for IQ k and larger than 0.93 for IQ η are also delineated.

1 -F (x) = 1 - x 0 PDF (IQ) dIQ = 1 x PDF (
The second expression, i.e., the one provided by Eq. ( 12), measures the probability of having a value of the quality index IQ larger than x. above is that the present set of computations indeed displays a high level of resolution. As summarized in section 2, this high level of resolution is combined with the use of high-precision numerical schemes and subgrid-scale models that satisfactorily recover the DNS limit, thus resulting in solid bases to perform high-fidelity numerical simulations. Finally, it must be underlined that the targeted objective is the simulation of the reactive flow stabilization and we are not really interested by the description of the flow inside the exit nozzle. Therefore, the mesh has been considerably coarsened in this region of the computational domain and no special effort has been spent to describe the flow in this part of the combustor. As a consequence, there are some pressure disturbances that appear at the nozzle wall at the level of each cell transition. The birth of such perturbations in the exit nozzle is visible on numerical schlieren images and has been already reported in the literature when coarse meshes are considered.

Analysis of computational results

Statistical convergence of the computational datasets

Before proceeding with a detailed inspection of the computational results, we proceed with a statistical convergence analysis. Therefore, in an attempt to verify the convergence of the second-order moments of the velocity field, eight probes are placed in the median plane of the computational domain (i.e., x 3 /D 1 = 0.0), as depicted in Fig. 9. Table 4 summarizes the various spatial filtering and temporal averaging operators. At this level, it should be emphasized that, for the present application to LES data, the temporal operator is applied to filtered quantities. For instance, the temporal average of the density is evaluated from the resolved density field using ρ . Recalling that the second-order centered moment of any quantity Φ is the expected value of the squared deviation from its mean, i.e., σ 2 Φ = Φ -Φ 2 = Φ 2 -Φ 2 , the temporal convergence of the second-order moments of the filtered velocity field u i is checked by evaluating the following expression:

R u i = u i u i -u i u i -u i -u i 2 (13)
The above quantity, i.e., the residual R u i , should be zero provided that u i u i = u i u i and u i u i = u i u i , which provides an excellent verification of the convergence of the temporal averaging operator.

Figure 10 displays the statistical convergence of the longitudinal and transverse components of the velocity field for cases RFSC-LST and RFSC-HST as obtained on the various probes defined in Fig. 9. From these results, one can notice that the solution convergence is obtained for t * ≥ 800.0 and t * ≥ 900.0 for cases RFSC-LST and RFSC-HST, respectively.

Therefore, the statistical post-processing of the computational results will be performed over snapshots obtained from these values up to the end of the simulation. The mixing process mainly takes place downstream of the barrel shock because there is a very high velocity gradient between the hydrogen jet and the crossflow. This triggers the birth of large-scale coherent structures induced by the Kelvin-Helmholtz (K-H) instabilities. Further downstream, the scale reduction processes come into play, thus leading to the breakup of these The corresponding fuel mixing degree is indeed a critical parameter to evaluate the supersonic combustor performance as a whole. In this respect, there exist many possible definitions of the mixing efficiency in the literature, see for instance [START_REF] Moule | Computational fluid dynamics investigation of a Mach 12 scramjet engine[END_REF] and [START_REF] Liu | Dynamics and mixing mechanism of transverse jet injection into a supersonic combustor with cavity flameholder[END_REF]. This quantity, i.e., the fuel mixing degree, can be defined as the mass flow rate ratio of reactants that would react to the total mass flow rate of reactants [START_REF] Liu | Dynamics and mixing mechanism of transverse jet injection into a supersonic combustor with cavity flameholder[END_REF]:

η m (x 1 ) = A(x 1 ) ρ u 1 Y H2,r dA / A(x 1 ) ρ u 1 Y H2 dA (14)
where, following [START_REF] Liu | Dynamics and mixing mechanism of transverse jet injection into a supersonic combustor with cavity flameholder[END_REF], the mass fraction of reactants that would be involved in the reaction Y H2,r is evaluated by assuming a complete oxidation of the available amount of hydrogen:

Y H2,r =    Y H2 Y H2 ≤ Y H2,st Y H2,st (1 -Y H2 )/(1 -Y H2,st ) Y H2 ≥ Y H2,st (15)
with Y H2,st the fuel mass fraction at stoichiometry. In Eq. ( 14), the elementary transverse surface element dA is evaluated from the product ∆x 2 • ∆x 3 and, at each location x 1 , the integral is performed over the whole transverse section A(x 1 ).

