state-to-state models for dissociation-recombination flow...

61st Course of Hypersonic Meteoroid Entry Physics3-8 October 2017, Ettore Majorana Foundation, Erice

State-to-state models fordissociation-recombination flow regimes

Elena KustovaSaint Petersburg State University, Russia

Outline

I IntroductionI Theoretical model of kinetics and transport propertiesI State-resolved cross sections and rate coefficients

I QCT-based simple theoretical modelsI QCT-based cross-sections of dissociation and vibrational

transitionsI State-resolved transport coefficients

I Influence of molecular sizeI Ro-vibrational couplingI Pr, Sc, Le numbers and heat/mass transfer

I New challengesI Conclusions

Motivation

Strongly non-equilibrium flows and media:

I Atmospheric entryI controlled vehicle descentI meteoroid fall

I Plasma-assisted technologiesI flow controlI discharges and their applicationsI combustionI ecological problems

State-to-state modeling is a powerful tool for in-depthstudies of strongly non-equilibrium situations.

State-to-state models

I The idea has originated in 1950sI E. Montroll, K. Shuler, J. Chem. Phys. 26 (3) (1957) 454

I First studies of non-equilibrium kinetics in inviscid flowsI Shock waves: E. Nagnibeda (1995), R. Brun et al (1995);

I. Adamovich, S. Macheret et al (1998)I Nozzles: S. Ruffin, C. Park (1992); G. Colonna et al (1999)I Stagnation line flows: I. Armenise et al (1996)

I State-to-state models for transition and reaction ratesI Theoretical: SSH, FHO, Treanor-MarroneI Molecular dynamics: G.Billing; Bari (F. Esposito, I. Armenise),

Perugia (A. Lagana, A. Lombardi); Barcelona (R. Sayos et al);I. Boyd et al; NASA Ames (R. Jaffe, D. Schwenke et al)

I State-resolved models for viscous flowsI Transport properties: E. Kustova, E. Nagnibeda (1998)I Heat transfer in different flows: E. Kustova et al (2000-2015)

Current state of the art

I ChallengesI Computational costsI Lack of data on the state-resolved rate coefficientsI Viscous flow simulationsI Polyatomic gases

I Groups involvedI CNR-Nanotec, ItalyI Michigan University, USAI University of Illinois Urbana-Champaign, USAI NASA Ames Research Center, USAI VKI, BelgiumI Saint Petersburg University, RussiaI and others...

Team of Saint-Petersburg University

I People:

I E. Nagnibeda, E. Kustova, O. Kunova, V. Istomin, G. Oblapenko,M. Mekhonoshina, A. Savelev, ...

I Recent papers:I O. Kunova, E. Nagnibeda. Chem. Phys. 2014, 441, 66.I O. Kunova, E. Nagnibeda. Chem. Phys. Lett. 2015, 625, 121.I E. Kustova, G.M. Kremer, Chem. Phys. Lett. 2015, 636, 84.I O. Kunova, E. Kustova, M. Mekhonoshina, E. Nagnibeda, Chem. Phys. 2015, 463, 70.I E. Kustova, G. Oblapenko. Phys. Rev. E, 2016, 93, 033127.I O. Kunova, E. Kustova, A. Savelev. Chem. Phys. Lett. 2016, 659, 80.I E. Kustova, M. Mekhonoshina, G. Oblapenko, Chem. Phys. Lett. 2017, 686, 161.I G.M. Kremer, O. Kunova, E. Kustova, G. Oblapenko. Physica A, 2017, 490, 92.I V. Istomin,E. Kustova, Chem. Phys. 2017, 485, 125.

Theoretical model

Collisional processesI Elastic collisions

I Translational energy exchangeI Inelastic non-reactive collisions

I Rotational energy exchangeI VV (vibration-vibration) energy exchangeI VT (vibration-translation) energy exchangeI Electronic excitation, VE and ET transitions

I Reactive collisionsI Dissociation-recombinationI Exchange reactionsI Ionization

