m. deepu, s. s. gokhale and s. jayaraj- numerical simulation of shock-free shear layer interaction...

8/3/2019 M. Deepu, S. S. Gokhale and S. Jayaraj- Numerical Simulation of Shock-Free Shear Layer Interaction in Reacting Flo

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International Journal of Dynamics of Fluids

ISSN 0973-1784 Vol.2, No.1 (2006), pp. 55-71

Research India Publications

http://www.ripublication.com/ijdf.htm

Numerical Simulation of Shock-Free Shear Layer

Interaction in Reacting Flows

M. Deepu1, S. S. Gokhale

2and S. Jayaraj

3

1Lecturer in Mechanical Engineering

N S S College of Engineering, Palakkad, Kerala-678 008, India

Email: [email protected] in Aerospace Engineering

Indian Institute of Technology Madras-600 036, India

Email: [email protected] in Mechanical Engineering

National Institute of Technology Calicut, Kerala-673 601, India

Email: [email protected]

Abstract

The performance of any supersonic combustor system depends on efficient

injection and complete burning. Computational analysis of the flow field

associated with supersonic combustors is presented. Results are obtained by

numerically solving unsteady, compressible, turbulent Navier-Stokes

equations, using Unstructured Finite Volume Method (UFVM) incorporating

RNG based - two equation model and time integration using three stageRunge-Kutta method. The developed numerical procedure is based on the

implicit treatment of chemical source terms by preconditioning. Reaction is

modeled using an eight step H2-air chemistry. The code is validated against

standard experimental data. The analysis could demonstrate the effect of

interaction of oblique shock wave with hydrogen stream in its mixing withcoaxially flowing air and subsequent reaction.

Key words: SCRAMJET, Supersonic combustor, Reduced chemistry, Point

implicit method, FVM.

IntroductionThe recent interest in single stage to orbit trans-atmospheric vehicle has lead to the

development of a hypersonic flight [1-5], which incorporates a supersonic combustor.

Supersonic Combustion Ramjet engine (SCRAMJET) benefits from the better

performance of air breathing propulsion system. Scramjets need a combustor that


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56 M. Deepu, S.S. Gokhale and S. Jayaraj

should have efficient mixing and combustion of fuel with air at supersonic speeds

without much pressure loss [5, 6]. Many experimental and numerical analyses have

been reported during the last few decades regarding the characteristics of the complex

flow field resulting due to fuel air mixing and combustion. Many fuel injection and

flame holding techniques which can have efficient fuel oxidant mixing with accurate

burning rate have been developed but there is always a trade-off between mixing

effectiveness and pressure drop occurring inside the combustor affecting the total

propulsive thrust available at the nozzle.

Performance of a supersonic combustor system depends on efficient injection and

complete burning. Future hypersonic vehicles are expected to require the performance

and operability benefits from air breathing propulsion systems as it is providing high

specific impulse [3]. The use of supersonic combustors in such vehicles requires

efficient supersonic combustion in combustor lengths short enough to be compatiblewith practical engine sizes.

Combustion modeling simulates the chemical and physical evolution of a complex

reactive flow system by numerically solving the governing time dependent

conservation equations of mass, momentum and energy. The reaction zone propagates

as a consequence of strong coupling between combustion chemistry and the

appropriate fluid mechanical process. The combustion model should accurately

predict the strong interaction between the energy released from chemical reactions

and the dynamics of fluid motion. The chemical reaction leads to the generation of

gradients in temperature, pressure and density, which in turn influence the transport of

mass, momentum and energy. Macroscopically these gradients may lead to the

development of vorticity or affect diffusion of mass and energy. On a microscopicscale, these gradients develop turbulence and thereby influence mixing and flame

velocities. The interaction of fuel with oxidizer is very difficult to predict in such

systems due to high non linear dependence of various flow parameters and chemical

source terms. Therefore predicting strong interplay between chemistry and fluid

dynamics in reacting flows is the real challenge in combustion modeling. The use of

supersonic combustors in atmospheric vehicles requires efficient combustion within

combustor lengths which is short enough for practical engine sizes. Supersonic

reacting flow field can be simulated by adding finite rate chemistry to standard

compressible Navier Stokes equations. Both turbulence and chemical kinetics are

important, since the residence time inside the combustor is much smaller. Explicit

treatment of all conservation terms with reaction chemistry results in stiff equationsand it will degrade the performance of numerical method as flow field and chemical

kinetics with differing time scales need to be solved simultaneously. Existence of

several non-equilibrium states creates challenges in solution procedure. Governing

equations of turbulent shear layer flows involving finite rate chemistry are often

difficult to solve due to stiffness (Ratio of largest time scale to smallest). Stiffness

will degrade the performance of numerical methods. While handling two differing

time scales, both can be advanced equally in pseudo time. In other words, it can be

treated as way of rescaling the equation in time such that both phenomena evolve at

comparable pseudo time scales resulting in higher time steps for integration.

