MACROSCOPIC AND MICROSCOPICMODELS FOR GAS-DYNAMIC

APPLICATIONS

41st Course: MOLECULAR PHYSICS ANDPLASMAS IN HYPERSONICS

E.NAGNIBEDA,University of Saint Petersburg, RUSSIA

ERICE-SICILY: 1–7 AUGUST 2005

CONTENT

1. Introduction• Old and new problems in modelling of nonequilibrium flows• Background

2. Kinetic theory models• Main principles• Weak and strong nonequilibrium flows• Comparison of different models:

• State-to-state (STS)• Multi-temperature• One-temperature

3. Application• behind shock waves• nozzles

4. Role of kinetic models in gas dynamicsEffects of

• different distributions• models of elementary processes

5. Mass action law in nonequilibrium flows

6. Conclusions

INTRODUCTION

Boltzmann (1872)Chapman-Enskog method (1916–1917):

• closing of fluid dynamics equations → transport coefficients

Polyatomic molecules with internal modes:

• WNE:• Eucken (1913)• Wang Chang, Uhlenbeck (1951)• Mason, Monchick (1958–1968)

• Since 1970th• SNE conditions in high temperature and high enthalpy flows• Coupling kinetics and gas dynamics• Importance of kinetics in gas dynamics• Results

• multi-temperature model• state-to-state approach in kinetics and transport theory• new state-to-state transport algorithms for reactive flows• vibrational-chemical coupling• CO2 flows (kinetic models)

• Now: models suitable for applications ?• model validation• implementation to CFD• Choice of a kinetic model depends on:

- relations between relaxation times- particular flow conditions

WEAK AND STRONG NON-EQUILIBRIUM(WNE AND SNE)

θ is macroscopic time of macroparameters changing; τ ismicroscopic characteristic time

1. τ � θ: WNE

2. τ ∼ θ: SNE

3. τ � θ: frozen

4. τ rap � τ sl ∼ θ: SNE, quasi-stationary models

1),4) — method of small parameter; 2) — DSMC method

IN REACTIVE FLOWS:

• τint ≤ τreact � θ: weak thermal and chemical non-equilibrium

• τint � τreact ∼ θ: weak thermal and strong chemicalnon-equilibrium

• τVV � τTRV ≤ τreact ∼ θ: strong thermal and chemicalnon-equilibrium, multi-temperature models

• τtr ≤ τrot � τvibr ≤ τreact ∼ θ: strong thermal and chemicalnon-equilibrium, state-to-state model (chemical-vibrationalcoupling)

KINETIC EQUATIONS

Dfi =1

εrapJ rapi +

1

εslJsli

Weak non-equilibrium (WNE):

εrap = τrap/θ � 1, εsl = τsl/θ � 1

Dfi =1

εJi , ε � 1

Ji = J rapi + Jsl

i

Strong non-equilibrium (SNE):

εrap � εsl ∼ 1

Dfi =1

εJ rapi + Jsl

i , ε � 1

ε ∼ τrap

θ∼ τrap

τsl

WNE case:

P = pI− 2µS− η∇ · vI

q = −(λtr + λint)∇T (one-component gas)

• µ is shear viscosity coefficient

• η is bulk viscosity coefficient

• λtr , λint are heat conductivity coefficients

SNE case:

P = (p − prel)I− 2µS− η∇ · vI

Harmonic oscillator model:

q = −(λtr + λint)∇T − λvibr∇Tv

Real vibrational spectra, anharmonic oscillator model:

q = −(λtr + λint + λvt)∇T − (λtv + λvv )∇T1

Transport coefficients depend on:

• WNE: elastic and all inelastic collision integrals

• SNE: elastic and inelastic collision integrals for rapidprocesses. Inelastic collision integrals for slow processesdetermine the reaction rates

