CHEMICAL EQUILIBRIUM INVISCID FLOW

OVER SARA RE-ENTRY VEHICLE

C.A. Rocha Pimentel∗ and Joao Luiz F. Azevedo†

Centro Tecnico Aeroespacial - CTA, Sao Jose dos Campos, SP, 12228-904, Brazil

H. Korzenowshi‡

Universidade do Vale do Paraıba -UNIVAP, Sao Jose dos Campos, SP, 12244-000, Brazil

Marcia B. H. Mantelli§

Universidade Federal de Santa Catarina -UFSC, Florianopolis, SC, 88040-900, Brazil

The present work performs an inviscid hypersonic flow simulations over a re-entry body.The flow over the small ballistic re-entry Brazilian vehicle SARA is assumed to be modeledby the planar two-dimensinal or the axisymmetric Euler equations. In the inviscid formula-tion one considers real gas flow simulations and, in order to analyze chemical effects in hightemperature flows, the Gardiner kinetic chemical mechanism of five sepcies is considered( N2, O2, O, N, NO ), and their reactions of combination and dissociation. The governingequations are discretized in conservative form in a cell centered, finite volume procedurefor unstructured triangular grids. Spatial discretization considers an upwind scheme. AMUSCL reconstruction of primitive variables is used in order to determine left and rightstates at interfaces. Time march uses an explicit, 2nd-order accurate, 5-stage Runge-Kuttatime stepping scheme. The results are presented for simulations to initial conditions of 79%

of Nitrogen and 21% of Oxygen in an altitude of 80 km in freestream Mach numbers 15 and18.

I. Introduction

The development of efficient numerical solvers is very important owing to the difficulties and high costsassociated with the experimental work at high speed flows. Typical hypersonic flows undergo chemical

and thermal processes that are very difficult to predict experimentally. Hence, the numerical simulation playsan important role in hypersonic vehicle design. The hypersonic fluid flow simulation over a blunt-nosed bodyis characterized by a strong detached shock ahead the body. This phenomenon is particularly interestingbecause the curved bow shock is a normal shock wave in the nose region, and away from this, one has allpossible oblique shock solutions for a given freestream Mach number.1

A finite volume formulation of compressible Euler equations in conservative form has been considered.2 Ahigh-resolution scheme is employed in order to obtain a good spatially resolution of the flow features. In thiswork the simulations are performed by using a second-order Liou flux-vector splitting scheme,3 implementedin an unstructured grid context. This scheme considers a MUSCL approach, that is, the interface fluxesare formed using left and right states at the interface, which are linearly reconstructed by primitive variableextrapolation on each side of the interface.4 A minmod limiter is used in order to avoid any undesiredoscillations in the solution. The equations are discretized in a cell centered based finite volume procedure ontriangular meshes. Time march uses an explicit, second-order accurate, five-stage Runge-Kutta time steppingscheme. For the gas flow simulations one considers that 79% of Nitrogen and 21% of Oxygen composes thegas. Real gas effects are considered for the reactive flow simulations.5 The chemical kinetic mechanism

∗Research Engineer, Instituto de Aeronautica e Espaco - IAE.†Senior Research Engineer. Currently, Director for Space Transportation and Licensing, Brazilian Space Agency, Instituto

de Aeronautica e Espaco - IAE, Senior Member AIAA.‡Professor, Faculdade de Engenharia e Urbanismo.§Professor, Departamento de Engenharia Mecanica.

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43rd AIAA Aerospace Sciences Meeting and Exhibit10 - 13 January 2005, Reno, Nevada

AIAA 2005-390

Copyright © 2005 by C.A.R. Pimentel, J.L.F. Azevedo, H. Korzenowski, M.B.H. Mantelli. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

considers five chemical species, with their combinatio and dissociation reactions.6 The hypersonic flowsimulations are performed over a blunt-nosed body. The freestream Mach numbers 15 and 18 were selectto conduct the numerical investigations. Results indicate that the scheme could adequately capture theflowfield features.

