Assessment of vibration–dissociation coupling models for hypersonic nonequilibrium simulations

Accepted Manuscript Assessment of Vibration-Dissociation Coupling Models for Hypersonic Nonequilibrium Simulations

Jiaao Hao, Jingying Wang, Chunhian Lee

PII: DOI: Reference:

S1270-9638(16)31360-8 http://dx.doi.org/10.1016/j.ast.2017.04.027 AESCTE 4010

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

3 January 2017 10 April 2017 26 April 2017

Please cite this article in press as: J. Hao et al., Assessment of Vibration-Dissociation Coupling Models for Hypersonic Nonequilibrium Simulations, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.04.027

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Assessment of Vibration-Dissociation Coupling Models for Hypersonic Nonequilibrium Simulations Jiaao Haoa, Jingying Wangb,*, Chunhian Leea a

School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China

b

School of Energy and Power Engineering, Shandong University, Jinan, Shandong, 250100, China

Abstract The fidelity of three widely-used two-temperature vibration-dissociation coupling models, including the Park model (1988), the MacheretíFridman model (1994), and the coupled vibration-dissociation-vibration (CVDV) model (1963), is numerically investigated via a comparison with state-specific results and existing shock tube data for oxygen flows. Under the hypothetical condition where a Boltzmann distribution corresponding to a vibrational temperature is assumed, it is found that the CVDV model with a proper parameter is capable of providing the most accurate results, whereas the Park and MacheretíFridman models present similar prediction accuracy for which the nonequilibrium dissociation rate coefficients could be as much as three orders of magnitude lower than state-specific values. However, for actual postshock flows with intensive vibrational excitation and dissociation processes, the CVDV model still shows significant discrepancies relative to state-specific results and experimental data, presenting a much lower vibrational temperature and a higher degree of dissociation. The essential cause for the disagreement lies in the existence of non-Boltzmann *

Corresponding author. Tel: +86 053188392890 E-mail address: [email protected]

distributions observed when the incubation period is over and dissociation starts to dominate. Recommendations on further modification of two-temperature vibrationdissociation coupling models are presented. Keywords: hypersonic; thermochemical nonequilibrium; vibration-dissociation coupling; state-specific method

1 Introduction Hypersonic vehicles like reentry capsules and cruisers would inevitably encounter intensive thermochemical nonequilibrium processes while flying at hypervelocity, such as vibrational excitation, dissociation, and even ionization. Further developments in hypersonic flight technologies demand a deep understanding of such postshock nonequilibrium phenomenon. Across a strong shock wave, a large amount of kinetic energy in the freestream is converted into the translational energy immediately, resulting in an extremely high translational temperature. Subsequently, the internal energy modes of the fluid are excited accompanied by nonequilibrium effects due to relatively low collision frequency. Meanwhile, chemical reactions such as dissociation and ionization occur to generate atoms and electrons. The thermal nonequilibrium and chemical reaction processes are coupled under most conditions of hypersonic flows. For instance, the nonequilibrium population of the vibrational mode may affect dissociation rates and dissociation, in turn, would cause the removal of the vibrational energy [1]. The above vibration-dissociation coupling effects can be described via two different approaches, namely, the state-specific method and the two-temperature model.

The former solves the master equation to trace the temporal and spatial variation of each vibrational level [2í5]. Unfortunately, it suffers from enormous computational costs, making the two-temperature model a more practical option. Up to date, various two-temperature vibration-dissociation coupling models have been developed in the literature on numerical simulations of hypersonic flows [6]. However, due to the lack of experimental and theoretical data, these models still remain untested. Recently, a new set of shock tube experiments was conducted by Ibraguimova et al. [7] in temperature range of interest for hypersonic applications, which has been used to verify different vibrational nonequilibrium models [8í10]. From these direct simulation Monte Carlo (DSMC) and master equation calculations, it was demonstrated that the state-specific method is capable of providing a good agreement with the experimental data. It is the object of this work to evaluate the adequacy of various two-temperature vibrationdissociation coupling models via high-fidelity state-specific results and shock tube measurements. The rest of this paper is organized as follows. Basic relations of vibrationdissociation coupling processes are derived in Sec. 2.1, followed by a brief description of existing two-temperature models in Sec. 2.2. Then, a state-specific method developed based on kinetic rates governing the vibrational excitation and dissociation processes is presented in Sec. 3. Finally, results obtained using existing twotemperature models are compared to state-specific results and shock tube data to evaluate their fidelity in Sec. 4. Conclusions are given in Sec. 5.