Comparison of the numerical result has been done with the standard axisymmetric


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Numerical Simulation of Shock-Free Shear Layer 57

reacting free shear flow experimental measurements of Cheng et al. [7] The predicted

heat release and species production rates are found to have reasonable agreement with

the experimental results. The analysis has been extended to study the mixing and

reacting behavior of hydrogen injection issuing into hot supersonic air stream with

angled inlet.

Governing Equations of Compressible Reacting FlowsThe conservation form of equations, which govern a two dimensional turbulent

compressible flow can be expressed in a generic form for axisymmetric flows as

1 ( )U F rGS

t x r r

+ + =

(1)

Where

( )

2

, F=

xx

r xr

rxx xr r eff

i

i

u

u

uvu

Tv E u v K xU E

ux

uYx

uY

+

+ + + =

( ) ( )

2

0

0

1

G= and0

r

r rx

r xr

rr r rx eff

r

r

i

r i

v

uv

v

T PE v u K rr S

v Hr

Hv

r

v Y

+

+ + + +

=

Here

2 2 12

3 3

rxx eff

u u rvP

x x r r

= + +

rxr rx eff

u v

r x

= = +


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2 2 12

3 3

r rrr eff

v u rvP

r x r r

= + +

2 12

3

r reff

v u rv

r x r r

= +

In the above versions of formulations, the effective viscosity of the flow field

tleff += (2)

From Sutherlands law laminar viscosity (Ns/m2) for different temperature (K) is

found as

l

=1.458

10-6

4.110

5.1

+TT

andt

is found from turbulence model

The effective thermal conductivity

eff l t K K K= + (3)In which

Pr

Cpk ll

=

t

t

t

Cpk

Pr

=

For the present analysis modified - model called Renormalisation Group (RNG)model was used. Yakhot et al. [8] proposed this model, which systematically removes

all the small scales of turbulence motion from the governing equation by expressing

their effects in terms of large scales and a modified viscosity. Following are the RNG

- model.

( ).( ) [ ]

k eff u div grad H

t

+ = +

(4)

( )( ) [ ]

k eff div U div grad H

t

+ = +

(5)

Here the turbulence source terms are defined as2

*

1 22 and H 2 .t ij t ij ijH E C E E C

= =

Turbulent viscosity is defined as2

tC

= (6)

Closure coefficients are evaluated as

C=0.0845, ==1.39, C1=1.42 , C2=1.68/

ij ij(2E .E ) =


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and C1*=

3

0

1

)/1(1

+

C

0=4.377, =0.012Value of constant is adjustable which is found from near wall turbulence data. Allother parameters are explicitly computed as part of RNG process. As noted in the

source term of the governing equations contains the chemical source terms.

. Theseterms represent the generation or destruction of species i due to chemical reaction. For

any chemical reaction with NRnumber of elementary reactions involving NSnumber

of species

1 1

.' ''

1 1

' ''

( ) ( '' ' )

S S

S S

ji ji

j j

N N

ji i ji i

i i

N N

i ji ji f bji i

C C

C k C k C

= =

= =

=

(7)

Where i=1,2,3.NS represents species and j=1,2,3NR represents reactions

Net change in concentration of any species can be found as. .

1

RN

i i

j j

C C=

=

(8)

And the net production of species is given by

iii WC..

= (9)

iW represents the molecular weight of individual chemical species.

Reaction rates in the above equation can be found from Arrhenius law

expj

j

j

N jf j j

f jb

eq

Ek A T

RT

kk

k

=

=(10)

Herejeqk is the equilibrium reaction constant.

An eight-step reaction mechanism proposed by Evans and Schexnayder [9] has been

used here for modeling hydrogen-air chemistry for which the reaction steps and

reaction rates are summarised in Table 1.


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Table 1: Reaction steps and reaction rates of eight step mechanism.

