Combustion devices operating at elevated pressures, such as liquid rocket engines (LRE), are usually characterized by supercritical thermodynamic conditions. Propellants injected into the combustion cham- ber experience real fluid effects on both their mixing and combustion. Transition through super-criticality implies abrupt variations in thermochemical properties which, together with chemical reactions and high turbulent levels introduce spatial and temporal scales that make these processes impractical to be simulated directly. Reynolds-Averaged Navier-Stokes (RANS) and Large Eddies Simulation (LES) equipped with suitable turbulent combustion modeling are therefore mandatory to attempt numerical simulation on real- istic length scales. In the present work, the building blocks for extending the unsteady/flamelet progress variable approach for turbulent combustion modeling to supercritical non-premixed turbulent flames are presented. Such approach requires a large number of unsteady supercritical laminar flamelet solutions at supercritical pressures, usually referred as flame structures, to be preliminarily established by solving the flamelet equations with suitable real fluid thermodynamics. Given such unsteady flame structures, flamelet libraries can then be generated for all thermochemical quantities. The explicit dependence on flamelet time is usually eliminated using mixture fraction, reaction progress parameter, and maximum scalar dissipation rate as independent flamelet parameters. Real fluid thermodynamics used for such unsteady supercritical laminar flamelet solutions, is taken into account by means of a computationally efficient cubic equation of state. In order to have a better handling of real gas mixtures, the real gas equation of state is written in a comprehensive three-parameter fashion. A priori analysis at supercritical pressures of transient flame structures is performed in order to study how solutions populate the flamelet state space which is usually characterized by the S-shape diagram representing a collection of steady solutions. High-pressure condi- tions ranging from 60 to 300 bar are chosen as representative of a methane/liquid-oxygen rocket engine operating condition

Creta, F.

Lapenna, P.E.