2 Vibration-dissociation coupling model 2.1 General relations Under the two-temperature assumption, the modeling of vibration-dissociation coupling processes simply consists of the following two factors: the nonequilibrium dissociation rate and the added or removed vibrational energy which is implemented as a part of the source term in the vibrational energy equation [1]. The dissociation/recombination reaction for diatomic molecule A2 in the i-th vibrational quantum state is A2 (i ) + M

A+A+M

(1)

where M is the third particle acting as a collision partner. The production rate of A2(i) can be expressed as d ª A 2 ( i ) ¼º =kb,i [ A ][ A ][ M ] − k f ,i ¬ª A 2 ( i ) ¼º [ M ] dt ¬

(2)

where [A] represents the number density of A; kf,i and kb,i denote the forward and backward reaction rate coefficients, respectively. According to the principle of detailed balance, the two coefficients are correlated as k f ,i kb,i

= Keq Q (Ttr ) exp ( ε i kTtr )

(3)

Here, Keq is the equilibrium constant for the global reaction; Q the vibrational partition function; Ttr the translational-rotational temperature; İi the vibrational energy of the ith level; and k represents the Boltzmann constant. Summing up all vibrational levels, the total production rate of A2 yields d [ A 2 ] = kb [ A ][ A ][ M ] − k f [ A 2 ][ M ] dt

(4)

where the total dissociation and recombination rate coefficients, kf and kb, are,

respectively, ª A 2 ( i ) º¼ k f = ¦ k f ,v ¬ and kb = [A2 ] i

¦ kb,i

(5)

i

Under equilibrium conditions, the total dissociation rate coefficient can be evaluated by k f ,eq = ¦ k f ,i

exp ( −ε i kTtr )

(6)

Q (Ttr )

i

Thus, a factor Z can be introduced to represent the degree of nonequilibrium as Z=

kf

(7)

k f ,eq

Following Eqs. (3) and (6), one has kb =

1 Keq

¦ k f ,i

exp ( −ε i kTtr ) Q (Ttr )

i

=

k f ,eq

(8)

Keq

which implies that the total recombination rate is independent of vibrational nonequilibrium. Furthermore, the added or removed vibrational energy can be easily obtained as ωvd = ¦ ε i i

d ª¬ A 2 ( i ) º¼ dt

(

= ¦ ε i kb,i [ A ][ A ][ M ] − k f ,i ª¬ A 2 ( i ) º¼ [ M ] i

)

(9)

If the forward and backward rates of mass production per unit volume are denoted by Ȧf and Ȧb, respectively, Ȧvd can be written as ωvd = ωb Eb − ω f E f

(10)

where the terms representing forward and backward weighted average vibrational energy, Ef and Eb, are given, respectively, by E f = ¦ εi i

k f ,i ª¬ A 2 ( i ) º¼ k and Eb = ¦ ε i b,i k f [ A2 ] kb i

It can be readily verified that Ef degenerates into Eb at an equilibrium state.

(11)

2.2 Existing two-temperature models Various vibration-dissociation coupling models have been developed for hypersonic numerical simulations.

In this section, three representative models are

presented including the Park model [11], the MacheretíFridman model [12], and the CVDV model [13]. 2.2.1 Park model Park [11] intuitively defined control temperatures to account for the influences of thermal nonequilibrium states on chemical reactions, which gives Tc = Ttrac Tvbc

(12)

The values of ac and bc differ for different reaction mechanisms. For dissociation reactions, ac and bc are commonly set to be 0.5. Following the modified Arrhenius form, the nonequilibrium dissociation reaction rate coefficient can be expressed by § C· k f = ATcn exp ¨ − ¸ © Tc ¹

(13)

where A, n, and C are reaction parameters. The nonequilibrium factor for the Park model can then be written as n

ZPark

§T · §C C· = ¨ c ¸ exp ¨ − ¸ © Ttr ¹ © Ttr Tc ¹

(14)

The source term in the vibrational energy equation, Ȧvd, is usually assumed to follow the non-preferential model [14] as ωvd = (ωb − ω f ) ev