Theoretical model

I Characteristic time relation

τtr ∼ τrot� τVV < τVT ∼ τVE < τreact ∼ θ

I Governing equations

dρ

dt+ ρ∇ · v = 0,

ρdvdt

+∇ · P = 0,

ρdU

dt+∇ · q+ P : ∇v = 0,

dncidt

+ nci∇ · v+∇ · (nciVci ) = Rvibrci + R react

ci i = 0, ..., Lc

Production terms

I Vibrational transitions, Rvibrci = RVV

ci + RVTci

I VV: RVVci =

∑dki ′k′

(nci ′ndk′kd, k′k

c, i ′i − ncindkkd,kk′

c,ii ′

)I VT: RVT

ci =∑

d nd∑

i ′

(nci ′k

dc, i ′i − ncik

dc,ii ′

)I Chemical reactions, R react

ci = Rdissci + Rexch

ci

I D-R: Rdissci =

∑d nd

(nc′nf ′k

drec, ci − ncik

dci, diss

)I EX: Rexch

ci =∑

dc′d′ki ′k′

(nc′i ′nd′k′kd′k′, dk

c′i ′, ci − ncindkkdk, d′k′

ci, c′i ′

)I kd ,kk

′

c,ii ′ , kdc,ii ′ , kdci , diss, k

drec, ci , k

dk, d ′k ′

ci , c ′i ′ are correspondingstate-resolved rate coefficients.

Transport terms

I Pressure tensor:

P = (p − prel) I− 2η S− ζ∇·v II prel , η, ζ are relaxation pressure, shear and bulk viscosity

coefficients.

I Diffusion velocities of diatomic species:

Vci = −∑k

Dcidkdk −∑d 6=c

Dcddd − DTc∇ lnT ,

I Dcidk are diffusion coefficients for different vibrational states, Dcd arespecies diffusion coefficients, DTc are thermal diffusion coefficients

I dci are diffusive driving forces for each vibrational state

Transport terms

I Heat flux:

q = −λ∇T − p∑c

DTcdc +∑c

Lc∑i=0

(52kT + 〈εcrot〉+ εci + εc

)nciVci

I λ = λtr + λrot is the heat conductivity coefficient of translationaland rotational degrees of freedom; 〈εcrot〉 is the mean rotationalenergy; εci , is the vibrational energy of the i-th state, and εc is thespecies formation energy.

I Contributions to the heat flux:

q = qF + qMD + qTD + qDVE

F: heat conduction (Fourier); MD: mass diffusion; TD: thermal diffusion;DVE: diffusion of vibrational states.

Transport Coefficients

I Transport coefficients can be found as solutions of transport linearsystems and thus expressed in terms of the bracket integralsdepending on the cross sections of rapid processes.

I Transport coefficients depend on the temperature, number densitiesof species and all vibrational level populations of molecular species.

I Dimension of transport linear systems is of the order of N(N = Nvibr + Nat , Nvibr is the total number of vibrational states inthe mixture, Nat is the number of atomic species).

I Total amount of diffusion coefficients is about N2.

I Transport linear systems should be solved at each time and spacestep of a CFD code. This is particularly complicated in polyatomicgas mixtures like CO2.

Heat fluxes. Examples

0 , 0 0 , 2 0 , 4

- 4 0 0 0

- 2 0 0 0

0

2 0 0 0

4 0 0 0

x , c m

q , k W / m 2 q H C , S T S q D V E , S T S q M D , S T S q H C , 1 - T q M D , 1 - T

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

- 5 0 0 00

5 0 0 01 0 0 0 01 5 0 0 02 0 0 0 0

x , c m

q , k W / m 2 q H C , S T S q D V E , S T S q M D , S T S q H C , 1 - T q M D , 1 - T

Contributions to heat flux behind the shock. M0 = 18. (a) – N2/N, (b) – O2/O

Figure 11. Contributions to the heat flux

Figure 12. Prandtl number and Fourier flux, catalytic wall

-5 104

0

5 104

1 105

1.5 105

2 105

2.5 105

3 105

0 1 2 3 4 5 6 7 8

Fourier

Therm.Diff

MassDiff.

Vibr.Exc.Molec.

HeatFl

q

(W m

-2)

N2+N = 78.58 %O2+O = 21.38 %NO = 0.04 %

1

2

3

4

5

no catalytic wall

= 5000 s-1

Te = 7000 K

pe = 1000 N/m

2

Tw = 1000 K

-5 103

0

5 103

1 104

1.5 104

2 104

2.5 104

3 104

0 1 2 3 4 5 6 7 8

Fourier

Therm.Diff

MassDiff.

Vibr.Exc.Molec.