Figure 13 displays the mixing efficiency profile obtained along the streamwise direction, zooming at the fuel injection and wall-mounted cavity, i.e., in a region delineated by -4.0 ≤

x 1 /D 1 ≤ 38.0. The obtained results display some similarities with those previously reported by [START_REF] Liu | Dynamics and mixing mechanism of transverse jet injection into a supersonic combustor with cavity flameholder[END_REF] who retained the same definition of the mixing efficiency. Thus, the mixing efficiency is found to be close to 100% in the direct vicinity of the fuel injection, where only a small amount of fuel diffused along the turbulent boundary layer. A minimum is reached ahead of the fuel jet injection because the mixing is dominated by the large scale vortices in the jet shear layer. Then, the mixing efficiency increases faster along the cavity, which means that mixing processes are enhanced, reaching finally almost 100% at the end of the cavity.

The present set of non-reactive results is now used to perform an analysis of ignition probabilities. Mixtures of fuel and oxidizer are indeed considered to be flammable only if a premixed flame is able to propagate into the corresponding fresh reactants, which happens only within given composition limits. The flammability limits of a reactive mixture determine the lower and upper fuel concentration so that combustion can proceed. These values are obtained from well-defined experimental procedures by varying the mixture composition and they may be altered not only by temperature and pressure but also by flame propagation direction under a gravitational field [START_REF] Zabetakis | Flammability characteristics of combustible gases and vapors[END_REF]. The lowest fuel concentration that may lead to flame propagation is hereafter denoted as the lower flammability limit (LFL), while the largest fuel concentration is called the upper flammability limit (UFL). Both LFL and UFL are expressed in terms of volume percentages in normal conditions, i.e., at 298 K and 1.0 atm. For instance, the LFL and UFL values for mixtures of hydrogen with air are 4% and 75% [START_REF] Zabetakis | Flammability characteristics of combustible gases and vapors[END_REF], which correspond to mixture fraction values equal to 0.003 and 0.167, respectively. At this level, it should be emphasized that there exist several possible ways to evaluate the mixture fraction. Discarding Lewis number effects, i.e., assuming the mixture fraction molecular diffusion coefficient equal to thermal diffusivity, this quantity may be thought as a fuel inlet tracer [START_REF] Gomet | A multiple-inlet mixture fraction model for nonpremixed combustion[END_REF], the value ξ f of which can be evaluated from a filtered passive scalar transport equation with boundary conditions set to zero ( ξ f = 0.0) at the oxidizer inlet and unity ( ξ f = 1.0) at the fuel inlet. The mixture fraction can also be deduced from atoms conservation. For instance, the nitrogen mass fraction Y N2 , which has been considered as a non-reactive scalar in the present set of computations3 , can be used to this purpose:

ξ N2 = 1.0 -( Y N2 /Y max N2 ) with Y max N2
the nitrogen concentration in the vitiated airstream inlet. Finally, it is also possible to evaluate the mixture fraction on the basis of the chemical reaction stoichiometry [START_REF] Peters | Turbulent Combustion[END_REF]:

ξ = φ Y H2 /Y ∞ H2 -Y O2 /Y ∞ O2 + 1 1 + φ (16)
where Y ∞ H2 is the hydrogen mass fraction issued from the fuel inlet (unity in the present case of pure fuel injection), Y ∞ O2 is the oxygen mass fraction in the vitiated airstream, and

φ = r st Y ∞ H2 /Y ∞ O2
denotes the equivalence ratio with r st the stoichiometric mass ratio. These various definitions are compared in Fig. 14, which reports the averaged profiles based on the use of each definition. For the present conditions, the obtained differences appear to be so slight that they can be neglected.

From the resolved mixture fraction PDF, the ignition probability P ign may be evaluated as follows:

P ign = 0.167 0.003 P ξ d ξ (17)
The field of the corresponding ignition probability P ign , obtained from the averaging of data obtained in the median plane (x 3 /D 1 = 0.0), is reported in Fig. 15 for cases RFSC-LST and RFSC-HST. This view is restricted to a zoom for a close-up of the fuel injection and cavity regions. One can verify that, even with the present set of values of the flammability limits, which are associated to normal conditions (i.e., moderate temperature levels in the fresh reactants), the highest levels of the ignition probability P ign follow quite closely the jet wake and the shear layer that develops above the cavity. It is interesting to notice that, in case RFSC-HST, there is a larger region featuring high values of P ign in the vicinity of the wall downstream of the hydrogen injection. As it will be shown later on, these locations correspond to those where the jet-wake and cavity stabilization modes develop. However, it should be fairly acknowledged that the consideration of the operating temperature level of the vitiated airstream T 0 , instead of the temperature 298 K associated to normal conditions, may significantly alter the results and more insights should be gained from the reactive flow simulations presented in the next section.