II. Mathematical Formulation

The axisymmetric time-dependent, compressible Euler equation may be described, in conservative vectorform, by

∂Q

∂t+

∂E

∂x+

∂F

∂y+ α H = 0 , (1)

where E, F and H are conservative flux vectors, Q is the vector of conservative quantities, and α = 0represents 2-D planar flow and α = 1 represents 2-D axisymmetric flow. If the equations are discretized in acell centered finite volume procedure, the discrete vector of conserved variables, Qi, is defined as an averageover the i-th control volume. In this context, the flow variables can be assumed as attributed to the cetroidof each cell.7 The Eq. (1) can be written in integral form for the i-th control volume as

∂

∂t(ViQi) +

∫

S

[Eedy − Fedx] +

∫

V

α HdV = 0 , (2)

where V represents the area of the control volume and S its boundary.The hypersonic reactive flow will be computed using the unsteady axisymmetric Euler equations, thus

neglecting molecular transport. These balance equations of mass, momentum, energy and species massfraction can be written as5

∂Q

∂t+

∂F

∂x+

∂G

∂y+ α H = Ω , (3)

where

Q = [ρ, ρu, ρv, ρE , ρY1, · · · , ρYK−1]T

, (4)

F =

ρu

ρu2 + p

ρuv

u(ρE + p)

ρY1u...

ρYK−1u

, G =

ρv

ρuv

ρv2 + p

v(ρE + p)

ρY1v...

ρYK−1v

, H =1

y

ρv

ρuv

ρv2

v(ρE + p)

ρY1v...

ρYK−1v

, (5)

and

Ω = [0, 0, 0, 0, ω1W1, · · · , ωK−1WK−1]T

, (6)

with p, YK and E given by

YK = 1 −

K−1∑

k=1

Yk , p = ρRT

K∑

k=1

Yk

Wk

, (7)

E = e +1

2(u2 + v2) =

K∑

k=1

Ykek +1

2(u2 + v2) , (8)

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where

ek = h0k +

∫ T

T0

cpkdT −

p

ρ. (9)

In these equations E is the total energy per unit of mass, e is the internal energy, R is the universalgas constant. The internal energy, the standard-state enthalpy and the specific heat at constant pressureper unit of mass of species k are noted ek, h0

k and cpk. Yk, ωk and Wk are the mass fraction, the molar

production rate and the molecular weight of chemical species k, respectively.

III. Kinetics Mechanism

The chemical kinetics mechanism for the reactive mixture of nitrogen and oxygen is due to Gardiner.6

This mechanism considers 5 species ( N2, O2, N, O, NO ) and 34 elementary reactions, and is given in Table 1.

Table 1. Reaction mechanism for N2-O2 : A in (cm,mol,s), E in (Kelvins).

Reactions A β E

N2 + N2 → 2N + N2 3.7e21 -1.6 113200.

2N + N2 → N2 + N2 1.08e13 -1.493 100.

N2 + O2 → 2N + O2 1.4e21 -1.6 113200.

2N + O2 → N2 + O2 3.07e13 -1.493 100.

N2 + NO → 2N + NO 1.4e21 -1.6 113200.

2N + NO → N2 + NO 3.07e13 -1.493 100.

N2 + N → 2N + N 1.6e22 -1.6 113200.

2N + N → N2 + N 3.51e14 -1.493 100.

N2 + O → 2N + O 1.4e21 -1.6 113200.

2N + O → N2 + O 3.07e13 -1.493 100.

O + O + M → O2 + M 3.64e18 -1.0 59380.

2O + N2 → O2 + N2 1.84e10 -0.714 109.

O2 + O2 → 2O + O2 1.64e19 -1.0 59380.

2O + O2 → O2 + O2 8.28e10 -0.714 109.

O2 + NO → 2O + NO 1.82e18 -1.0 59380.

2O + NO → 02 + NO 9.19e09 -0.714 109.

O2 + N → 2O + N 1.82e18 -1.0 59380.

2O + N → O2 + N 9.19E09 -0.714 109.

O2 + O → 2O + O 4.56e19 -1.0 59380.

2O + O → O2 + O 2.3e11 -0.714 109.

NO + N2 → N + O + N2 4.0e20 -1.5 75500.

N + O + N2 → NO + N2 2.16e19 -1.3217 97.

NO + O2 → N + O + O2 4.0e20 -1.5 75500.

N + O + O2 → NO + O2 2.16e19 -1.3217 97.

NO + NO → N + O + NO 8.0e20 -1.5 75500.

N + O + NO → NO + NO 4.32e19 -1.3217 97.

NO + N → N + O + N 8.0e20 -1.5 75500.

N + O + N → NO + N 4.32e19 -1.3217 97.

NO + O → N + O + O 8.0e20 -1.5 75500.

N + O + O → NO + O 4.32e19 -1.3217 97.

N2 + O → NO + N 1.82e14 0.0 38370.

NO + N → N2 + O 7.35e13 -0.07083 666.

NO + O → O2 + N 3.8e09 1.0 20820.

O2 + N → NO + O 4.07e10 0.886 4689.

