State to state models for dissociation behind shock waves Minelli Pierpaolo IMIP-CNR Bari, Italy

State to state models for State to state models for dissociation behind shock wavesdissociation behind shock waves

Minelli PierpaoloMinelli Pierpaolo

IMIP-CNRIMIP-CNR

Bari, ItalyBari, Italy

Boundary conditions for the kinetic Boundary conditions for the kinetic simulation of a shock wavesimulation of a shock wave

00 11

pp00 pp11

uu00 uu11

hh00 hh11

upstreamupstream downstreamdownstream

€

p=kTm

Perfect gas equationPerfect gas equation

€

1u1 =0u0

€

p1 +1u12 =p0 +0u0

2

€

h1 +u12

2=h0 +

u02

2

Rankine-Hugoniot equationsRankine-Hugoniot equations

We use the We use the Rankine-HugoniotRankine-Hugoniot conditions, obtained numerically, to implement conditions, obtained numerically, to implement the the EulerEuler code (which use the full state-to-state kinetic scheme). These results code (which use the full state-to-state kinetic scheme). These results allow us to solve the shock wave problem neglecting transport phenomena allow us to solve the shock wave problem neglecting transport phenomena (viscosity, heat conduction, etc.). These last outcomes, obtained in a region (viscosity, heat conduction, etc.). These last outcomes, obtained in a region where transport coefficients are negligible and atomic recombination is where transport coefficients are negligible and atomic recombination is irrelevant, are assumed as boundary conditions in the data file of our irrelevant, are assumed as boundary conditions in the data file of our DSMC DSMC codecode..

Collision ModelsCollision Models

• Elastic collisions: VSS modelElastic collisions: VSS model• RT collisions: Larsen-Borgnakke methodRT collisions: Larsen-Borgnakke method• Vibrational and dissociative collisionVibrational and dissociative collision

A-M collisionsA-M collisions

M-M collisionsM-M collisions

€

A2 ν v( )+ A→ A2 ′ ν v( )+ A

A2 ν v( )+ A→ 2A+ A

⎧ ⎨ ⎪

⎩ ⎪

€

mono- quantum transitions

A2 ν v( )+ A2 ζv( ) → A2 ν v ±1( )+ A2 ζv( )

A2 ν v( )+ A2 ζv( ) → A2 ν v ±1( )+ A2 ζv m1( )

multi- ( )quantum transitions dissociation

A2 ν v( )+ A2 ζv( ) → 2A+ A2 ζv( )

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

Detailed balance principleDetailed balance principle

When a couple of selected particles has been accepted for the collision, the probability When a couple of selected particles has been accepted for the collision, the probability

associated to the particular state, described by associated to the particular state, described by νν’’AA and and νν’’

BB, will be given by:, will be given by:

For definition of probability must be valid the normalization condition:For definition of probability must be valid the normalization condition:

In the case of study of the relaxation phenomena, the principle of the detailed balanceIn the case of study of the relaxation phenomena, the principle of the detailed balance

leads to the following expression:leads to the following expression:

where where ii and and f f represents initial and final states of the molecules. represents initial and final states of the molecules.

€

σ i→ f ;g( )g2 =σ f → i; ′ g( ) ′ g 2

€

p ′ ν A, ′ ν B( ) =σ ν A,νB → ′ ν A, ′ ν B;g( )

σ TOT

€

p ′ ν A, ′ ν B( ) =1 All states

′ ν A and ′ ν B

∑

Parker ModelParker Model

Parker has used an empirical non-impulsive potential model that incorporatesParker has used an empirical non-impulsive potential model that incorporates

a small degree of asymmetry to derive an expression for the rotational a small degree of asymmetry to derive an expression for the rotational relaxationrelaxation

time time rotrot. The following approximate expression is obtained:. The following approximate expression is obtained:

where T where T ** is the characteristic temperature of the intermolecular potential and is the characteristic temperature of the intermolecular potential and