Reactive flow analysis

According to the studies of [START_REF] Micka | Dual-mode combustion of a jet in cross-flow with cavity flameholder[END_REF], for the range of conditions reported in Micka's Thesis manuscript [START_REF] Micka | Combustion stabilization, structure, and spreading in a laboratory dual-mode scramjet combustor[END_REF], the vitiated air-stream temperature T 0 plays an important role in determining the combustion stabilization mode. For inlet temperature such that T 0 > 1350 K, combustion stabilization occurs through a jet-wake mode, whereas for T 0 < 1150 K, a cavity-stabilized regime is expected. In the intermediate range (1150 K < T 0 < 1350 K) combustion oscillates between the two stabilization regimes. Equation ( 18) may be used to approximate the fraction of time f the combustion takes place in jet-wake stabilized mode [START_REF] Micka | Dual-mode combustion of a jet in cross-flow with cavity flameholder[END_REF]:

f = 1 2 + |T 0 -1250.0| 2 erf T 0 -1250.0 75.0 (18)
where the value of the vitiated airstream temperature T 0 must be provided in Kelvin S.I. unit.

The corresponding function is displayed in Fig. 16 together with the two conditions studied in the present work.

From a practical viewpoint, the flame will remain in the cavity-stabilized mode for moderate values of the temperature T 0 except if a large enough fluctuation flashes it forward to a relatively stable location in the jet-wake stabilization region. If this occurs, the flame will then remain in the jet-wake stabilized mode until it becomes unstable due to another fluctuation, forcing it to flash back to a cavity-stabilized regime. The magnitude of the fluctuations required for the flame to flash back and forth becomes smaller as the temperature T 0 rises.

Moreover, as T 0 is increased, the magnitude of the fluctuations that may cause the flame to flash back increases also. Thus, with the rise of the temperature T 0 , combustion spends more time in the jet-wake stabilized mode, until it reaches a sufficiently high level of temperature where it cannot be destabilized by any fluctuation. In the experiments, for obvious safety reasons, it is necessary to ensure that the ignition of the mixture will be successful and a spark plug located in the center of the cavity bottom wall is used. In the computations, ignition is simply obtained from the temperature increase on the rear wall of the cavity. As far as the use of adiabatic wall condition is concerned, it seems worth to emphasize that, according to the information provided by [START_REF] Micka | Combustion stabilization, structure, and spreading in a laboratory dual-mode scramjet combustor[END_REF], after the end of the main fuel injection (end of a given run), all flow streams except the main airstream were terminated and the electric heater was turned off. The main cold stream of air continued to flow through the combustor between two consecutive runs to provide cooling before the next test. According to [START_REF] Micka | Combustion stabilization, structure, and spreading in a laboratory dual-mode scramjet combustor[END_REF], eight combustion tests can be performed in a day. In this manner, the possible influence of experimental condition variations between the runs remains limited. Finally, it is noteworthy that the same adiabatic condition has been retained in the other (seldom) computational studies of this experimental setup, see for instance [START_REF] Zettervall | A computational study of ramjet, scramjet and dualmode ramjet combustion in combustor with a cavity flameholder[END_REF].

In a first step of the reactive flow analysis, a comparison is performed between computational results obtained using either the PSR or the U-PaSR subgrid-scale combustion models.

Such a comparison may be useful to further assess the level of computational resolution. The PSR model corresponds to the quasi-laminar approximation and consists in neglecting the possible influence of subgrid-scale fluctuations at the resolved level: the filtered reaction rates are evaluated directly from the detailed chemical scheme applied to the filtered composition.

