Flow, Turbulence and Combustion manuscript No.(will be inserted by the editor)

LES to ease understanding of complex unsteady combustionfeatures of ramjet burners

L.Y.M. Gicquel · A. Roux

Received: date / Accepted: date

Abstract Ramjet burners are known to produce highly unsteady operating conditions withstrong couplings between combustion, acoustics and flow dynamics. Predicting such op-erating limit-cycles still remains a difficult task for Computational Fluid Dynamics (CFD)although recent use of Large Eddy Simulation (LES) clearly opens new possibilities. Themain difficulties for LES are to properly address numerically specific flow features at thesame time. For example, a proper representation of the acoustic ramjet eigenmodes necessi-tates for the solver to be able to treat shocks often present at the inflow conditions withoutinterfering with the low Mach number flow in the region of combustion. Chemistry mod-elling is another difficulty and it is still not clear what level of description is sufficient toreproduce the unsteady coupling between heat release and acoustics. Turbulent combustionmodelling is an additional problem. Finally, such confined burners are strongly influencedby heat losses and although such issues are rarely addressednumerically, the burner stabil-ity is expected to be strongly dependent on its thermal equilibrium. Despite these difficul-ties and as discussed in this work, LES can greatly contribute in the understanding of thekey mechanisms at play in the expression of the ramjet oscillatory operation. Prior to thisdemonstration, key elements needed for such LES are detailed: shock capturing techniquesand chemical modelling. The effect of the chemistry model onthe LES results is brieflydiscussed. The proposedstrategy is then confronted to three operating conditions of an ex-perimentally measured ramjet burner. Althoughthe schemeis quite simple and essentiallyrelies on the dynamical response of combustion,i.e.competition between the local flow andflame speeds, most of the mean flow features are well reproduced for the three operatingconditions. Peak pressure spectra are well predicted indicating the proper relative energycontent distribution of the unsteady LES predictions.

Keywords LES · Ramjets· Thermo-acoustic instabilities· Reduced chemistry

L.Y.M. GicquelCERFACS, 42 avenue G. Coriolis, 31057 Toulouse Cedex 1, FranceTel.: +0033-5-61-19-30-46Fax: +0033-5-61-19-30-00E-mail: lgicquel@cerfacs.fr

A. RouxCenter for Turbulence Research, Stanford University, Stanford, CA 94305-3030, USA

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1 Introduction

Flow instabilities [1] and most importantly thermo-acoustic instabilities [2–6] are critical inramjet engines: they can lead to vehicle damage and cannot bepredicted in the early con-ception stages. Recent progresses in Computational Fluid Dynamics (CFD), such as reactingLarge EddySimulation (LES) [7–10], and the continuously increasing computer power ofparallel architectures (www.top500.org), allow to investigate such problems by applyingLES to ramjet burners [11–13]. However, proper assessment of LES, its modelling and lim-its are needed prior to the understanding of the flow physics.Validations of the differentsimplifications and modelling hypotheses are key steps. In an attempt to illustrate the po-tential of LES in such a context, the current work proposes toillustrate key elements anddevelopments that are necessary to start investing in LES oframjet burners.

Recent numerical predictions obtained by LES for turbulentreacting flows [14–19] un-derline the power of the approach for laboratory and industry like configurations. Most re-cent LES’s have focused on gas turbine configurations because these systems exhibit a va-riety of difficult problems that are welladdressedthrough that fully unsteady approach: ig-nition, quenching, thermo-acoustic instabilities... Ramjets have received less attention sincethe pioneering work of Kailasanath [20]. Only a few studies have been devoted specificallyto LES of ramjets [12,20–24,13].Among the identified difficulties to be addressed byCFD and more specifically LES, one needs to look into the modelling of turbulence inhighly compressible flows, turbulent combustion and the possible change of regimesdue to compressibility or numerics [25]... Such issues are critical if the method de-vised is extrapolated to supersonic ramjet burners where shocks and combustion arestrongly interacting [25–29]. Ramjet combustors exhibit significant differences com-pared to gas turbine flows or scramjets. Compared to turbines, the velocities are muchhigher, combustion is not stabilized by swirl, nozzles usually start and terminate thechamber. Contrarily to scramjets, combustion does not proceed in a fully supersonicflow since the combustor Mach number remains below one. Shocks need however to bedealt with if the inflow and outflow nozzle that decelerate andaccelerate the velocityfield are to be computed by LES. Finally and for both ramjet applications, chemistryis expected to play a determining role [13,25,29]. Numerically, large Mach number flowsinduce large density gradients which are difficult to handlewith low-dissipation numericalschemes as commonly used in LES [8,30]. The strategy devisedhere to address this diffi-culty is discussed and validated to ensure that it does notinterfere with combustion. As oftoday, many reactive LES’s are limitedto simplechemical schemes [31–37] and the impactof suchsimplifications on the reliability of LES predictionsis anopen issue. The procedureimplemented in [13] is retained here and its validation is extended to three operating condi-tions of the same configuration. Results prove the approach quite satisfactory and yield meanflow fields that are in agreement with experimental findings [12,22,38]. Unsteady featuresthat are experimentally observed to depend on the ramjet operating condition, are properlypredicted as discussed below.

The document is organized as follows. The ramjet configuration is presented in section 2along with the operating conditions simulated by LES and forwhich experimental data areavailable. The LES solver is presented in section 3. Section4 discusses the LES results forthe various cases and the procedure validation (sub-section 4.2) is obtained through com-parisons with measurements of the mean velocity fields for the hot flow configuration. Abrief synopsis (sub-section 4.3) recapitulates the methodology andexploits the three un-steady predictions to underline the potential limitationsand use ofthe proposed strategyto understand theunsteadyflow response of ramjet burners.

