On the kinetic formulation of vibrationally coupled

diatomic dissociation and monatomic recombination

Ryosuke Yano, Kojiro Suzuki∗

Dept. of Advanced Energy, Graduate school of Frontier Sciences, Univ. of Tokyo,

5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan

In this paper, we formulate the diatomic dissociation-recombination by the kinetic the-ory. The collision term of the Wang-Chang-Uhlenbeck equation is replaced by the extendedMorse model. The diatomic molecular dissociation is formulated by using the truncatedmodel of the vibration mode. The recombination reaction is formulated by defining thedistribution function of the intermediate. On the basis of formulated kinetic model, thecatalysis on the wall is investigated using two kinetic models. Finally the ablation on thewall is formulated using the finite catalytic wall. The formulated kinetic models are nu-merically investigated by solving shock layer problem around a circular cylinder with thefinite catalytic wall or ablation.

Nomenclature

f Distribution function, m−6s3

n Number density, m−3

vi Molecular velocity, ms−1

T Temperature, Kui Flow velocity, ms−1

xi Cartesian body axes, mKn Knudsen nunberDt Fluid derivative, ∂/∂t + vi∂/∂xi

ν Collision frequency, s−1

r Mass fractionD Threshold of dissociation

Subscriptstr Translational moderot Rotational modevib Vibrational mode∞ Freestreamw Wall

Superscriptsel Elastic collisiontr Translational-rotational relaxationtrv Translational-rotational-vibrational relaxationN Molecules produced by the dissociaiontr Translational moderot Rotational modevib Vibrational modeemit emission from the wall

∗Professor, AIAA senior member, Dept. of Advanced Energy, Graduate school of Frontier Sciences, Univ. of Tokyo,kjsuzuki@k.u-tokyo.ac.jp

1 of 13

American Institute of Aeronautics and Astronautics

10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4334

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

Hypersonic rarefied flows around the space vehicle or reentry body have been analyzed using the DirectSimulation Monte Carlo (DSMC) method1 since 1970s. In such analyses, the Boltzmann equation withchemical reactions is solved, whereas the complex chemcal reactions involved with the polyatomic moleculeor multiple bodies are solved by assuming the reactive cross section without couting the detailed quantumtransitions of polyatomic molecules via inelastic collisions. On the other hand, the direct evaluation of theBoltzmann equation2 beocomes rigorous, whereas the development of the solver of the Boltzmann equationwith chemical reactions on the basis of the quantum transitions remains as a difficult issue. The Boltzmannequation with quantum transitions is called as the Wang-Chang-Uhlenbeck (WCU) equation.3 The directformulation of the chemical reactions by the WCU equation is still difficult theme for the kinetic theory.Yano et al. succeeded in formulating the diatomic dissociation4 and monatomic recombination5 coupled tothe vibrational mode by using the extended Morse model. In this paper, we review the formuation by Yanoet al.45 and extend our kinetic model to describe the catalysis and ablation on the wall. Especially, theaccurate kinetic model of the catalysis on the wall has not been obtained yet, whereas the catalysis is thesignificant parameter to determine the critical heat flux on the wall. Additionally, the ablation is expectedto decrease such a critical heat flux on the wall. The intermediate is significant chemical species, which leadsto the recombination. In this paper, the catalysis for the recombination reaction on the wall is exclusivellyconsidered. Especially, we focus on the interactions between the intermediate and wall. The deactivation ofthe vibrational energy of the intermediate via the catalysis on the wall promotes the recombination reactions,when we use the truncated harmonic oscillator model for the vibrational mode. As another approach to thecatalysis on the wall, the enhance of the dissociation energy on the wall is considered.In Sec. II, the Bhatnagar-Gross-Krook (BGK) model for the gas mixture is described. In Sec. III, theextended Morse model for the gas mixture is described. In Sec. IV, the dissociation-recombination modelis formulated. In Sec. V, the catalytic model on the wall is formulated using proposed kinetic model andthe ablation-model on the wall is formulated. In Sec. VI, we numerically analyze the shock layer problemwith the dissociation-recombination around a circular cylinder with finite catalytic wall. Obtained numericalresults indicate that proposed kinetic model can demonstrate the dissociation-recombination reaction in thehypersonic flow. Additionally, numerical results obtained by the proposed finite catalytic wall-model confirmthe increase of the recombination by the enhance of the threshold of the dissociation or the deactivation ofthe vibrational energy of the intermadiate or diatomic molecule via the catalysis on the wall. The effects ofthe ablation on the wall are numerically confirmed by changing the number density of the emitted diatomicmolecule. Surely, the increase of the number density of the emitted diatomic molecule leads to the decreaseof the heat flux.