The PSR, i.e., well-stirred reactor (WSR), and PaSR representations are standardly retained as elementary building blocks of supersonic combustion modelling [START_REF] Gomet | Influence of residence and scalar mixing time scales in non-premixed combustion in supersonic turbulent flows[END_REF][START_REF] Potturi | Large-eddy/Reynolds-averaged Navier-Stokes simulation of cavity-stabilized ethylene combustion[END_REF][START_REF] Fulton | Turbulence/chemistry interactions in a ramp-stabilized supersonic hydrogen-air diffusion flame[END_REF][START_REF] Candler | Wall-modeled large-eddy simulation of autoignition-dominated supersonic combustion[END_REF]. In the present study, the U-PaSR modelling framework has been also considered (Moule et al., 2014a). Like the eddy-dissipation concept (EDC) model [START_REF] Ertesvag | The eddy dissipation turbulence energy cascade model[END_REF], where chemical reactions take place, the field of γ * -not reported for the sake of conciseness -is indeed found to approach unity and it remains larger than 0.85 as a whole. The PSR or At this level, it is noteworthy that the two figures discussed above, i.e., Figs. 17 and 18, reflects the quality of the computational resolution with respect to chemical processes relevant to auto-ignition and flame propagation. This is confirmed by Figs. 19 which reports the viscosity ratio µ SGS / µ plotted versus the heat release rate (HRR). Figure 19 clearly shows that the most important part of the HRR takes place in well-resolved regions that correspond to values of µ SGS / µ smaller than 0.5. In these two scatterplots, more than 99% of the points are indeed associated to values of µ SGS / µ smaller than 0.5. Added to Figs. 17 and 18, and to the previous inspection of resolution criteria reported in section 4, it brings some additional support in favor of computational mesh adequacy so as to capture the dynamics of the turbulent flame. the temperature and inducing a deflection of the shear layer due to the thermal expansion induced by the heat release. The conditions then become more favorable to the spreading of chemical reactions as the combustion process develops, thus increasing the temperature and allowing the reaction zone to extend over the whole cavity, up to a point where the heat release rate becomes sufficient to stabilize the combustion process. Depending on the operating conditions, the combustion can spread upstream of the cavity, as it can be seen in case RFSC-HST. It is noteworthy that, as expected, the vitiated air-stream temperature significantly influences the combustion development. The combustion spreads significantly faster at a higher temperature. Significant H 2 O production takes place along the upper part of the cavity for case RFSC-LST (cavity stabilized mode), whereas for case RFSC-HST (jet-wake stabilized mode) combustion spreads downstream of the fuel injection and within the cavity.

Figure 21 reports the averaged heat release rate issued from the computations, presented in a way similar to the flame luminosity images of [START_REF] Micka | Dual-mode combustion of a jet in cross-flow with cavity flameholder[END_REF]. To favor the comparison with the images reported in the experimental work, the HRR has been integrated

along the spanwise direction. The computational results confirm that the cavity-stabilized combustion mode is characterized by a reaction zone anchored at the leading edge of the cavity, spreading into the main flow at an approximately constant angle. In the jet-wake stabilized mode the reaction zone is stabilized upstream of the cavity and the corresponding leading edge is curved. This is fully consistent with the experimental observations. Nevertheless, it should be emphasized that such a comparison remains purely qualitative since the relation that may exist between flame luminosity and HRR remains by far unknown. It is however quite interesting to see that the global change of behaviour that is observed between case RFSC-LST and case RFSC-HST is captured by the present set of computations. As underlined at the end of section 4, the mesh has been considerably coarsened downstream of the wall-mounted cavity and there are some pressure disturbances that appear at the nozzle wall. Their imprint is clearly visible on the right side of the HRR fields reported in Fig. 21. Whatever the leading mechanism of combustion stabilization, e.g., self-ignition or flame propagation phenomena, the temperature of the fresh mixture T u appears as a critical parameter. Figure 23 presents filtered temperature scatterplots issued from the non-reactive flow simulations. It is plotted against the filtered mixture fraction ξ together with a standard linear approximation (dashed line) and a polynomial approximation (continuous line). The linear approximation is expressed thanks to the following expression T u (ξ) = ξ T f uel + 1ξ T ox with T f uel the hydrogen inlet stream temperature and T ox the vitiated air inlet stream temperature, the values of which are provided in Table 1.

In fact, it must be emphasized that the use of a linear expression to express the temper- ature of the fresh mixture as a function of the mixture fraction is highly questionable. First, temperature is not an extensive quantity and, for such supersonic flow conditions, it would be more relevant to express the total enthalpy as a function of the mixture fraction [START_REF] Mura | Numerical simulation of supersonic nonpremixed turbulent combustion in a scramjet combustor model[END_REF]. As previously emphasized by [START_REF] Izard | A Lagrangian model of combustion in high-speed flows: application to scramjet conditions[END_REF], a linear relationship between total enthalpy and mixture fraction may hold if the Lewis and Prandt numbers are unity and the effects induced by temporal variations of pressure remains negligible. In addition to this, it seems worth recalling that, in supersonic flow regimes, compressibility effects may play an important role, in such a manner that the linear representation may no longer be adequate because of the presence of shock and expansion waves. In the present simulation, the fuel jet ( ξ = 1) strongly expands, causing a significant temperature decrease. The temperature then increases with the Mach disk compression, whereas the oxidizer stream ( ξ = 0) characteristics are altered through the bow shock compression and wall friction. Considering all these complex effects, which are difficult to include within a simple mathematical expression, it is proposed instead to fit the results of the non-reactive numerical simulation using a fourthorder polynomial approximation so as to express the temperature as a function of the mixture fraction. The corresponding evolution T u (ξ) is depicted in Fig. 23. Since chemical reactions take place far downstream of the fuel injection, it seems worthwhile to attempt to discriminate between premixed and non-premixed (i.e. diffusive) combustion contributions. A premixedness index ζp is therefore considered. It is defined as follows:

ζp = 1 2 (1 + n F • n O ) (19)
with n F and n O being normal unit vectors associated to the direction of the molecular diffusion fluxes of the fuel and oxidizer, respectively, which are evaluated from

n F = V F / V F and n O = V O / V O
where V F and V O denote the filtered diffusion velocity of the fuel and oxidizer, respectively. As it is defined, this index is expected to approach zero for diffusive combustion and unity for premixed combustion. It should be fairly recognized that its representativity can be questioned in certain situations but it remains widely used in preliminary inspections of partially-premixed combustion [START_REF] Martínez-Ferrer | Compressibility and heat release effects in high-speed reactive mixing layers: structure of the stabilization zone and modeling issues relevant to turbulent combustion in supersonic flows[END_REF]. Figure 24 displays the PDF of this premixedness index obtained in a volume restricted to 0.01 ≤ ξ ≤ 0.99 at several times for cases RFSC-LST and RFSC-HST4 . The peak associated to a premixedness index equal to zero is much higher than the peak associated to unity indicating mostly diffusive combustion. However, the contribution of the premixed mode to the HRR is more significant and corresponds to a premixed flame structure (in red) that develops in the vicinity of the cavity leading edge, as it can be seen in the top part of Fig. 24. Since a non-negligible amount of chemical reaction takes place in a premixed combustion mode, attention is now focused on this contribution. To this purpose, some preliminary computations are performed with the Cantera software, considering mixtures of fresh reactants at various composition with the temperature of the fresh mixture set according to the polynomial expression T u (ξ) discussed above. The chemistry description used to perform this set of computations is the same as the one retained to perform the whole set of simulations. From the obtained results, it is noteworthy that, as the mixture fraction is increased from the left to the right side of Fig. 25, the self-ignition delay first decreases and displays a local minimum for the most reactive conditions ξ mr and then increases, until it reaches very large values for rich mixtures. As expected, the most reactive state, which corresponds to a minimal value of the self-ignition delay, does not correspond to stoichiometry. This is a direct outcome of the vitiation of the oxidizer inlet stream, which features a temperature that is significantly larger than the one of the fuel inlet stream. It is also remarkable that the self-ignition delay displays a sharp increase for rich conditions, in such a manner that spontaneous ignition regimes are less likely for these mixtures. More precisely, the corresponding auto-ignition regimes require more time to develop, which may favor premixed flame propagation. We will see below that the characteristic time scales of premixed flame propagation indeed remain significantly smaller than the ignition delays for such rich mixtures. However, as long as lean conditions are considered, only the auto-ignition regimes are expected to occur. Turbulent premixed combustion regimes are now analyzed using the coordinates introduced by Barrère and Borghi [START_REF] Borghi | Sur la structure des flammes turbulentes[END_REF][START_REF] Borghi | On the structure and morphology of turbulent premixed flames[END_REF]. The data are thus plotted in Fig. 28 using a normalized length scale ratio t /δ 0 L and velocity ratio u rms /S 0 L as the set of coordinates. In these expressions, the turbulence integral length scale is approximated from t = u 3 rms /ε with ε the resolved molecular dissipation rate, which has been evaluated as the trace of the second rank resolved molecular dissipation rate tensor, i.e.,

ε ij = 1 ρ τ ik -τ ik ∂ ∂x k ρ u j -ρ u j ρ + τ jk -τ jk ∂ ∂x k ρ u i -ρ u i ρ (20)
The velocity fluctuation RMS is approximated from u rms = 2k/3 with k the resolved turbulent kinetic energy, which is evaluated from k = ρ u i u iρ u i u i /(2 ρ ). The flame characteristics are estimated from the local conditions associated to the corresponding unburnt mixtures. Quantities S 0 L and δ 0 L are thus computed at each computational point that fulfills the following two conditions: (i) the mixture fraction remains within the lean and rich flammability limits, i.e., within a range where premixed flame propagation may occur, and

(ii) the considered point lies inside a zone where the probability to obtain a premixed flame structure remains larger than a given threshold value, which is presently set to 40%.