��������������� �������������

�����������������

��������������������

�������������������

���������������

������������������������������

�������� ����������

!������������"����������#���������

��������������������

!������������"��������������������

��$���%&'(����������������)�*��

��$���%&'(������������������

• Weak non-equilibrium regime

τ � θ

Closed system of conservation equationsNon-equilibrium effects in- transport coefficients

• Strong non-equilibrium regime

τrap � τsl ∼ θ

Extended system of macroscopic parametersNon-equilibrium effects in- transport coefficients- relaxation equations- coupling gas dynamics and kinetics

• Adequate kinetic model- accuracy- computation cost

• Comparison of STS and QS models- shock heated flows- expanding flows

REACTING GAS MIXTURESSTATE-TO-STATE APPROACH:

• Macroscopic parameters: nci , v,T

• Governing equations:

dnci

dt+ nci∇ · v +∇ · (nciVci ) = Rvibr

ci + R reactci ,

c = 0, . . . , L, i = 0, ..., Lc ,

ρdv

dt+∇ · P = 0,

ρdU

dt+∇ · q + P : ∇v = 0.

c is a chemical species number, c = 1, ..., Li is a vibrational level number, i = 0, 1, ..., Lc

TRANSPORT TERMSfirst order (N.-S.) approximation:

• Pressure tensor

P = (p − prel)I− 2µS− η∇ · vI

• Vci — state dependent diffusion velocity

• Heat fluxq = −λ

′∇T − p∑ci

DTcidci+

+∑ci

(5

2kT + 〈εci 〉rot + εc

i + εc)nciVci ,

λ′= λtr + λrot

prel = protrel , η = ηrot

zero-order: P(0) = nkT I, q(0) = 0, V(0)ci = 0

PRODUCTION TERMS:

• Rvibrci =

∑dki ′k ′

(kd ,k ′kc,i ′i nci ′ndk′ − kd ,kk ′

c,ii ′ ncindk)

• R22ci =

∑dkc ′d ′i ′k ′

(kd ′k ′,dkc ′i ′,ci nc ′i ′nd ′k ′ − kdk,d ′k ′

ci ,c ′i ′ ncindk)

• R23ci =

∑d

nd(kdrec,cinc ′nf ′ − kd

ci ,dissnci )

• VV, VT exchanges

Aci + Adk Aci ′ + Adk′

• Exchange reactions

Aci + Adk Ac′i ′ + Ad′k′

• Dissociation–recombination

Aci + Ad Ac′ + Af ′ + Ad

• kd ,kk ′

c,ii ′ , . . . — state specific rate coefficientsfirst order:

kγ = k(0)γ (T ) + k(1)

γ (nci ,T ) +∇ · vk(2)γ (nci ,T )

QUASI-STATIONARY MODELS:

• Multi-temperature approach• Macroscopic parameters: nc , v,T ,T c

1

dnc

dt+ nc∇ · v +∇ · (ncVc) = R react

c ,

ρdWc

dt= Rw

c −mcWcRreactc ,

ρdv

dt+∇ · P = 0,

ρdU

dt+∇ · q + P : ∇v = 0.

ρcWc(T ,T c1 ) =

∑i

inci (nc ,T ,T c1 )

• Heat flux:

q = −λ′∇T −

∑c

λcv∇T c

1 + qMD + qTD

λ′= λtr + λrot + λvt

• Production terms:

• R22c =

∑dc′d′

(kd′,dc′,c nc′nd′ − kd,d′

c,c′ ncnd)

• R23c =

∑d

nd(kdrec,cnc′nf ′ − kd

c,dissnc)

• kd ′,dc ′,c — multi-temperature reaction rate coefficients

kdd ′cc ′ = k

dd ′(0)cc ′ (T ,T c

1 ,T d1 ) + k

dd ′(1)cc ′ (n1, ..., nL,T ,T c

1 ,T d1 )+

+kdd ′(2)cc ′ (n1, ..., nL,T ,T c

1 ,T d1 )∇ · v

• One-temperature approachNon-equilibrium chemical kinetics, thermal equilibriumvibrations