HeatFl

q

(W m

-2)

N2+N = 78.58 %O2+O = 21.38 %NO = 0.04 %

1

2

3

4

5

no catalytic wall

= 500 s-1

Te = 3000 K

pe = 1000 N/m

2

Tw = 500 K

7.5 10-1

8 10-1

8.5 10-1

9 10-1

9.5 10-1

0 1 2 3 4 5 6 7 8

Pr

Pr

Prandtl Number

N2+N = 78.58 %O2+O = 21.38 %NO = 0.04 %

= 5000 s-1

Te = 7000 K

pe = 1000 N/m

2

Tw = 1000 K

partially catalytic wall

NN

= 3.88 10-3

ON = 7.60 10

-3

OO

= 1.00 10-2

NO = 6.73 10

-3

-4 104

0

4 104

8 104

1.2 105

1.6 105

2 105

2.4 105

0 1 2 3 4 5 6 7 8

qF

qF from Pr=const

q (W

m-2

)

1

2

N2+N = 78.58 %O2+O = 21.38 %NO = 0.04 %


= 5000 s-1

Te = 7000 K

pe = 1000 N/m

2

Tw = 1000 K

NN

= 3.88 10-3

ON = 7.60 10

-3

OO

= 1.00 10-2

NO = 6.73 10

-3

-2 105

-1 105

0

1 105

2 105

3 105

4 105

5 105

6 105

0 1 2 3 4 5 6 7 8

Fourier

Therm.Diff

MassDiff.

Vibr.Exc.Molec.

HeatFlq

(W m

-2)

η

1

2

3

4

5

N2+N = 78.58 %O2+O = 21.38 %NO = 0.04 %


β = 5000 s-1

Te = 7000 K

pe = 1000 N/m2

Tw = 1000 K

γNN

= 3.88 10-3 γON

= 7.60 10-3

γOO

= 1.00 10-2 γNO

= 6.73 10-3

Heat flux along the stagnation line in air near non-catalytic (a) and partially catalytic (b) surface.

State-to-state dissociation models

I Ladder-climbing model

I Modifications of the Treanor–Marrone model suitable forstate-to-state kinetics:

I Meolans & Brun (1995);I Kustova & Savelev (2016);I Andrienko & Boyd (2016).

I Molecular dynamics (QCT simulations):

I Esposito, Capitelli et al (2006);I Open DataBase: Planetary entry integrated models,

http://phys4entrydb.ba.imip.cnr.it/Phys4EntryDB/.I NASA AMES (Jaffe, Schwenke, Chaban et al), no open data;I Boyd et al (2016);I Pogosbekian et al (2008).

Treanor–Marrone model

The state-specific dissociation rate coefficient kMi,diss from the vibrational

state i after a collision with a partner M:

kMi,diss = ZM

i kMdiss, eq(T ),

kMdiss, eq(T ) is the thermal equilibrium dissociation rate coefficient, ZM

i isthe non-equilibrium factor:

ZMi = Zi (T ,U) =

Zvibr(T )

Zvibr(−U)exp

(εik

(1T

+1U

)),

εi is the vibrational energy of the ith state, Zvibr(T ) is the equilibriumvibrational partition function

Zvibr(T ) =∑i

exp(− εikT

).

Parameter U

I Commonly used parametersI U =∞ (non-preferential dissociation);I U = D/6k =const;I Linear function of T , U = 3T ;

I Recent modelsI Andrienko & Boyd:

U is a weighted linear combination of U = D/(6k) andU = 3T ;

I Pogosbekian U = D(0.5+ T

20000

)I Kustova & Savelev [O. Kunova, E. Kustova, A. Savelev.

Chem. Phys. Lett. 2016, 659, 80]

Parameter U

State-dependent parameter

U(i ,T ) =N∑

n=0

anεni exp

(T

K∑k=0

bk εki

)

I obtained by fitting QCT data of Phys4Entry database; thecoefficients are given in [Kustova et al, Chem. Phys. 2016]

I requires modification of the original formula for the state-specificnon-equilibrium factor

ZMi =

Zvibr(T )exp(− D

kUi

)∑j

exp(−D − εj

kUj

) exp(εik

(1T

+1Ui

))

STS dissociation rate coefficients

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 01 0 - 1 6

1 0 - 1 4

1 0 - 1 2

1 0 - 1 0

1 0 - 8

O 2

k i , m 3 / s

T , K

i = 1 , U ( i , T ) i = 1 , D B i = 1 0 , U ( i , T ) i = 1 0 , D B i = 2 5 , U ( i , T ) i = 2 5 , D B i = 4 0 , U ( i , T ) i = 4 0 , D B