Figure 28 shows that, from one location to another, the turbulent combustion regimes are highly variable. Some points correspond to quasi-laminar conditions, while others feature strong turbulence-chemistry interactions (TCI). It should be emphasized that, with thick flames, thickened-wrinkled flames, and even laminar ones, almost all possible turbulent combustion regimes may be found. In condition RFSC-LST, most of the points are located in the vicinity of thickened-wrinkled flame regime, between the line Da = 1.0 and the Klimov- [START_REF] Borghi | On the structure and morphology of turbulent premixed flames[END_REF] Williams limit5 [START_REF] Williams | Criteria for existence of wrinkled laminar flame structure of turbulent premixed flames[END_REF], around the horizontal line corresponding to u rms /S 0 L = 10.0.

10 -1 10 0 10 1 10 2 10 -1 10 0 10 1 10 2 PSR R e t = 1 R e t = 1 0 0 D a = 1 D a = 1 0 0 K a = 1 K a =
These regimes are consistent with those previously reported by [START_REF] Quinlan | A priori analysis of flamelet-based modeling for a dual-mode scramjet combustor[END_REF] for scramjet conditions. There are also some points featuring a non-negligible heat release rate located in the thick flame region. It is noteworthy that the corresponding points are located in the vicinity of the hydrogen injection, where the reactive mixture remains highly segregated and velocity fluctuations are quite large. On average, the case RFSC-HST displays significantly larger heat release rates, with a statistics spread towards the thickened and thickened-wrinkled flame regimes, which corresponds to smaller values of the Damköhler number. Also, chemical reactions take place at locations where velocity fluctuations may reach extremely large values and, as a consequence, non-negligible finite-rate chemistry effects come into play.

It is noteworthy that the above evaluation of the dissipation rate ε does not account for the unresolved (i.e., modelled) contribution. The relevance of this approximation can be evaluated through a complementary analysis of the computational results. The objective of this analysis is to provide an additional assessment of the LES resolution level by investigating the contribution of the SGS model to the turbulent energy dissipation. To quantify the SGS contribution, one can define two components ε SGS and ε SGS , which are acting on the resolved turbulent kinetic energy and SGS turbulent kinetic energy k SGS , see for instance [START_REF] Ben-Nasr | Assessment of subgridscale modeling for large-eddy simulation of a spatially-evolving compressible turbulent boundary layer[END_REF]. Following [START_REF] Davidson | Transport equations in incom-pressible urans and les[END_REF], these two components can be formulated in such a manner that

ε SGS = ε SGS + ε SGS (21)
where the two contributions are evaluated from

ε SGS = 1 ρ T * ij ∂ u j ∂x k (22)
and Finally, the ratio of the SGS dissipation ε SGS to the resolved viscous dissipation ε SGS is plotted as a function of y + in Fig. 30. It is remarkable that, in a wide range of the boundary layer, from about y + = 15 to y + = 200, this ratio takes values smaller than unity and, even at larger distances from the wall, it does exceed 1.5. This confirms that the resolved dissipation rate ε, as given by Eq. 20, provides a rather satisfactory estimate of the TKE dissipation rate, which can be used to evaluate characteristic turbulence scales so as to proceed with a qualitative inspection of turbulent combustion regimes.

725 ε SGS = 1 ρ T * ij -T * ij ∂ ∂x k ρ u j -ρ u j ρ (

Summary and conclusions

A new set of highly-resolved large-eddy simulations is presented. It corresponds to the geometry of a jet in a supersonic crossflow (JISCF) of vitiated air delivering hydrogen upstream of a squared cavity. The conditions are relevant to experiments previously conducted at the University of Michigan. To the best of the authors' knowledge, this is the first detailed computational investigation of this experimental test case. The computations are performed with the computational solver CREAMS and make use of a recent immersed boundary method (IBM) algorithm. The wall-adapting local eddy (WALE) model is retained as the subgrid-scale viscosity closure. Several criteria are used to check the computational resolution, especially near the walls, which are represented through an IBM, where some efforts have been spent

to capture the flow as accurately as possible. The use of the WALE model allows to satisfactorily recover the behavior in the buffer layer that separates the logarithmic zone from the viscous sublayer. The filtered chemical rates are represented using either the PSR or the U-PaSR closures and it is remarkable that, for the present level of computational resolution, the results obtained using the two distinct frameworks are very similar.