(15)

which implies that molecules are created or destroyed at the average vibrational energy denoted by ev. Hence, the Park model is, in essence, a completely phenomenological

simplification of the general vibration-dissociation coupling process, which may introduce significant inaccuracy. 2.2.2 MacheretíFridman model The MacheretíFridman model [12] is established for homonuclear molecules by assuming impulsive collisions and two dissociation regimes stemmed from upper and lower vibrational states. The resulting nonequilibrium factor of the MacheretíFridman model is given by ZM-F = Zl + Zh

(16)

ª § 1 1 ·º Zl = L × exp « −θ d ¨ − ¸ » T T » tr ¹ ¼ © a ¬«

(17)

where

Zh =

1 − exp ( − θ v Tv )

1 − exp ( − θv Ttr )

ª

§ 1 1 ·º − ¸» © Tv Ttr ¹ ¼»

(1 − L ) × exp « −θ d ¨ ¬«

(18)

θv and θd denote the characteristic vibrational temperature and dissociation temperature, respectively; 2

§ mA · Ta = αTve + (1 − α ) Ttr , α = ¨ ¸ © mA + mB ¹

(19)

and the parameter L depends on the type of collision processes. For molecule-atom (A2íB) collisions, 1−n

L=

9 π (1 − α ) § Ttr · ¨ ¸ 64 © θd ¹

ª 5(1− α ) Ttr º «1 + » 2θd ¬ ¼

(20)

and for molecule-molecule (A2íB2) collisions, 3 2− n

L=

2 (1 − α ) § Ttr · ¨ ¸ π 2α 3 4 © θd ¹

(

)

ª 7 (1 − α ) 1 + α Ttr º «1 + » « » 2θd ¬ ¼

The source term, Ȧvd, is also expressed as

(21)

ωvd = ωb Eb − ω f E f

(22)

where the forward and backward weighted average vibrational energy are specified, respectively, as 2 § § Tv · ¨ E f = Z lα kθ d ¨ ¸ + Z h kθ d ¨ © Ta ¹ ©

· ¸ Z M-F and Eb = (1 − L + α L ) kθd ¸ ¹

(23)

According to Eq. (23), the energy removed from the upper vibrational states is assumed to be equal to the dissociation energy, while that from the lower states is set to be a portion of the dissociation energy. Hence, the MacheretíFridman model can be regarded as semi-empirical. 2.2.3 CVDV model The CVDV model [13] assumes that truncated harmonic oscillators relax through Boltzmann distributions corresponding to the vibrational temperature, Tv, and that the probabilities of dissociation scale exponentially with vibrational levels. The resulting nonequilibrium factor is given by ZCVDV =

Q (Ttr ) Q (TF )

Q (Tv ) Q ( −U )

(24)

where TF is defined by 1 1 1 1 = − − TF Tv Ttr U

(25)

with U being a parameter in unit of temperature. The source term, Ȧvd, can be obtained by ωvd = ωb E ( −U ) − ω f E (TF )

where the function of weighted average vibrational energy is defined by

(26)

E (T ) =

1 ¦ εα exp ( − εα kT ) Qv (T ) α

(27)

Following the assumption of truncated harmonic oscillators, the expression can be simplified into E (T ) =

kθ v , s

−

kθ d , s

exp (θ v , s T ) − 1 exp (θ d , s T ) − 1

(28)

In the CVDV model, U is the only semi-empirical parameter, whose value may significantly influence the nonequilibrium dissociation rates. As U increases, the probability decreases that a dissociating molecule comes from an upper vibrational energy level [15]. For U ĺ ’, dissociation is equiprobable from all levels, which is also called the Hammerling model [16]. It can be easily proven that the CVDV model strictly satisfies the general relations of the vibration-dissociation coupling process discussed in Sec. 2.1.

3 Flow solver 3.1 Two-temperature implementation It is assumed that the translational and rotational energy modes are in equilibrium corresponding to a translational-rotational temperature Ttr and that the vibrational energy of molecules is described by a vibrational temperature Tv. Only the ground electronic state is considered for O2 and O. The thermochemical nonequilibrium flow downstream of a normal shock is governed by the following one-dimensional compressible flow equations

­∂ ° ∂x ( ρ s u ) = ωs , s = 1, 2, , ns ° ° ∂ ρu2 + p = 0 °° ∂x ® ° ∂ ª ρ u § h + 1 u 2 ·º = 0 ¸ ° ∂x «¬ ¨© 2 ¹ »¼ ° ° ∂ ( ρ ue ) = ω + ω v t −v vd °¯ ∂x