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 01 0 - 2 4

1 0 - 2 2

1 0 - 2 0

1 0 - 1 8

1 0 - 1 6

1 0 - 1 4

1 0 - 1 2

1 0 - 1 0

1 0 - 8

N 2

k i , m 3 / s

T , K

i = 1 , U ( i , T ) i = 1 , D B i = 1 0 , U ( i , T ) i = 1 0 , D B i = 3 0 , U ( i , T ) i = 3 0 , D B i = 6 0 , U ( i , T ) i = 6 0 , D B

Parameter U

0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 00

2 0 0 0 04 0 0 0 06 0 0 0 08 0 0 0 0

1 0 0 0 0 01 2 0 0 0 01 4 0 0 0 01 6 0 0 0 01 8 0 0 0 02 0 0 0 0 0 ( a )

T , K

U , K

i = 3 i = 1 5 i = 3 5 U = 3 T U = D / 6 k

O 2

0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 00

2 0 0 0 0

4 0 0 0 0

6 0 0 0 0 ( b )N 2U , K

T , K

i = 3 i = 2 0 i = 4 0 U = 3 T U = D / 6 k

0 1 0 2 0 3 0 4 00

5 0 0 0 0

1 0 0 0 0 0

1 5 0 0 0 0

2 0 0 0 0 0

2 5 0 0 0 0

3 0 0 0 0 0 ( a )

i

U , K O 2

T = 2 0 0 0 K T = 1 0 0 0 0 K T = 2 0 0 0 0 K U = D / 6 k

0 1 0 2 0 3 0 4 0 5 0 6 0 7 00

1 0 0 0 0

2 0 0 0 0

3 0 0 0 0

4 0 0 0 0

5 0 0 0 0

6 0 0 0 0

7 0 0 0 0 ( b )N 2

i

U , K T = 2 0 0 0 K T = 1 0 0 0 0 K T = 2 0 0 0 0 K U = D / 6 k

Nonequilibrium factors, O2+O

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0

1 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1Z i

D / 6 k 3 T U ( i , T ) A n d r i e n k o P o g o s b e k y a n

T , K

O 2 , i = 0

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0

1 0 - 2

1 0 - 1

1 0 0

Z i D / 6 k 3 T U ( i , T ) A n d r i e n k o P o g o s b e k y a n

T , K

O 2 , i = 5

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0

1 0 - 1

1 0 0

1 0 1

1 0 2

Z i


T , K

O 2 , i = 1 0

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 01 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

Z i


T , K

O 2 , i = 2 0

Nonequilibrium factors, N2+N

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 01 0 - 9

1 0 - 8

1 0 - 7

1 0 - 6

1 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2Z i

D / 6 k 3 T U ( i , T )

T , K

N 2 , i = 0

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 0

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0Z i

D / 6 k 3 T U ( i , T )

T , K

N 2 , i = 5

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 01 0 - 2

1 0 - 1

1 0 0

1 0 1

1 0 2

1 0 3

Z i

D / 6 k 3 T U ( i , T )

T , K

N 2 , i = 1 0

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 2 0 0 0 01 0 01 0 11 0 21 0 31 0 41 0 51 0 61 0 71 0 81 0 9

1 0 1 01 0 1 11 0 1 21 0 1 31 0 1 41 0 1 5 Z i

D / 6 k 3 T U ( i , T )

T , K

N 2 , i = 3 0

Nonequilibrium factors, O2+O

0 2 41 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

Z i

ε, e V


O 2 , T = 1 5 0 0 0 K

0 2 4 6 8 1 0

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

1 0 1

1 0 2

1 0 3 Z i

D / 6 k 3 T U ( i , T )

ε, e V

N 2 , T = 1 5 0 0 0 K

State-specific non-equilibrium factors in O2 (a) and N2 (b) as functions of vibrational

energy at temperature 15000 K for different parameter U.

STS dissociation rate coefficients

I An easy-to-implement generalization of the widely knownTreanor–Marrone model suitable for the state-to-state andmulti-temperature flow simulations is proposed.

I The dependence of the parameter U on the vibrational state of adissociating molecule is obtained by fitting QCT results.

I The model can be applied for any vibrational ladder.

I For O2 and N2 dissociation U(i ,T ) provides a good agreement withQCT rate coefficients in the whole range of i and T .

I Work is in progress on O2+O2, N2+N2, O2+N2 collisions.