II. Gas mixture model

The construction of gas mixture model for the linear Boltzmann equation is difficult, because nonequi-librium between different species generates diffusions. Various gas mixture models of the BGK (Bhatnager-Gross-Krook)-type6 are described by Hamel,7 Oguchi9 (-Abe10), and Garzo.11 In this paper, we used themethod of Oguchi (-Abe), which determines the relaxational rate to give transport coefficients obtained bythe Chapman-Enskog expansion.13 We assume that intermolecular forces follow the inverse power law, whichrelates the intermolecular force, F to the intermolecular distance, r, with an inverse power law as F = κ/rη,where the values of the constant κ and the exponent η are a function of the colliding molecular species.For η = 5, molecules are called Maxwellian molecules, and for η = ∞, molecules are called hard-spheremolecules. In this section, we only consider the elastic term Qel

ab and in the next section we consider theinelastic term. For simplicity, the description of the quantum space is omitted. The elastic collision termbetween species a and b can be expressed as

Qelab = νaa

(f (0)

aa − fa

)+ νab

(f

(0)ab − fa

). (1)

2 of 13

American Institute of Aeronautics and Astronautics

where νaa = 1/τaa and νab = 1/τab. Equilibrium distribution functions f(0)aa and f

(0)ab are given by

f(0)aa = na

(ma

2πkTa

)3/2

exp[−ma(va−ua)2

2kTa

], (2)

f(0)ab = na

(ma

2πkTab

)3/2

exp[−ma(va−uab)

2

2kTab

]. (3)

where Ta = Taa and ua = uaa. In eq. (2) and (3), uab, uba, and Tab are evaluated with the exchangecoefficient δ as

uab = ua + δ (ub − ua) , (4)uba = ub + δ ma

mb(ua − ub) , (5)

Tab = Ta + 2maδma+mb

[Tb − Ta + (ub−ua)2

6k

2mb − δ (ma + mb)]. (6)

where the exchange coefficient δ is given by9 as

δ =ma + mb

2ma

2α

naνab. (7)

α =16nanb

3 (ma + mb)2

×√

2k

πmamb (mbTa + maTb)Ωab. (8)

Currently, general expressions for νaa and νab are not introduced. Hamel focused on two limits to determineνab and δ: he sets ua = ub at ma = 0 and he sets Ta = Tb at ma = mb. For these two conditions, δ and νab

are given by7 as

δ =mb

ma + mb, (9)

νab =mb

2ma

α

na. (10)

Using the expressions from eqs. (1) and (9), the self-diffusion coefficient Dab is obtained by multiplying eq.(1) va and integrating in the va velocity space, and executing Maxwell’s first iteration12 to yield

Dab =nb (ma + mb) kTa

(na + nb) mambνab. (11)

[Dab]1 is the first approximation of the Sonine polynomial in the Chapman-Enskog expansion, and can beexpressed as

[Dab]1 =3E

2 (na + nb) (ma + mb). (12)

E is defined by the Chapman-Cowling13 for inverse power law molecules as

E(T ) =

(2πkTm0MaMb

) 12

8Γ(3− 2

η−1

)A1(η)Ωab

. (13)

where Ma = ma/m0, Mb = mb/m0,m0 = ma + mb, Ωab = πd2ab = π

(κab

2kT

)2/(η−1), and Γ is the gammafunction. A1(η) is the function defined by Chapman-Cowling in.13 For Dab = [Dab]1, νab is given by

νab = γDnbkTa

E(T ), (14)

γD =2 (ma + mb)

2

3mamb. (15)

3 of 13

American Institute of Aeronautics and Astronautics

This implies that the formulation of νab depends on which transport coefficient is chosen to be equal to thatby Chapman-Enskog expansion of Boltzmann equation among viscosity coefficient, thermal conductivity, self-diffusion coefficient, or thermal diffusion coefficient. In this paper, we use an expression for νab correspondingto the self-diffusion coefficient. In eq. (14), we substitute Θab for T, Ta to correspond to the formulation byBergers,14 which yields

νab = γDnbkΘab

E(Θab),

Θab = Ta +nb

na + nb(Tb − Ta)

+nanb

3k/ma (na + nb) (nama + nbmb)

× (ub − ua)2 .

(16)

We use the formulation given by eq. (9) to represent δ.

III. Extended Morse model for binary mixture gas

Here, we consider the dissociation-recombination reaction (I):

A2 + M ↔ A + A + M,

where A is a hypothetical monatomic molecule, A2 is a hypothetical diatomic molecule and M is either A,A2 or a hypothetical intermediate A∗2.The terms for the collision between A2 and M or A and M are formulated in this section. The Boltzmannkinetic equation for a gas mixture of the hypothetical diatomic molecules A2 and the monatomic molecules Awas previously formulated4 on the basis of the BGK formalism.6 The collision term for a diatomic moleculewas previously formulated using the extended Morse model for a gas mixture.4 As with the collision termfor a monatomic molecule, the collision term for a diatomic molecule is modified in this study to satisfy theenergy conservation law. The concrete formulation for a gas-mixture model is described in Appendix A.First, the conventional Boltzmann kinetic equation for A2 is formulated as4

DA2t fA2 =

∑

M=A,A2,A∗2

νelA2M

(Mel

A2M − fA2

)

+νtrvA2M

(M trv

A2M − fA2

)

+νtrA2M

(M tr

A2M − fA2

), (17)

where fA2 is the distribution function of the diatomic molecule A2; DA2t ≡ ∂/∂t + vA2

i ∂/∂xi, where vA2i is

the i component of the molecular velocity of A2 and xi is the i component of the physical space; and thesubscript M corresponds to either A, A2 or the intermediate A∗2. In eq. (17), νA2M is the collision frequencyof A2 − M collisions defined in eq. (16), where the superscripts trv and tr indicate the translational-rotational-vibrational relaxation and translational-rotational relaxation, respectively.The gain terms Mel

A2M , M trvA2M , and M tr

A2M in eq. (17) are defined using the extended Morse model4 as

MelA2M = f

(0)t (nA2 ,mA2 ,uA2M , TA2M ) f

(0)rot (l, Trot) fvib (s) , (18)

