Development and Implementation of Explicit Reduced

Reaction Models in Supersonic Reacting Shear Flow

Simulations

A. C. Zambon ∗A. T. Sriram † and H. K. Chelliah ‡

Mechanical and Aerospace Engineering, University of Virginia, Charlottesville VA 22904, USA

Starting from any detailed reaction model, an automatic reduction procedure basedon quasi steady-state approximation for selected chemical species and on elimination ofthe fast reactions has been developed. The reduced reaction models developed on thebasis of ignition delay of a fuel-air mixture in a zero-dimensional configuration and laminarflame configurations are then extended to simulations of supersonic reacting shear layers.Predictions of a hydrogen-air shear layer obtained with a detailed kinetic model and reducedreaction models developed are presented indicating good quantitative agreement at variouscross sections in the two-dimensional flow field considered. Further analysis also shows thatthe models can adequately explain basic combustion characteristics associated with thelocation of detached reaction fronts and ignition delay of zero-dimensional configuration.The present simulations are also being extended to shear flows comprising of ethylene-airsystem, including vitiation effects.

I. Introduction

Development of Scramjet combustors requires fundamental understanding of fuel-air mixing, ignition,flame propagation and extinction. Although the flow dynamics in an actual combustor is rather complexdue to various flow structure and shock wave interactions, a reacting mixing layer established betweentwo parallel streams is quite attractive for fundamental studies aimed at validating mixing and finite-ratechemical kinetic models. In particular, this simple flow configuration offers an excellent mechanism forvalidating reduced reaction models currently being developed for supersonic reacting flow simulations, asdescribed below.

In Mach 6-8 flight regime, storable jet fuels or hydrocarbon fuels become a viable alternative to cryo-genic hydrogen fuel systems. However, the performance of hydrocarbon fuel Scramjet combustors can beadversely affected by longer ignition delays associated with relatively slower chemical reaction rates. Inorder to overcome the longer reaction time of hydrocarbon fuels, careful vehicle design optimizations mustbe performed based on accurate, but realistic chemical kinetics models. This is a rather challenging taskconsidering that a typical jet fuel can contain over 300 different hydrocarbon components. Detailed kineticmodels that describe the oxidation of a single component large hydrocarbon fuel (eg. dodecane, methylcy-clohexane, etc.) are rather large (100’s of species in 1000’s of elementary reactions) and invariably containsignificant uncertainties with regard to rate constants. On the other hand, because of the fast thermaldecomposition of larger hydrocarbon molecules, only sub-models containing rate controlling C1-C3 speciesin 100’s of elementary reactions with somewhat well established rate constants may be needed.1–5 In thenear term, even these simple sub-models cannot be implemented in multi-dimensional turbulent reactingflow simulations because of the tremendous computational effort. This is particularly true for flows withlarge Damkohler numbers.6 Fortunately, by considering the characteristic flow and chemical time scales inthe flow field of interest, these elementary reaction models can be systematically “reduced.”7,8

The reduction procedure is based on the implementation of quasi steady-state (QSS) approximation andretains all the species and reactions of the starting elementary mechanism (identified here as the skeletal

∗Research Associate.†Research Associate, Member, AIAA.‡Associate Professor, Associate Fellow AIAA.

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AIAA 2007-772

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mechanism), but species associated with QSS are solved explicitly via algebraic relationships.9 Moreover,knowing that the elementary reaction models and rate constants are continuously being updated as newchemical kinetic data become available, the reduction procedure developed here has been automated usinga MATLAB program, requiring minimal intervention.

II. Reduced mechanisms

Several detailed kinetic models for C1-C3 hydrocarbons have been reported in the literature recently, forexample Wang and co-workers,1 Lawrence Livermore National Laboratory,2 UC San Diego,3 GRI-3.0,4 andothers.5 After considering the applicability of these detailed models in predicting 119 sets of ignition delaydata for H2, CH4, C2H4 and C3H8, the detailed model by Wang and co-workers1 was selected as an examplefor testing the reduction approach. It should be pointed out that the MATLAB based computational toolsdeveloped here can be easily adapted for any changes introduced to the detailed reaction models.