• Macroscopic parameters: nc , v,T• Heat flux:

q = −λ′∇T + qMD + qTD

λ′= λtr + λint

η = ηint

prel = pintrel

• Production terms contain one-temperature reaction rates

kdd′

cc′ = kdd′(0)cc′ (T ) + k

dd′(1)cc′ (T , n1, ..., nL)+

+kdd′(2)cc′ (T , n1, ..., nL)∇ · v

• WNE• Macroscopic parameters: Nλ, v,T• Nλ — elements (atoms) number densities

λ′= λtr + λint + λchem

η = ηint,chem, prel = 0

no production terms — only conservation equations• Limit passage from one-temperature to WNEregime

APPLICATIONS:

• Behind shock waves

• In nozzles

N2/N (L = 46) and O2/O (L = 33) mixtures;VV, VT exchange, dissociation, recombination;Anharmonic oscillators;STS and QS approximations

• Rate coefficients:• VV,VT — Billing, Capitelli; SSH models, Macheret,

Adamovich• Dissociation — Treanor-Marrone model, ladder climbing,

trajectory (Capitelli, Esposito)• Recombination — detailed balance

• Comparison:• distributions• gas dynamic parameters• reaction rates

• Distributions:• STS• QS:

- complex strong nonequilibrium distribution (Treanor +plateau + Boltzmann a.o.)- nonequilibrium Boltzmann distribution (h.o.)- Treanor distribution (a.o.)- thermal equilibrium Boltzmann distribution

SHOCK HEATED GAS:T0 = 293 K, p0 = 100 Pa, M0 = 15

0 5 10 15 2010-7

10-6

1x10-5

1x10-4

10-3

10-2

10-1

2''2'2

1''

1'1n i/n

i

Figure: Reduced level population of N2 behind a shock wave. Solid lines:x = 0.03 mm; dashed lines: x = 0.8 mm. 1,1’: state-to-state approach;2,2’: two-temperature approach; 1”,2”: one-temperature approach.

Temperature

0,0 0,5 1,0 1,5 2,07000

8000

9000

10000

11000

12000

13000

14000

3

2

1

T,

K

�����

Figure: Temperature behind a shock wave as a function of x , (N2,N). 1:state-to-state approach; 2: two-temperature approach; 3:one-temperature approach.

Molar fraction of atoms

0.0 0.5 1.0 1.5 2.00.20

0.25

0.30

0.35

0.402'

2

1'1

nat/n

X, CM

Figure: Molar fraction of atoms behind a shock wave as a function of x ,M0 = 5. 1,1’: (N2,N)); 2,2’: (O2,O). Solid lines — with recombination,dashed lines — without recombination

Heat flux

0,0 0,5 1,0 1,5 2,0-500

-400

-300

-200

-100

0 3

21

��������

�

�����

Figure: Heat flux behind a shock wave as a function of x , (N2,N). 1:state-to-state approach; 2: two-temperature approach; 3:one-temperature approach.

NOZZLE FLOWConic nozzle, 21◦, T∗ = 7000 K, p∗ = 100 atm.

Vibrational distributions

Figure: Vibrational distributions, x/R = 50, (N2,N). 1: STS model; 2:two-temperature, anharmonic oscillator; 3: two-temperature, harmonicoscillator; 4: one-temperature.

Vibrational distributions(state-to-state approach)

Vibrational distributions(state-to-state approach)

Vibrational distributions

Figure: Vibrational distributions, x/R = 50, (N2,N). 1: dissociation andrecombination; 2: dissociation; 3: recombination; 4: no dissociation andrecombination.

Temperatures

Figure: Temperature in a nozzle, (N2,N). 1,1’: T and T1, STSapproach; 2,2’: T and T1, two-temperature, anharmonic oscillator; 3,3’:T and Tv , two-temperature, harmonic oscillator; 4: T, one-temperature.