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Table 1 List of flow conditions simulated by LES and for which experimental diagnostics are available.

Case Flight condition Operating conditions

I Low speed Φ = 0.35,T inlett = 750 K,

Qf uel = 0.02 kg/s,Qair = 0.9 kg/s

II Medium speed Φ = 0.5,T inlett = 750 K,

Qf uel = 0.03 kg/s,Qair = 0.9 kg/s

III High speed Φ = 0.75,T inlett = 750 K,

Qf uel = 0.044 kg/s,Qair = 0.9 kg/s

2 Target configuration and scope of the study

The configuration of interest corresponds to the ONERA’s experimental ramjet burner forwhich two geometries are available. The first large scale burner is diagnosed for mixingproblems at low speed, while the second real scale burner is devoted to reacting conditions.Details on the different experimental data sets are accessible in [23,38–41]. The burningconfiguration can operate on liquid-gas or gas-gas injection systems. For our study, only thepurely gaseous propane and air cases are considered. Fuel isinjected through a tranquilizingor pre-injection chamber connected to the aft part of the main square pipe combustionchamber throughtwo holes, Fig. 2. For practical reasons detailed below, the entire set-upis computed by LES. Note that previous numerical studies on the reacting configuration areavailable in [12,13,22,24]. Contrarily to these initial LES computations that treat a sin-gle operating condition, the work presented here,deals with the entire operating range forwhich data is available. These three high speed conditions are representative of the real con-ditions encountered with real ramjet applicationsat high altitude. Experimentally, thesethree operating points exhibit distinct mean flow features with different mean flame posi-tions as well as different pressure spectra indicating a wide range of oscillatory operatingregimes of the burner. The list of all three target conditions is provided in Table 1:i.e.Φ , theequivalence ratio,T inlet

t , the inlet total temperature,Qf uel, the fuel mass flow rate andQair ,the air mass flow rate.

Figure 2 presents the computational domain along with its characteristic dimensionsas retained for this work. Based on previous investigations, all the nozzles present in theexperimental test facility are kept within the computational domain in order to reduce theuncertainties potentially introduced by the acoustic treatment of the numerical inflow andoutflow boundary conditions. The main implication of such a choice is the need for the LESsolver to treat adequately shocks present in the two air inlet pipes after the inlet nozzles.Before to proceed with the computation, note that several hypotheses have been called uponand can clearly impact the numerical predictions, Fig. 2:

– In the context of such geometrically constrained configurations, flame / wall interac-tions, radiation and wall heat transfer is expected to play arole in the determinationof the operating limit-cycle selected by the burner for the various operating conditions.Although of major importance, such physics imply the need for accurate estimates ofthe burnt gas composition and temperature over a wide range of conditions which isdifficult to meet with the strategy devised here:i.e. reduced chemical scheme. Currentinvestigations [10,42–44] and turbulent combustion model[43,45–49] open new per-

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(a)

(b)

Fig. 1 Computational domain shape treated by LES: (a) specificities of the configuration and (b) combustorcharacteristic dimensions in millimeters. Note that for the three LES computations discussed in this docu-ment, the question of wall interaction and impact on the prediction remains open.

spectives and may resolve such a need. However and within ramjet burners such meth-ods or needs also call for a solution to the conjugate heat transfer problem [50] which isnot yet mature for turbulent reacting flows.Note that such issues can not be neglectedsinceexperimental facility is heavily cooled to ensure sustainable conditions over a longduration.

– Predicting oscillatory limit-cycles of ramjet burners requires to properly simulate allinflow and outflow responses to internal changes. The selection of the limit-cycle alsodepends on the ability of the LES model to reproduce the driving mechanism. Withthe proposed strategy, this driving force is believed to be thermo-acoustics [51,52] andto a lesserextent the inflow outflow boundary conditions. In that case, it is postulatedand partially demonstrated here that only the local flow and combustion speeds needto be accurate while the adiabatic flame temperature does notdominate. If so, reducedchemistry is applicable with some restrictions as detailedin [13].

Although such hypotheses are of strong implications especially in the context of ramjetburners, preliminary developments and applications of LES[14–19] confirm the potential of

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the approach for gas turbine applications for example. The aim of the remaining part is thusto gauge such hypotheses by directly comparing the LES unsteady and mean flow structureagainst experimental observations and point to the numerical prediction limitations.

3 LES modelling

The parallel LES code [30,53–55] solves the full compressible Navier-Stokes equationsusing a cell-vertex approximation. The numerical integration uses Taylor-Galerkin weightedresidual central distribution schemes. This explicit scheme, which providesup to third-orderaccuracy on hybrid meshes, is particularly adequate for low-dissipation requirements of LESapplications. LES of reacting flows involves the spatial Favre filtering operation that reducesfor spatially, temporally invariant andlocalizedfilter functions [56] to:

f (x, t) =1

ρ(x, t)

∫ +∞

−∞ρ(x′, t) f (x′, t)G(x′−x)dx′, (1)

whereG denotes the filter function.In the mathematical description of compressible turbulentflows with chemical reactions

and species transport, the primary variables are the species volumic mass fractions1 ρk(x, t),the velocity vectorui(x, t), the total energyet(x, t) ≡ es+1/2 uiui and the densityρ(x, t) =

∑Nk=1 ρk(x, t).