The detailed kinetic model described by Wang and co-workers consisting of roughly 70 species in 500reactions for C2-C3 hydrocarbons (eg. ethylene, propane), can be further stripped down to skeletal kineticmodels which contain only the essential rate controlling reactions. For example, a 31 species in 128 reactionskeletal model was recently developed by Zambon and Chelliah9 which is consistent in complexity to otherskeletal models reported recently. A typical comparison of the ignition delay predictions of a premixed stoi-chiometric ethylene-air mixture using the detailed and the skeletal model derived is shown in Fig. 1. Thesepredictions are obtained by integrating the transient species and energy equations of a zero-dimensional vol-ume. Furthermore, Fig. 2 shows a comparison of the non-premixed counterflow flame extinction predictionsof ethylene-air system using this skeletal model and the detailed model.

3 4 5 6 7 8 9 10 11 12

100

102

104

106

104 / T ( K −1 )

Igni

tion

Del

ay T

ime

( µ

s )

0.1 atm

1 atm

10 atm

Detailed ModelSkeletal Model18−Step Reduced Model15−Step Reduced Model

0 1000 2000 3000 4000 5000 6000 7000 8000 90001400

1500

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2300

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Strain Rate ( s −1 )

Max

imum

Tem

pera

ture

(

K )

Detailed ModelSkeletal Model18−Step Reduced Model15−Step Reduced Model

0.5 atm

1 atm2 atm

5 atm10 atm

Figure 1: Comparison of ignition delay data of a stoi-chiometric mixture of C2H4-air using detailed, skeletaland reduced reaction models (Zambon and Chelliah9).

Figure 2: Comparison of extinction prediction of anon-premixed C2H4-air counterflow flame using de-tailed, skeletal and reduced reaction models (Zambonand Chelliah9).

As mentioned in the introduction, in multi-dimensional, turbulent, transient reacting flow simulations,implementation of such “skeletal” models is not feasible because of the large number of scalar variablesinvolved and the stiffness associated with the “fast” chemical reactions. The only viable option is to usesystematically developed reduced reaction models, where fast reversible chemical reactions are set to partialequilibrium or a set of species is identified in QSS based on groups of fast chemical reactions. For ethylene-airmixtures, previous work has shown that such reduced reaction models can be developed for a narrow rangeof physical-chemical conditions.10,11 Recent work by Zambon and Chelliah9 has shown that by relaxing QSSof a few species in the system, robust 15-18 step reduced reaction models can be developed to completelydescribe the ignition, propagation, and extinction of C2H4-air mixtures, for a wide range of pressures,temperatures, and mixture compositions. A sample of these results is shown in Figs. 1 and 2 indicating thelevel of agreement possible, as well as some deficiencies of the 15-step model at low temperatures and of theskeletal model at high pressures.

The implementation of reduced reaction models in multi-dimensional computational codes with implicittreatment of chemical source terms requires determination of the Jacobian matrix, either analytically ornumerically. As part of the MATLAB reduction program developed, analytical Jacobian matrix is automat-

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ically developed and can be readily implemented in multi-dimensional reacting flow simulations, such as theshear flow simulations described in this paper.

III. Reacting shear flow simulations

Considerable effort has been devoted to simulation of reacting shear flows by numerous authors,12–16

often using various turbulence-chemistry models and simplified chemical reaction models. In contrast, thefocus here is on implementation of systematically reduced reaction models without turbulence-chemistrymodels, but under numerical resolutions approaching DNS like simulations where only molecular transportand chemistry are important. Although grid resolution of the order of 20µm was considered, because theresolution of all spectral features as well as higher-order characteristic boundary conditions were not ad-dressed in this study, we will refrain from identifying the present work belonging to the same class as DNSsimulations.17–19

III.A. Governing Equations

For the two-dimensional flow field considered, the conservative form of mass, x-momentum, y-momentum,total energy and species equations can be written as

∂U

∂t+

∂F

∂x+

∂G

∂y= H, (1)

where the state vector U is represented as U = [ρ, ρu, ρv, ρE, ρY1, ρY2, ...ρYN−1]T . Here ρ is density, u and vare velocity components in x and y directions, respectively, E is total energy and Yi (i = 1, ..., N − 1) is themass fraction of i-th species, with N as the total number of species. For the laminar formulation consideredhere, the flux vectors F and G in x− and y-direction, respectively, are evaluated based on the moleculartransport properties. For highly diffusive hydrogen, the diffusion velocity is obtained either using multi-component diffusion coefficient formulation21 or iteratively solving the full Stefan-Maxwell equation. Thechemical source term H is also evaluated using laminar kinetics, described either using elementary (detailedand skeletal) or reduced kinetic models.