M trvA2M = f

(0)t

(nA2 ,mA2 ,uA2M , T ∗A2M

)f

(0)rot

(l, T ∗A2M

)f

(0)vib

(s, T ∗A2M

), (19)

M trA2M = f

(0)t

(nA2 ,mA2 ,uA2M , T ∗∗A2M

)f

(0)rot

(l, T ∗∗A2M

)fvib (s) . (20)

where nA2 is the number density of A2, mA2 is the mass of A2, TA2M is defined using eq. (6) and uA2M

is defined in eqs. (4) and (5). ft(0), frot

(0), and fvib(0) correspond to the Maxwell-Boltzmann distributions

for the translational, rotational, and vibrational modes, l and s are the quantum numbers of the rotationaland vibrational modes, and Trot and Tvib are the temperatures of the rotational and vibrational modes,

4 of 13

American Institute of Aeronautics and Astronautics

respectively. Here, f(0)rot and f

(0)vib are given by

f(0)rot(l.Trot) =

(2l + 1)exp (−l(l + 1)θr/Trot)∑∞l=0(2l + 1)exp (−l(l + 1)θr/Trot)

,

f(0)vib (s, Tvib) = e−sθv/Tvib

(1− e−θv/Tvib

), (21)

where θr and θv are the characteristic temperatures of the rotational and vibrational modes, respectively.T ∗A2M and T ∗∗A2M in eqs. (19) and (20) are defined as

T ∗A2M = 3RA2/2TA2M+Cv,rot(Trot)Trot+Cv,vib(Tvib)Tvib

3RA2/2+Cv,rot

“T∗A2M

”+Cv,vib

“T∗A2M

” , (22)

T ∗∗A2M = 3RA2/2TA2M+Cv,rot(Trot)Trot

3RA2/2+Cv,rot

“T∗∗A2M

” . (23)

where Cv,rot and Cv,vib are the specific heats of the rotational and vibrational modes, respectively, andRA2 = k/mA2 is the gas constant. From eqs. (17), (18), (19), (20), (22) and (23), the nonequilibriumbetween the velocities of A2 (uA2) and M ( uM ) and nonequilibrium between the translational temperaturesof A2 (TA2) and M (TM ) are relaxed by uA2M and TA2M in the gain terms Mel

A2M , M trvA2M and M tr

A2M .Kolobov2 reported that the conventional formulation for A4 does not satisfy the energy conservation law inthe nonequilibrium regime. To satisfy the energy conservation, the Boltzmann kinetic equation is formulatedas

DAt fA =

∑

B=A,A2,A∗2

νAB (MAB − fA) , (24)

where fA is the distribution function of the monatomic molecule A and νAB is the collision frequency ofA−B collision defined in eq. (16). The gain term MAB in eq. (24) is defined as

MAB = f(0)t (nA,mA,uAB , TAB) , (25)

where nA is the number density of A, mA is the mass of A and TAB is obtained by setting Ta = TA,ma =mA,mb = mA2 and replacing Tb with TB in eq. (6), where TB is the translational temperature of the moleculeB. From eqs. (24) and (25), the nonequilibrium between the velocity of A (uA) and the velocity of B ( uB)and nonequilibrium between the translational temperature of A (TA) and the translational temperature ofB (TB) are relaxed by uAB and TAB in the gain term MAB .In the next section, eq. (24) is divided into two-stage collisions to formulate the recombination.

IV. Formulation of Diatomic molecular dissociation and monatomicrecombination

The transition of the vibrational quantum state is assumed to follow Landau-Teller-type transitions,implying that the largest change in the vibrational quantum number is ±1. In this section, dissociation andrecombination are formulated on the basis of the truncated harmonic oscillator model.15

A. Formulation of diatomic molecular dissociation

In this subsection, we consider the dissociation of a hypothetical diatomic molecule, A2, which dissociatesaccording to the following reaction:

A2 + M → A + A + M ,

where M represents either A, A2 or A∗2. By setting the vibrational quantum number that induces dissociationto D, the number density, flow velocity and temperature of the molecule A generated by the dissociationcan be derived from the mass, momentum and energy conservations as follows:4

nNA = 2×∑∞

s=D

∑∞l=0

∫fA2 dv, (26)

uNA =

∑∞s=D

∑∞l=0

∫vA2fA2 dv/nN

A , (27)

TNA = 2mA2

3nNA k

∑∞s=D

∑∞l=0

∫ 12

(v2

A2− uN

A

2)

+eA2,rot + (eA2,vib − eD)

fA2 dv. (28)

5 of 13

American Institute of Aeronautics and Astronautics

Macroscopic quantities generated by the dissociation are identified by the superscript N . eA2,vib is thevibrational energy of the molecule A2 and eD is the vibrational energy of the molecule A2 whose vibrationalquantum number is equal to the quantum number D responsible for inducing dissociation. eA2,rot is therotational energy of molecules. The distribution function of the molecule A generated by dissociation isassumed to follow the Maxwell-Boltzmann distribution, whose macroscopic quantities are defined usingeqs. (26)-(28). This model, in which molecules above the vibrational threshold D dissociate, is called thetruncated harmonic oscillator model.15 A more detailed dissociation model coupled with the vibrationalmode has been constructed on the basis of state-to-state kinetics.16

The dissociation rate kfA2

by counting molecules that exceed threshold D in the gain term is

kfA2M =

νtrvA2M

nM

∞∑

s=D

f(0)vib

(s, T ∗A2M

)

=νtrv

A2M

nMe− Dθv

T∗A2M

= C (ΘA2M )e− EA2

T∗A2M , (29)

where ΘA2M is given by eq. (16). C (ΘA2M ) and E are defined by

C (ΘA2M ) ≡ νtrvA2M

nM, (30)

EA2 ≡ Dθv. (31)

Finally, we obtain a modified Arrhenius rate equation for the dissociation of A2 from eq. (29).