The multi-species fluid follows the ideal gas law,p = ρ r T andes =∫ T

0 Cp dT− p/ρ ,wherees is the mixture sensible energy,T the temperature,p the pressure, Cp the fluid heatcapacity at constant pressure andr is the mixture gas constant. The LES solver takes intoaccount changes of heat capacity with temperature and composition using tabulated valuesof individual species heat capacities,which varies with composition and is obtained byr = R

W = R∑Nk=1

YkWk

, whereR= 8.314 [kg m2 / (s2 K)] and Wk is the molecular weight ofthe speciesk. The viscous stress tensor, the heat diffusion vector and the species moleculartransport use classical gradient approaches. The fluid viscosity follows Sutherland’s law, theheat diffusion coefficient follows Fourier’s law, and the species diffusion coefficients are ob-tained using a species Schmidt number along with the Hirschfelder Curtis approximation [7]and velocity corrections for mass conservation.

The application of the filtering operation to the instantaneous set of compressibleNavier-Stokes transport equations with chemical reactions yields the LES equations,Eqs. (2)-(4), which need modelling for the system to be closed [57,58]:

∂ ρ ui

∂ t+

∂∂x j

(ρ ui u j) = − ∂∂x j

[Pδi j − τi j − τi jt ], (2)

∂ ρ E∂ t

+∂

∂x j(ρ E u j) = − ∂

∂x j[ui (Pδi j − τi j )+q j +q j

t ]+ ωT , (3)

∂ ρ Yk

∂ t+

∂∂x j

(ρ Yk u j) = − ∂∂x j

[Jj,k +Jj,kt]+ ωk. (4)

1 Throughout the manuscript Einstein’s rule of summation applies to repeated indices except for indexkwhich is reserved to refer to thekth species.

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3.1 SGS models

The unresolved Sub-Grid Scale (SGS) stress tensorτi jt , is modelled using the Boussi-

nesq assumption [59–61] :

τi jt − 1

3τll

t δi j = −2ρ νt Si j , with, Si j =12

(∂ ui

∂x j+

∂ u j

∂xi

)− 1

3∂ ul

∂xlδi j . (5)

In Eq. (5), Si j is the resolved strain rate tensor andνt is the SGS turbulent viscosity. TheWall Adapting Linear Eddy (WALE) model [62] is chosen to model the SGS viscosity:

νt = (Cw∆)2(sd

i j sdi j )

3/2

(Si j Si j )5/2 +(sdi j s

di j )

5/4, with sd

i j =12

(gi j

2 + g ji2)

+13

gll2δi j . (6)

In Eq. (6), ∆ denotes the filter characteristic length (approximated by the cubic-rootof the cell volume),Cw is a model constant equal to 0.49 andgi j is the resolved velocitygradient.

In Eqs. (3) and (4), the SGS species fluxJi,kt= ρ (uiYk− uiYk), and the SGS energy

flux qit = ρ (uiE− ui E), are respectively modelled by use of the species SGS turbulent

diffusivity Dt,k = νt/Sct,k, whereSct,k is the turbulent Schmidt number (Sct,k = 0.9 forall k). The eddy diffusivity is also used along with a turbulent Prandtl number Prt = 0.6,so that λt = ρ νt Cp/Prt :

Ji,kt= −ρ

(Dt,k

Wk

W∂ Xk

∂xi−YkVc

i

)and qi

t = −λt∂ T∂xi

+N

∑k=1

Ji,kths,k. (7)

In Eq. (7) the mixture molecular weight W and the species molecular weightWk can becombined with the species mass fraction to yield the expression for the molar fractionof speciesk: Xk = YkW/Wk. In expression (7),Vc

i is the diffusion correction velocity re-sulting from the Hirschfelder Curtis approximation [7] and T is the modified filteredtemperature which satisfies the modified filtered state equation [63–65], p = ρ r T. Fi-nally, hs,k stands for the enthalpy of speciesk. Although the performance of the modelscould be improved through the use of a dynamic formulation [66,63,67–69], they areconsidered sufficient to address the present investigation. Note also that throughoutthe work, the variations of the molecular coefficients resulting from the unresolvedfluctuations are neglected so that the various expressions for the molecular coefficientsbecome only functions of the filtered field [7]. Likewise higher order correlations orpressure/ velocity SGS terms [27] appearing in the filtered governing equations are ne-glected.

SGS combustion terms are modelled using the Dynamic Thickened Flame (DTF) model [70].Following the theory of laminar premixed flames [71], the flame speedS0

L and the flamethicknessδ 0

L may be expressed as:

S0L ∝

√λ A and δ 0

L ∝

λS0

L

=

√λA

, (8)

whereλ is the thermal diffusivity andA the pre-exponential constant. Increasing the thermaldiffusivity by a factorF , the flame speed is kept unchanged if the pre-exponential factor is

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decreased by the same factor [72]. This operation leads to a flame thickness which is multi-plied byF and easily resolved on a coarser mesh. Additional information needs however tobe supplied so as to properly reproduce the effect of theSGSinteraction between turbulenceand chemistry [73–75]. This is the intent of the so-called efficiency functionE [70]. Whenthickening is applied everywhere in the flow, the model is limited to fully premixed com-bustion. To compute partially premixed or non-premixed flames [7], a modified version ofthe Thickened Flame model (DTF) isused [75–77]. In this evolution thelocal thickeningfactor depends on the local mesh size: typically thickeningmust ensure that enough pointsare present within the flame zone and the thickening factorF is given by:

F = 1+(Fmax−1)S and Fmax=Nc

∆xδ 0

L , (9)

whereNc is the number of points used to resolve the flame front (typically Nc = 5 to 10).Although this approach is still being developed and furthervalidations are needed, such amodel retains valuable properties which make it suitable for complex applications:

– when used along with Arrhenius type chemical schemes, finiterate chemical effects areretained (ignition, extinction...),

– the balance between chemistry and diffusive effects is preserved within the limit ofvalidity of the efficiency function,

– as such it will produce premixed, partially premixed and diffusion flames, the formerstill needing in-depth validation in the context of turbulent diffusion flames.