III.B. Numerical method

The above system of equations is integrated using the SPARK2D code developed at NASA Langley Re-search Center,12,14 which is an extension of MacCormak-like schemes with compact difference operators toobtain a higher level of numerical accuracy (formally fourth-order accurate). In addition, we have shownthat SPARK2D code can accurately predict the unsteady detonation phenomena, i.e. galloping detonationwaves.20

The chemical source term H in Eq. (1) is computed implicitly because of the stiffness associated with fastchemical reaction rate terms. Such an implicit scheme requires the determination of the Jacobian matrix,∂H/∂U , which is determined analytically in SPARK2D. In the present implementation, this Jacobian matrixis generated automatically as part of the MATLAB reduction program which saves considerable effort indeveloping alternate reduced reaction models. The thermodynamic, chemical kinetic and transport dataare implemented using the Sandia transport and CHEMKIN codes,21,22 with modifications to include fullStefan-Maxwell equation for species diffusion. Implementation of Sandia chemistry and transport codes inSPARK2D has also greatly facilitated testing of different chemistry models.

In the present integrations, a global computational time step is chosen based on the minimum of the fluid-dynamical time step or the chemical relaxation time step. With the specified initial and boundary conditions,the governing equations are then marched forward in time until steady vortex shedding frequency is attainedor some predetermined integration step is reached (typically 20,000 iterations, followed by another 8,000iterations to obtain average flow features for comparison of chemical kinetic models).

III.C. Computational domain and initial/boundary conditions

The computational domain of the two-dimensional supersonic reacting shear layer considered is shown in Fig.3, with dimensions of 16 mm in stream-wise direction and 8 mm in transverse direction. A finite thicknesssplitter plate of 0.5mm located at y=3.75mm separates the incoming parallel supersonic fuel and air streams

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(unless otherwise mentioned the fuel stream is pure hydrogen). At the inflow, on either side of the splitterplate, a finite-thickness boundary layer is imposed, namely 1.5 mm in air side and 0.8 mm in hydrogen side.While other computational domains and splitter plate thicknesses were considered, the results presented hereare all based on the above values.

Air is assumed to enter through the upper inflow region with free-stream Mach number of Mair,∞=1.5,pressure pair,∞= 0.1013 MPa and temperature Tair,∞=2000 K, and hydrogen is assumed to enter throughthe lower inflow region with free-stream Mach number MH2,−∞=1.1, pressure pH2,−∞=0.1013 MPa andtemperature TH2,−∞=900K. The corresponding convective Mach number is 0.37. We also prescribe a smoothvariation in velocity (from zero to free-stream value) and pressure and temperature (from stagnation to free-stream value) on either side of the splitter plate for the selected boundary layer thicknesses. The splitter plateis assumed to be adiabatic and satisfy no-slip condition. While these inflow boundary conditions correspondto a limiting case, for present reaction model validation purposes exactly the same conditions are used.

At the inflow boundary, the pressure and species mass fractions are prescribed by setting the stream-wisegradient to be zero. At the other three boundaries, all the variables are extrapolated form the interior.Although non-reflective boundary conditions must be prescribed at the exit, usually one-order less than theinterior domain, as mentioned above, the main focus here is on the implementation and validation of reducedreaction models. Hence the outflow boundary condition implementation is treated as simple as possible.

For the hydrogen-air case, the above physical conditions were selected such that a detached, vigorouslyreacting region is established near the mid point of the computational domain considered as seen in Fig. 4,yielding a truly kinetic controlled flame holding conditions. In future investigations, effects of inflow andoutflow boundary conditions based on Navier-Stokes characteristic boundary conditions will be investigated,including any effects of the computational domain selected on the predicted flame standoff distance.

0 16

8

Figure 3: Schematic of the two-dimensional com-putational domain with a splitter plate located aty =3.75mm.