B. Formulation of monatomic molecular recombination

The recombination reaction among three bodies is written as

A + A + M → A2 + M.

The above recombination occurs in two ways. In the first way, the recombination occurs via pure threebody collisions. In the other way, the recombination occurs when the intermediate generated as a result of arotational barrier (i.e., quantum mechanical tunnel effect)17 at the A−A collision inelastically collides withM .Here, the recombination among three bodies is assumed to occur only in the latter way, namely,

A + A → A∗2, First-stage,A∗2 + M → A2 + M, Second-stage.

The first-stage reaction is formulated using kinetic formalism as5

∂fA

∂t= νAA (1− α)MAA − fA , (32)

∂fA∗2

∂t= νAAαMA∗2 , (33)

where a new parameter α in eqs. (32) and (33) is the ratio of A generating the intermediate A∗2 in the A−Acollision. Here, α is determined not by using kinetic theory but by using quantum mechanics.From eq. (33), the distribution function of the intermediate A∗2 generated at the calculated time interval εis5

fA∗2 =∫ T+ε

T

dtνAAαMA∗2 , (34)

where fA∗2 |t=T = 0.Here, we assume that α is related to the lifetime of the intermediate A∗2. α = 1 holds, when ε ¿ TL, in which

6 of 13

American Institute of Aeronautics and Astronautics

TL is the lifetime of the intermediate A∗2. When TL ¿ 1, however, such a constraint for the time intervalby α = 1 induces stiffness in the calculation via a small time interval. When TL ≤ ε, α must be less thanunity, because the intermediates A∗2 dissociate into A before collisions with the third body M . α decreaseswith exponential dependence on the ratio ε/TL.18 The actual relation between α and the calculated timeinterval ε must be determined not in the framework of the kinetic theory but in the framework of quantummechanics. Simulations based on the rotationally favored recombination model for a real molecule wereperformed by Colonna et. al.1819

The generated intermediates A∗2 collide with A,A2, and A∗2, which correspond to M in the second-stepcollision of the recombination.Finally, the second-stage collision is formulated as5

∂fA∗2

∂t=

∑

M=A,A2,A∗2

νelA∗2M

(Mel

A∗2M − fA∗2

)

+νtrvA∗2M

(M trv

A∗2M − fA∗2

)

+νtrA∗2M

(M tr

A∗2M − fA∗2

). (35)

After the calculation of eq. (35), the intermediates A∗2, whose vibrational quantum state is s < D, recombineinto A2, and those whose vibrational state is D ≤ s dissociate into A. The distribution function of Agenerated by the dissociation of the intermediates A∗2 is formulated by replacing A2 with A∗2 in eqs. (26)-(29).In eqs. (33) and (34), MA∗2 is determined by momentum energy conservation given by5

mAvA + mAvA = mA2vA∗2 , (36)12mAv2

A +12mAv2

A = eA∗2 ,tr + eA∗2 ,rot. (37)

Assuming the translational mode is in equilibrium with the rotational mode, we obtain the following MA∗2by taking moments for both sides of eqs. (36) and (37) similarly to that for both sides of eqs. (27) and (28):5

MA∗2 = f(0)tr (

nA

2,mA2 , uA, TA∗2 )f

(0)rot(l, TA∗2 )fvib(s = D). (38)

The vibrational quantum number of the intermediates A∗2 is set as s = D.The order of calculation is as follows:(i) Calculation of the A−A collision from eq. (32).(ii) Calculation of the distribution function of intermediate A∗2 from eqs. (34) and (38).(iii) Calculation of the (A2−A,A, A∗2), (A−A2, A

∗2), and (A∗2−A,A2, A

∗2) collisions from eqs. (17) and (24)

and (35).(iv) Calculation of the dissociation of A2 and A∗2 from eqs. (26)-(28). A∗2, whose vibrational quantumnumber, s, is s < D, becomes A2.As indicated in procedures (i) and (iii), the A−B collision in eq. (24) is divided into two steps: (1) A−Acollision (first step) and (18) A−A2, A

∗2 collisions (second step).

The recombination rate of A2 is derived from eq. (35) as5

kbA∗2M =

νtrvA∗2M

nM

D−1∑s=0

f(0)vib

(s, T ∗A∗2M

)

=νtrv

A∗2M

nM

(1− e

− DθvT∗

A∗2M

)

= C (ΘA∗2M )

(1− e

− EA2T∗

A∗2M

). (39)

Finally, the recombination rate of A + A + M → A2 + M is obtained from eqs. (34) and (39) as:

kbAM =

νtrvA∗2M

nM

(1− e

− DθvT∗

A∗2M

)×

∫ T+ε

T

dtανAA

2nA︸ ︷︷ ︸=ϑ

,

7 of 13

American Institute of Aeronautics and Astronautics

=νtrv

A∗2M

nM

(1− e

− DθvT∗

A∗2M

)× ϑ

= C (ΘA∗2M )

(1− e

− EA2T∗

A∗2M

)× ϑ, (40)

∼ C (ΘA∗2M )× ϑ for T ∗A∗2M ¿ EA2 , (41)

where ϑ =∫ T+ε

TdtανAA/2nA.