The chemical production rates are given by the Arrhenius law :

kf = AT β exp(−E/RT ). (10)

The calculation of kf and of the molar production rates ωk are performed using the CHEMKIN-IIpackage.8 The thermodynamic properties are calculated according to the procedures developed by Kee.8

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IV. Spatial Discretization Algorithm

The implementation of the 2nd-order Liou scheme is based on an extension of the Godunov approach. Theprojection stage of the Godunov scheme, in which the solution is projected in each cell on piecewise constantstates, is modified. This constitutes the so-called MUSCL approach for the extrapolation of primitivevariables (p, u, v, T, Yk). By this approach, left and right states at a given interface are linearly reconstructedby primitive variable extrapolation on each side of the interface, together with some appropriate limitingprocess in order to avoid the generation of new extrema.

The convective operator, C(Qi), which discretize the surface integral of Eq. (2), can be written in thepresent cell centered case by the expression

C(Qi) =

3∑

k=1

(Eik∆yik − Fik∆xik) . (11)

The interface fluxes, Eik and Fik , are defined as

Eik = E+(QL) + E−(QR) ,

Fik = F+(QL) + F−(QR) , (12)

where QL and QR are the left and right states at the ik interface obtained by linear extrapolation process.Even considering a 2nd-order flux vector splitting scheme with a MUSCL approach, it is possible to

obtain oscillations in the solution. Therefore one must use nonlinear corrections, namely limiters, to avoidany oscillations. In this work a simple minmod limiter was adopted.

V. Time Discretization Algorithm

The Euler equations, fully discretized in space by an upwind method and assuming a stationary mesh,can be written as

d Qi

d t= −

1

Vi

C (Qi) + Ω (Qi) . (13)

Time advancement of the solution from time step n to n + 1 is achieved by the use of Strang’s tiem-stepspliting procedure.9

Qn+1i = L

(

∆ t

2

)

C (∆ t)L

(

∆ t

2

)

Qni , (14)

that separately integrates the fluid dynamics operator L and the chemistry operator C at each cell. Thisprocedure is 2nd-order accurate and gives the flexibility of choosing specialized integrators for the chemicalkinetics and the fluid dynamics. A more detailed discussion on this subject can be found in the work ofLeVeque.10

The present work uses a fully explicit, 2nd-order accurate, 5-stage Runge-Kutta time-stepping scheme11

to advance the fluid dynamics part of the governing equations in time. The time integration scheme can,therefore, be written as

Q(0)i = Q

(n)i ,

Q(l)i = Q

(0)i − αl

∆ tiVi

C

(

Q(l−1)i

)

, l = 1, 2, · · · , 5 (15)

Q(n+1)i = Q

(5)i ,

where the superscripts n and n+1 indicate that these are property values at the beginning and at the end ofthe n-th time step, and the particular values of the α coefficients used are those suggested by Mavriplis.11 Forsteady state, inert gas problems, a local time stepping option has been implemented in order to accelerate

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convergence.12 In the reactive flow case, one cannot use a space-varying time step. However, some conver-gence acceleration can be achieved by recalculating a global time step at each iteration as the minimum ofthe local ∆ ti’s obtained from the CFL condition.

The use of a time-step splitting procedure allows the adoption of a specialized solver for the integrationof the chemistry operator C in the Eq. (14). This corresponds to a separate integration of the ODE

d Qi

d t= Ω (Qi) . (16)

It can be noticed from Eq. (16) and (3)-(6) that such a time-step splitting procedure results in a constantdensity thermal explosion problem at each computational cell. The authors have chosen to perform theintegration of Eq. (16) using VODE,13 which is an ODE solver tailored for the solution of problems whiochinclude stiff source terms. VODE uses a variable time-step, variable order, backward differentiation formula,together with a modified Newton method whose Jacobian matrix is evaluated numerically. This last featureis of particular value when using different chemical kinetics schemes. A further advantage of using this stiffODE solver is the fact that it has no stability limits on the choise of the time step. Therefore, only the fluiddynamics requirements constrain the choise of ∆ t.

VI. Numerical Results

The small ballistic re-entry vehicle SARA configuration was used to perform the numerical simulations,assuming freestream Mach number 15 and 18. For the gas flow one considers 79% of Nitrogen and 21%of Oxygen composes the gas. Pressure and temperature initial conditions to an altitude14 of 80km are0.000010387atm and 198.639K, respectively. The numerical simulations was obtained by using 2nd-orderLiou flux vector splitting scheme. Although this scheme could capture strong discontinuities without smear-ing the solution, one observes the generation of spurious numerical oscillations behind the shock wave. Thedevelopment of the carbuncle problem is visible in Fig. 1.

X, cm

Y,c

m

1.5 2 2.5 3 3.5-0.25

0

0.25

0.5

0.75

1

Carbuncle Problem

Figure 1. Streamtraces showing carbuncle problem development.