(

)

(29)

where the index ns represents the number of chemical species; ȡs and ȡ are the density of species s and gas mixture, respectively; h and ev denote the specific enthalpy and vibrational energy of the gas mixture; Ȧs represents the species mass production rate due to chemical reactions. The truncated harmonic oscillator model is used to evaluate the vibrational energy. Park’s 1990 chemical reaction model [1] is utilized with forward rate coefficients expressed in the generalized Arrhenius form and backward reaction rates calculated via equilibrium constants. The energy exchange between the translational and vibrational energy modes, denoted by Ȧtív, is modeled using the LandauíTeller model [17]. For the O2íO2 interaction, the vibrational relaxation time is calculated by the MillikaníWhite expression [18] with Park’s high temperature correction [1]. For the O2íO interaction, it is evaluated using the curve fit given by Ref. [5]. Ȧvd represents the added or removed vibrational energy, whose specific expression has been given earlier. The initial condition is obtained by the RankineíHugoniot relations assuming that chemical composition and the vibrational mode are frozen. The conservation equations are then integrated using the explicit fourth-order RungeíKutta scheme. 3.2 State-specific implementation Similar to the two-temperature implementation, it is also assumed that a single

translational-rotational temperature Ttr exists and that the high-lying electronic energy levels can be neglected. However, no assumption is made for the vibrational level distribution. The vibrational elementary processes considered in this study include vibrational-vibrational-translational (V-V-T) bound-bound transitions by O2 impacts, vibrational-translational (V-T) bound-bound transitions by O impacts, and vibrationaldissociation (V-D) bound-free transitions by O2 and O impacts. The resulting master equation for the number density of O2 at the i-th vibrational level can be written as ∂ ª¬O2 ( i ) º¼ ∂t

­° ∂ ªO2 ( i ) º¼ ½° ­° ∂ ªO2 ( i ) º¼ ½° ­° ∂ ª¬O2 ( i ) º¼ ½° =® ¬ +® ¬ ¾ ¾ +® ¾ ∂t ∂t ∂t ¯° ¿°V-V-T ¯° ¿°V-T ¯° ¿°V-D

(30)

where °­ ∂ ª¬ O 2 ( i ) º¼ °½ ® ¾ ∂t °¯ °¿V-V-T

{

}

(31)

= ¦¦¦ kV-V-T ( m, n → i, j ) ¬ª O 2 ( m ) ¼º ¬ª O 2 ( n ) ¼º − kV-V-T ( i, j → m, n ) ª¬ O 2 ( i ) º¼ ª¬O 2 ( j ) º¼ m

n

j

°­ ∂ ¬ª O 2 ( i ) ¼º °½ = ¦ kV-T ( j → i ) ¬ªO 2 ( j ) ¼º − kV-T ( i → j ) ¬ªO 2 ( i ) ¼º ® ¾ ∂t ¯° ¿°V-T j ≠ i

{

}

(32)

­° ∂ ¬ªO 2 ( i ) ¼º ½° O2 O2 = kV-D ( c → i ) [O ]2 [ O2 ] − kV-D ( i → c ) ¬ªO 2 ( i ) ¼º [O2 ] ® ¾ ∂ t ¯° ¿°V-D

{

{

}

}

(33)

O O + kV-D ( c → i ) [O] − kV-D ( i → c ) ª¬O2 ( i )º¼ [O ] 3

In the expressions, kV-V-T(i,jĺm,n) and kV-T(iĺj) denote the rate coefficients of V-V-T and V-T transitions, respectively; kV-D(iĺc) and kV-D(cĺi) are the rate coefficients of dissociation and recombination with superscripts representing the collision type. The nonreactive O2íO2 V-V-T transition rates are generated by the authors using the forced harmonic oscillator (FHO) model. The theoretical expressions for the V-V-T transition probabilities are given by Ref. [2]. The governing parameters for a fully repulsive potential are taken from Ref. [4], which can yield reliable O2íO2 V-V-T rates at

temperatures higher than 1000 K. In order to reduce the computational cost, only the multiquantum V-V-T transitions with the jump less than or equal to 5 are taken into consideration according to the results of heat bath calculations from Ref. [19]. The O2íO2 V-D rates are taken from the FHO analysis of Ref. [20], whereas the reactive O2íO V-T and V-D rates are taken from the quasi-classical trajectory (QCT) calculations [5,21]. The number density of O can then be determined by ª § ∂N · º ∂N O = ¦ « −2 ¨ i ¸ » ∂t © ∂t ¹ V-D ¼ i ¬

(33)

The equations are coupled with the one-dimensional compressible flow equations to fully resolve the vibrational level populations of O2 behind a normal shock. The RankineíHugoniot relations are utilized to provide the initial condition and the explicit fourth-order RungeíKutta scheme is employed for the numerical integration.