3.2 Shock capturing technique

Since the ramjet flow contains shoked nozzles, specific shockcapturing techniques areneeded for a centered numerical LES scheme to preserve the positivity of the solution in re-gions where strong gradients exist [78,79]. There, the methodology of Cook and Cabot [80,81] is used to thicken the shock front by introducing ahyper-viscosity β in the viscousstress tensorτi j equivalent to an additional pressure term,

τi jmodi f ied= (β − 2

3µ)

∂ ul

∂xlδi j + 2 µ Si j = 2 µ

[Si j − 1

3∂ ul

∂xlδi j

]− PCook δi j , (10)

where µ is the dynamic viscosity andSi j is the symmetric strain rate tensor. The bulkviscosity,β , is modelled as,

β = C(∆x)4∇2‖S‖ and ‖S‖ =(

Si j Sji

)1/2, (11)

whereC is fixeda posteriorito 5 according to [80]. This viscosity acts on very sharp velocitygradients characterizing shocks but goes back to zero wherethe velocity evolves smoothly.Note that the resulting viscous stress is equivalent to the original model with an addedartificial pressure term also notedPCook hereinafter.

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3.3 Chemical scheme

Two simplified one-step chemical schemes are tested. They both take into account fivespecies (C3H8, O2, CO2, H2O and N2) and are described by the global one-step irreversiblereaction,

C3H8 +5O2 −→ 3CO2 +4H2O. (12)

The reaction rate for this reaction reads,

q = A

(ρYC3H8

WC3H8

)0.856(ρYO2

WO2

)0.503

exp

(− Ea

RT

), (13)

with a pre-exponential factorA = 3.2916×1010 [cgs] and an activation energy Ea = 31,126cal.mol−1. Note that this expression is the result of a one-step irreversible Arrheniusfit [76] of the detailed mechanism proposed by Peters [82]. The exponents used in Eq. (13)areobtained to get the right laminar flame speed in lean conditions. This first scheme willbe called in the following ”standard one-step scheme”.

Figure 2 shows the comparison between the detailed chemistry given by Peters [82]and the standard one-step scheme for a given range of equivalence ratio. The latter over-estimates flame speeds for large values of the equivalence ratio (Fig. 2 -a-). To circum-vent this well-known and improper behaviour of single-stepschemes, a modified simplifiedscheme (named ”fitted one-step scheme”) is introduced. For this fitted one-step scheme, thepre-exponentialA is adjusted to yield laminar flame speeds over an extended range of equiv-alence ratioφ . Indeed, laminar flame speed is proportional to the square root of A. It is thuspossible to alter the behaviour of the burning velocity by imposing a functional dependencyof A on the equivalence ratio. Note that such a reasoning infers that dynamics is to be privi-leged and that flame positioning is governed by a competitionbetween chemistry and localflow speed. Equation (13) is thus rewritten as,

q = f (φ)A

(ρYC3H8

WC3H8

)0.856(ρYO2

WO2

)0.503

exp

(− Ea

RT

), (14)

where,

f (φ) = 12

[1+ tanh

(0.8−φ

1.5

)]+ 2.11

4

[1+ tanh

(φ−0.11

0.2

)][1+ tanh

(1.355−φ

0.24

)]. (15)

Note that adiabatic flame temperatures are not modified byf (φ) and are still over-estimated with an error of 7% atφ = 1 (Fig. 2 -b-).Such a behaviour is expected sincetemperature can only be affected by a change of composition of the hot gases whichis not possible with the proposed modification. Use of a detailed scheme (if possible inthis context) would be the ultimate alternative since not only would the laminar flame speedbe correct for rich conditions but also would be the adiabatic flame temperature. Finally,for the standard one-step scheme, Schmidt numbers are constant but differ from one speciesto another while the fitted one-step scheme uses equal Schmidt numbers for all species toevaluate the local equivalence ratioφ required in Eq.(15) [83].

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(a) (b)

Fig. 2 One-step chemical scheme validations against the detailedmechanism of [82]: (a) laminar flame speedand (b) adiabatic flame temperature as a function of equivalence ratio.

4 Results and discussion

The LES methodology detailed above is applied for the three operating conditions represen-tative of the real range of operation of a ramjet burner, Table 1. All conditions are simulatedwith the same mesh composed of 5000000 tetrahedral cells with imposed local refinementin the regions of interest:i.e. at the inflow and outflow nozzles as well as in the main pipecombustion chamberso as to reduce grid dependency of the predictions. Note thatef-fects of the initialization procedure were observed and essentially resulted in the fullextinction of the burner if the head-end of the combustor didnot contain any hot prod-ucts. Only the stabilized and fully reacting cases are discussed in this work. Resultsissued for the first operating conditions but obtained by thetwo one-stepschemesare firstconfronted to illustrate the impact of this model on the meanflow predictions. The fittedscheme is then used for the remaining condition and all predictions are detailed to assess thesuitability of the models for a large range of conditions. This validation is obtained first bycomparing mean flame position against experimental findingsand by the identification ofthe main frequencies present in the fully unsteady LES simulations and reported experimen-tally. Prior to these specific analyses in the ramjet burner,a validation of the shock capturingscheme is proposed for a simpler reacting case.

