DIRECT MONTE CARLO SIMULATION OF OXYGEN DISSOCIATION BEHIND SHOCK WAVES

D. Bruno*, F. Esposito#*Istituto di Metodologie Inorganiche e dei Plasmi del C.N.R., sez. Bari, Italy

M. Capitelli°, S. Longo†, P. Minelli¥†Dep. Chermistry, University of Bari, Italy

A study is conducted on the nonequilibrium dissociation of Oxygen behind shock waves. Numerical simulations are performed by means of the DSMC method. A state-to-state vibrational kinetic model is adopted specified by a set of microscopic cross sections. The cross sections for atom-molecule processes are derived from QCT trajectory calculations and include multiquantum transitions whereas for the molecule-molecule processes semiclassical rate constants for monoquantum transitions have been inverted numerically. Results for a strong shock in Oxygen are reported.

INTRODUCTION

The characterization of the flow around space vehicles under reentry conditions into terrestrial atmosphere has been the subject of several investigations. The aim of these studies is to predict the aerodynamic (e.g.: the drag) and the thermal (e.g.: the heat flux) properties of the flow. Despite many efforts, a complete picture is still lacking. The reason is that, at the hypersonic velocities typical of these flows, many nonequilibrium processes play an important role. Therefore, kinetic modeling is needed whose results crucially depend on the details with which each elementary process is described. This, in turn, calls for a careful description of many high temperature energy exchange and chemical reaction processes whose knowledge is far from complete.In particular, a live issue is the role of the nonequilibrium vibrational kinetics of the molecular species in affecting the degree of dissociation in the flow. A study has been undertaken by the authors to produce a fully consistent simulation of air flow behind strong shock waves.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _*PhD, researcher#PhD, researcher°AIAA member, full professor†AIAA member, full professor¥PhD student

The idea is to use the Direct Simulation Monte Carlo method, which enables to couple consistently several nonequilibrium phenomena, and to feed in up-to-date information about the elementary processes obtained from molecular dynamics calculations. The idea has been applied to the simulation of shock waves in Nitrogen1. Now, the same scheme is applied to the study of shock waves in Oxygen. After a brief description of the simulation method, the kinetic model adopted is discussed. Sample calculations are reported and discussed.

THE DSMC METHOD

The DSMC method2 is a simulation tool well known to the rarefied gas community. It has been successfully applied to kinetic studies of reacting gas mixtures1. The modeling is done much along the lines of the previous work. The simulation tool is a one dimensional, Direct Simulation Monte Carlo code with the majorant frequency scheme3 for efficient sampling of collision pairs. The kinetic description is obtained by assigning to each simulated particle, beside chemical species, position and velocity, its internal state. The molecules are described as vibrating rigid rotors. The rotational degrees of freedom have a continuous spectrum whereas the vibrational levels are discrete; they are derived from

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AIAA 2003-4059

Copyright © 2003 by Domenico Bruno. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

calculations on the ab initio potential energy surface4. This analysis gives 47 bound vibrational levels for the oxygen molecule.

KINETIC MODEL

The kinetic model in DSMC simulation is determined by specifying a set of microscopic cross sections for all processes described. In particular, the transport properties of the gas mixtures are determined by the features of the total cross section. In this work, since the internal state kinetics is the main concern, we adopt a hard sphere model. However, since molecules in different vibrational levels are treated as different chemical species, we assign to each a different hard sphere diameter. These diameters are assigned by summing a fixed elastic contribution to the maximum integral inelastic cross section of the molecule. The elastic contribution is chosen such that, at room temperature, the measured viscosity coefficient is reproduced. This somewhat artificial procedure stems from the lack of data on total or elastic cross sections for the high lying vibrational levels. Diameters based on viscosity measurements reflect the leading contribution of the most populated low lying levels. These diameters, however, are much too small to account justonly for the total inelastic cross section of the molecules in high lying vibrational levels. Also the QCT calculations for the atom-diatom system, being focussed on energy exchage and reactive processes, do not give any reliable information on the actual total cross section. The model used in this work will therefore not be able to reproduce the viscosity and diffusion coefficients of the real gas at high temperatures even though the differences are expected to be small.Given the high temperatures typical of the flows under study, a reasonable assumption is to treat the rotational degrees of freedom as classical instead of quantised and to model the nonequilibrium kinetics of these modes by the phenomenological Larsen-Borgnakke method5 where the expression for the rotational relaxation number6 with numerical parameters from Boyd7 is used.The attention is focussed on the careful modeling of the state-to-state vibrational kinetics and its bearing on the dissociation process. This entails the description of three

types of elementary processes: atom-molecule and molecule-molecule vibrational energy exchanges and dissociation processes.The diatom-diatom energy transfer processes are modeled by fitting directly the rate coefficients of Billing and Kolesnick8 by a nonlinear symplex method. The rate coefficients are determined from refined semiclassical trajectory calculations. Mono-quantum transitions are included along with quasi-resonant VV (vibration/vibration) energy transfers; i.e. the following processes are taken into account:

†

O2 v( ) + O2 ´ O2 v - 1( ) + O2 (1)

†

O2 v + 1( ) + O2 w( ) ´ O2 v( ) + O2 w - 1( ) (2)

These rate coefficients were originally derived by considering the oxygen molecule as a Morse oscillator with 33 bound levels. They have, therefore, been fitted on the energy scale in order to be applied to the 47 levels that our PES supports.The dissociation rate calculated with this kinetic model does not compare very well with semiempirical chemical models that fit the available experimental data9,10: see fig. 1.The cross sections for these processes are assumed to be of the form:

†

s = aEb exp -gE( ) (3)

E being the collision energy and a, b, g the fitting parameters for each transition. The cross sections for the backward transitions are determined by application of the detailed balance principle. For transitions between high levels, the exponential damping of eq. (3) is not very effective; therefore we decided to cut the cross sections at Emax=4 eV. Collisions at energies higher than this do not contribute significantly to the rate coefficients up to temperatures of 10000 K.

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10-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

2 103 4 103 6 103 8 103 1 104

Kd, c

m3 /s

*par

t

T, K

Figure 1. Molecule-molecule dissociation rate constant of oxygen as a function of temperature for different kinetic models. Continuous line: Gardiner9; Dashed line: Shatalov10; Points. Billing8.

Nonetheless, we note that in the DSMC simulation, even if the temperature does not go much beyond 9000 K, many collisions occur in the shock front at energies much bigger than Emax. This problem awaits definitive solution: the calculated rates cannot be extrapolated to much higher temperatures and the fitted formulas have no meaning at energies that do not contribute to the rate coefficient. Anyway, the best fit parameters are able to reproduce the calculated rate coefficients very well in the temperature range [300K, 10000K]. Some representative cross sections for VT and VV processes are reported in figs. 2 and 3, respectively.Detailed cross sections for the atom-diatom collision process involving oxygen have been calculated by the quasiclassical trajectory method4. The following processes have been considered:

†

O + O2 v, j( ) Æ 3O (4)

†

O + O2 v, j( ) Æ O + O2 ¢ v , ¢ j = all( ) (5)

with the translational energy in the interval 0.001-3 eV. Atomic and molecular species are considered in their respective electronic ground state. All ro-vibrational states (about 6400) supported by the diatomic potential have been considered for initial and final

states, except for final rotation j' which is summed up. Also rotational quasibound states have been considered, for two purposes: a) to estimate their influence on dissociation process; b) for future application in recombination calculations by means of orbiting resonance theory. Concerning reactants, the resolution on initial rotation j allows to choose at will the rotational temperature. A sample of dissociation cross sections is shown in fig. 4 for the rotational temperature of 2000 K, from initial vibrational states indicated in the legend. An important non-adiabatic correction factor has been applied to dissociation results, with good global agreement with experimental results. For use in DSMC calculations the detailed QCT cross sections have been rotationally averaged at a rotational temperature Tr=10000 K.

10-5

10-3

10-1

101

103

0 1 2 3 4 5

111213141

s

(v->

v-1)

, A

2

E, eV

Figure 2. Cross sections used in this work for the VT process eq. (1).

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10-3

10-1

101

103

0 1 2 3 4 5

1,010,930,291,91,29

s

(v,w

->v-

1,w

+1),

A

2

E, eV

Figure 3. Cross sections used in this work for the VV process eq. (2).

CALCULATIONS

In order to produce a shock in the numerical domain, we let a supersonic flow enter the domain from the left side (x=0) and fix the steady-state boundary conditions (density, temperature, flow speed, molar fraction of atoms, vibrational distribution function) on the right side (x=L) according to the results of a fluid dynamic simulation of the shock which uses the Euler equations and the full state-to-state kinetic scheme as the DSMC simulation.

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5 3

v=15v=20v=25

v=30v=35v=40

Figure 4. Dissociation cross sections in

†

A2 as obtained from QCT calculations. The rotational temperature is 2000 K; the legend indicates the initial vibrational state.