4 Results 4.1 Comparison with state-specific method assuming Boltzmann distribution In this section, the dissociation rate coefficients and added or removed vibrational energy calculated using the three vibration-dissociation models are compared with the results computed by the state-specific rates. The test case is performed by assuming that the population of vibrational levels satisfies a Boltzmann distribution in terms of Tv. The nonequilibrium factors computed with different models at Ttr = 4000 K, 7000 K, and 10,000 K are depicted in Fig. 1. 4000í10,000 K corresponds to the postshock temperature range of the shock tube experiments of Ibraguimova et al. [7]. For the CVDV model, results using different values of U are presented. It is seen that the factor Z is exactly equal to one at the equilibrium state and drops as the vibrational temperature

Tv decreases. Among the three coupling models, the Park model predicts the fastest decline at low Tv resulting in unreasonable dissociation rate coefficients, which could be as much as three orders of magnitude lower than the state-specific data; while the MacheretíFridman model gives the lowest nonequilibrium factor at higher Tv, also failing in capturing the behavior of the state-specific results; the CVDV model, in contrast, can agree well with the accurate results, provided that a proper value of U is chosen. It can be seen that a larger value of U leads to a larger nonequilibrium factor. For the O2íO2 interaction, the dependence of U on the translational-rotational temperature Ttr is much stronger than that for the O2íO interaction. In general, dissociation due to the O2íO2 interaction dominates in the region immediately after the shock, where Ttr is high, but Tv is relatively low. Therefore, based on the results shown in Fig. 1, U = θd/3 would be a reasonable choice for both O2íO2 and O2íO collisions within the temperature range of 4000í10,000 K. 10

0

10

10

-2

10

-3

10-1

Ttr = 4000 K

Park model M-F model CVDV, U = θd/2 CVDV, U = θd/3 CVDV, U = θd/6 FHO

10-4 10

Z

Z

10-1

-5

10

-2

10

-3

Ttr = 4000 K

Park model M-F model CVDV, U = θd/2 CVDV, U = θd/3 CVDV, U = θd/6 QCT

10-4 10

-6

10 1000

0

-5

-6

1500

2000

2500

Tv (K)

3000

3500

4000

10 1000

1500

2000

2500

Tv (K)

3000

3500

4000

10

0

10

10

-2

10

-3

10

-4

10

-5

10-1

Z

Z

10-1

Ttr = 7000 K

Park model M-F model CVDV, U = θd/2 CVDV, U = θd/3 CVDV, U = θd/6 FHO

-6

10 1000

0

10

-2

10

-3

10

-4

10

-5

Ttr = 7000 K

-6

2000

3000

4000

Tv (K)

5000

6000

7000

10 1000

100

100

10-1

10-1

-2

Ttr = 10,000 K

10-3

10

10-5

2000

4000

6000

Tv (K)

(a) O2íO2 collisions

8000

3000

4000

Tv (K)

6000

7000

Ttr = 10,000 K

10

10000

5000

-2

10-3

Park model M-F model CVDV, U = θd/2 CVDV, U = θd/3 CVDV, U = θd/6 FHO

-4

2000

Z

10

Z

10

Park model M-F model CVDV, U = θd/2 CVDV, U = θd/3 CVDV, U = θd/6 QCT

Park model M-F model CVDV, U = θd/2 CVDV, U = θd/3 CVDV, U = θd/6 QCT

-4

10-5

2000

4000

6000

Tv (K)

8000

10000

(b) O2íO collisions

Fig. 1 Nonequilibrium factors at Ttr = 4000 K, 7000 K, and 10,000 K

The vibrational energy removed due to dissociation using different models are compared in Fig. 2. Again, the CVDV model with U = θd/3 provides the most accurate prediction among the three models. Apparently, the non-preferential model used in the Park model significantly underestimates the vibrational energy removal, especially at high Tv, which suggests that dissociation proceeds mostly from the high vibrational levels as Tv approaches Ttr. The MacheretíFridman model acknowledges this variation tendency and assumes that the removed energy is close to the dissociation energy at high Tv. However, it can be seen that the state-specific results are much lower than the dissociation limit.