From eq. (41), the recombination rate kbAM is independent of the threshold temperature, EA2 = Dθv, when

TA∗2M ¿ EA2 .The equilibrium constant KA2M ≡ kb

AM/kfA2M is introduced from eqs. (29) and (40) as follows when the

dissociation is the reverse path of the recombination by assuming translational-rotational equilibrium:

KA2M =

ϑC (ΘA∗2M )

(1− e

− EA2T∗

A∗2M

)

C (ΘA2M )e− EA2

T∗A2M

(42)

∼ ϑC (ΘA∗2M )eEA2

T∗A2M

C (ΘA2M )for T ∗A∗2M ¿ EA2 .

(43)

In eq. (43), TA2 ' TA∗2 , where TA2 is the translational temperature of A2, and thus, C (ΘA2M ) ' C (ΘA∗2M ).Finally, we obtain the following form of the equilibrium constant:5

KA2M = ϑe

EA2T∗

A2M for T ∗A∗2M ¿ EA2 ∧ TA2 ' TA∗2 . (44)

Because ϑ is the function of temperature TA (i.e., ϑ = ϑ(TA)), the equilibrium constant defined using eq.(44) is equivalent to that defined by Vincenti and Kruger,20 which is introduced from statistical mechanics.

V. Kinetic model for finite catalytic wall and ablation on wall

The catalysis on the wall plays an important role, because the recombination rate on the wall is thesignificant parameter to determine the heat flux, which is the significant design parameter of space vehicle.On the basis of the kinetic formulation of the dissociation-recombination reaction, the kinetic model for thefinite catalytic wall is formulated. Additionally, the ablation on the wall is formulated on the basis of theproposed kinetic model for the finite catalytic wall. The interaction between A∗2 and wall is significant inthe formulation of the finite catalytic wall. In procedures shown in (i)-(iv) in Sec. IV, such interactionsare ignored. Then, the interaction between A∗2 and the wall is inserted between the items (iii) and (iv).Figure 1 (Left) shows the interactions between the intermediate and wall. As shown in Fig. 1 (Left), thedeactivation of the vibrational energy of A∗2 or A2 increases the recombination rate. The deactivation ofthe vibrational energy is done by decreasing the vibtational temperature on the wall. For example, A∗2 aftercollisions decribed in eq. (35) expriences collisions with the wall. The distribution function of A∗2 aftercollisions with molecules is obtained from eq. (35) as follows:

fA∗2 |T+ε = fA∗2 +∫ T+ε

T

dt

∑

M=A,A2,A∗2

νelA∗2M

(Mel

A∗2M − fA∗2

)

+νtrvA∗2M

(M trv

A∗2M − fA∗2

)+ νtr

A∗2M

(M tr

A∗2M − fA∗2

). (45)

8 of 13

American Institute of Aeronautics and Astronautics

The production rate of the number density of A2, which is generated by the recombination of A∗2 via thecatalytic collisions with the wall, is given by:

1ε

∑l,s

∫vη<0

|vη|fA∗2 |T+εdvA∗2∑l,s

∫0≤vη

vηf(0)t

(1,mA∗2 ,0, TA∗2 ,tr,w

)f

(0)rot

(l, TA∗2 ,rot,w

)f

(0)vib

(s, TA∗2 ,vib,w

)dvA∗2︸ ︷︷ ︸

nA∗2 ,w

×D−1∑s=0

f(0)vib

(s, TA∗2 ,vib,w

)

=1εnA∗2 ,w

(1− e

− DθvTA∗2 ,vib,w

), (46)

where vη is the normal vector to the wall as shown in Fig. 1 (Left). From eq. (46), we can easily understandthat more A∗2 recombines into A2 by decreasing TA∗2 ,vib,w on the wall.As another approach to the finite catalytic wall, the change of threshold of the dissociation on the wall isconsidered. The larger the threshold of the dissociation (Dw) on the wall becomes, the more A recombinesinto A2 on the wall as shown in Fig. 1 (Right). Such two types of recombination models are numericallyinvestigated in Sev. VI.The ablation is modeled by assuming that A2 is constantly emitted from the wall. The state of emitted A2

is fixed to the Maxwell-Boltzmann distribution function with the constant number density, translational-rotational temperature and vibrational temperature. The effects of the number density emitted on the wallare numerically investigated in Sec. VI.