4.1 Shock capturing scheme validation

Many shock capturing techniques are suitable for non-reacting flows computed by LES.However, their extension to LES of reacting flows is not direct and require validationprior to their use in the ramjet configuration. Indeed typical combustion models aredesigned for low Mach number flows and their use in compressible flows with shocks isacceptable provided that combustion occurs in low Mach number zones (M < 0.5) andthat the shock capturing technique does not interact with combustion [81]: i.e if thetwo models are active at distinct places and do not interfere. A validation scheme thatis relevant to the ramjet configuration in terms of complexity and difficulty is the Billetet al. [84] shock hydrogen bubble interaction. In this two-dimensional DNS problem, apure hydrogen bubble interacts with a shock delimiting two air conditions as presented onFig. 3(a). In suchproblem, reflections occur as the bubble proceeds through the shock andmixing between air and hydrogen eventually yields the auto-ignition of the mixture and heatreleaseas illustrated on Fig. 3(b).

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(a) (b)

Fig. 3 Shock-bubble interaction: (a) sketch of the configuration and (b) instantaneous field of temperatureand iso-levels of reaction rate.

Fig. 4 Shock capturing validation: instantaneous view of the heatrelease rate, top part, and shock capturingsensor, bottom part, as issued by the proposed scheme.

Based on such DNS, the activation of the shock capturing scheme and combustion canbe confronted in space at a given instant as shown in Fig. 4. For this diagnostic, the in-stantaneous flow field being fully symmetric with respect to the central axis, the top viewcorresponds to the heat release field while the bottom part indicates regions where theshock-capturing scheme is triggered. Clearly, both modelsare activated in distinctzonesand only a very small volume of artificial pressure,Eq. (10), coincides with reacting zones.Where combustion occurs, the additional pressure added through the use of the shock-capturing scheme is lower than one thousandth times the meanpressure which furtherconfirms the suitability of the modelling approach. Most of the physics is therefore re-spected and both models work adequately without strong interactions. For the ramjet

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burner, since the regions of combustion and where shocks arepresent, are also dis-tinct: i.e. shocks appear after the inflow air nozzles (top and bottom pipes of Fig. 2)while reaction proceeds in the chamber main pipe as confirmedby Fig. (5).

Fig. 5 Instantaneous view of the Mach number and heat release distributions as obtained for the ramjet burnerby LES using the shock capturing technique presented previously.

4.2 Chemical scheme impact

The current section focuses on the effect of the chemical model for LES of the ramjet burnerby use of the DTF model, numerical issues and potential turbulent combustion andchemistry effects being investigated in [24]. Only Case I of Table 1 is used for the compar-ison: i.e. two LES with the same numerical parameters and models exceptfor the chemicalscheme. The first set of results relies on the ”standard one-step scheme” while the secondone uses the ”fitted one-step scheme”. Details on this specific comparison are also availablein [24] where specific diagnostics are provided in terms of the thermo-acoustic responseof the two simulations. Complementary mean field diagnostics are given here and allow tofurther exploit the limitations of bothapproachesin the context of the ramjet burner beforean extensive validation of the methodology to the entire range of operating conditions.

Figure 6 presents the mean axial velocity component in the main chamber as issued bythe two computations, the top part for the ”standard one-step scheme” and the bottom partfor the ”fitted one-step scheme”. Experimental fields in two restrained regions of the mainpipe are also provided for a direct comparison between the two computations and measure-ments. Most of the differences observed with the two LES and induced by the differentchemical models concentrate in the downstream and upstreamregions of the two air feedingjets that impact each other when entering the main combustion pipe. Downstream the im-pacting jets, two recirculation bubbles appear on the top and bottom walls of the combustor.Clearly and depending on the chemistry used, theextent of these recirculation zones differ.With the ”standard one-step scheme” (top view of Fig. 6) the low speed zone near the topwall is highly constrained by the main pipe flow stream issuedby the impacting jets. Whenthe ”fitted one-step scheme” is used (bottom view of Fig. 6), this low speed region is muchwider and the main air stream jet expands further downstreamthe main pipe. This last be-havior seems to be in better agreement with experimental observations [23] as shown in thelower part of Fig. 6.

More quantitative comparisons are given in Fig. 8 where the axial component of the ve-locity vector is compared to measurements for three cross stream stations in the combustoras displayed on Fig. 7.Although all numerical predictions can be improved, accountingfor the rich side behavior of the flame speed seems mandatory in this ramjet burner.

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Fig. 6 Chemical model effect on the mean axial velocity field in the main ramjet combustion pipe. The tophalf of the computational domain corresponds to the numerical predictions obtained using the ”standard one-step scheme” while the bottom half relies on the ”fitted one-step scheme”. Experimental views are added forpart of the domain (additional windows in dotted lines).

Indeed the two measured downstream profiles, positioned after the jet impact, are bet-ter reproduced by the ”fitted one-step scheme” LES. The ”standard one-step scheme”LES predicts a much larger flow speed at these stations. Thesedifferences, inducedby the change of chemistry only, indicate that rich combustion occurs in this burnerwhere fuel is injected pure the head-end. Reproducing adequately the competition be-tween the flow and flame speeds in rich conditions is hence recommended especiallyif flame stabilization and flow path is impacted by such rich flames (as suspected inthis specific application). Note that such a flow/chemistry competition mechanism is ofprime importance for a thermo-acoustic analysis of ramjetsas discussed in [13].