Figure 4: For the 20 µm grid resolution, the H2O massfraction contours showing a detached reaction regionoccurring within the vortical structures.

III.D. Grid resolution study

For the purpose of validation of chemical kinetic models, accurate numerical resolution of the flame front isabsolutely critical. The numerical grids are treated uniformly in the stream-wise direction, while they arestretched in the transverse direction with fairly uniform grid near the center of the mixing region (see Fig.3). Three different grid resolutions were considered so far, namely 100 µm, 40 µm and 20 µm. While allsimulations capture the formation of large-scale vortical structures, there are clear differences in the details.For example, in the present computational domain of 8x16mm, the fine grid of 20µm show about 10 coherentvortical structures, while this number drops to 9 for 40 µm and 7 for 100 µm. Besides such global featuresof the flow, the grid resolution effects were addressed by averaging simulations over 8,000 iterations (aftersteady vortex shedding frequency is attained — after 20,000 iterations) and by performing detailed compar-isons. Figure 5 shows an example of the averaged water vapor mass fraction for a grid resolution of 20 and100 µm, respectively, showing distinct differences.

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(a) (b)

Figure 5: Averaged H2O mass fraction for a computational grid of (a) 20 µm and (b) 100 µm.

Subsequent comparison of the transverse profiles of water vapor are shown in Fig. 6, for two differentcross sections (x=10 mm and x =15 mm downstream of the inflow boundary). Such comparisons clearlyindicate that even with a 20 µm grid resolution, the numerical grid adopted may be not adequate. Fur-ther higher resolution investigations are currently underway to address the resolution limits needed for thepresent fourth-order scheme implemented. Besides water vapor comparisons, other flow and scalar variablesalso show similar discrepancies with varying grid resolution. In all these results, the wider reaction zonepredicted with the coarser 100 µm grid is a direct consequence of increased numerical dissipation.

(a)

0 0.05 0.1 0.15 0.20

2

4

6

8

H2O Mass fraction

y −

dis

tanc

e (m

m)

100 µm grid40 µm grid20 µm grid

(b)

0 0.05 0.1 0.15 0.2 0.250

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4

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8

H2O Mass fraction

y −

dis

tanc

e (m

m)

100 µm grid40 µm grid20 µm grid

Figure 6: Comparisons of water vapor mass fraction for three different grid resolutions at x=10 mm and x=15 mm.

In the remainder of this paper, we use 20µm grid resolution in validation of our implementation of reducedreaction models for H2-air case, while 40 µm grid is used for C2H4-air case.

IV. Results and discussion

When the lip thickness of the splitter plate is sufficiently large, the flow experiences a recirculation aroundthe lip, which develops into a classical vortex shedding phenomena, enhancing the mixing of the initiallyseparated supersonic fuel and air streams. Subsequent chemical reaction is localized within the vorticalstructures with longer residence times. With characteristic flow residence times of the order of 10-30 µs,this supersonic shear layer configuration offers an excellent mechanism for validation of molecular mixingand finite-rate chemical kinetic models, including reduced reaction mechanisms appropriate for hypersoniccombustion applications.

IV.A. Hydrogen-air supersonic shear layers

The simplicity of hydrogen-air detailed reaction model (typically 9 species and 18-25 elementary reactions)offer an excellent system for validation of reduced reaction modelling approach, as described here. For

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the inflow free-stream boundary conditions described in section III.C (i.e. MH2,−∞ = 1.1, Mair,∞ = 1.5,TH2,−∞ = 900K, Tair,∞ = 2000K, δsplitter=0.5mm), the resulting contour plots of Mach number, tempera-ture, and mass fraction of H2, and OH after 20,000 iterations (roughly 60 µs) are shown in Fig. 7. Theseresults are using a detailed mechanism for hydrogen-air consisting of 9 species in 18 step elementary reactionmodel.

Mach number Temperature

H2 OH

Figure 7: Contour plots of Mach number, temperature, mass fraction of H2 and OH, using the detailed kinetic model.

Furthermore, as discussed in section III.D, for detailed comparison purposes the simulation results up to8,000 iterations were averaged as shown in Fig. 5a (H2O) and in Fig. 8 (temperature and H2). In particular,in the well resolved simulations, the averaged H2O mass fraction shows two-distinct layers of reaction whereintense H2O production is occurring (see Fig. 5a). The origin of these two layers may be related to thereaction occurring within the pairs of alternating vortical structures being shed from the splitter plate.