State-resolved exchange reactions

NO formation: N2(i) + O = NO(k) + N

I Theoretical models:

I Rusanov & Fridman (1984);I Polak (1984);I Warnatz (1992);I Aliat (2008).

I Molecular dynamics (QCT simulations):

I Bose & Candler (1996);I Esposito & Armenise (2017);I Open Stellar DataBase:

http://esther.ist.utl.pt/pages/stellar.html.I NASA AMES (?) no open data;

I Dependence on NO vibrational state is not taken into account intheoretical models.

STS exchange rate coefficients

Aliat model (generalization of the Treanor-Marrone model)

kexchci,d (T ,U)=

C (T ,U)kexch

eq exp(− Ea

kU

)exp

[εcik

(1T

+1U

)], εci < Ea

C (T ,U)kexcheq exp

(Ea

kT

), εci > Ea

C (T ,U)=

∑i(1)

1Z vibrc (T )

exp(−Ea − εci

kU

)+∑i(2)

1Z vibrc (T )

exp(Ea − εcikT

)−1

In the original paper [A. Aliat, Physica A, 387:4163-4182, 2008], there isa misprint in the last term.


0 2 4 6 8 1 01 0 - 2 0

1 0 - 1 9

1 0 - 1 8

1 0 - 1 7

1 0 - 1 6

1 0 - 1 5

R u s a n o v P o l a k W a r n a t z A l i a t , U = 3 T A l i a t , U = D / 6 k S t e l l a r

ε , e V

T = 1 0 0 0 0 Kk i , m3 / s

0 2 4 6 8 1 0

1 0 - 1 9

1 0 - 1 8

1 0 - 1 7

1 0 - 1 6

1 0 - 1 5


ε , e V

T = 2 0 0 0 0 Kk i , m3 / s

5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 01 0 - 2 6

1 0 - 2 5

1 0 - 2 4

1 0 - 2 3

1 0 - 2 2

1 0 - 2 1

1 0 - 2 0

1 0 - 1 9

1 0 - 1 8

1 0 - 1 7

1 0 - 1 6

1 0 - 1 5


T , K

i = 0k i , m3 / s

5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 01 0 - 2 1

1 0 - 2 0

1 0 - 1 9

1 0 - 1 8

1 0 - 1 7

1 0 - 1 6

1 0 - 1 5

1 0 - 1 4


T , K

i = 2 0k i , m3 / s

Exchange reaction rate coefficients for N2+O reaction for different models with

corrected Aliat model.


Generalized Aliat model

kexchN2(i),NO(k)(T ,U)=

C kexch

eq exp(−Ea + εNOk

kU

)exp

[εN2i

k

(1T

+1U

)], εci < Ea + εNO

k

C kexcheq exp

(Ea + εNOk

kT

), εci > Ea + εNO

k

C(T ,U) =

∑i(1)

1Z vibrN2

(T )exp

(−Ea + εNOk − εN2

i

kU

)+

+∑i(2)

1Z vibrN2

(T )exp

(Ea + εNO

k − εN2i

kT

)−1

Generalized model accounts for the dependence on final NO vibrational state.


0 1 2 3 4 5 6 7 8 9 1 01 E - 2 8

1 E - 2 6

1 E - 2 4

1 E - 2 2

1 E - 2 0

1 E - 1 8

M o d e l , i = 0 M o d e l , i = 5 M o d e l , i = 1 0 M o d e l , i = 4 0 S t e l l a r , i = 0 S t e l l a r , i = 5 S t e l l a r , i = 1 0 S t e l l a r , i = 4 0

T = 5 0 0 0 Kk i , c m 3 / s

ε , e V0 1 2 3 4 5 6 7 8 9 1 0

1 E - 2 0

1 E - 1 9

1 E - 1 8

1 E - 1 7

ε , e V

T = 1 5 0 0 0 Kk i , c m 3 / s

M o d e l , i = 0 M o d e l , i = 5 M o d e l , i = 1 0 M o d e l , i = 4 0 S t e l l a r , i = 0 S t e l l a r , i = 5 S t e l l a r , i = 1 0 S t e l l a r , i = 4 0

State-specific exchange reaction rate coefficients in N2+O collision as functions from

N2 for different NO levels for Stellar and our models.


I Discrepancy between rate coefficients of N2(i) + O = NO(k) + Nreaction obtained by QCT and theoretical models is demonstrated.