Forward Rates Backward Rates

Reaction A N E A N E

H2+M=H+H+M 5.51018 -1.0 51987.0 1.81018 -1.0 0.0O2+M=O+O+M 7.21018 -1.0 59340.0 4.01017 -1.0 0.0H2O+M=H+OH+M 5.21021 -1.5 59386.0 4.41020 -1.5 0.0OH+M=O+H+M 8.51018 -1.0 50830.0 7.11018 -1.0 0.0H2O+O=OH+OH 5.81013 0.0 9059.0 5.31012 0.0 503.0H2O+H=OH+H2 8.41013 0.0 10116.0 2.01013 0.0 2600.0O2+H=OH+O 2.21014 0.0 8455.0 1.51013 0.0 0.0H2+O=OH+H 7.51013 0.0 5586.0 3.01013 0.0 4429.0Here M is a third body. The reaction rates are expressed in Arrhenius law form.

Units of A are multiple of cm3.

mole-1.

s-1.

E is in Joules

Evaluation of thermodynamic properties specific heat at constant pressure, enthalpy

and entropy are respectively found from standard thermodynamic data from Mc Bride

et al. [10] as.

2 3 4ip

i i i i i

cA B T C T DT E T

R= + + + + (11)

2 3 4

2 3 4 5i i i i i i

iH B C D E F A T T T T RT T

= + + + + + (12)

2 3 4ln

2 3 4

i i i ii i i

S C D E A T BT T T T G

R= + + + + + (13)

For each species two sets of coefficients (Ai-Gi) are used for the temperature intervals,

one applicable from 300K up to 1000K and the other applicable from1000K up to

3000K. Total energy of flow field is given by

( )221

2

1vu

phYE

iN

i

ii ++= =

(14)

Temperature is worked out from the above equation using the Newton-Raphsonmethod from the modified energy equation. The pressure is calculated from the

resulting temperature as follows

1

iN

i

i i

Yp R T

W

=

= (15)

Numerical ModelingTo analyse the SCRAMJET combustor flow field unstructured finite volume method

is employed here. The unstructured grid techniques are able to idealise and analyse

geometrically complex problems with more ease compared to conventional structured


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grid method. The finite volume Runge-Kutta scheme suggested by Jameson et al. [11]

is used in the present analysis.

The conservation equation for a control volume can be written as

.. 0i

i i i

faces

dUV F ds S V

dt+ = (16)

Vi is the cell volume and ds is the area of elemental sides. This is a system of ordinary

differential equation. For obtaining the solution this has to be integrated with respect

to time.

For a control cell ABCD as shown in Fig. 4.1 above FVM formulation can be writtenas

( )ij ABCD ij

Ud fdy gdx Sd t

+ =

(17)

Here f and g are the Cartesian components of flux vector. The area is evaluated from

the vector product of the diagonals as

1

2ABCD AC BD

x x = (18)

[ ]1

( )( ) ( )( )

2

ABCD C A D B D B C Ax x y y x x y y = (19)

The cell averages of the derivative of different flow variables for the surface

described by a quadrilateral ABCD can be evaluated as follows.

( )( ) ( )( )1

2A C B D B D A C

ABCDUdy U U y y U U y y= (20)

Relation for the x derivative is

1U Ud

x x

=

(21)

( )( ) ( )( )

( )( ) ( )( )A C B D B D A C

A C B D B D A C ABCD

U U U y y U U y y

x x x y y x x y y

=

(22)

A

B

C

D

Fig. 4.1. An arbitrary shaped plane quadrilateral


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Relation for the r derivative is

1U Udr r

= (23)

( )( ) ( )( )

( )( ) ( )( )A C B D B D A C

A C B D B D A C ABCD

U x x U U x x U U

r x x y y x x y y

=

(24)

In order to avoid oscillations near shocks and other uncoupled errors, the numerical

solution procedure needs to incorporate artificial dissipation. For the present analysis

the numerical dissipation terms are formed with Laplacian and Biharmonic operators.

The dissipation operation can be written as

(2) (4)

( ) ( ) ( )i i iD U d U d U = + (25)where d

(2)(Ui) represents the contribution of the undivided Laplacian operator and

d(4)

(Ui) is the contribution of the Biharmonic operator. Biharmonic operator does the

background dissipation to damp high frequency uncoupled error modes.