Temperature H2

Figure 8: Contour plots of averaged temperature and mass fraction of H2, using the detailed kinetic model.

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For the present hydrogen-air shear layer simulations, starting with the 9 species, 18-step elementaryreaction mechanism, a 5-step reduced reaction model was developed by selecting HO2 to be in steady-state.This 5-step model can be represented by

O2 + H2O2 = 2 HO2, (I)

HO2 = O2 + H, (II)

H2 = 2 H, (III)

H2O + O = H2 + O2, (III)

2 H2O + 2 H = 3 H2 + O2. (V )

Together with the analytical Jacobian matrix derived, implementation of this 5-step reduced reaction modelfor hydrogen-air in SPARK2D yields a similar mixing layer flow structure as the detailed model shown inFigs. 5,7-8. Instead of comparing such contour plots obtained with the detailed and reduced reaction model,quantitative comparisons can be made based on line plots of flow variables across the mixing layer as shownin Figs. 9-12, for two different down stream locations, namely x = 10mm and x = 15mm. The favorablecomparisons between the detailed mechanism and the 5-step reduced mechanism developed based on zero-dimensional ignition delay, laminar flame propagation, and counterflow flame extinction phenomena clearlyindicate the robustness of the mechanism reduction approach. Further parametric investigations are neededto ensure that these comparisons are truly rate dependent.

x = 10mm

500 1000 1500 2000 2500 30000

2

4

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8

u − velocity (m/sec)

y −

dis

tanc

e (m

m)

5 − stepDetailed

x = 15mm

500 1000 1500 2000 25000

2

4

6

8

u − velocity (m/sec)

y −

dis

tanc

e (m

m)

5 − stepDetailed

Figure 9: Comparison of the averaged velocity u across the mixing layer obtained using the detailed and 5-step reduced model

for hydrogen-air, for (a) x = 10mm and (b) x = 15mm.

x = 10mm

800 1000 1200 1400 1600 1800 20000

2

4

6

8

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y −

dis

tanc

e (m

m)

5 − stepDetailed

x = 15mm

800 1000 1200 1400 1600 1800 20000

2

4

6

8

Temperature (K)

y −

dis

tanc

e (m

m)

5 − stepDetailed

Figure 10: Comparison of the averaged temperature across the mixing layer obtained using the detailed and 5-step reduced

model for hydrogen-air, for (a) x = 10mm and (b) x = 15mm.

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x = 10mm

0 0.01 0.02 0.03 0.04 0.05 0.060

2

4

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8

H2O Mass fraction

y −

dis

tanc

e (m

m)

5 − stepDetailed

x = 15mm

0 0.05 0.1 0.15 0.20

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4

6

8

H2O Mass fraction

y −

dis

tanc

e (m

m)

5 − stepDetailed

Figure 11: Comparison of the averaged H2O mass fraction across the mixing layer obtained using the detailed and 5-step

reduced model for hydrogen-air, for (a) x = 10mm and (b) x = 15mm.

x = 10mm

0 2 4 6 8

x 10−3

0

2

4

6

8

OH Mass fraction

y −

dis

tanc

e (m

m)

5 − stepDetailed

x = 15mm

0 0.005 0.01 0.0150

2

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OH Mass fraction

y −

dis

tanc

e (m

m)

5 − stepDetailed

Figure 12: Comparison of the averaged OH mass fraction across the mixing layer obtained using the detailed and 5-step

reduced model for hydrogen-air, for (a) x = 10mm and (b) x = 15mm.

IV.B. Analysis of flame standoff and ignition delay

Based on the average velocity in x-direction, the flow residence time can be evaluated as a function of thedistance measured from the inflow boundary, as shown in Fig. 13. This integrated time depends somewhaton the inflow y-location (or the j-th grid value), especially in the vicinity of the splitter plate. Consideringthe induction length of about 8mm for the present physical parameters selected (see OH contours in Fig. 7),the integrated flow residence time is of the order of 20 µs, which should correspond to the ignition time ofa fluid element being transported. For a fluid element containing a stoichiometric mixture of hydrogen-airat atmospheric pressure, the zero-dimensional ignition model described in section II yields an ignition delayof about 20 µs at a temperature of about 1350 K, as shown in Fig. 14. This temperature of 1350 K isessentially slightly below the average temperature of the mixing layer with free stream temperatures of 900and 2000K.