Depending on the inlet vitiated airstream temperature, two stabilization modes are recovered: cavity-stabilized regime and jet-wake stabilized regime. In the cavity-stabilized combustion mode, combustion is anchored at the leading edge of the wall-mounted cavity and spreads into the main flow at an approximately constant angle, whereas jet-wake stabilized combustion takes place directly downstream of the fuel injection, i.e., upstream of the cavity.

In this respect, it seems worth mentioning that different averaged streamline flowfields -not reported for the sake of conciseness -have been obtained for cavity and jet-wake stabilization modes. For high operative temperatures, combustion primarily takes place in the jet-wake stabilization mode while, for lower operative temperatures, the cavity stabilization mode is obtained. This is consistent with the experimental results of [START_REF] Micka | Dual-mode combustion of a jet in cross-flow with cavity flameholder[END_REF]. In this respect, the present set of computations constructively complements the experimental data, e.g., the corresponding computational data are used to discriminate the contributions of premixed and non-premixed combustion modes. The turbulent combustion regimes are also inspected on the basis of classical combustion diagrams for the two levels of inlet temperature that have been considered. This analysis confirms the occurrence of highly turbulent premixed flame conditions. For intermediate values of the operative temperature, some oscillations between the two modes are expected and it would be interesting to run complementary numerical simulations at these intermediate operative temperatures so as to understand how these oscillations may happen and how long each stabilization mode is active. This offers an interesting perspective for future works.

Acknowledgments

This study is a part of the Ph.D. A part of this work has been presented during the invited plenary talk entitled "High-fidelity numerical simulation of reactive high-speed flows" given by Arnaud Mura at the tenth International Conference on Computational Fluid Dynamics (ICCFD10) held in Barcelona (Spain).

  Figure 2: Normalized mesh size distributions with the cavity geometry delineated with a white line

  0 without any fuel injection and are run without any chemical reaction until the fuel injection is developed in the whole computational domain. A typical snapshot depicting an iso-surface ( ξ = 0.5) of the fuel inlet tracer 1 is presented in Fig.3. Chemical reactions are then activated at t * = 300.0, i.e., after a fuel particle got enough time to cross the entire computational domain, and the simulation is run up to t * = 1200.0. Datafiles are stored at a frequency 1.0 MHz. During the numerical simulations, the CFL number value has been varied between 0.3 and 0.7, while the Fourier number Fo has been set constant and equal to 0.9.

Figure 3 :

 3 Figure 3: Iso-surface ( ξ = 0.5) of the fuel inlet tracer at time t * = 350.0 (case RFSC-LST)

Figure 4 Figure 4 :

 44 Figure4reports the distributions of the dimensions of the computational cells present at the wall location, expressed in wall units. The top subfigure, i.e., Fig.4a, corresponds to distributions obtained for case RFSC-LST, while the distributions obtained for case RFSC-HST have been gathered in the bottom subfigure, i.e., Fig.4b. Since the focus of the present study is placed on the reactive flow development, these distributions correspond to values obtained in cells where the filtered mixture fraction ξ is such that 0.001 ≤ ξ ≤ 0.999, thus excluding non-reactive mixtures associated to pure fuel or pure oxidizer. The inspection of these distributions confirms that the wall-resolved LES criteria of Table3are quite satisfactorily fulfilled.

  Figure 5: Non dimensional velocity profile in wall units and viscosity ratio profile at several locations x1/D1 for cases RFSC-LST (left) and RFSC-HST (right)

Figure 6

 6 Figure 6 displays the probability density function (PDF) of the two quality indexes ob-

  Figure 8: Mean fields of the index IQ η obtained in the median plane along the spanwise direction (x3/D1 = 0.0)

Figure 9 :

 9 Figure 9: Probes location for the second-order moments convergence verification

Figure 10 :

 10 Figure10: Convergence of the second-order moments of filtered velocity at the probe locations defined in Fig.9

  Figure 12: Instantaneous contours of fuel mass fraction together with the sonic iso-line (in black) in several spanwise planes

Figure 12 Figure 13 :

 1213 Figure12reports instantaneous contours of fuel jet mass fractions together with the sonic iso-line in several spanwise planes. The mixing between fuel and oxidizer starts at the fuel jet injection with most of the fuel spread into the supersonic flow. The fuel jet is however slightly torn up by the cavity low-speed flow, with a small part of the hydrogen flowing into the wall-mounted cavity, which should result in some mixing enhancement due to recirculation zones present in the cavity. This may be checked through a mixing efficiency analysis.