(Z(Zrotrot))∞∞ is the limiting value. The value employed in our simulation are: is the limiting value. The value employed in our simulation are:

€

Zrot=Zrot( )∞

1+π

32

2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟T∗

T

⎛

⎝ ⎜

⎞

⎠ ⎟

12

+π 2

4+ π

⎛

⎝ ⎜

⎞

⎠ ⎟T∗

T

€

Nitrogen: T∗ =91.5 ,K Zrot( )∞ =16

Oxygen: T * =90 ,K Zrot( )∞ =14.4

QCT model for QCT model for A-MA-M collisions (1) collisions (1)

In the DSMC simulation, rotation and vibration have been uncoupled. QCT In the DSMC simulation, rotation and vibration have been uncoupled. QCT cross sections must be mediated on the rotational spectrum in such a way to cross sections must be mediated on the rotational spectrum in such a way to obtain, at every fixed reference average rotational temperature, cross sections obtain, at every fixed reference average rotational temperature, cross sections that depend only from the vibrational level of the molecule. If we consider a that depend only from the vibrational level of the molecule. If we consider a generic atom-molecule transition:generic atom-molecule transition:

We can write:We can write:

andand

where where

€

A2 ν v , j( ) + A→ A2 ′ ν v, ′ j( ) + A

€

σ j ′ j = σ ν v, j → ′ ν v , ′ j( )′ j =0

jmax

∑

€

σ ν v → ′ ν v( ) =

wjσ j jj=0

jmax

∑

wjj=0

jmax

∑

€

wj =e−E νv , j( ) kTrot 2 j +1( )

QCT model for QCT model for A-MA-M collisions (2) collisions (2)

Let consider the following atom-molecule transition:Let consider the following atom-molecule transition:

with with νν’’vv = = 0,1,…, 0,1,…, ννmaxmax, , ννmaxmax+1.+1.

Generally, it is valid the law of conservation of energy, which, in this case, Generally, it is valid the law of conservation of energy, which, in this case, can be written as:can be written as:

From this equation, it is possible to obtain the only unknown variable From this equation, it is possible to obtain the only unknown variable EE’’kinkin, ,

and so and so gg’’ modulus. modulus.

€

A2 ν v( )+ A→ A2 ′ ν v( )+ A

€

ETOT =Ekin +Ev = ′ Ekin +E ′ v

Pseudo-level Pseudo-level ((ννmaxmax+1+1)) models the continuum: if the molecule reaches the models the continuum: if the molecule reaches the

pseudo-level pseudo-level ((ννmaxmax+1+1)), then the two bounded atoms dissociate according to:, then the two bounded atoms dissociate according to:

€

A2 ν v( ) + A→ A2 νmax+1( ) + A≡3A

QCT and Detailed balance principleQCT and Detailed balance principle

We define the following two types of transitions according to the We define the following two types of transitions according to the corresponding cross sections:corresponding cross sections:

But:But:

For this reason, we take into account only QCT results regarding direct For this reason, we take into account only QCT results regarding direct collisions and dissociation processes. For cross sections regarding indirect collisions and dissociation processes. For cross sections regarding indirect processes, we impose the principle of detailed balance:processes, we impose the principle of detailed balance:

€

σ i→ i+k; ′ g( ) ′ g 2 =σ i+k→ i;g( )QCT

g2

€

σ i→ i+k; ′ Ekin( ) =Ekin

′ Ekin

σ i+k→ i;Ekin( )QCT

€

σ i→ f( )QCT

g2 ≠σ f → i( )QCT

′ g 2

€

Direct transitions: σ i→ i−k( ) withk=1,..., i

Indirect transitions:σ i→ i+k( ) withk=1,...,vmax−i

Molecule-Molecule collision models (1)Molecule-Molecule collision models (1)

Two difficulties:Two difficulties:

• There are not detailed database for these kind of processes!There are not detailed database for these kind of processes!

• It is preferred to publish rate coefficients or fit rather than cross section data!It is preferred to publish rate coefficients or fit rather than cross section data!