Laplacian operator for the ith

cell is given by

=

=6

1

2)(

k

iki UUU (26)

where, the summation in k is taken over all control volumes which have the common

interface with ith

cell.

The Biharmonic operator is given by

)(2

1)( 22

6

1

)4()4(

ik

k k

k

i

i

iki UUt

V

t

VUd

+

= =

(27)

The Biharmonic operator is not operated near shock as this will produce pre and

post shock oscillations.

Here

)(,0max)2()4()4(

ikik k =

With the addition of artificial dissipation terms the space discretisation will become

( )( ) ( ) 0i

i i i

d UV R U D U

dt

+ = (28)

Explicit treatment of all conservation terms with reaction chemistry results in stiff

equations and it will degrade the performance of numerical method. Flow field and

chemical kinetics with differing time scales need to be solved simultaneously.

Bussing and Murman [12] introduced the method of preconditioning the

conservation equations in conjunction with chemical source terms alone being treated

implicitly. Such a method has the advantage of both explicit and implicit methods.

While handling two differing time scales, both can be advanced equally in pseudo

time. In other words, it can be treated as way of rescaling the equation in time such

that both phenomena evolve at comparable pseudo time scales resulting in higher time

steps for integration.


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Thus the modified equation is

1 ( )

U F rGSJ St x r r

+ + = (29)

The point implicit formulation of the time stepping can be written as

( )

( )

( )

(0) ( )

0 (1) (0) (0) (0)

1

1 (2) (0) (1) (0)

2

3 (3) (0) (3) (0)

3

1 (3)

( )

( )

( )

n

i i

i i i i

i i i i

i i i i

n n

i i i

U U

tiSJ U U R D

Vi

tiSJ U U R D

Vi

ti

SJ U U R DVi

U U U

+

=

=

=

=

= +

(30)

The point implicit scheme in which all the six chemical species terms are treated

implicitly and all other terms explicitly. The preconditioning Matrix SJ used for this

purpose is given by

=

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

. . .

. . .

. . .

1 . . . . .

1 . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . 1

H H H

i i i

H O H O

O O O

i i i

H O H O

H O H O H O

i i i

H O H

t t tY Y Y

t t tSJ

Y Y Y

t t tY Y Y

O

(31)

Code Validation and Test ResultsCoaxial Hydrogen-Air Supersonic Mixing and Reaction

Numerical simulations were performed on an axisymmetric configuration consisting

of coaxial jets (Hydrogen and vitiated air) exiting at supersonic velocities intoambient air. Cheng et al. [7] made measurements of temperature and composition

using ultra-violet Raman scattering and laser induced predissociative fluorescence

techniques. These experimental results are used for validation. The flow conditions

have been summarized in Table 2.

The Details of computational domain is given in Fig. 1. The left face of the

computational domain is given with a supersonic inflow condition in the region of

supersonic air stream. The burner lips were assumed to non-catalytic. Hydrogen jet

with above mentioned conditions is introduced at respective position as separate

boundary condition. Remaining portion in left face is considered as no slip wall. Top

face is a no slip wall. A supersonic out flow condition is maintained at outlet. Since


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the entire flow field is symmetrical about axis of the coaxial jet, a symmetry condition

is utilized there. The computational domain was discretised in to 104000 control

volumes. The initial condition for entire domain is ambient conditions. The grid was

added to allow specification of the entrainment boundary condition well upstream of

the coaxial jets. The grid is clustered to give more resolution near the lip regions in

the transverse direction, and near the burner exit in the streamwise direction. The

criteria chosen for steady state solution is the density error in successive steps falls

less than 1.010-06.

Table 2: Coaxial burner exit conditions.

H2 Jet Air Jet Ambient

Air

Diameter(mm)

2.36 17.8 120

Mach

Number

1.0 2.0 0

Pressure

(MPa)

0.112 0.107 0.101

Temperat

ure (K)

545 1250 300

Mass

Fraction

21.0

HY =

2

2

0.245,

0.175

O

H O

Y

Y

=

=

2

2

0.233,

0.01

O

H O

Y

Y

=

=

Figure 2 shows comparison between experimental data and numerical prediction of

temperature at exit which is 101 mm from exit. Reasonable match has been obtained

inside core region of the jet and a slight difference in the mixing layers.