VI. Numerical study on the catalysis and ablation on wall

To solve eqs. (17) and (24), Yee’s second-order numerical scheme, namely, the Total Variable Diminishing(TVD) scheme,21 is used for the left-hand side of eqs. (17) and (24), and a second-order Runge-Kutta timeintegration method is used for the right-hand side. Formulation for either curvilinear coordinates or reduceddistribution functions is shown in22 and is used to nondimensionalize the direction in which the flow isuniform.To validate our model, we simulated diatomic molecular dissociation and monatomic molecular recombinationfor the hypothetical molecules A and A2. For the problem description, hypersonic flow around a cylinder wasanalyzed. For the properties of the hypothetical diatomic molecules A2, the molecular mass was 28[kg/kmol],and the characteristic temperature of the vibrational mode was θv = 3390[K]. The vibrational relaxationcollision model used here was that given by Millikan and White23 and Bird.1 The properties of A2 weresimilar to those of N2 molecules. The diameter ratio of monatomic molecules to diatomic molecules was setas dA/dA2 = 0.5. The fractions of monatomic and diatomic molecules in the free stream, where quantitieswere denoted by the subscript ∞, were calculated as rA∞ = nA∞/(nA∞ + nA2∞) = 0.01 and rA2∞ =nA2∞/(nA∞ + nA2∞) = 0.99, respectively. The relative gas concentrations for monatomic and diatomicmolecules were nA∞/n∞ = 0.005 and nA2∞/n∞ = 0.495, respectively. The temperature of the free streamwas T∞ = 1000[K]. The temperature of the wall was set to Tw = 5000[K]. Molecular scatter is isotropicfrom the wall and the molecules are assumed to be in thermal equilibrium. The catalicity of the wall wasnot considered. For this calculation, the rotational mode was assumed to be always in equilibrium with thetranslational mode. The velocity of the free stream was set to 5268[m/s] and KnAA = 10 (see the definitionof Kn in Appendix B). We used a 64 × 64 grid for the velocity space ((vx, vy) = (64, 64)), an 81 × 54 gridfor the physical space ((x, y) = (81, 54)), and 60 grid points for the vibrational quantum space (s = 60).As a result, the distribution function fA2(vx, vy, s, x, y) of the molecule A2 requires 1074954240 grid points.The velocity space has a range of −20

√2kT∞/m∞ ≤ vx, vy ≤ 20

√2kT∞/m∞, and mA/m∞ = 0.5 and

mA2/m∞ = 1.0, where m∞ is the representative mass. Preliminary numerical calculations confirm thatthese grids are sufficiently dense to minimize numerical error.In the calculation of hypothetical molecular dissociation and recombination, α was set to unity and ε/t∞(t∞ is the representative time) was set as 10−3, which is assumed to be much shorter than the lifetime ofthe intermediate and the threshold of the dissociation is fixed to D = 5.At first, we consider the catalysis on the wall. Figures 1 shows profiles of the mole fraction and temperaturealong the stagnation streamline (SSL), when Dw = 20 on the wall. In Fig. 1, the interaction between A∗2and the wall is ignored. As shown in Fig. 1, A in the case of Dw = 20 drastically recombines into A2 on thewall in comparisons with Dw = 5. Such a drastic recombination on the wall pushes the shock toward thewall. As shown in Fig. 2, the vibrational temperature of A2 drastically increases on the wall, because the

9 of 13

American Institute of Aeronautics and Astronautics

vibrational energy of A∗2 increases by increasing Dw.Next, we consider on the interaction between A∗2 and the wall. In our numerical study, following four casesare considered.

Case A: TA2,vib,w = Tw.Case B: TA2,vib,w = T∞.Case C: TA2,vib,w = Tw, TA∗2 ,vib,w = T∞.Case D: TA2,vib,w = TA∗2 ,vib,w = T∞.

Figures 2 shows the profiles of the mole fraction and temperature along the stagnation streamline. Incases C and D, the recombination rates are larger than those in cases A and B because of TA∗2 ,vib,w =T∞. Consequently, the extent of decrease of TA∗2 ,vib,w is the most dominant parameter to determine therecombination rate of A on the wall. As shown in Fig. 2, the vibrational temperature of A2 decreases owingto the decrease of TA∗2 ,vib,w.At last, the effect of the ablation on the wall is considered. In this calculation, the interaction between A∗2and the wall is ignored. A2 emitted from the wall via the ablation is given by the following two types ofconditions, namely nemit

A2= 2.5n∞ and nemit

A2= 5.0n∞. T emit

A2,tr = 5T∞ (Translational-rotational temperature)and T emit

A2,vib = T∞. Figure 3 shows profiles of the mole fraction, velocity and temperature along the stagnationstreamline for nemit

A2= 0.0, nemit

A2= 2.5n∞ and nemit

A2= 5.0n∞. The more A2 is emitted from the wall,

the shock wave is pushed toward the wall. rA2 ' 0.8 ∧ rA ' 0.2 is obtained on the wall in the case ofnemit

A2' 2.5n∞. rA2 ' 1.0 ∧ rA ' 0.0 is obtained on the wall in the case of nemit

A2= 5.0n∞. Figure 6 shows

the profiles of the velocity along the stagnation streamline. As shown in Fig. 3, the velocity of A2 in thecases of nemit

A2= 2.5n∞ or nemit

A2= 5.0n∞ has the negative values around the wall due to the emission of A2

from the wall. As a result, the velocity of A in the cases of nemitA2

= 2.5n∞ or nemitA2

= 5.0n∞ also has thenegative values around the wall due to the exchanges of their momentum via collisions between A and A2,which are given by eqs. (4) and (5). As shown in Fig. 3, the more A2 are emitted from the wall, the smallerinclination of the temperature (∂T/∂x) on the wall is obtained. The smaller inclination of the temperatureis equivalent to the smaller heat flux on the wall. Surely, the inclination of the temperature in the case ofnemit

A2= 5.0n∞ approximates to the zero, whereas the inclination of the temperature on the wall in the case

nemitA2

= 2.5n∞ is the finite value, which is smaller than that in the case of nemitA2

= 0.0. Consequently, themore emissions of A2 decrease the heat flux on the wall.