The Euler equations for the post shock

relaxation are integrated with an adaptive spatial step; the state-to-state kinetics, including recombination processes, is linearised and then solved in matrix form by a Gaussian method. The rate coefficients are obtained from the set of cross sections by integrating numerically over a Maxwell distribution function at the relative temperature.Results are presented for the flow parameters behind a shock wave in oxygen. The parameters of the flow are givenbelow:

T0=300 KP0=100 PaMa0=11l0=4e-3 cm

The downstream boundary condition is set 250 mfp units downstream of the shock where all the gradients have become very small (Kn<10-2). However, at this stage the recombination process does not affect the results. The recombination to dissociation ratio is less than 10-6 at the downstream boundary and, consistently, the Euler results without recombination coincide with those previously obtained.

RESULTS

Let us start by examining some macroscopic quantities: in figs. 5, 6 we report, respectively, the total number density and the atomic molar fraction. The spatial scale is always expressed using the upstream mean free path l0 as a unit. By inspecting fig. 5 we can observe the effect of the thermal relaxation on the gas number density. The compression ratio behind the shock wave reaches rapidly the value of 6 valid for strong shocks in a gas of rigid rotors. From here it grows slowly to the value of 8, following the slow vibrational relaxation. Further increase of the compression ratio is due to the increase of the atom fraction as reported in fig. 6.In fig. 7 we report the three 'temperatures' which are relevant for this study: i.e. the static translational temperature T, the rotational temperature Tr and the T01 vibrational temperature based on the O2(v=1)/O2(v=0) population ratio.

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0 100

1 1017

2 1017

0 100 200 300 400 500

n, cm-3

x/l

Figure 5. Plot of the number density along the flow. The abscissa is expressed in upstream mean free path units.

0 100

1 10-2

2 10-2

0 100 200 300 400 500

cat

x/l

Figure 6. Plot of oxygen atomic fraction.

It can be seen that the sudden jump in T is rapidly followed by slower jump for Tr. At the final stage of rotational relaxation energy is drawn from the translational to the rotational degrees of freedom, thereby producing the overshoot in T vs. x which can be seen in the figure.

0

2000

4000

6000

8000

180 200 220 240 260

T, K

x/l

Figure 7. Translational, rotational and vibrational temperatures along the flow. The rotational temperature is obtained under the classical rigid rotor hypothesis. The vibrational temperature is T01.

The vibrational relaxation follows with a longer relaxation time. This slower relaxation is also affected by its mesoscopic structure in terms of the vibrational distribution functions (vdf). In fig. 8 we report some of these vdf's at different positions along the flow. It can be noticed that the nonequilibrium affects only the early stages of the relaxation

10-5

10-3

10-1

0 1 2 3 4 5 6E

v, eV

0

2000

4000

6000

8000

180 200 220 240 260

Figure 8. Vibrational distribution function of O2 molecules for different positions along the flow.

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10-5

10-3

10-1

0 1 2 3 4 5E

r, eV

0

2000

4000

6000

8000

195 205 215

Figure 9. Distribution of rotational energy of O2 molecules for different positions along the flow.

In fig. 9 the distribution of (classical) rotational energy is plotted. The different curves refer to different positions along the shock front and during the rotational relaxation. It can be seen the strong deviation from the equilibrium distribution during relaxation.In fig. 10 the translational distribution functions at different positions along the shock front are reported. At the two extreme points the distribution are equilibrium distributions at the upstream and downstream temperatures, respectively. At the points in between, a sensible deviation of the translational distribution from the Maxwell one is observed, in the form of a 'tail'.

10-4

10-2

100

0 100 2 105 4 105 6 105 8 105

v, cm/s

0

2000

4000

6000

8000

195 205 215

Figure 10. Translational distribution functions for different positions along the flow.

ACKNOWLEDGMENTS

This work was supported by M.I.U.R. (Contract 2001031223_009) and by ASI (Contract CSPA.ATD.SC.02.03).

REFERENCES

[1] D. Bruno, M. Capitelli, F. Esposito, S. Longo, P. Minelli, Chem. Phys. Lett. 360 31 (2002); D. Bruno, M. Capitelli, F. Esposito, S. Longo, P. Minelli, AIAA 2001-2761.[2] G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, 1994.[3] M. S. Ivanov, S. V. Rogasinsky, Soviet J. Numer. Anal. Matyh. Modelling, 3 453 (1988).[4] F. Esposito, M. Capitelli, Chem. Phys. Lett. 364 180 (2002)[5] C. Borgnakke, P. S. Larsen, J. Comput. Phys., 18 405 (1975).[6] J. G. Parker, Phys. Fluids, 2 449 (1959).[7] I. D. Boyd, Phys. Fluids A, 2 447 (1990).[8] G. D. Billing, R. E. Kolesnick, Chem. Phys. Lett., 200 382 (1992)[9] W. C. Gardiner, Combustion Chemistry, Springer, 1984.[10] O.P. Shatalov, S. A. Losev, AIAA 97-2597 (1997)..

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