Figure 1: Details of computational domain


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The difference is due to initial turbulence level specification given at the inlet and also

due to the numerical dissipation effects. Figure 3 gives the quantitative comparison of

water mole fraction profiles at exit. The non-interference of fuel core to the outer

region can be noticed. The species production rates in the mixing region agree well

with experimental values. The water mole fraction values at core region could not

established. The prediction capability of the code developed is proven by this

comparison.

Figure 2: Temperature profile at exit.

Figure 3: Water mole fraction profile at exit.

The temperature plot for the entire domain is given in Fig. 4. The mixing becomes

more predominant in the region far from the jet outlets and heat release gradually

increases in the mixing layer between hydrogen and air. Non reacting fuel core

extends up to 30 mm from jet outlets. Outer layer between supersonic air stream and


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ambient has almost same temperature through out the length, beyond this non reacting

core region.

Figure 4: Field view of temperature for the entire domain.

Though the typical eight step chemical kinetic mechanism employed here is

performing well when coupled with two dimensional turbulent compressible Navier-

Stokes equation; the introduction of seven species has created difficulty in

computational procedure. The total number of conservation variables increased to

thirteen including the chemical species which will pose a real challenge while

applying for scramjet combustors of larger geometry.

Shock-free shear layer interaction in reacting flows

In real scramjet combustor, the flow filed is extremely complex. Various mixing

enhancement studies [13,14] used strut based fuel injectors. The parallel injection hasthe advantage of momentum addition with minimum pressure loss. But the mixing

efficiency of such a system is very low when compared to transverse injections. The

mixing behavior of the planar jet can be enhanced by the artificial production of

recirculation zone due to shock wave-expansion wave pattern. For the present analysis

an oblique shock is created in the inlet of the flow passage by a 80

wedge. The oblique

shock emanated from the inlet will interact with coaxially injected hydrogen stream.

The interaction of shock with shear layer formed between air and hydrogen streams

will enhance mixing and thereby promotes reaction. The details of computational

domain are given in Fig. 5. The flow conditions have been summarized in Table 3.

Figure 5: Details of computational domain.


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Table 3: Flow conditions.

H2 Jet Air Jet

Diameter (mm) 6 60

Mach Number 1.0 3.0

Pressure (MPa) 0.2 0.107

Temperature (K) 400 900

Mass Fraction 21.0HY = 2 0.245OY =

The computational domain is discretised in to 54,000 control volumes. Figure 6

gives Mach contour plot. Low velocity regions can be identified along the path of

progress of the hydrogen jet. Alternate compression and expansion takes place for the

jet and is not enough to perturb the flow field much in the region near to the jet

outlets. But the shock wave reflections interfere with the upcoming jet and localised

low velocity regions are produced. Though these regions are responsible for pressure

loss of the jet, certainly enhances mixing and reaction. The computed pressure

contours are depicted in Fig. 7. Due to chemical reaction enormous amount of heat is

released at mixing layer. The variation of temperature along radial direction for

various axial locations is plotted in Fig. 8. The mixing and reaction is promoted by

shock induced vortex generation. Figure 9 shows the variation of the water massfraction along radial direction for various axial locations.

Figure 6: Mach contour plot.

Figure 7: Computed pressure contours.


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Figure 8: Variation of temperature for various axial locations.

Figure 9:Variation of water mass fraction for various axial locations.

The effect of mixing and subsequent reaction is more down stream areas. The shock

reflections are responsible for blocking the development of jets in the shear layer and

thereby creating the low velocity regions. Since the residence time associated with the

flow field in such areas is more, the reaction is more predominant in such areas

leading to enhanced heat release. The numerical scheme could demonstrate such a

complex flow field.


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ConclusionA point implicit finite volume solver has been developed and tested for the supersonic

mixing and reaction of parallel streams of hydrogen and air. The implicit treatment of

the chemical species in the source term reduces the stiffness considerably and higher

CFL almost equal to that of non-reacting situations has been obtained. Reasonable

match has been obtained with experiment for the temperature inside core region of the

jet and a slight difference in the mixing layers. Though the species production in the

core of the jet is not predicted well, but it could almost accurately follow that in the

mixing layer.

Future works aims at the influence of initial level of turbulence in controlling

reaction. Though the typical eight step chemical kinetic mechanism employed here is

performing well when used along with two dimensional turbulent compressible

Navier-Stokes equation; the introduction of seven species has created difficulty incomputational procedure. The total number of conservation variables has risen to

thirteen including the chemical species. It will set a real challenge while applying for

scramjet combustors of larger three dimensional geometry with more number of grids.