Heat flux

Figure: Heat flux, (N2,N). 1: STS model; 2: two-temperature,anharmonic oscillator; 3: two-temperature, harmonic oscillator; 4:one-temperature.

CONCLUSIONS 1.

• Vibrational distributions, macroscopic parameters and heattransfer are studied in the STS and QS approximations inreacting gas flows behind shock waves and in nozzles

• In nozzles• QS model can be used with a good accuracy for practical

calculations of gas dynamic parameters and heat fluxes• STS model is needed for the calculation of the vibrational

distributions and reaction rates

• Behind shock waves• STS model gives a much better accuracy compared to the

QS ones not only for distributions and reaction rates but alsofor macroscopic parameters

REACTION RATES IN NONEQUILIBRIUM FLOWS

THEORETICAL MODELS

• ReactionsA2(i) + M A + A + M

AB(i) + M A + B + M

AB(i) + C AC (i ′) + B

AB(i) + CD(k) AC (i ′) + BD(k ′)

• Global reaction rate coefficients

k(M)diss (T , n0, n1, . . . , nL) =

1

nmol

L∑i=0

nik(M)diss,i (T )

k(M)rec (T ) =

L∑i=0

k(M)rec,i (T )

k(M)diss,i (T ), k

(M)rec,i (T ) are state-to-state rate coefficients of

dissociation from the i th level and recombination to the i th level,ni are the vibrational distributions

• Detailed balance principle

k(M)rec,i (T ) = kM

diss,i (T )si

(mmol

m2at

)3/2

×

×h3(2πkT )−3/2(T )Zrot exp

(−εi − D

kT

)Zrot is the molecular rotational partition function, D is thedissociation energy.

• State-to-state non-equilibrium factor:

Z(M)i =

k(M)diss,i

k(M)diss,eq

k(M)diss,eq follows the generalized Arrhenius law:

k(M)diss,eq = AT n exp

(− D

kT

)A, n are tabulated for many reactions

• Global dissociation rate coefficients:

k(M)diss = k

(M)diss,eqZ

(M)

Z (M) is the averaged non-equilibrium factor

Z (M) =∑

i

niZ(M)i

Z(M)i (T ,U) =

Zvibr (T )

Zvibr (−U)exp

(εi

k(1

U+

1

T)

)

DEVIATION FROM MASS ACTION LAW

• In a thermal equilibrium gas (τint � τreact ∼ θ)

k(M)diss,eq =

1

Zvibr

∑i

si exp(− εi

kT

)k

(M)i ,diss(T )

k(M)diss,eq

k(M)rec

=n2at,eq

nmol ,eq= Keq(T )

nat,eq, nmol ,eq are the equilibrium species concentrations

• The equilibrium constant Keq(T ):

Keq(T ) =(Z at)2

Zmolexp

(D

kT

)Z at = Z at

tr , Zmol = Zmoltr ZrotZvibr

• In non-equilibrium gases:

K(M)diss−rec =

k(M)diss (T , n0, n1, . . . , nL)

k(M)rec (T )

= ?

• K(M)diss−rec = Keq(T )Z (M)

Factor Z (M) defines deviation from the Mass Actions Law innon-equilibrium gases

Calculation of Z (M) in non-equilibrium flows

• Dissociation models:• Ladder-climbing• Treanor-Marrone• Trajectory calculations (Esposito, Capitelli)

• Vibrational transition models:• Generalized SSH model• Billing, Capitelli

• Vibrational distributions:• State-to-state model:

ni ,T — solution of equation of state-to-state kinetics coupledto gas dynamic conservation equations

• Two-temperature models:• Harmonic oscillator (Boltzmann distribution)• Anharmonic oscillator (Treanor distribution)• Anharmonic oscillator (Treanor–plateau–Boltzmann; strongly

non-equilibrium distribution)

ni (T ,T1),T ,T1 — solution of equations of two-temperaturekinetics coupled to gas dynamic equations