Fig. 7 View of the head-end of the combustor. Position of the different cross stream stations for the profilesshown on Fig. 8

A second main difference between the two LES, is the actual frequency content of theunsteady simulation, Fig. 9.Pressure records, for a probe located 10 mm downstream inthe upper air feeding jet and provided on Fig. 9, show that high frequency perturbationsdominate (around 4 000 Hz) the ”standard one-step scheme” LES when only low frequency

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Fig. 8 Chemical model effect on the mean axial velocity profiles in the main ramjet combustion pipe. Eachprofile is taken along cross stream lines at three axial stations and whose coordinate is given and measuredfrom the head-end wall of the main pipe combustion chamber.

peaks are present with the ”fitted one-step scheme” LES.The origin of such differentenergetic contents is attributed to the triggering of an acoustic transverse eigenmode ofthe system when the ”standard one-step scheme” is used whichis not the case with the”fitted one-step” model (cf. [13] for more details on the different mechanisms involved).Experimental records confirm the presence of low frequency oscillations (mainly around 400Hz) but do not show any high frequency oscillations predicted by the ”standard one-stepscheme” LES.Only the “fitted one-step scheme” properly reproduces the different peaksexhibited by the experiment.

Preliminary validations and confrontations of the proposed LES strategy for ramjet burn-ers indicate that a fully detailed kinetic scheme may not be required to apprehend combus-tion instabilities of such burners.The only pre-requisite for the simpler chemical ap-proaches to be applicable is that they should provide a good estimate of the local flamespeed:i.e. for a wide range of equivalence ratios including rich combustion. This lastpoint is of particular importance since the configuration is partially premixed and canthus exhibit rich combustion regimes. Note that the main disadvantage of reduced schemesis the adiabatic flame temperature which is badly predicted for the rich side. Use of otherapproaches would be clearly needed to improve the numericalpredictions. These methodshowever require additional validations to deal with partially premixed combustion. More-over, that need is not justified by the validation or the regimes to be simulated here. Indeed,all cases of Table 1 correspond to lean mean equivalence ratiosso errors on the tempera-ture field are potentially reduced. That specific issue is of importance and will be specifi-cally addressed in the following section.

4.3 Application to the three flight conditions

Having partially validated the approach for a lean mean operating condition, LES is ex-tended to the following cases that are representative of twoother flight operating points.The changes from one operating condition to the next correspondto an increase of the meanequivalence ratio obtained by an increased fuel mass flow rate.

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Fig. 9 Spectra of the pressure fluctuations 10 mm downstream the upper air feeding jet.

(a) (b)

(c)

Fig. 10 Non-dimensionalized axial velocity component and its RMS in the symmetry plane of the main pipecombustion chamber as obtained by LES with use of the ”fitted one-step scheme” for the three operatingconditions of Table1. Each subplot is divided in two parts: the top part for the mean axial velocity field andthe bottom part for its RMS. All quantities are non-dimensionalized by the mean bulk velocity value in theair feeding pipe.

15

Based on the LES mean flow predictions, top views of Fig. 10, the axial component ofthe velocity field (non-dimensionalized by the mean bulk velocity in the air feeding pipeof each flight condition) remains quite similar for the threecases. For all results, the al-ready identified recirculation bubblesare locateddirectly downstream of the jet air streamentering the main pipe chamber is present. Its extent increases with the mean equivalenceratio indicating different level of interactions between the air jets and combustion. Note alsothat the downstream part of the main chamber pipe reaches mean axial velocity values thatincrease from Case I to III which agrees with the increasing mean equivalence ratio andthe corresponding increasing mean adiabatic flame temperature. The associated RMS fieldsdiffer slightly from one case to the next. For Case III, most of the RMS activity is observedin the shear layer present in the air feeding pipe (first bend of the air feeding pipe prior toits connection with the main combustion chamber pipe).Such activity is representative ofa wall flow separation point. The second shear layer which generates large values of RMSlocates in the main chamber pipe at the junction between the high velocity jet entering thechamber and the lower wall separation bubble discussed in sub-section 3.2. As the flightspeed increases, these two regions have RMS levels which also increasepointing to an in-creased fluctuating activity. Note that the air stream shearlayer RMS level has a relativeincrease that is more important than the second location. The flow is expected to change alot in this stream when comparing Case I with Case III.

Fig. 11 Representation of the mean reaction zones as seen in the experiment (left column) and obtained byLES (right column) for the three cases of Table 1. The experimental data corresponds to a direct view of themean OH emission field while LES is obtained by a transverse integration of the heat release weighted bytemperature.

16

A first and indirect confirmation of the good quality of the LESpredictions is providedby Fig. 11. In this figure, the mean OH emission observed experimentally (direct viewthrough a side window) is shown on the left for the three cases. On the right side andfor all three cases, a cross-stream integrated LES mean heatrelease field is shown. Ofcourse a direct comparison between the experiment (left column) and LES (right column) ishere not possible since the reduced scheme includes no OH radical. Qualitative comparisonsare however acceptable. Based on this diagnostic targeted toward combustion, all three LESlocate the mean reaction regions in agreement with the experimental findings. More par-ticularly, the experimental observation which differentiates the three conditions, is properlyreproduced by LES:i.e. reduced combustionthe head-endas the mean equivalence valueincreases. The main consequence of this weaker flame with increasing mean equivalenceratio is the extinction of the dome combustion for Case IIIand an enhanced and continu-ous flame front after the impact of the two air feeding jets. For Cases I and II, that mainchamber flame front is not continuous but rather located behind and within the recirculationbubbles issued by the incoming air stream on the top and bottom walls of the main chamber.Although very complex, this mean flow and combustion response to the mean equivalenceratio seems to be very well reproduced by the proposed LES strategy. Such a comment isalso confirmed by spectral analyses of pressure probes in thecomputational domain and mi-crophone time-series obtained experimentally (not shown here) on all three cases. Furtherinvestigations are nonetheless required to fully validatethe approach. This requires accessto more in depth measurements and is the subject of ongoing work. These preliminary LESpredictions are nonetheless very encouraging and are detailed below to exploit the potentialof LES to understand the unsteady nature of such flows. The aim of the following is thusto analyze the unsteady response of the flows to the operatingcondition or more specifi-cally mean equivalence ratio. The large scale LES flow features are of particular interest toqualify the oscillating operation of the burner and its thermo-acoustic stabilityfor differentoperating conditions.