0 5 10 150

10

20

30

40

50

x − distance (mm)

Res

iden

ce T

ime

(µ s

)

y = 4.00 mmy = 4.25 mm

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−1

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Igni

tion

Del

ay T

ime

( µ

s )

0.1 atm

1 atm

10 atm

10 atm

Detailed ModelSkeletal Model18−Step Reduced Model15−Step Reduced Model

Figure 13: Estimated flow residence time in the shearlayer based on averaged u velocity component in thevicinity of the splitter plate.

Figure 14: Predicted ignition delay for a stoichiomet-ric mixture of H2-air at p=1 atm based on the zero-dimensional model.

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Such favorable comparisons between the ignition delay of a zero-dimensional problem and the induc-tion time of a multi-dimensional shear flow problem lends support to the idea of mechanism reduction viasimplified reacting flow geometries.

IV.C. Ethylene-air supersonic shear layers

Unlike the hydrogen-air system, simulation of two-dimensional shear layers with a detailed reaction model forethylene involving 31 species is computationally expensive. Fortunately, the 18-step reduced reaction modeldescribed in section II, together with the analytical Jacobian matrix derived, can be readily implemented inthe present shear layer configuration.

Because the ignition delay of stoichiometric C2H4-air mixture is at least an order of magnitude greaterthan that of a H2-air mixture (see Figs. 1 and 14), establishing a strong reacting front within the small com-putational domain considered (8mm×16mm) requires use of elevated inflow temperatures (Tair,∞ = 2200Kand TC2H4,−∞ = 2000K). Predictions under such conditions has been initiated and are shown in Fig. 15. Fortemperatures over 2000K considered here, the definition of ignition location becomes somewhat uncertainbecause of the substantial radical pool formed at the vicinity of the inflow boundary. In the near future,parallelized version of the code will be used to extend above simulations to lower inflow temperatures andinvestigation of vitiation effects. Clearly, as the inflow temperatures is reduced, the large computational do-mains required necessitate the use of well validated turbulence-chemistry models, including validation withexperiments.

C2H4 H2O

Figure 15: Contour plots of mass fraction of C2H4 and H2O of a reacting ethylene-air shear layer using the 18-step reduced

reaction model.

V. Conclusions

Reduced reaction models developed for hydrogen-air mixtures have been implemented in simulation of asupersonic reacting shear layer established behind a finite-thickness splitter plate. Because of the simplicity ofhydrogen-air detailed reaction model, this system has facilitated the validation of reduced reaction modellingapproach by making extensive comparisons with the detailed model predictions. Besides the qualitativecomparisons of the shear flow structure, averaged flow variables obtained with reduced and detailed modelsindicate excellent agreement across the shear layer at various downstream locations. The flame standoffdistance (or the induction time) demonstrate the excellent agreement with the ignition delay data obtainedusing the zero-dimensional ignition simulations under similar pressure, temperature, and concentrations,allowing efficient development of reduced reaction models.

Grid resolution studies indicate that the integration of laminar conservation equations with the fourth-order accurate algorithm of SPARK2D code does not quite attain the desired grid independence limit for∆x ≈ ∆y = 20µm. Higher resolution studies are currently underway to establish this limit. Furthermore, inthe boundary condition implementations, especially in the subsonic outflow regions, uncertainties associatedwith the predicted results are being addressed via implementation of non-reflective boundary conditions.Irrespective of the above deficiencies, with judicious selection of chemical and physical parameters, thepresent flow configuration offers an excellent mechanism of validating mechanism reduction approaches,

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especially fuels containing large hydrocarbon molecules.

Acknowledgments

We wish to express our appreciation to Dr. Phil Drummond for encouragement of this work and to Mr.George Rumford, program manager of the Defense Test Resource Management Center’s (DTRMC) Test andEvaluation/Science and Technology(T&E/S&T) program, for funding of this effort under the High SpeedHypersonic Test focus area.

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