It is well known that the rate constants are related to the reaction cross It is well known that the rate constants are related to the reaction cross sections through the sections through the LaplaceLaplace transforms in situations where thermal transforms in situations where thermal equilibrium may be assumed for the translational degrees of freedom of the equilibrium may be assumed for the translational degrees of freedom of the reactant molecules.reactant molecules.

€

k= πmr( )−1

2 2 kT( )32 σ E( )0

∞

∫ Ee−E kTdE

It is not simple to invert this It is not simple to invert this Laplace transformLaplace transform, because this operation is , because this operation is ill-ill-conditionedconditioned and, moreover, when a mathematical solution is obtained, it and, moreover, when a mathematical solution is obtained, it presents, sometimes, presents, sometimes, strong oscillation or non-physical behaviourstrong oscillation or non-physical behaviour..

Molecule-Molecule collision models (2)Molecule-Molecule collision models (2)

In order to overcome these problems, we have used two types of approach, one In order to overcome these problems, we have used two types of approach, one for for nitrogennitrogen case study, and another one for case study, and another one for oxygenoxygen and and hydrogenhydrogen..

Despite the potential capabilities of state-to-state direct description of the Despite the potential capabilities of state-to-state direct description of the chemical kinetics offered by DSMC, phenomenological model for internal state chemical kinetics offered by DSMC, phenomenological model for internal state kinetics have been applied to this method as a rule (e.g. Larsen-Borgnakke kinetics have been applied to this method as a rule (e.g. Larsen-Borgnakke model). We therefore have developed a simple model for collision between model). We therefore have developed a simple model for collision between vibrationally excited molecules. Following the approach of vibrationally excited molecules. Following the approach of AndersonAnderson, we use a , we use a simple flexible statistical model for translational and internal energy exchange simple flexible statistical model for translational and internal energy exchange that strictly satisfies detailed balance and incorporates some features of the that strictly satisfies detailed balance and incorporates some features of the real cross section.real cross section.

NitrogenNitrogen

Molecule-Molecule: Nitrogen (1)Molecule-Molecule: Nitrogen (1)

In the simulation we take into account two kinds of mono-quantum energy In the simulation we take into account two kinds of mono-quantum energy exchange collisions:exchange collisions:• Vibration-translation (VT) energy exchangeVibration-translation (VT) energy exchange

€

σν v → ν v −1ζ v → ζ v

⎛

⎝ ⎜

⎞

⎠ ⎟=cVTν v ′ Ekin

σν v → ν v +1ζ v → ζ v

⎛

⎝ ⎜

⎞

⎠ ⎟=cVT ν v +1( ) ′ Ekin

Detailed balance principleDetailed balance principle

€

A2 ν v( ) + A2 ζ v( ) → A2 ν v +1( ) + A2 ζ v( )

€

σ ν v ,ζ v → ν v +1,ζ v;g( )g2 =σ ν v +1,ζ v → ν v ,ζ v; ′ g( ) ′ g 2

€

cVT ν v +1( )12

mr ′ g 2 ⎡

⎣ ⎢

⎤

⎦ ⎥g2 = cVT ν v +1( )

12

mrg2

⎡

⎣ ⎢

⎤

⎦ ⎥ ′ g 2

Molecule-Molecule: Nitrogen (2)Molecule-Molecule: Nitrogen (2)

• Vibration-vibration (VV) energy exchangeVibration-vibration (VV) energy exchange

€

σν v → ν v +1ζv → ζv −1

⎛

⎝ ⎜

⎞

⎠ ⎟=cVV ν v +1( )ζv ′ Ekin

For example, in this processFor example, in this process

€

A2 ν v( ) + A2 ζ v( ) → A2 ν v +1( ) + A2 ζ v −1( )

cross sections are proportional, through an adjustable factor cross sections are proportional, through an adjustable factor ccVVVV, to the product , to the product

between the two higher vibrational numbers in the considered transition and between the two higher vibrational numbers in the considered transition and the kinetic energy of the couple of colliding particles after hit.the kinetic energy of the couple of colliding particles after hit.