This instability forms as a protuber-ance in the nose region. Starting sim-ulations, a very small spurious vortexis still visible closest to the stagnationpoint. Since the magnitude of vortic-ity produced by a shock dependes largelyon the magnitude of the tangencial ve-locity component, one will expect vortexseverity to increase with increasing lev-els of velocity generated inside the struc-ture of captured shock waves.15 Moreartificial dissipation can avoid this car-buncle instability. The pressure numbercontours, obtained with the second-orderLiou scheme are presented in Fig. 2 and 5to Mach number 15 and 18, respetive, inan angle of attack θ = 0 and 10 degrees.The contours indicate that the flow fea-tures are well captured by this solution,the bow shock and the flow expansionover the body are well represented. Onecan see that at the stagnation region the shock is normal, and away from this the shock wave graduallybecomes curved and weaker. The hypersonic flow ahead the shock becomes subsonic behind this one, thatis, there is a strong compression of the flow in this region. Slightly above and down of the stagnation region,the shock is oblique and pertains to the strong shock-wave solution. As we move further along the shock,the wave angle becomes more oblique, and the flow deflection decreases until reach the maximum deflectionangle. From the stagnation region until this point the flow is subsonic. Above this one, all points on theshock correspond to the weak shock solution. This region is characterized by supersonic flow.

If one considers real gas effects, the stagnation temperature would decay due the formation of species

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dissociation. The maximum temperature value in the stagnation region reaches to Mach number 15 in angleof attack θ = 0 and 10 degrees are 5450K and 4800K, and to Mach number 18 in angle of attack θ = 0 and10 degrees, 6581K and 6200K, respective. The temperature contours are plotted in Fig. 3 and 6.

The Fig. 4 and 7 show the mass fractions of the major mixture constituents along stagnation line to theMach number 15 and 18 to θ = 0 and 10 degrees, respective. Observe that, the oxygen molecules dissociatesinside the shock region and remain dissociated next to the wall whereas the nitrogen dissociates within theshock region because the high temperatura and partially recombines next to the wall.

VII. Conclusions

The present work performed an inviscid flow simulations over a small ballistic re-entry vehicle, the SARAconfiguration. For the inviscid case, the fluid was treated as a real gas. The governing equations arediscretized in a cell centered finite volume algorithm. Unstructured meshes are used to obtain the numericalsolutions. The equations are advanced in time by an explicit, 5-stage, 2nd-order accurate, Runge-Kuttatime stepping procedure. The spatial discretization scheme considers a 2nd-order Liou flux-vector splittingscheme. A MUSCL reconstruction of primitive variable extrapolation is performed in order to obtain leftand right states at interfaces. The results indicate that the scheme is able to reproduce some phenomenapresent in hypersonic flow.

VIII. Acknowledgments

The authors thanks the support of Fundacao de Amparo a Pesquisa do Estado de Sao Paulo, FAPESP,which provided a post-doctoral scholarship to the first author under the Project Research Grant No. 00/1351-2 − 0. The authors also gratefully acknowledge the partial support of Fundacao Coordenacao de Aper-feicoamento de Pessoal de Nıvel Superior, CAPES, under the Project PROCAD No. 0130/01− 0.

References

1Anderson, J. D. J., Hypersonic and High Temperature Gas Dynamics, McGraw Hill, 1989.2Hirsh, C., Numerical Computation of Internal and External Flows. Vol. 2 Computational Methods for Inviscid and

Viscous Flows, Wiley, 1990.3Liou, M. S., “A Sequel to AUSM: AUSM+,” Journal of Computational Physics, Vol. 129, No. 2, 1996, pp. 364–382.4Azevedo, J. L. F. and Korzenowski, H., “Comparison of Unstructured Grid Finite Volume Methods for Cold Gas Hyper-

sonic Flow Simulation,” Aiaa 16th applied aerodynamics conference albuquerque, June 1998.5Pimentel, C. A. R., Etude Numerique de la Transition Entre Une Onde de Choc Oblique Stabilise Par Un Diedre et Une

Onde de Detonation Oblique, Ph.D. thesis, Universite de Poitiers, Poitiers, Oct. 2000.6Hachemin, J. V., “Development d’un code Navier-Stokes parabolise pour des ecoulements tridimensionnels en desequilibre

thermochimique,” Note tecnique 1995-8, FR ISSN 0078-3781, 1995.7Korzenowski, H., Tecnica em Malhas Nao-Estruturadas para Simulacao de Escoamento a Altos Numero de Mach, Ph.D.