LAPENNA , PASQUALE EDUARDO

CRETA, Francesco

Archivio della ricerca- Università di Roma La Sapienza

XXXVIII Meeting of the Italian Section of the Combustion InstituteTowards an Unsteady/Flamelet Progress Variable method for non-premixedturbulent combustion at supercritical pressuresP.E. Lapennaa, F. CretaaaDepartment of Mechanical and Aerospace Engineering, Sapienza University, Rome, ItalyAbstractCombustion devices operating at elevated pressures, such as liquid rocket engines (LRE), are usuallycharacterized by supercritical thermodynamic conditions. Propellants injected into the combustion cham-ber experience real fluid effects on both their mixing and combustion. Transition through super-criticalityimplies abrupt variations in thermochemical properties which, together with chemical reactions and highturbulent levels introduce spatial and temporal scales that make these processes impractical to be simulateddirectly. Reynolds-Averaged Navier-Stokes (RANS) and Large Eddies Simulation (LES) equipped withsuitable turbulent combustion modeling are therefore mandatory to attempt numerical simulation on real-istic length scales. In the present work, the building blocks for extending the unsteady/flamelet progressvariable approach for turbulent combustion modeling to supercritical non-premixed turbulent flames arepresented. Such approach requires a large number of unsteady supercritical laminar flamelet solutions atsupercritical pressures, usually referred as flame structures, to be preliminarily established by solving theflamelet equations with suitable real fluid thermodynamics. Given such unsteady flame structures, flameletlibraries can then be generated for all thermochemical quantities. The explicit dependence on flamelet timeis usually eliminated using mixture fraction, reaction progress parameter, and maximum scalar dissipationrate as independent flamelet parameters. Real fluid thermodynamics used for such unsteady supercriticallaminar flamelet solutions, is taken into account by means of a computationally efficient cubic equationof state. In order to have a better handling of real gas mixtures, the real gas equation of state is writtenin a comprehensive three-parameter fashion. A priori analysis at supercritical pressures of transient flamestructures is performed in order to study how solutions populate the flamelet state space which is usuallycharacterized by the S-shape diagram representing a collection of steady solutions. High-pressure condi-tions ranging from 60 to 300 bar are chosen as representative of a methane/liquid-oxygen rocket engineoperating conditions.1. IntroductionHigh performance liquid rocket engines (LRE) widely utilize liquid oxygen (LOx) together with liquidhydrogen (LH2) and, more recently liquified natural gas (LNG) as propellants, because of their high specificimpulse and non-toxic combustion. Despite the experimental and theoretical effort in the past decades, in-depth understanding of the complex phenomenology taking place in the thrust chamber of a typical LRE, isstill to be fully achieved [1].Email addresses: pasquale.lapenna@uniroma1.it (P.E. Lapenna), francesco.creta@uniroma1.it (F. Creta)Operating conditions of LRE are mainly determined by chamber pressure pC and injection temperatureof the fuel T f and oxidizer Tox, and are usually classified with respect to the critical properties of the pro-pellants. In most of the practical cases methane is injected in supercritical conditions (T > Tcr = 190.8K,p > pcr = 4.599MPa) while liquid oxygen (LOx) is injected in a transcritical state (T < Tcr = 154.6K,p > pcr = 5.043MPa). The injection temperature of LOx can be significantly lower than the critical tem-perature of oxygen, typically between 80K and 120K, enhancing real fluid effects on both the mixing andcombustion processes, thereby exhibiting sharp gradients of density and abrupt changes of mixture thermo-dynamical properties. In order to characterize combustion chambers at supercritical pressures [2], unifiedtreatment of fundamental multi-species real gas thermodynamics is mandatory in numerical simulations,as well as detailed chemical kinetic mechanismssuitable for high pressure conditions. Experiments in thepast decade [3, 4, 5] have shown that supercritical mixing and combustion can be conceptually based on asingle-phase mixture. Thus, numerically, a general fluid is modeled as a ”dense” gas with liquid like densityand gas like diffusivities that does not experience any droplet formation with a vanishing surface tension.Turbulence/chemistry interaction models in supercritical reacting flows, both in LES and RANS, aregenerally based on steady state flamelet assumptions [6, 7]. The aim of the present work, on the otherhand, is to establish a framework for the extention to supercritical combustion of unsteady flamelet progressvariable (UFPV) approaches [8, 9].An a priori study of unsteady flamelet solutions, prototypical of transient flame configurations charac-terized by time dependent features such as re-ignition and quenching, are investigated numerically by meansof a general fluid formulation for the unsteady laminar flamelet equations.2. Supercritical Unsteady FlameletIn this work, real fluid thermodynamic properties are taken into account by means of a computationallyefficient cubic equation of state (EoS) written in a general three-parameter fashion [10] which reads, for apure species:p =ρRuTW − bρ −aα(T ) ρ2(W + δ1bρ) (W + δ2bρ)(1)where p, ρ, T are fluid pressure, density and temperature, W is the molecular weight, Ru the universal gasconstant and α(T ) is a temperature correction factor. The three parameters characterizing the EoS are a, band δ1 ( whereas δ2 = (1 − δ1)/(1 + δ1)). The choice of parameters in Eq. (1) determines which particularcubic EoS has been chosen. The classical Peng-Robinson form is recovered for δ1 = 1 +√2 while theSoave-Redlich-Kwong form for δ1 = 1. A complete three parameter EoS, such as the RK-PR, is obtainedfor δ1 = δ1(Zc), where Zc is the critical compressibility