We have adjusted the model parameters to reproduce the rate coefficients We have adjusted the model parameters to reproduce the rate coefficients calculated by calculated by Billing and FisherBilling and Fisher in the temperature range of interest. in the temperature range of interest.

Molecule-Molecule: Oxygen and HydrogenMolecule-Molecule: Oxygen and Hydrogen

For For oxygenoxygen and and hydrogenhydrogen, we have used a different approach. Practically, the , we have used a different approach. Practically, the diatom-diatom energy transfer processes are modelled by fitting directly the diatom-diatom energy transfer processes are modelled by fitting directly the rate coefficients, of rate coefficients, of BillingBilling and and KolesnickKolesnick for oxygen and of for oxygen and of MatveyevMatveyev and and SilakovSilakov for hydrogen, by for hydrogen, by downhill simplex methoddownhill simplex method..

This method is due to This method is due to NelderNelder and and MeadMead and it requires only function and it requires only function evaluations. A simplex is the geometrical figure consisting, in N dimensions, of evaluations. A simplex is the geometrical figure consisting, in N dimensions, of N+1 points (or vertices) and all their interconnecting line segments, polygonal N+1 points (or vertices) and all their interconnecting line segments, polygonal faces, etc.faces, etc.

For multidimensional minimization, the best we can do is give our algorithm a For multidimensional minimization, the best we can do is give our algorithm a starting guess, that is, an N-vector of independent variables as the first point to starting guess, that is, an N-vector of independent variables as the first point to try. The algorithm is then supposed to make its own way downhill through the try. The algorithm is then supposed to make its own way downhill through the unimaginable complexity of an N-dimensional topography, until it encounters unimaginable complexity of an N-dimensional topography, until it encounters a (local, at least) minimum. The downhill simplex method must be started not a (local, at least) minimum. The downhill simplex method must be started not just with a single point, but with N+1 points, defining an initial simplex.just with a single point, but with N+1 points, defining an initial simplex.

Downhill simplex method (2)Downhill simplex method (2)

Mono-quantum transitions (Mono-quantum transitions (VTVT) are included along with quasi-resonant ) are included along with quasi-resonant VV VV energy transfer.energy transfer.

The cross sections for these processes are assumed to be of the form:The cross sections for these processes are assumed to be of the form:

€

σ =αEβexp−γE( )

EE being the collision energy and being the collision energy and αα,, ββ and and γγ the fitting parameter for each the fitting parameter for each transition. The cross sections for the backward transitions are determined by transition. The cross sections for the backward transitions are determined by application of the detailed balance principle.application of the detailed balance principle.

We decide to cut the cross sections at We decide to cut the cross sections at EEmaxmax==3eV3eV..

The best fit parameters are able to reproduce the calculated rate coefficients The best fit parameters are able to reproduce the calculated rate coefficients very well in the temperature range very well in the temperature range ((300-10000K300-10000K).).

Downhill simplex method results (1)Downhill simplex method results (1)

(a)(a) VT cross sections obtained by Downhill Simplex Method for a process in which molecule, that do not VT cross sections obtained by Downhill Simplex Method for a process in which molecule, that do not change vibrational level, is at the ground state.change vibrational level, is at the ground state.

(a)(a)

10-8

10-6

10-4

10-2

100

102

0 0,5 1 1,5 2 2,5 3

H2(v)+H

2(0) ---> H

2(v-1)+H

2(0)

v=1v=5v=10v=14

( )E eV

(c)(c) VV cross sections obtained by Downhill Simplex Method for various processes. VV cross sections obtained by Downhill Simplex Method for various processes.

(c)(c)

10-5

10-4

10-3

10-2

10-1

100

0 0,5 1 1,5 2 2,5 3

H2(v)+H

2(w+1) ---> H

2(v+1)+H

2(w)

v=1 ; w=0v=5 ; w=4v=10 ; w=9v=14 ; w=13

( )E eV

(b)(b) VT cross sections obtained by Downhill Simplex Method for a process in which molecule, that do not VT cross sections obtained by Downhill Simplex Method for a process in which molecule, that do not change vibrational level, is not at the ground state.change vibrational level, is not at the ground state.