AcknowledgmentsThe first author thanks T. Jayachandran, Head of Fluid Mechanics and Thermal

Analysis Division of Propulsion group, Vikram Sarabhai Space Centre,

Thiruvananthapuram, India, and his team members for their valuable suggestions for

this work.

Nomenclature

c Sound velocity

Ci Concentration of a species

Cp Specific heat at constant pressure

Cv Specific heat at constant volume

D Artificial dissipation

E Total energy

Eij Deformation tensor

Ej Threshold energy of a reaction F&G Flux vectors

G Gibbs free energy (kJ/kg)

H Enthalpy

K Thermal conductivity

k Reaction rate

p Pressure

Q Heat flux

r Radial direction

S Source term

Si Entropy

t Time

T Temperature

U Vector of conservative variables


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V Volume of the control volume

u Favre average velocity in axial direction

vr Favre average velocity in radial direction

Yi Mass fraction of species

Adjustable closure coefficient

Kinetic energy dissipation rate

Molecularity of a reaction

Kinetic energy of turbulence

Viscosity

Kinematic viscosity

Density

Shear stress Surface area

Cell volume

Rate of production of species

Subscripts

f Forward reaction

b Backward reaction

i Chemical species

j Reaction step

lLaminar

r Radial direction

t Turbulent

Superscripts

Product

Reactant

References

[1] Seiner, J, M., Dash, S. M., and Kenzakowski, D. C., (1999), HistoricalSurvey on Enhanced Mixing in Scramjet Engines, AIAA Paper 99- 4869.

[2] Tishkoff, J. M., Drummond, J. P., Edwards, T. and Nejad, A. S., (1997),

Future direction of supersonic combustion research: Air Force/NASA

Workshop on Supersonic Combustion, AIAA paper 97-1017.

[3] Voland, R.T., Rock, K.E., Huebner, L.D., Witte, D.W., Fischer, K.E., and

McClinton, C. R. (1998), Hyper-X Engine Design and Ground Test

Program, AIAA Paper 98-1532.

[4] Mercier, R. A. and Ronald, T. M. P., (1996), U S Air force hypersonic

technology Program, AIAA Paper, AIAA 7th International Space planes and

Hypersonics Systems and Technology Conference, Plenary Session.


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[5] Gruber, M. R., and Nejad, A. S., (1995) New Supersonic Combustion

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[6] Tomioka, S., Murakami, A., Kudo, K., and Mitani, T., (1998), Combustion

Tests of a Staged Supersonic Combustor with a Strut, AIAA paper 98-3273.

[7] Cheng, T. S., Wehrmeyer, J. A., Pitz, R. W., Jarret, O., and Northam, J. B.,

(1991), Finite Rate Chemistry Effects in Mach 2 Reacting Flow, AIAA

paper 91-2320.

[8] Yakhot, V., Orzag, A., Gatski, T. B., and Speziade, C. B., (1992),

Development of Turbulence Model for Shear Flows by a Double Expansion

Technique, Physics of Fluids, 4, pp. 1510-1520.

[9] Evans, J. S., and Schexnayder, C. J., (1980), Influence of Chemical Kinetics

and Unmixedness on Burning Supersonic Hydrogen Flames, AIAA Journal,

18, pp. 188-193.[10] Mc Bride, B. J., and Gordon, S., (1971), Computer Program for Calculation

of Complex Chemical Equilibrium Compositions, Rocket Performance,

Incident and Reflected Shocks, and Chapman-Jouget Detonations, NASA SP-

273.

[11] Jameson, A., Schmidt, W., and Turkel, A., (1981), Numerical simulation of

Euler equation by finite volume method using Runge-Kutta scheme, AIAA

Paper 81-1258.

[12] Bussing, T. R. A., and Murman, E. M., (1988), Finite Volume Method for the

Calculation of Compressible Chemically Reacting Flows, AIAA Journal, 26,

pp.1070-1078.

[13] Menon, S., (1989), Shock-Wave- Induced Mixing Enhancement in ScramjetCombustors, AIAA Paper No. 89-0104.

[14] Fernando, E.M., and Menon, S., (1991), Mixing Enhancement in

Compressible Mixing Layers - An Experimental Study, AIAA, Paper No. 91-

1721.

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