thesis, Instituto Tecnologico de Aeronautica, Sao Jose dos Campos, June 1998.8Kee, R. J., Rupley, F. M., and Miller, J. A., “CHEMKIN-II: A Fortran Chemical Kinetics Package for Analysis of Gas

Phase Chemical Kinetics,” SAND86-8009B/UC-706, 1991.9Strang, G., “On the Construction and Comparison of Difference Schemes,” SIAM J. Num. Anal., Vol. 5, 1968, pp. 506–

517.10LeVeque, R. J. and Yee, H. C., “A Study of Numerical Methods for Hyperbolic Conservation Laws with Stiff Source

Terms,” Journal of Computational Physics, Vol. 86, No. 1, 1990, pp. 187–210.11Mavriplis, D. J., “Multigrid Solution of the Two-Dimensional Euler Equations on Unstructured Triangular Meshes,”

AIAA Journal , Vol. 26, No. 7, 1988, pp. 824–831.12Azevedo, J. L. F. and da Silva, L. F. F., “The Development of an Unstructured Grid Solver for Reactive Compressible

Flow Applications,” Aiaa paper 97–3239, July 1997.13Byrne, G. D. and Dean, A. M., “The Numerical Solution of Some Kinetics Models with VODE and CHEMKIN II,”

Computers Chem., Vol. 17, No. 3, 1993, pp. 297–302.14Bertin, J. J., Hypersonic Aerothermodynamics, AIAA - Education Series, 1994.15Johnston, I. A., Simulation of Flow Around Hypersonic Blunt-Nosed Vehicles for the Calibration of Air Data Systems,

Ph.D. thesis, University of Queensland, Queensland, Jan. 1999.

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X, cm

Y,c

m

1 2 3 4 5 6 7 8 9-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

P ( atm )0.003500.003210.002920.002620.002330.002040.001750.001460.001170.000870.000580.00029

(a)

X, cm

Y,c

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18-6

-5

-4

-3

-2

-1

0

1

2

3

4P ( atm )0.002720.002500.002270.002050.001820.001590.001370.001140.000920.000690.000470.00024

(b)

Figure 2. Pressure contours to freestream Mach number 15 : a) Axisymmetric case : θ = 0 and b) 2-D planarcase : θ = 10 degrees.

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X, cm

Y,c

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

1

2

3

4

5

6

7

8

T ( K )5450498245144045357731092641217317051236768300

(a)

X, cm

Y,c

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18-6

-5

-4

-3

-2

-1

0

1

2

3

4T ( K )4800440940183627323628452455206416731282891500

(b)

Figure 3. Temperature contours to freestream Mach number 15 : a) Axisymmetric case : θ = 0 and b) 2-Dplanar case : θ = 10 degrees.

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X, cm

Mas

sfr

actio

n

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

N2

O2

O

N

NO

(a)

X, cm

Mas

sF

ract

ion

4.0 4.5 5.0 5.5-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

NO

N2

O2 O

N

(b)

Figure 4. Species mass fractions along stagnation line to freestream Mach number 15 : a) Axisymmetric case :θ = 0 and b) 2-D planar case : θ = 10 degrees.

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X, cm

Y,c

m

1 2 3 4 5 6 7 8-0.5

0

0.5

1

1.5

2

2.5

3P ( atm )0.005000.004580.004170.003750.003340.002920.002500.002090.001670.001260.000840.00043

(a)

X, cm

Y,c

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18-6

-5

-4

-3

-2

-1

0

1

2

3

4P ( atm )0.003720.003410.003100.002790.002480.002170.001870.001560.001250.000940.000630.00032

(b)

Figure 5. Pressure contours to freestream Mach number 18 : a) Axisymmetric case : θ = 0 and b) 2-D planarcase : θ = 10 degrees.

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X, cm

Y,c

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

1

2

3

4

5

6

7

8

T ( K )6581601054394868429737263155258420131442871300

(a)

X, cm

Y,c

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18-6

-5

-4

-3

-2

-1

0

1

2

3

4T ( K )62005682516446454127360930912573205515361018500

(b)

Figure 6. Temperature contours to freestream Mach number 18 : a) Axisymmetric case : θ = 0 and b) 2-Dplanar case : θ = 10 degrees.

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X, cm

Mas

sF

ract

ion

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

N2

O2 O

NNO

(a)

X, cm

Mas

sF

ract

ion

2.5 3.0 3.5 4.0 4.5 5.0 5.5-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

NO

N2

O2 O

N

(b)

Figure 7. Species mass fractions along stagnation line to freestream Mach number 18 : a) Axisymmetric case :θ = 0 and b) 2-D planar case : θ = 10 degrees.

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