I An error in the original Aliat model is corrected; the correctedexpressions provide much better agreement with the QCT data thanother theoretical models.

I Generalization of the Aliat model is proposed allowing to take intoaccount the vibrational level of the reaction product.

I The resulting model is quite simple and accurate and can be easilyimplemented to CFD solvers.

I Work is in progress on other exchange reactions.

QCT-based cross sections

I State-resolved cross sections of inelastic collisions are required forcalculations of STS transport coefficients and DSMC simulations.

I The following algorithm is proposed to extract cross sections fromQCT data on the rate coefficients:

I Rate coefficient

ki (T ) =

(8kTπmcd

)1/2 ∫ ∞0

exp(− E

kT

)E

kTσi (E )d

E

kT

I QCT data are approximated by functions suitable for applyinginverse Laplace transform (ILT)

I Analytical expressions for the cross sections are derived usingthe ILT

I Cross sections of VV, VT transitions and dissociation in N2 and O2have been obtained [B. Baikov, D. Bayalina, E. Kustova, G. Oblapenko,AIP Conf. Proc. 1786, 090005 (2016)].

QCT-based cross sections

4 0 0 0 0 6 0 0 0 0 8 0 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 1 4 0 0 0 0 1 6 0 0 0 00

1 0

2 0

3 0

4 0

� � �

σ,� �

� � � � � � ��

0 2 0 0 0 0 4 0 0 0 0 6 0 0 0 0 8 0 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 1 4 0 0 0 0 1 6 0 0 0 00

5 0

1 0 0

1 5 0

2 0 0

2 5 0

3 0 0 σ, � �

� � �

� � � � � � ��

Dissociation cross sections in oxygen as functions of translational energy t calculated

for selected levels using ILT, the hard spheres model (HS), and the hard spheres model

with a contribution of only the radial component of the energy (HS*)

STS transport coefficients

Simplified transport algorithms

I Simplified algorithm for calculation of state-resolved transportcoefficients yields significant reduction of computational costs.

I It is based on the assumptions:

I elastic cross sections are independent of the vibrational level;I molecules are rigid rotators, that is, their rotational energy is

independent of the vibrational state.

I The validity of these assumptions has been studied recently[E. Kustova, G.M. Kremer, Chem. Phys. Lett., 2015, 636, 84-89],[E. Kustova, M. Mekhonoshina, G. Oblapenko, Chem. Phys. Lett., 2017,686, 161-166].


Effect of molecular diameters

I Diameters of vibrationally excited states are calculated using Morseor Tietz-Hua potentials.

I Collision integrals are calculated using the hard sphere model.

I Under thermal equilibrium conditions, the effect of increasingmolecular size on the viscosity and thermal conductivity of airspecies is small but can be noticeable for H2, Cl2, I2 at hightemperatures.

I Under non-equilibrium conditions, the effect is negligible for bothshock heated gases (with low populations of high states) andexpanding flows (low-temperature).


1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 20 . 9 0

0 . 9 2

0 . 9 4

0 . 9 6

0 . 9 8

1 . 0 0

x , m

O 2

N 2

O 2

N 2

η/η0

Ratios of the shear viscosity coefficients η/η0 for shock heated flows of N2 and O2 as

functions of the distance x from the shock front. Solid lines refer to the state-to-state

distributions (STS), dashed lines to the thermal equilibrium Boltzmann distributions

(TE) calculated at the same gas temperature.


Coupled effect of molecular diameters and non-rigidity

0 5000 10000 15000 20000 25000 30000 35000 40000T, K

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

λ/λRR

N2/N

λRR, xN = 0, crot = k/m

λSTS, xN = 0

λSTS,xN = 20%

λSTS, xN = 50%

0 5000 10000 15000 20000 25000 30000 35000 40000T, K

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

ζ/ζ R

R

N2/N

ζRR, xN = 0, crot = k/m

ζSTS, xN = 0

ζSTS, xN = 20%

ζSTS, xN = 50%

Ratio of thermal conductivity coefficients λ/λRR (left) and bulk viscosity coefficients

ζ/ζRR (right); N2/N mixture. Black line: rigid rotator model with constant crot ;

colored lines: full state-specific model; solid and dashed lines correspond to models

with constant and variable molecular diameters respectively.