Figure 14 :

 14 Figure14: Averaged mixture fraction profiles obtained using three distinct definitions

  Figure 15: Ignition probability Pign in the median plane x3/D1 = 0.0

Figure 16 :

 16 Figure 16: Combustion stabilization mode versus vitiated air-stream temperature

  Figures 17 and 18 report, respectively, the averaged cross stream profiles of OH and HO 2 mass fractions obtained at several locations x 1 /D 1 , considering the PSR (blue line) and the U-PaSR (red line) SGS model, for cases RFSC-LST (continuous line) and RFSC-HST (dashed line). From this figure, it is clear that the PSR and U-PaSR models lead to quite similar results. This is an outcome of the well-resolved computational meshes, which lead the subgrid-scale U-PaSR model to behave like a perfectly-stirred reactor (PSR). In the regions

Figure 17 :Figure 18 :

 1718 Figure 17: PSR (blue) and U-PaSR (red) OH averaged mass fraction profile comparison, for cases RFSC-LST (continuous line) and RFSC-HST (dashed line) x 1/D 1 =-1.0

µ

  Figure 19: Scatter plot of µSGS/ µ versus heat release rate for (a) case RFSC-LST and (b) case RFSC-HST

Figure 20 :

 20 Figure 20: Temporal evolution of OH mass fraction for case RFSC-LST (left) and case RFSC-HST (right) in the median plane x3/D1 = 0.0

Figure 21 :

 21 Figure 21: Averaged heat release rate (HRR) issued from the computations (arbitrary scaled units)

  Figure 22: Isovalue surface of the OH filtered mass fraction colored by the normalized filtered temperature T /T0 and fuel inlet tracer iso-surface ξ = 0.5. Back side: OH mass fraction. Top: grayscale levels associated to YOH variations, colormap associated to normalized temperature variations.

  Figure 23: Scatterplots of the fresh reactant temperature

  Figure 24: Premixedness index distribution: field in the median plane at time t * = 1100.0 (top) and associated PDF at various times (bottom)

Figure 25

 25 Figure 25 displays the self-ignition delay of the corresponding mixtures as a function of the mixture fraction (top) or equivalence ratio (bottom) for cases RFSC-LST and RFSC-HST. It is plotted together with the polynomial expression retained for the fresh reactant temperature.

Figure 25 :Figure 26 :

 2526 Figure 25: Self-ignition delay τign as a function of the mixture fraction

  Figure 28: Premixed turbulent combustion diagram based on the Borghi-Barrère coordinates[START_REF] Borghi | On the structure and morphology of turbulent premixed flames[END_REF]

  Figure 29: SGS Activity parameters as functions of y +

Figure 30 :

 30 Figure 30: Ratio of the SGS dissipation to the viscous dissipation, i.e., εSGS/ε, as a function of y +

  ). These two SGS closures make use of a detailed description of chemistry and the mechanism of O'Conaire et al. (2004), consisting of 9 chemical species

(H 2 , O 2 , H 2 O, H, O, OH, HO 2 , H 2 O 2 and N 2 ) and 21 elementary reaction steps, is used to represent H 2 -air chemistry. Additional details about these two SGS turbulent combustion closures will be provided in the next sections of the manuscript.

Table 1 :

 1 Characteristic dimensions of the computational grid

	• 10 -5

Table 4 :

 4 Expressions of spatial filtering and temporal averaging operators

  Thesis of F.H. Eugênio Ribeiro, which is financially supported by the Brazilian program Ciência sem Fronteiras of CNPq and Brazilian Air Force. F.H. Eugênio Ribeiro is indebted to Dr. Anthony Techer and Aimad Er-Raiy for many stimulating discussions on tricky technical issues. This work was granted access to the HPC resources of CINES, IDRIS, and TGCC under the allocations x20152b7456 and i20152b7251.

This quantity will be further discussed in the next section.

The nozzle pressure ratio (NPR) is larger than ten.

It is indeed worth noting that nitrogen is not involved in the detailed chemistry description considered in the present study.

Only the samples associated to non-negligible values of the HRR are considered

It may be worthwhile to recall that this limit corresponds to Ka = 1.0.


Mark Kellett

Laurent Vio

Christophe Bobin

Laurine Brondeau

Mikael Cardot-Martin

Hélène Isnard

Lacour Daniel

Marie-Christine Lépy

Valérie Lourenço

M Marie

Measurement of the absolute gamma-ray emission intensities from the decay of 147 Nd

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