• N2/N mixture flows:• Behind shock waves• In nozzles

Averaged factor Z

2000 4000 6000 8000 1000010-4

10-2

100

102

104

106

108

1010

(a)Z

2'2

1'

1

T, K

Figure: Averaged factor Z . 1,2: Treanor distribution; 1’,2’: Boltzmanndistribution; 1,1’: Trajectory calculations; 2,2’: Treanor-Marrone model,U = D/6k

Averaged factor Z

2000 4000 6000 8000 1000010-4

10-1

102

105

108

1011

1014

1017

(b)

2'

2

1'

1

Z

T, K

Figure: Averaged factor Z . 1,2: Treanor distribution; 1’,2’: Boltzmanndistribution; 1,1’: Trajectory calculations; 2,2’: Treanor-Marrone model,U = D/6k

Averaged factor Z

2000 4000 6000 8000 10000

100

103

106

109

1012

1015

1018

1021

(c)

2'2

1'1

Z

T, K

Figure: Averaged factor Z . 1,2: Treanor distribution; 1’,2’: Boltzmanndistribution; 1,1’: Trajectory calculations; 2,2’: Treanor-Marrone model,U = D/6k

SHOCK HEATED GASN2/N mixture, T0 = 293 K, p0 = 100 Pa, M0 = 15

Figure: Averaged factor Z behind the shock front. 1: state-to-stateapproach, Treanor-Marrone (U = D/6k); 2: state-to-state approach,ladder-climbing model; 3: two-temperature approach

two-temperature model overestimates k(M)diss and Z

Z < 1 behind a shock, K(M)diss−rec < Keq(T )

EXPANDING NOZZLE FLOWN2/N mixture, conic nozzle, 21◦, T∗ = 7000 K, p∗ = 100 atm

Figure: Averaged factor Z in nozzle. 1: state-to-state approach; 2:two-temperature approach, anharmonic oscillator model; 3:two-temperature approach, harmonic oscillator

T1 > T : strong deviation of K(M)diss−rec from Keq(T )

Z � 1 in a nozzle, K(M)diss−rec � Keq(T )

Dissociation rate coefficient

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

2x10-20

4x10-20

6x10-20

8x10-20

1x10-19

32

1

��������

���

����

Figure: Dissociation rate coefficient k(N2)diss , m3s−1, as a function of x . 1:

state-to-state approach; 2: two-temperature approach; 3:one-temperature approach

Dissociation rate coefficients in different approaches

Figure: Averaged dissociation rate coefficient k(mol)diss m3s−1 versus x/R

in a conic nozzle. O2/O, T ∗ = 4000 K, p∗ = 1 atm. Curves 1:state-to-state model; 2: two-temperature anharmonic oscillator model; 3:two-temperature harmonic oscillator model; 4: one-temperature model.

Dissociation rate coefficients in different approaches

Figure: Averaged dissociation rate coefficient k(mol)diss m3s−1 versus x/R

in a conic nozzle. N2/N, T ∗ = 7000 K, p∗ = 1 atm. Curves 1:state-to-state model; 2: two-temperature anharmonic oscillator model; 3:two-temperature harmonic oscillator model; 4: one-temperature model.

CONCLUSIONS 2.

• The influence of different vibrational distributions on averageddissociation rates is found to be important

• In a non-equilibrium gas the ratio of forward and backwardreaction rate coefficients K deviates noticeably from theequilibrium rate constant Keq(T ). Using Keq(T ) instead of Kin practical calculations can lead to a significant error

• The effect of anharmonicity on K increases in a highly excitedgas

• The results justify using a simple Treanor-Marrone model fordissociation behind a shock wave

THE INFLUENCE OF A MODEL OF ELEMENTARYPROCESSES on:

• distributions

• macroscopic parameters

Different models for vibrational transitions:

• SSH for h.o.

• SSH for a.o.