0 5000 10000 15000 20000 25000 30000 35000 40000T, K

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

λ/λRR

O2/O

λRR, xO = 0, crot = k/m

λSTS, xO = 0

λSTS, xO = 20%

λSTS, xO = 50%

0 5000 10000 15000 20000 25000 30000 35000 40000T, K

0.5

1.0

1.5

2.0

2.5

ζ/ζ R

R

O2/O

ζRR, xO = 0, crot = k/m

ζSTS, xO = 0

ζSTS, xO = 20%

ζSTS, xO = 50%


ζ/ζRR (right); O2/O mixture. Black line: rigid rotator model with constant crot ;

colored lines: full state-specific model; solid and dashed lines correspond to models

with constant and variable molecular diameters respectively.



0 5000 10000 15000 20000 25000 30000 35000 40000T, K

0.80

0.85

0.90

0.95

1.00

λ/λ

RR

H2/H

xH = 0

xH = 20%

xH = 50%

0 5000 10000 15000 20000 25000 30000 35000 40000T, K

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ζ/ζ R

R

H2/H

xH = 0

xH = 20%

xH = 50%


ζ/ζRR (right); H2/H mixture. Solid and dashed lines correspond to models with

constant and variable molecular diameters respectively.


I Thermal conductivity and bulk viscosity coefficients are compared tothose computed using a rigid rotator model with the fixed size.

I Simplified algorithm yields overestimation of the thermalconductivity coefficient. The effect is especially prominent inhydrogen and to a lesser extent in oxygen.

I Accounting for increasing diameters of vibrationally excitedmolecules leads to a further decrease in thermal conductivity.

I Using the rigid rotator model causes a two-fold increase in the bulkviscosity coefficient; accounting for variable molecular diametersdoes not have any significant effect on the bulk viscosity coefficient.

I Assumption of a constant rotational specific heat breaks down attemperatures higher than 10000 K and leads to an overestimation oftransport coefficients, especially in oxygen flows.

Pr, Sc numbers in STS model

I Modified Fick’s law for a binary mixture:

ρmiVmi = −ρDeff,mi∇ymi , Deff,mi = − (1− xmi )

∑k 6=i

xmk

Dmimk+

xaDma

−1,Dma, Dii , Dmm = Dmimk are binary and self-diffusion coefficients.

I Schmidt numbers in the state-to-state approach:

Sci = Scmi =η

ρDeff,mi, Sca =

η

ρDeff,a.

I Prandtl number

Pr =cpη

λ, λ = λtr + λrot , cp = cp,rot .