(b)(b)

10-8

10-6

10-4

10-2

100

102

0 0,5 1 1,5 2 2,5 3

H2(v)+H

2(5) ---> H

2(v-1)+H

2(5)

v=1v=5v=10v=14

( )E eV


(a)(a)

5 10-15

6 10-15

7 10-15

8 10-15

9 10-15

10-14

0 2000 4000 6000 8000 10000

H2(1)+H

2(1) ---> H

2(2)+H

2(0)

Downhill Simplex MethodMatveyev and Silakov

T(K)

(b)(b)

3 10-13

4 10-13

5 10-13

6 10-13

0 2000 4000 6000 8000 10000

H2(10)+H

2(10) ---> H

2(11)+H

2(9)


T(K)

(a)(a), , (b)(b), , (c)(c) Comparison between rate coefficients obtained by Downhill Simplex Method and that Comparison between rate coefficients obtained by Downhill Simplex Method and that obtained by fit from Matveyev and Silakov.obtained by fit from Matveyev and Silakov.

(c)(c)

6 10-13

7 10-13

8 10-13

9 10-13

10-12

0 2000 4000 6000 8000 10000

H2(14)+H

2(14) ---> 2H+H

2(13)


T(K)


(a)(a) Comparison between cross sections obtained by Downhill Simplex Method and by QCT Comparison between cross sections obtained by Downhill Simplex Method and by QCT

(a)(a)

10-1

100

101

102

0 0,5 1 1,5 2 2,5 3

H2(14)+H ---> 3H

Downhill Simplex MethodQCT

( )E eV

(b)(b) Comparison between rate coefficients, relative to the same process of Comparison between rate coefficients, relative to the same process of (a)(a), obtained by Downhill , obtained by Downhill Simplex Method and by QCT.Simplex Method and by QCT.

(b)(b)

10-10

10-9

10-8

0 2000 4000 6000 8000 10000

H2(14)+H ---> 3H


T(K)



(a)(a)

10-6

10-5

10-4

10-3

0 0,5 1 1,5 2 2,5 3

H2(0)+H ---> 3H


( )E eV


(b)(b)

10-17

10-16

10-15

10-14

0 2000 4000 6000 8000 10000

H2(0)+H ---> 3H


T(K)



(a)(a)

100

101

102

0 0,5 1 1,5 2 2,5 3

H2(14)+H ---> H

2(13)+H


( )E eV


(b)(b)

2 10-10

3 10-10

4 10-10

5 10-10

0 2000 4000 6000 8000 10000

H2(14)+H ---> H

2(13)+H


T(K)

Molecule-Molecule: dissociationMolecule-Molecule: dissociation

• Multi-quantum VT transitionMulti-quantum VT transition

Dissociation cross sections from lower levels are equal to mono-quantum cross Dissociation cross sections from lower levels are equal to mono-quantum cross section from the last bounded level vsection from the last bounded level vmaxmax to the pseudo-level v to the pseudo-level vdissdiss, calculated at , calculated at

the same total energy, dampened from a negative exponential factor.the same total energy, dampened from a negative exponential factor.

€

σ ν v ,ζv → νdiss,ζv;Etot( ) =σ νmax,ζv → νdiss,ζv;Etot( )

Such damping factor is called Such damping factor is called vibrational favouringvibrational favouring..