5.5 5

5.5 Ttr = 4000 K

4

3.5

2.5 2

2 1.5

1

1

0.5

0.5

Tv (K)

3000

0 1000

4000

5.5

4

4

3

4000

3 2.5

2

2

1.5

1.5

1

1

0.5

0.5 2000

3000

4000

Tv (K)

5000

6000

0 1000

7000

5.5

2000

3000

4000

Tv (K)

5000

6000

7000

5.5 Ttr = 10,000 K

5

Park model M-F model CVDV, U = θd/3 FHO

4.5 4

4

3

3 2.5

2

2

1.5

1.5

1

1

0.5

0.5 4000

Park model M-F model CVDV, U = θd/3 QCT

3.5

2.5

2000

Ttr = 10,000 K

4.5

Ef (eV)

3.5

0

3000

Park model M-F model CVDV, U = θd/3 QCT

3.5

2.5

5

Tv (K)

Ttr = 7000 K

4.5

Ef (eV)

3.5

Ef (eV)

5

Park model M-F model CVDV, U = θd/3 FHO

4.5

0 1000

2000

5.5 Ttr = 7000 K

5

Ef (eV)

3 2.5

1.5

2000

Park model M-F model CVDV, U = θd/3 QCT

4

3

0 1000

Ttr = 4000 K

4.5

Ef (eV)

3.5

Ef (eV)

5

Park model M-F model CVDV, U = θd/3 FHO

4.5

6000

Tv (K)

(a) O2íO2 collisions

8000

10000

0

2000

4000

6000

Tv (K)

8000

10000

(b) O2íO collisions

Fig. 2 Removed vibrational energy due to dissociation at Ttr = 4000 K, 7000 K, and 10,000 K

Figure 3 shows the comparison of the vibrational energy added due to recombination among the three models. Note that the added energy in the CVDV model, determined by Eq. (27), is a constant independent of Ttr. The MacheretíFridman model also predicts almost constant results, which are very close to the dissociation limit.

According to the analysis in Sec. 2.1, the added vibrational energy is equal to the removed energy evaluated at equilibrium conditions and thus is independent of vibrational nonequilibrium, which implies that the Park model with the non-preferential assumption is inconsistent with the physical relations. The state-specific results indicate that the added vibrational energy due to recombination relies on Ttr to some extent, which is closer to the MacheretíFridman result at low Ttr, but approaches the CVDV value at higher Ttr. 5.5 5 4.5 4

Eb (eV)

3.5 3 2.5

Park model M-F model CVDV, U = θd/3 FHO

2 1.5 1 0.5 0

2000

4000

6000

Temperature (K)

8000

10000

8000

10000

(a) O2íO2 collisions 5.5 5 4.5 4

Eb (eV)

3.5 3 2.5

Park model M-F model CVDV, U = θd/3 QCT

2 1.5 1 0.5 0

2000

4000

6000

Temperature (K)

(b) O2íO collisions Fig. 3 Added vibrational energy due to recombination

Previous analysis demonstrates that the CVDV model with U = θd/3 is capable of

providing the most accurate results among the three models. In the next section, only the Park and CVDV models will be utilized to calculate the postshock flowfields. 4.2 Comparison with state-specific method behind normal shock The vibrational temperature profiles of O2 were measured recently by Ibraguimova et al. [7] behind the front of shock waves with the translational-rotational temperature ranging from 4000 K to 10,800 K. Two cases are considered in this paper with freestream conditions listed in Table 1. In the table, V’ is the shock velocity; p’ and T’ are the freestream pressure and temperature, respectively. Case 1 corresponds to a mild nonequilibrium condition without notable dissociation, whereas Case 2 possesses a higher postshock translational-rotational temperature and a lower pressure resulting in much more intensive vibrational excitation and dissociation processes. In Case 2, the postshock species distributions of O2 and O were also derived using Beer’s law and the absorption coefficients. Table 1 Freestream conditions Case #

V’ (km/s)

p’ (Pa)

T’ (K)