Pr, Sc numbers in shock waves

0,000 0,002 0,004 0,006

0,66

0,68

0,70

x, m

M=15, 1T M=10, 1T M=15, STS

Pr N2/N

0,000 0,001 0,002

0,60

0,65

0,70

x, m

M=15, 1T M=10, 1T M=15, STS

O2/OPr

0,000 0,002 0,004 0,0060,46

0,48

N2/N

x, m

M=15, 1T M=10, 1T M=15, STS

Sca

0,000 0,001 0,002

0,5

0,6

x, m

O2/O

M=15, 1T M=10, 1T M=15, STS

Sca


Pr, Sc numbers in conic nozzle expansion

0 10 20 30 40 500,68

0,69

0,70

0,71

0,72

0,73

N2/N, P*=1atm N2/N, P*=100atm O2/O, P*=1atm

Pr

x/R0 10 20 30 40 50

0,5

0,6

0,7

0,8

x/R

N2/N, P*=1atm N2/N, P*=100atm O2/O, P*=1atm

Scc

0 10 20 30 40 50

0,55

0,60

0,65

0,70

0,75

x/R

N2/N, STS N2/N, 1T O2/O, STS O2/O, 1T

Pr

0 10 20 30 40 50

0,48

0,50

0,52

0,54

0,56

0,58

0,60

x/R

N2/N, STS N2/N, 1T O2/O, STS O2/O, 1T

Scc


Sc numbers for different states in shock wave and nozzle

0 , 0 0 0 0 , 0 0 2 0 , 0 0 4 0 , 0 0 60 , 7 1 3

0 , 7 1 4

m o l e c u l e s i = 0 i = 1 i = 5 i = 1 0

S c

x , m

N 2 N

0 , 0 0 0 0 , 0 0 1 0 , 0 0 20 , 6 6

0 , 6 7

0 , 6 8

0 , 6 9

0 , 7 0

0 , 7 1

0 , 7 2

m o l e c u l e s i = 0 i = 1 i = 5 i = 1 0

x , m

S c O 2 O

0 1 0 2 0 3 0 4 0 5 0

0 , 7 0

0 , 7 2

0 , 7 4

0 , 7 6

x / R k r

m o l e c u l e i = 0 i = 1 i = 5 i = 1 0

S c N 2 N , P k r = 1 a t m

0 1 0 2 0 3 0 4 0 5 00 , 6 6

0 , 6 8

0 , 7 0

0 , 7 2

0 , 7 4

0 , 7 6

0 , 7 8

0 , 8 0

0 , 8 2 O 2 O , P k r = 1 a t mS c

x / R k r

m o l e c u l e i = 0 i = 1 i = 5 i = 1 0


I In the state-to-state approach, the Schmidt numbers are introducedfor each vibrational state. The Prandtl number is determined onlyby translational and rotational energies.

I In shock heated flows

I For M=10, Pr ≈ const.I For M=15, 1-T and STS models yield different results; in the

1-T case, Pr decreases with x ; in the STS case it varies weakly.I In N2, Pr and Sc do not change considerably, in O2 their

variation is significant.I Different qualitative behavior of Sci is found in N2 and O2.

I In expanding flows

I 1-T and STS models yield different behavior of Pr;I Sc numbers are similar;I Sci decrease with i .

New challenges

Electronically excited gases

I In high-temperature flows, electronic excitation plays significant rolein heat transfer.

I Problems in STS modelling

I High number of atomic excited states located below ionizationpotential(170 for N atoms and 204 for O atoms).

I Lack of data on the electronic energy transitionsI Strong variation of atomic size for highly excited states

I Work in progress (V. Istomin)

I One temperature model and its application for strong shockwaves

I State-to-state model of transport processes

Electronically excited gases

1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 01

1 0

1 0 0

1 0 0 0C o n f i n e d a t o m a p p r o x i m a t i o n

T e m p e r a t u r e [ K ]

p = 2 0 0 0 P a ( H e r m e s ) p = 4 2 0 0 P a ( F i r e I I ) p = 1 0 1 3 2 5 P a ( 1 A t m )

r m a x [ A ]ο

1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 01 0

1 0 0

1 0 0 0

1 0 0 0 0

D e b y e - H u c k e l c r i t e r i a


p = 2 0 0 0 P a ( H e r m e s ) p = 4 2 0 0 P a ( F i r e I I ) p = 1 0 1 3 2 5 P a ( 5 % i o n i z a t i o n )

r m a x [ A ]ο

Maximum of atomic radius for electronically excited species.

1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 00 , 0

0 , 5

1 , 0

1 , 5

2 , 0 N

g r o u n d e l e c t r o n i c s t a t e 0 . 7 c m , c o m b i n e d

λ [ W / m / K ]

B o l t z m a n n ( l e v e l s ) : ( 6 ) ( 1 3 ) ( 5 0 ) ( 1 7 0 )

T e m p e r a t u r e [ K ]1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0

0 , 0

0 , 5

1 , 0

1 , 5

2 , 0 O

g r o u n d e l e c t r o n i c s t a t e 0 . 7 c m , c o m b i n e d

λ [ W / m / K ]

B o l t z m a n n ( l e v e l s ) : ( 6 ) ( 1 3 ) ( 5 0 ) ( 2 0 4 )


Thermal conductivity λ of N and O as a function of T .

New challenges

Polyatomic gases

I Studying of CO2 is important for Mars entry, greenhouse gasconversion, ecological problems.

I Problems in STS modelling

I Strong coupling of different vibrational modes.I Multiple channels of vibrational relaxation.I Huge amount of vibrational states due to the modes coupling.I Lack of data on the electronic energy transitions.

I Work in progress (I. Armenise)

I Full and reduced model of state-to-state kinetics and transportprocesses.

I Sensitivity analysis of various kinetic processes for 1-Dstagnation line problem.

I Estimates for different contributions to the heat flux.

Polyatomic gases

Typical vibrational distribution (left) and heat flux along stagnation line (right) in

CO2.

Conclusions

I A lot of work has been done in developing STS models of kineticsand transport properties.

I Due to increasing computer power, STS models represent nowefficient tool for in-depth studies of various non-equilibrium flows.

I Imlementation of STS models to engineering solvers for 3-D viscousflow simulations is still questionable.

I Developing of reduced-order models keeping the main advantages ofthe STS ones is a challenging problem in modern non-equilibriumfluid dynamics.

state-to-state models for dissociation-recombination flow...

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