Fig. 12 Schematic representation of line for which unsteady diagnostics are provided below. The bottomscale corresponds to the axial coordinate (in meter) presented in the following analysis.

In order to proceed with the unsteady analysis of the three LES predictions, the numer-ical data is reduced to point-wise temporal records of pressure along the axis pictured onFig. 12. Typical spectral maps (pressure amplitudes as a function of its frequency and po-sition in the burner) obtained on this axis for the three simulations are provided on Fig. 13.For Case I, the dominant frequency seen throughout the computational domain (not includ-ing the inlet nozzle) corresponds to a pressure oscillationof ≈ 100 Hz and will be referredto as Mode 1 in the following discussion. For Case III, the dominant oscillation is around400 Hz, named Mode 3 afterwards. This last mode is present in the air feeding jet pipes

17

and the first part of the combustion chamber. The two modes appear in Case II along withpotential combinations as observed on Fig. 13.Indeed the spectral map of Case IIshowsmuch broader spectra and a much more complex response of the burner. For clarity, all ofthe potential oscillatory motions are noted on Fig. 13. However and in order to proceed withthe analysis, only Modes 1 and 3 are discussed and detailed below as they are present in allthree LES and seem at the root of the observed limit-cycles.

(a) (b)

(c)

Fig. 13 Spatial evolution of the pressure spectral content in the ramjet burner for the three operating condi-tions and obtained by LES. The abscissa is the frequency and the ordinate corresponds to the axial distancealong the burner axis, Fig. 12. Gray scales are here to denotepressure levels reported numerically. (a) Case I,(b) Case II and (c) Case III

Reconstruction of the space and time variation of the pressure field as issued by Mode1 and 3 are provided on Fig. 14. The reconstruction is obtained based on the amplitude andphase relationship contained in the FFT analysis of all the points of Fig. 13and filteringout the signal of no interest. Thanks to this manipulation, the spatial pressure evolutionat given successive time instants and for a given frequency of interest is accessible aspictured on Fig. 14. On this figure, each line corresponds to the pressure fluctuationdistribution within the burner and at a given time. For Mode 1, Case I is used to extractthe spatial dependency of the phase and amplitude at100 Hz. Case III provides thesame information for Mode 3: i.e. for a signal at 400Hz. Thanks to these diagnostics,the specific nature of the oscillations is partly illustrated if looking for the following :

18

– The crossing of all the time lines with the zero amplitude axis underlines the presenceof a pressure node itself indicative of an acoustic eigenmode.

– The crossing of the different time lines at different axial position and for different pres-sure amplitudes is indicative of convected pressurefluctuation in the computationaldomain. The speed at which such information travels can be related to the nature ofthatperturbation:(u+ c) or (u− c) for purely acoustic information and(u) for entropicperturbations.

(a) (b)

Fig. 14 Space and time evolution of the two main modes observed numerically: (a) Mode 1 and (b) Mode 3.

The pure acoustic nature of Mode 1 or 3 is not evidenced by Fig.14. Instead, both modeis more likely to be a combination of convected information and pure acoustic eigenmodesof specific parts of the burner. For example, Mode 3 has a clearacoustic node in the mainchamber pipe (aroundx ≈ 0.5 m), Fig. 14(b). The shape of the envelop in this part of theburner points to the expression of the half wave main chamberpipe acoustic eigenmodewhich then propagates into the air feeding pipes (x < 0 m). This information is eventuallyreflected by the inflow nozzlewhich sends back pressure perturbations traveling down-stream toward the main chamber pipe at the flow velocity. Mach number effects as wellas the conversion of upstream traveling pressure waves (acoustic fluctuations) into down-stream traveling pressureperturbations by the nozzle seem of importance in the determi-nation of the flow operating limit cycle. Combustion is also expected to play a determiningrole as it is the sole mechanism by which acoustic energy is force-fed to the flow. Mode1, Fig. 14(a), has a less obvious nature. No clear acoustic eigenmode is observed from thereconstructed envelop. Convection of the pressure perturbations in the air feeding pipes isclearly illustrated. The main chamber pipe on the other handseems to have a different typeof response. Indeed all the points positioned atx> 0 m produce a uniform pressure variationat all instants indicating that the mean chamber pressure orcontained mass strongly varieswith time. Further investigations of the numerical predictions are clearly needed to con-clude on the exact nature of the two Modes. However first indications point to mixed modesresulting from the expression of eigenmodes and convected perturbations. Their effect oncombustion is illustrated in the following and underlines the complexity of the couplingsinvolved in the determination of the operatinglimit-cycle in ramjet burners.