1

3.07

266.644

295

2

4.44

106.658

295

Postshock profiles of Ttr and Tv calculated using the Park model, the CVDV model, and the state-specific method for Cases 1 and 2 are presented in Fig. 4. Also shown in the figure are the experimental data of Tv. For Case 1, the results from all the models agree well with the experiment data due to the fact that it is dominated by the O2íO2 vibrational excitation process accompanying with a low degree of dissociation, which

can be adequately described by the LandauíTeller model and the MillikaníWhite expression. For Case 2, results computed by the state-specific method may follow the experiment data especially after t = 0.2 ȝs, whereas the results provided by the twotemperature models show large deviations. Although there are distinct differences on the resulting nonequilibrium factors and added or removed vibrational energy between the Park and CVDV models, the predictions of the two models are unexpectedly similar, which will be explained in detail in the following section. 6000

Temperature (K)

5000

4000

3000 Ttr, Park Tv, Park Ttr, CVDV, U = θd/3 Tv, CVDV, U = θd/3 Ttr, State-specific Tv, State-specific Experiment

2000

1000

0

0

0.2

0.4

0.6

Time (μs)

0.8

1

(a) Case 1 12000 Ttr, Park Tv, Park Ttr, CVDV, U = θd/3 Tv, CVDV, U = θd/3 Ttr, State-specific Tv, State-specific Experiment

Temperature (K)

10000

8000

6000

4000

2000

0

0

0.2

0.4

0.6

Time (μs)

0.8

1

(b) Case 2 Fig. 4 Postshock temperature profiles predicted using different models

Postshock mass fraction profiles of the atomic oxygen O calculated using different

models together with the experiment data are shown in Fig. 5. Again, the Park model and the CVDV model produce similar results, whose values are much larger relative to the mass fraction computed by the state-specific method and the measurement. The discrepancy between the state-specific results and the experimental data in the vicinity of t = 0.2 ȝs could be attributed to the modeling errors of vibrational kinetic rates and the measurement uncertainties. In general, the state-specific method is capable of making an accurate prediction on the postshock vibration-dissociation coupling effects. 1 O , Park O , CVDV, U = θd/3 O , State-specific Experiment

Mass fraction

0.8

0.6

0.4

0.2

0

0

0.2

0.4

Time (μs)

0.6

0.8

1

Fig. 5 Postshock mass fraction profiles of atomic O predicted using different models for Case 2

4.3 Further discussions on two-temperature models This section is devoted to a detailed discussion of the inaccuracy produced by the two-temperature models using the state-specific results as a benchmark. Figure 6 compares the equilibrium dissociation rate coefficients between the state-specific data and Park’s 1990 chemical reaction model, with the temperature ranging from 1000 K to 20,000 K. It is seen that Park’s 1990 model shows a good agreement with the FHO results for the O2íO2 interaction, but deviates from the QCT curve for the O2íO interaction at temperatures larger than 10,000 K with a difference less than one order

of magnitude. Nevertheless, Park’s 1990 model proves to be adequate for current simulations. 15

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(b) O2íO collisions Fig. 6 Equilibrium dissociation rate coefficients calculated using different models

Postshock profiles of nonequilibrium factors predicted using different models for Case 2 are presented in Fig. 7. Note that the Park model produces identical results for different types of collisions and the same is true of the CVDV model. The nonequilibrium factors are relatively small due to the large gap between Ttr and Tv immediately after the shock and rise rapidly as the vibrational excitation process proceeds. After t = 0.05 ȝs, the Park model predicts the largest value, presenting a

completely different variation trend as compared to the remaining two models. The nonequilibrium factor of the Park model is even larger than one near t = 0.1 ȝs. This overshoot phenomenon is caused by the inconsistency of the Park model against the general physical relations of the vibration-dissociation coupling. In contrast, the factors given by the CVDV model and the state-specific method share a similar variation trend and keep lower than one throughout the process. However, the state-specific results approach one much more slowly than the CVDV result, which are still less than 0.3 at t = 1 ȝs. 2 1.8

Park CVDV, U = θd/3 O2−O2, State-specific O2−O , State-specific

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Fig. 7 Postshock nonequilibrium factor profiles predicted using different models for Case 2

Postshock profiles of the vibrational energy removal given by different models for Case 2 are shown in Fig. 8. It can be seen that the CVDV model predicts larger vibrational energy removal, while the non-preferential Park model significantly underestimates the vibrational energy as compared to the state-specific method after t = 0.05 ȝs. Larger nonequilibrium factor and removed vibrational energy of the CVDV model lead to a larger degree of dissociation, lower specific vibrational energy, as depicted in Fig. 9, and thus lower Tv (cf. Fig. 4). In addition, the combined effect of the nonequilibrium factor and removed vibrational energy coincidently results in similar

results between the Park and CVDV models. 5.5 Park CVDV, U = θd/3 O2−O2, State-specific O2−O , State-specific