factor [11], and represents an additional dependenceon Zc. Complete definitions for a, b, α and δ1 as well as their fitting parameters can be found in work ofCismondi and Mollerup [10].The validity of the real fluid EoS is extended to an arbitrary number of components, considering amixture as a single phase and unique pure hypothetical fluid. The parameters required by the EoS arecalculated from conventional molar fraction based mixing rules [11] on the basis of the critical state of eachpure component, such as critical temperature and pressure as well as acentric factor.Given the EoS parameters (aα,b,δ1) of a mixture, every thermodynamical relation can be expressed interms of a reference ideal low-pressure property and a real gas departure function [12] derived from the realgas EoS. In the present case of RK-PR EoS departure functions assume the form derived by Kim et al. [13].2 0 200 400 600 800 1000 1200 1400 100  200  300  400  500  600  700 ρ [Kg/m3 ]T [K]pRflameletChemkinRefprop 0 2000 4000 6000 8000 10000 12000 100  200  300  400  500  600  700C p [KJ/KgK]T [K]pRflameletChemkinRefprop 0 100 200 300 400 500 100  200  300  400  500  600  700 ρ [Kg/m3 ]T [K]pRflameletChemkinRefprop 0 5000 10000 15000 20000 100  200  300  400  500  600  700C p [KJ/KgK]T [K]pRflameletChemkinRefpropFigure 1: Validation of principal thermodynamic properties: density and constant pressure specific heat of pure oxygen (top) andpure methane (bottom) as a function of temperature at three different pressures, respectively 50bar, 100bar and 200bar. Solid lines(−) are the in-house Rflamelet code results, dashed lines (−−) are Chemkin ideal gas EoS results [14], circles are Refprop softwareresults from the NIST database [15]In Fig. 1, the thermodynamical properties for both the fuel and the oxidizer, as estimate by the abovemodel, are compared against NIST data. In the proximity of the critical point, thermodynamic and transportproperties exhibit anomalies in their behavior, usually referred as near-critical enhancement.The local flame structure of a turbulent non-premixed flame can be described, in the high Damkoho¨lerlimit, by one dimensional flamelet equations in mixture fraction space. Such equations highlight the un-steady competition between chemical kinetics and molecular diffusion processes enhanced by turbulentmixing. The diffusion parameter of the flamelet equations is the scalar dissipation rate χ of the mixturefraction z defined as χ = 2D(∇z)2, where D is the diffusion coefficient of z. Assuming unitary Lewis-number and using the energy equation in terms of enthalpy derivatives in order to avoid numerical issuesdue to constant pressure specific heat, cp, enhancement, flamelet equations read [16]:∂Yi∂t=12χ∂2Yi∂z2+ω˙iρ;∂T∂t=12χ1cp[∂2h∂z2+ns∑k=1hk∂2Yk∂z2]+ω˙Tcpρ(2)The solution of the flamelet system, equations (2), defines the flame structure in mixture fraction z-space. The functional dependence of the scalar dissipation rate on mixture fraction can be assumed as thatstemming from a temporal mixing layer in physical space and reads χ(z, t) = χmax(t) exp(−2erfc−1(2z))2.We solve the time dependent flame structure in a fully coupled fashion, thus no operator splitting tech-nique was adopted, the whole system being integrated using a stiff solver for ordinary differential equations.This unsteady flamelet system has been recently used to analyze supercritical methane/LOx auto-ignitingflame structures [17].33. Results and DiscussionAs previously mentioned, state-of-art supercritical combustion CFD simulations currently rely on sim-plified steady state flamelet models where chemistry is assumed to respond infinitely fast to perturbationsfrom the turbulent flow field. In a high pressure environment the reduces kinematic viscosity and thuselevated Reynolds number, can lead to rapid and intense scalar dissipation rate variation, due to its inter-mittent nature [18]. Therefore a turbulent combustion model, such as the UFPV method, that can representunsteadiness in the flame structure arising from scalar dissipation rate fluctuations can become crucial [8, 9].The response of laminar, methane/LOx, supercritical flamelets to turbulent perturbations is investigatedin a a priori fashion by means of representative synthetic signals. Turbulent effects are modeled as abruptchanges or sharp signals in time of the scalar dissipation rate [19]. Following [20] the scalar dissipation rateprofile fluctuates with a factor φ(t) that can vary from 0.5 to 2:χ(z, t) = φ(t)χmax exp(−2erfc−1(2z))2 (3)In the above scalar dissipation rate signal, φ(t) is chosen as a triangular wave [19]. This signal iscompletely characterized by a peak amplitude Aχ and a duration τχ as shown in Fig. 2. The signal φ(t)is imposed to the supercritical flamelet system, starting from a steady state solution found for a scalardissipation rate near the quenching value.Realistic LOx-Methane LRE combustion chamber conditions for supercritical flamelet calculations areemployed, with a background pressure of 60 bar. Flamelet boundary conditions are pure oxygen at theoxidizer side (Tox = 120 K) and pure methane at the fuel side (T f uel = 300 K). A detailed chemical kineticmechanism for methane oxidation at elevated pressures, referred to as Ramec [21], is used throughout thisstudy. The scalar dissipation rate model is expressed as in eq. (3), and the initial condition is a steady stateflamelet solution obtained for χmax = 50000s−1.For a given perturbation time τχ it is possible to find a critical perturbation amplitude Aχcr which marksthe bifurcation between quenching and re-ignition of a steady state burning flame structure. The charac-teristic time τχ has been chosen of the order of a Kolmogorov timescale τK on the basis of experimentaland numerical data for hydrogen/LOx combustion at elevated pressure [22], in which such a timescale wasestimated at τK ≈ 1 µs. For the near critical pressure of 60 bar, the critical perturbation amplitude is foundto be Aχcr = 1.329. The bifurcational behavior of the flame structure subject to scalar dissipation ratesignals around the critical perturbation amplitude, is shown in terms of heat release rate (HRR) and peaktemperature in time Fig. 2.    