Instantaneous views of the heat release rate (dark grey iso-surface) and of the verticalvelocity component (light grey iso-surface for negative and black iso-surface for positivevalues) obtained from Case I provide a view of the coupling between the flow and combus-tion if Mode 1 dominates the limit-cycle, Fig. 15. The two instants of Fig. 15 respectivelycorrespond to the minimum and peak values of the pressure oscillation within T1, the period

19

corresponding to Mode 1. The main effect of that Mode shape, Fig. 14(a), is to influencethe flame position within the main pipe. When pressure is minimum within the chamber,Fig. 15(a), the two air streams are at their maximum speed imposing large velocities whichpush the flame front downstream the main pipe. Half a period later, Fig. 15(b), the air feed-ing jets velocity is greatly reduced and the flame has propagated upstream almost enteringthe dome. Similar views are shown in Fig. 16 for Case III for four distinct instants: (a)T3,(b) T3 + T3/4, (c) T3 + T3/2, (d) T3 + 3T3/4 within Mode 3. Similarly to Case I, Mode 3induces large air feeding speed fluctuations, the flame frontis however less affected by thisoscillation at least in its axial position. At all instants of Fig. 16, hot gases remain in thenear impacting jet field. The main differencewith Case I is observed in the dome whichproduces strong axial velocity fluctuations that are not observed when Mode 1 dominates.That specific difference induces differences for the main chamber fuel alimentation processand the combustion intensity. Such changes of flow dynamics in this specific region are ex-plained by the different spatial gradient of pressure resulting from Mode 1 or 3:i.e. zeropressure gradient in the dome for Mode 1 and large variationsfor Mode 3.

(a) (b)

Fig. 15 Instantaneous views of the heat release rate (dark grey iso-surface) and vertical velocity component(light grey iso-surface for negative and black iso-surfacefor positive values) within the burner and at twoinstants of Mode 1: (a) when pressure within the main pipe is minimum and (b)T1/2 later when pressure ismaximum.

The following conclusions are underlined thanks to the detailed analysis of the flow LESpredictions obtained for all three cases:

– First, the proposed LES strategy to address ramjet reactingconfigurations seems verypromising and qualitative comparisons between mean experimental observation andLES predictions are satisfactory for all three conditions.

– Second, all three cases exhibit distinct operating limit-cycles with two extreme cases(Case I and III) whose spectral content is dominated by two modes whose frequencieswere reported experimentally.

– Mode 1: at≈ 100 Hz dictates the limit-cycle in the LES of Case I. LES allows toidentify two components: purely propagated pressure oscillations in the air feedingpipe accompanied by a bulk pressure rise or drop of the main pipe chamber pressure.The net result of the oscillation is strong variations of theflame front position in themain chamber pipe.

– Mode 3: at≈ 400 Hz dictates the limit-cycle in the LES of Case III. LES allowsto identify two components: a purely acoustic eigenmode of the main pipe chamber(half-wave eigenmode) coupled to propagated pressureoscillations in the air feed-ing pipes. The net result of this oscillation seems associated with no reaction in thedome and a mean flame front located downstream the air feedingjets.

– Third, the determination of the limit-cycle is observed to be strongly linked to thethermo-acoustic stability of the main chamber acoustic eigenmodes which involves the

20

(a) (b)

(c) (d)

Fig. 16 Instantaneous views of temperature (black iso-surface), negative axial velocity component (grey iso-surface) and vertical velocity component (light grey for negative and white iso-surface for positive values)within the burner and at four instants of Mode 3: (a)T3, (b) T3 +T3/4, (c)T3 +T3/2, (d)T3 +3T3/4.

coupling between heat release fluctuations and pressure fluctuations. The end nozzleimpedances are also suspected to play a determining role in the establishment of thelimit-cycle. In particular, the inlet air inflow nozzle response to acoustic fluctuations andthe potential conversion of acoustic energy into purelyconvected pressure fluctuations(traveling at a the speed(u)) while acoustic waves travel at the speed(u+c)) needsto be clarified.

5 Conclusion

Ramjet burners are known to produce highly unsteady operating conditions with strong cou-plings between combustion, acoustics and flow dynamics. Predicting such operating limit-cycles still remains a difficult task for Computational Fluid Dynamics (CFD) although recentuse of Large Eddy Simulation (LES) clearly opens new possibilities. Despite clear difficul-ties a simplified numerical and modelling strategy is discussed and validated for LES of theONERA ramjet experimental burner. In particular, it is shown that the choice of the reducedchemical kinetics model can be critical although simple modification may be sufficient.Likewise and in order to reduce the modelling effort, inlet and outlet nozzles were chosento be included in the computational domain which implied theproper capture of shocks.The implemented shock capturing technique isvalidated in the context of a highly com-pressible reacting flowand proves to be adequate for the ramjet LES’s . The resultingnumerical LES tool is then confronted to three operating conditions of an experimentallymeasured ramjet burner and LES results are clearly promising. Indeed and despite the factthat the proposed scheme is quite simple since it essentially relies on the dynamicalresponse of combustion,i.e. competition between the local flow and flame speeds; forall cases, most of the mean flow features are well reproduced.For the three operatingconditionsthe mean flow featuresand peak pressure spectra are in agreement with exper-imental findings. Preliminary investigations relying on the spatial and temporal analyses of

21

the numerical results indicate that for this configuration two oscillating modes prevail. Thefirst mode at≈ 100 Hz involves the propagation in the air feeding alimentation ducts ofpressureperturbations along with a bulk pressure response of the main pipe combustionchamber. The main effect of this mode is the large axial variations of the flame position inthe main pipe. The second mode at≈ 400 Hz, involves the expression of the half-wave maincombustion pipe acoustic eigenmode and the propagation in the air feeding pipe of the pres-sureperturbations. To conclude, the determination of the ramjet limit-cycle as a functionof the mean operating condition seems to depend on the thermo-acoustic response of themain combustion pipe as well as the inflow response to pressure oscillationsproduced bycombustion.

Acknowledgements This research used resources of the Argonne Leadership Computing Facility at Ar-gonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy un-der contract DE-02-06CH11357 and access to the Centre Informatique National de l’Enseignement Superieur(CINES) in Montpellier, France, under the project FAC2554.A. Roux is supported financially by the FrenchDelegation General pour l’Armement (DGA). Special thanks go to the ONERA team for sharing their dataand observations on their installation as well as to the COCCFEA organization team for their interest in thiswork.

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