• FHO (Macheret, Adamovich)

• Billing, Capitelli

For Dissociation:

• Treanor-Marrone model

• Ladder climbing model

N2(v) + N2 = N2(v − 1) + N2

0 10 20 30 40

10-18

10-17

1x10-16

10-15

(a)

4

3

2

1

kVTv v-1

, m3/s

v

Figure: Rate coefficients of VT transitions. T = 6000 K. 1: SSH modelfor anharmonic oscillators; 2: SSH model for harmonic oscillators; 3:FHO model; 4: BC model.

N2(v) + N = N2(v − 1) + N

0 10 20 30 40

10-18

10-17

1x10-16

SSH, anharmonic osc. SSH, harmonic osc. FHO BC

(b)

4

3

2

1

kVTv v-1

, m3/s

v

Figure: Rate coefficients of VT transitions. T = 6000 K. 1: SSH modelfor anharmonic oscillators; 2: SSH model for harmonic oscillators; 3:FHO model; 4: BC model.

APPLICATION FOR SHOCK HEATED GAS

0 0.5 1.0 1.5 2.0

0.01

0.02

0.03

0.04

0.05

1 - Treanor-Marrone model, U=3T2 - ladder-climbing model

nat / n

x, cm

2

1

Figure: Molar fraction of atoms behind a shock wave. 1:Treanor-Marrone model, U = 3T ; 2: ladder-climbing model.

0 10 20 30 4010-20

10-15

1x10-10

1x10-5

1x100

x=2cm

x=0.01cm

(a)ni / n

v

3'

2'1'

32

1

Figure: Vibrational distributions behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel. Curves 1–3 correspond to x = 0.01 cm; 1’–3’ to x = 2 cm;

0.0 0.5 1.0 1.5 2.0

9000

10000

11000

12000

13000

(b)

3

2

1

T, K

x, cm

Figure: Temperature behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel. 1: SSH model for anharmonic oscillators; 2: SSH model forharmonic oscillators; 3: FHO model.

0.0 0.5 1.0 1.5 2.0

0.02

0.04

0.06

0.08

0.10

0.12 (c)

3

2

1

nat / n

x, cm

Figure: Atomic molar fractions behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel.

0.0 0.5 1.0 1.5 2.0-500

-400

-300

-200

-100

0

SSH, anhharmonic oscillator SSH, harmonic oscillator FHO

(d)

3

2

1

q, kW/m2

x, cm

Figure: Total heat flux behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel.

CALCULATIONS WITH DIFFERENT MODELS OFENERGY EXCHANGES

T ∗ = 4000 K, p∗ = 1 atm, (O2,O)

T ∗ = 7000 K, p∗ = 1 atm, (N2,N)

T ∗ = 7000 K, p∗ = 1 atm, (N2,N)

CONCLUSIONS 3.

• The choice of vibrational energy exchange model is importantfor values of level populations and macroscopic parameters

• Theoretical information approach and Treanor-Marrone modelwith correct values of the model parameter give a goodagreement with QST calculations

• Employing ladder-climbing model for dissociation and SSHfor harmonic oscillator can cause a noticeable error in gasparameters in high temperature flows

• For (N2,N) FHO for vibrational transitions andTreanor-Marrone with U = 3T for dissociation can berecommended

PROBLEMS to BE SOLVED:

• Implementation of developed models directly to CFD codes(air-mixtures, CO2, . . .) (under development)

• State-dependent rate coefficients for exchange reactions.Available data for calculations of kinetics, gas dynamics andtransport in multi-component reacting mixtures. Analyticalmodels

• Non-maxwellian reaction rates (under development)

• Role of rotational-vibrational coupling in kinetics and transport

• Estimation of bulk viscosity and prel in reacting flows

• Extending the number of processes under consideration inhigh-temperature flows

ACKNOWLEDGEMENTS

This work is supported by INTAS (Grant N 03-51-5204)

The presented results have been obtained in collaboration withProf. E.V. Kustova and Dr. T. Alexandrova.

Presentation have been prepared with help of graduate studentK. Karakulko.