VII. Conclusions

A diatomic molecular dissociation-recombination model is formulated here using the extended Morsemodel for a gas mixture. In this model, the vibrational quantum states of diatomic molecules and interme-diates are used to determine the dissociation and recombination. The obtained equilibrium constant underthe assumption of translational-rotational equilibrium coincides with that derived from statistical mechanics,when the intermolecular nonequilibrium at the temperature between a diatomic molecule and its interme-diate is negligible and when the temperature defined in the gain term by the intermediate’s collisions ismuch lower than the dissociation temperature. Two models are proposed to express the wall catalysis onthe recombination reaction. One is the model, in which the recombination rate increases by increasing thethreshold of dissociation on the wall. The other is the model, in which the intermediates interact with walland the recombination rate increases due to the decrease of the vibrational temperature of the intermediateson the wall. Numerical study on the shock layer around the circular body with the finite catalytic wallconfirms that our two models increase the recombination on the wall. Additionally, the ablation on the wallsurely works to decrease the heat flux, when diatomic molecules are emitted enough to dismiss the heat flux.

Acknowledgements

This work is supported by Grant-in-Aid for Scientific Research (B) No. 21360413 of Japan Society forthe Promotion of Science.

10 of 13

American Institute of Aeronautics and Astronautics

Wall

A

A

A2*

A2

A2*A2*

A

A

(D vibrational quantum number)

(vibrational quantum number < D)

TA2*,vib,w

M=A, A2, A2*

CollisionsDeactivation of the vibrational energy of A2* or A2

leads to the increase of recombination on the wall

Wall

A

A

A2

A2*

A2*

A

A

(Dw vibrational quantum number)

(vibrational quantum number < Dw)

TA2*,vib,w

(vibrational quantum number = D)

M=A, A2, A2*

CollisionsD<Dw leads to the increase

of the recombination

v

Figure 1. Catalysis on the wall via deactivation of the vibrational energy of A∗2 or A2 (Left) or via the variationof Dw (Right).

-X/R

MO

LE

FRA

CT

ION

1 1.1 1.2 1.3 1.40.2

0.3

0.4

0.5

0.6

0.7

-X/R

MO

LE

FRA

CT

ION

1 1.5 2 2.5 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

rA2 (Dw=20)

rA (Dw=5)

rA2 (Dw=5)

rA (Dw=20)

Mole fraction profiles along SSL

D=5-X/R

NO

RM

AL

IZE

DT

EM

PER

AT

UR

E

1 1.1 1.2 1.34

4.5

5

5.5

6

6.5

7

TA2vib(Dw=5)

TA2vib(Dw=20)

-X/R

NO

RM

AL

IZE

DT

EM

PER

AT

UR

E

1 2 3 42

4

6

8

10

12

14

16

18

TA2tr (Dw=20)

TA (Dw=5)

TA2 (Dw=20)

TA2tr (Dw=5)

Temperature profiles along SSL

D=5TA2vib(Dw=5)

TA2vib(Dw=20)

Figure 2. Mole fraction profiles along the stagnation streamline (SSL) for Dw = 5 and Dw = 20 (Left),Temperature profiles along the stagnation streamline (SSL) for Dw = 5 and Dw = 20 (Right)

References

1G. A. Bird, Molecular Gas Dynamics and Direct Simulation of Gas Flows, Oxford Science, (1994).2V. I. Kolobov, R. R. Arslanbekov, V. V. Aristov, A. A. Frolova and S. A. Zabelok, Unified solver for rarefied and

continuum flows with adaptive mesh and algorithm refinement, J. Comp. Phys., Vol. 223, 2, pp589-608, (2007).3C. S. Wang Chang and G. E. Uhlenbeck, Transport Phenomena in Polyatomic Molecules, Univ. of Michigan Publication

CM-681, (1951).4R. Yano, K Suzuki, H. Kuroda, Formulation and numerical analysis of diatomic molecular dissociation using Boltzmann

kinetic equation, Phys. Fluids, 19, 017103 (2007).5R. Yano, K. Suzuki and H. Kuroda, Formulation and numerical analysis of vibrationally coupled recombination of

monatomic molecules using Boltzmann kinetic equation, Phys. Fluids 21, 127101 (2009);6P. L. Bhatnagar, E. P. Gross and M. Krook, A Model for Collision Process in Gases.I.Small Amplitude Processes in

Charged and Neutral One-Component System, Phs. Rev. Vol.94, No.3, (1954).7B. B. Hamel, Kinetic Model For Binary Gas Mixtures, Phys. of Fluids Vol.8-3 pp418-425, (1965).8T. F. Morse, Kinetic Model for Gases with Internal Degrees of Freedom, Phys. of Fluids, 6, pp159-169, (1964).9H. Oguchi, A Kinetic Model For a Binary Mixture and Its Application to a Shock Structure, Advan. Appl. Mech., Suppl.