€

e−Eνmax

−Eνv

kU

If we consider the following VT reaction:If we consider the following VT reaction:

€

A2 νmax( ) + A2 ζ v( ) → 2A+ A2 ζ v( )

Cross section can be written as:Cross section can be written as:

€

σ νmax,ζ v → ν diss,ζ v;Etot=Ekin∗ +Eνmax( ) =cVTν diss Ekin

∗( )

′

Finally, we can write for dissociative reactions, through multi-quantum Finally, we can write for dissociative reactions, through multi-quantum transition, the following expression of cross section:transition, the following expression of cross section:

€

σ ν v ,ζ v → ν diss,ζ v;Etot=Ekin∗ +Eνv( ) =cVTν diss Ekin

∗( )

′e−

Eν max−Eν v

kU

Shock wave produced in pure Nitrogen (I)Shock wave produced in pure Nitrogen (I)

(a)(a)

0 100

2 1015

4 1015

6 1015

8 1015

1 1016

-200 0 200 400 600 800 1000 1200x/λ

(b)(b)

0 100

2 10-2

4 10-2

6 10-2

8 10-2

1 10-1

-200 0 200 400 600 800 1000 1200/x λ

In Fig. In Fig. (a)(a) and and (b)(b) we report, the total number density and the atomic molar we report, the total number density and the atomic molar fraction. The spatial scale is always expressed using the upstream mean free fraction. The spatial scale is always expressed using the upstream mean free path path λλ as a unit. We can observe the effect of the thermal relaxation on the as a unit. We can observe the effect of the thermal relaxation on the gas number density. The gas number density. The compression ratiocompression ratio behind the shock waves reaches behind the shock waves reaches rapidly the value of 6 valid for strong shocks in a gas of rigid rotors. From 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 here it grows slowly to the value of 8, following the slow vibrational slow vibrational relaxationrelaxation. Further increase of the compression ratio is due to the increase of . Further increase of the compression ratio is due to the increase of the atom fraction.the atom fraction.

Shock wave produced in pure Nitrogen (II)Shock wave produced in pure Nitrogen (II)

(c)(c)

0,0 100

4,0 103

8,0 103

1,2 104

-20 0 20 40 60 80

TT

rot

T01

x/λ

(d)(d)

0,0 100

4,0 103

8,0 103

1,2 104

-200 0 200 400 600 800 1000 1200

TT

rot

T01

x/λ

In Fig. In Fig. (c)(c) and and (d)(d) we report the three ‘temperature’ which are relevant for we report the three ‘temperature’ which are relevant for this study: i.e. the static translational temperature this study: i.e. the static translational temperature TT, the rotational , the rotational temperature temperature TTrotrot and the and the TT0101 vibrational temperature vibrational temperature

€

T01 =−Δε01 kln n1 n0( )[ ]−1

It can be seen that the sudden jump in T is rapidly followed by slower jump for It can be seen that the sudden jump in T is rapidly followed by slower jump for TTrotrot. The vibrational relaxation follows with a longer relaxation time.. The vibrational relaxation follows with a longer relaxation time.

Shock wave produced in pure Nitrogen (III)Shock wave produced in pure Nitrogen (III)

(a)(a)(a)(a)

10-5

10-4

10-3

10-2

10-1

100

0 2 4 6 8

x/λ=-138/x λ=17/x λ=26/x λ=47/x λ=74/x λ=145/x λ=850

, Energy eV

(b)(b)

10-5

10-4

10-3

10-2

10-1

100

0 1 2 3 4 5 6 7

x/λ=-138/x λ=5/x λ=11/x λ=17/x λ=23/x λ=32

, Energy eV

This slower relaxation is also affected by its mesoscopic structure in terms of This slower relaxation is also affected by its mesoscopic structure in terms of the vibrational distribution functions (vdf). In Fig. the vibrational distribution functions (vdf). In Fig. (a)(a) we report some of these we report some of these vdf’s at different positions along the flow. It can be noticed that the vdf is vdf’s at different positions along the flow. It can be noticed that the vdf is sensibly non-Boltzmann due to the highly effective VV/VT processes.sensibly non-Boltzmann due to the highly effective VV/VT processes.In Fig. In Fig. (b)(b) is plotted the distribution of (classical) rotational energy. The is plotted the distribution of (classical) rotational energy. The different curves refer to different positions along the shock front and during different curves refer to different positions along the shock front and during the rotational relaxation. It can be seen the strong deviation from the the rotational relaxation. It can be seen the strong deviation from the equilibrium distribution during relaxation.equilibrium distribution during relaxation.