φ(t) [-]t [s]τχAχ4.5E055.5E056.5E057.5E050 1.0E-06 2.0E-06 500 1000 1500 2000 2500 3000 3500 4000χ max [s-1 ]T max [K]t [s]Aχ=1.30Aχ=1.354.5E055.5E056.5E057.5E050 1.0E-06 2.0E-06 0 2e+13 4e+13 6e+13 8e+13 1e+14χ max [s-1 ]HRR [W/m3 ]t [s]Aχ=1.30Aχ=1.35Figure 2: Scalar dissipation rate signal (left), characterized by function φ(t), a triangular wave defined by an amplitude Aχ and aduration τχ . Time evolution of peak temperature (middle) and heat release rate (right) during signals around the critical amplitude.4The signal characterized by Aχ = 1.30 will allow the re-ignition of the flame structure with the flameletpeak temperature promptly recovering the initial steady state value. On the other hand the signal charac-terized by Aχ = 1.35, because of the increased heat losses will cause the ultimate quenching of the flamestructure with rapid decrease of the peak flamelet temperature toward the mixing lines values. Similar obser-vations can be made for the heat release rate, with the only difference that during the first part of the signalwhen the scalar dissipation rate is increasing, HRR is also increasing since it is still diffusion controlled.Figure 3: Real gas flame structure results for a scalar dissipation rate signal characterized by Aχ = 1.35 and τχ = 1 µs leading toquenching: Temperature T (z, t) (left) and constant pressure specific heat of the mixture Cmixp (z, t) (right)Figure 3 exhibits the unsteady quenching solution signature in (z, t) space. In particular T (z, t) is seen toabruptly relax towards the quenched, adiabatic mixing line solution. The cp(z, t) solution, on the other hand,gradually recovers a clear peak due to near-critical enhancement, which moves to richer values of mixturefraction. In the steady ignited solution, such a peak was confined far nearer the LOx side, permanentlyacting as a a heat sink [17] for the energy equation (2), thus enhancing the quenching phenomena.123450 100 150 200Aχ crp [bar]Figure 4: Critical amplitude of the scalar dissipation rate triangular signal Aχcr at various supercritical pressures.Figure 4 shows how pressure influences the crital amplitude Aχcr for a given perturbation time. In asimilar manner to the quenching value of the scalar dissipation, Aχcr is obseved to increase almost linearlywith pressure. While this hints at a greater absolute resilience of high pressure flames to scalar dissipationperturbations - essentially due to a greater heat release rate - it is equally true that at high pressures, enhancedkinetic timescales are expected to reduce the extent to which scalar dissipation can be increased relative tothe quenching value, for the same perturbation time τχ.54. Conclusion and perspectiveAn a priori analysis of transient LOx-Methane flame structures has been carried out at supercritical pres-sures, utilizing an unsteady laminar flamelet model as a building block of an UFPV method. High-pressureconditions have been chosen as representative of a LOx-Methane rocket engine operating conditions. Realgas effects have been taken into account in unsteady multi-component reacting calculations by means of acomprehensive three parameter equation of state. Re-ignition and quenching of the flame structure undersynthetic turbulent perturbations has been studied using scalar dissipation rate signals. A critical scalar dis-sipation amplitude has been observed to increase with pressure. Transient flamelet solutions are expectedto populate the flamelet state space plane and can thus be representative of flamelet libraries for an unsteadyflamelet progress variable approach. From the combustion modeling point of view additional effort is stillneeded in order to take into account radiative heat losses, differential diffusion and soot formation.References[1] G.P. Sutton and O. Biblarz. Rocket Propulsion Elements. John Wiley and Sons, New Jersey, 8th Edition, 2010.[2] V. Yang. Modeling of supercritical vaporization, mixing, and combustion processes in liquid-fueled propulsion systems.Proceedings of the Combustion Institute, 28(1):925–942, 2000.[3] P. Scouflaire, J-C. Rolon, S. Candel, M. Juniper, and a. Tripathi. 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SandiaNational Laboratories, 1993.[15] Refprop. http://webbook.nist.gov/chemistry/fluid. National Institute of Standards and Technology Webbook, 2014.[16] T. Kim, Y. Kim, and S.K. Kim. Numerical analysis of gaseous hydrogen/liquid oxygen flamelet at supercritical pressures.International Journal of Hydrogen Energy, 36(10):6303–6316, May 2011.[17] P. E. Lapenna, P. P. Ciottoli, M. Valorani, and F. Creta. Numerical investigation of unsteady laminar methane/lox flamelet atsupercritical pressures. European Combustion Meeting 2015, Budapest, Hungary.[18] Stephen B Pope. Turbulent flows. Cambridge university press, 2000.[19] John C. Hewson. An extinction criterion for nonpremixed flames subject to brief periods of high dissipation rates. Combustionand Flame, 160(5):887 – 897, 2013.[20] Y. Xuan and G. Blanquart. A flamelet-based a priori analysis on the chemistry tabulation of polycyclic aromatic hydrocarbonsin non-premixed flames. Combustion and Flame, 161(6):1516 – 1525, 2014.[21] E.L. Petersen, D.F. Davidson, and R.K. Hanson. Kinetics Modeling of Shock-Induced Ignition in Low-Dilution CH 4 / O 2Mixtures at High Pressures and Intermediate Temperatures. 290:272–290, 1999.[22] B Ivancic and W Mayer. Time-and length scales of combustion in liquid rocket thrust chambers. Journal of propulsion andpower, 18(2):247–253, 2002.6

Towards an Unsteady/Flamelet Progress Variable method for non-premixed turbulent combustion at supercritical pressures

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Towards an Unsteady/Flamelet Progress Variable method for non-premixed turbulent combustion at supercritical pressures

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