4,1, (1967).10K. Abe, Shock wave structures in binary gas mixtures with regard to temperature overshoot, Phys. of Fluids, 17, 1333,

(1974).11V. Garzo, A. Santos and J.J. Brey, A Kinetic Model For a Multicomponent Gas, Phys. Fluids, A1(2), (1989).12E. Ikenberry and C. Truesdell, On the Pressures and the Flux of Energy in a Gas according to Maxwell’s Kinetic Theory,

Part I, II, J. Rational Mech. Anal., Vol. 5, No.1, pp1-128, (1956).13S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge Univ. Press, 3rd edition,

(1970).14D. Bergers, Kinetic Model Solution for Axisymmetric Flow by the Method of Discrete Ordinates, J. of Comp. Phys. 57,

285-302, (1985).15I. Rusinek and R. E. Roberts, Semiclassical calculation for collision induced dissociation. II. Morse oscillator model, J.

of Chem. Phys. 68. 3. pp. 1147-1152, (1978).

11 of 13

American Institute of Aeronautics and Astronautics

-X/R

MO

LE

FRA

CT

ION

1 1.5 2 2.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

rA (Case A)rA (Case B)rA (Case C)rA (Case D)rA2 (Case A)rA2 (Case B)rA2 (Case C)rA2 (Case D)

Mole fraction profiles along SSL

-X/R

NO

RM

AL

IZE

DT

EM

PER

AT

UR

E

1 1.5 2 2.5 3 3.5 4 4.52

4

6

8

10

12

14

16

18 TA (Case A)TA (Case B)TA (Case C)TA (Case D)Ttr

A2 (Case A)Ttr

A2 (Case B)Ttr

A2 (Case C)Ttr

A2 (Case D)Tvib

A2 (Case A)Tvib

A2 (Case B)Tvib

A2 (Case C)Tvib

A2 (Case D)

Temperature profiles along SSL

Figure 3. Mole fraction profiles along the stagnation streamline (SSL) for Cases A, B, C and D (Left),Temperature profiles along the stagnation streamline (SSL) for Cases A, B, C and D (Right)

16G. Colonna, I. Armenise, D. Bruno and M. Capitelli, Reduction of state-to-state kinetics to macroscopic models inhypersonic flows, J. of Thermophyics and heat transfer, 20, pp477-486, (2006).

17R. E. Roberts, R. B. Bernstein and C. F. Curtiss, Resonance theory of termolecular recombination kinetics: H+H+M →H2 + M, J. Chem. Phys., 50, 12, pp5163-5176, (1969).

18G. Colonna, F. Esposito and M. Capitelli, The Role of Rotation in State-to-State Vibrational Kinetics, 9th AIAA/ASMEJoint Thermophysics and Heat Transfer Conference, San Francisco, California, AIAA 2006-3423, (2006).

19G. Colonna, F. Esposito, M. Capitelli, H2 state-to-state kinetics in nozzle expansion, in XXIV International Symposiumon Rarefied Gas Dynamics, AIP Conference Proceedings 762, Portogiardino, Monopoli, Bari, Italy, July 10-16 2004, Ed. M.Capitelli, Melville, New York, pp1007-1012, (2005).

20W. G. Vincenti and C. H. Kruger, Introduction to Physical Gas Dynamics, Krieger Publishing Co., Florida, (1965).21H. C. Yee, A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods, von Karman Institute for Fluid

Dynamics, Lecture Series, Vol.4, (1989).22K. Morinishi and H. Oguchi, A computational method and its application to analysis of rarefied gas flows, Proc. 14th

Int. Rarefied Gas Dynamics symp. 1, pp149-158, (1984).23R. C. Millikan and D. R. White, Systematics of Vibrational Relaxation, J. of Chem. Phys. Vol.39-12, pp3209-3213, (1963).

12 of 13

American Institute of Aeronautics and Astronautics

-X/R

MOLEFRACTIO

N

1 2 3 40

0.2

0.4

0.6

0.8

1rA2(nemit

A2/n =0.0)

rA2(n

emit

A2/n =5.0)

rA2(nemit

A2/n =2.5)

rA(n

emit

A2/n =0.0)

rA(n

emit

A2/n =5.0)

rA(n

emit

A2/n =2.5)

Mole fraction profiles along SSL

-X/R

NORMALIZEDVELOCITY

1 2 3 4

0

1

2

3

4

5

6

7U

A2(nemit

A2 /n =0.0)

UA2(n

emit

A2/n =5.0)

UA2(n

emit

A2/n =2.5)

UA(n

emit

A2/n =0.0)

UA(n

emit

A2/n =5.0)

UA(n

emitA2/n =2.5)

Velocity profiles along SSL

-X/R

NORMALIZEDTEMPERATURE

1 2 3 4

5

10

15

TA2

vib(n

emit

A2/n =0.0)

TA2

vib(n

emit

A2/n =5.0)

TA2

vib(n

emit

A2/n =2.5)

TA(n

emit

A2/n =0.0)

TA(n

emitA2/n =5.0)

TA(n

emit

A2/n =2.5)

Temperature profiles along SSL

TA2

tr(n

emit

A2/n =5.0)

TA2

tr(n

emit

A2/n =0.0)

TA2

tr(n

emit

A2/n =2.5)

Figure 4. Mole fraction profiles along the stagnation streamline (SSL) for nemitA2

/n∞ = 0.0, 2.5 and 5.0(Left Top),

Velocity profiles along the stagnation streamline (SSL) for nemitA2

/n∞ = 0.0, 2.5 and 5.0 (Right Top), Temperature

profiles along the stagnation streamline (SSL) for nemitA2

/n∞ = 0.0, 2.5 and 5.0 (Left Bottom)

13 of 13

American Institute of Aeronautics and Astronautics