Shock wave produced in pure Nitrogen (IV)Shock wave produced in pure Nitrogen (IV)

(a)(a)

In Fig. In Fig. (c) (c) we report the we report the velocity velocity along the flow, while in Fig. along the flow, while in Fig. (d)(d) we report we report the the translational distribution functiontranslational distribution function at different positions along the shock at different positions along the shock front. At the two extreme points the distribution are equilibrium distributions front. At the two extreme points the distribution are equilibrium distributions at the upstream and downstream temperatures, respectively. At the points in at the upstream and downstream temperatures, respectively. At the points in between a sensible deviation from the Maxwell one is observed, in the form of between a sensible deviation from the Maxwell one is observed, in the form of a a ‘tail’‘tail’. Even if the effect of such a deviation on macroscopic quantities is . Even if the effect of such a deviation on macroscopic quantities is probably small under typical conditions, these plots show the degree of detail probably small under typical conditions, these plots show the degree of detail which our model can achieve in describing the shock thickness.which our model can achieve in describing the shock thickness.

0 100

1 105

2 105

3 105

4 105

5 105

-200 0 200 400 600 800 1000 1200x/λ

(c)(c)

10-4

10-3

10-2

10-1

100

0 100 2 105 4 105 6 105 8 105

x/λ=-138/x λ=5/x λ=11/x λ=14

/x λ=17/x λ=20/x λ=32/x λ=1146

, /v cm s

(d)(d)

ConclusionsConclusions

In our work, we have studied by direct simulation the vibrational and In our work, we have studied by direct simulation the vibrational and dissociation kinetics in a strong normal shock wave in molecular gases. The dissociation kinetics in a strong normal shock wave in molecular gases. The main achievement of our work in comparison with previous studies is the self-main achievement of our work in comparison with previous studies is the self-consistent modelling of translational, rotational and vibrational kinetics of a consistent modelling of translational, rotational and vibrational kinetics of a diatomic gas which is undergoing dissociation in the shock wave. Furthermore, diatomic gas which is undergoing dissociation in the shock wave. Furthermore, we used only data in form of cross sections from the literature, or calculated by we used only data in form of cross sections from the literature, or calculated by molecular dynamics methods. molecular dynamics methods. Results show that the vibrational, rotational and Results show that the vibrational, rotational and translational distribution functions can significantly deviate from the equilibrium translational distribution functions can significantly deviate from the equilibrium one in the shock region.one in the shock region. Such deviations affect in turn the rate of elementary Such deviations affect in turn the rate of elementary processes.processes.

ImprovementsImprovements

Critical points:Critical points:

• Computational costComputational cost

• There are few data for molecule-molecule There are few data for molecule-molecule interactioninteraction

In order to overcome the first problem we are led to consider a In order to overcome the first problem we are led to consider a parallelization of the code. This will also permit to extend the calculations to parallelization of the code. This will also permit to extend the calculations to 2D.2D.

This is necessary in order to couple the DSMC with fluid-dynamics codes for This is necessary in order to couple the DSMC with fluid-dynamics codes for the simulation of near-continuum flows.the simulation of near-continuum flows.

A hybrid particle-continuum computational method would be more accurate A hybrid particle-continuum computational method would be more accurate in order to describe re-entry phenomena, paying attention to the information in order to describe re-entry phenomena, paying attention to the information exchange at the interface between the particle and the continuum domain.exchange at the interface between the particle and the continuum domain.

Finally we are studying how to couple DSMC with a PIC method in order to Finally we are studying how to couple DSMC with a PIC method in order to reproduce plasma phenomena or shock waves in ionized gases.reproduce plasma phenomena or shock waves in ionized gases.

Documents

State to state models for dissociation behind shock waves Minelli Pierpaolo IMIP-CNR Bari, Italy