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Fig. 8 Postshock removed vibrational energy profiles predicted using different models for Case 2 2 Park CVDV, U = θd/3 State-specific

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Fig. 9 Postshock specific vibrational energy profiles predicted using different models for Case 2

Although the CVDV model with an appropriate value of U shows a good agreement on both nonequilibrium dissociation rate coefficients and added or removed vibrational energy with the data obtained by the state-specific rates with an assumption of a Boltzmann distribution corresponding to Tv, there are still significant discrepancies against the state-specific results and experimental data for postshock calculations. The essential cause for the disagreement lies in the existence of non-Boltzmann

distributions. Figure 10 presents the vibrational level distributions of O2 at t = 0, 0.01, and 0.1 ȝs. At t = 0 ȝs, the state-specific and Boltzmann distributions are completely overlapped due to the assumption of frozen vibrational energy mode. At t = 0.01 ȝs, it can be observed that the higher levels are significantly overpopulated as a result of the vibrational excitation. Then, notable dissociation occurs leading to an underpopulation of the higher levels at t = 0.1 ȝs. Although not shown here, the distribution similar to that at t = 0.1 ȝs maintains until t = 3 ȝs when the equilibrium state is finally reached. This phenomenon reduces the nonequilibrium factor and removed vibrational energy relative to the CVDV model and thus leads to a higher Tv and a lower degree of dissociation. 10

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Fig. 10 Postshock vibrational distribution evolution of O2 for Case 2

5 Conclusions The assessment of existing two-temperature vibration-dissociation coupling models is numerically investigated in the present paper for oxygen flows. Three widelyused models are considered including the Park model, the MacheretíFridman model, and the CVDV model. A quantitative comparison is firstly made among these models against the state-

specific data under the assumption of a Boltzmann distribution in terms of Tv. The results demonstrate that the Park and MacheretíFridman models significantly underpredict the nonequilibrium dissociation rate coefficients at low Tv, while the CVDV model may agree well with the accurate results, provided that the value of the parameter U is chosen properly. As for the vibrational energy removed due to dissociation, the state-specific data indicate that dissociation proceeds mostly from high vibrational levels as Tv approaches Ttr. However, the Park model fails in capturing such trend variation and the results obtained by the MacheretíFridman model is too close to the dissociation limit. Again, the CVDV model provides the best prediction. Furthermore, the state-specific results indicate that the dependence of the vibrational energy added due to recombination on the temperature is weak, which is closer to the MacheretíFridman model at low Ttr, but approaches the CVDV model at higher Ttr. Among the three models, the CVDV model is capable of providing the most accurate results, where U = θd/3 is recommended for O2íO2 and O2íO collisions within the temperature range of 4000í10,000 K. Postshock calculations are then performed using the Park and CVDV models. The resulting temperature and species mass fraction profiles are compared to those predicted using the state-specific method and experimental data. It is found that, although the CVDV model with an appropriate value of U shows a good agreement on both the nonequilibrium dissociation rate coefficients and added or removed vibrational energy with the state-specific data assuming a Boltzmann distribution corresponding to Tv, there are still significant discrepancies compared to the state-specific results and

experimental data under the condition with intensive nonequilibrium processes. The essential cause for the disagreement lies in the existence of non-Boltzmann distributions. The state-specific results demonstrate that a prominent underpopulation of high vibrational levels observed when the incubation period is over and dissociation starts to dominate leads to a higher vibrational temperature and a lower degree of dissociation relative to the CVDV model. Strictly speaking, the aforementioned conclusions are only valid for oxygen flows. For the case of nitrogen and air flows, the behaviors of the various models could differ significantly. Nevertheless, it is reasonable to suggest that further modification of the two-temperature vibration-dissociation model should be focused on the vibrational distribution functions. Once the non-Boltzmann distributions are appropriately accounted for, one may expect that the CVDV model would reproduce state-specific results and experimental data.

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[13] P. V. Marrone, C. E. Treanor, Chemical relaxation with preferential dissociation from excited vibrational levels, Physics of Fluids, 6(9) (1963) 1215í1221. [14] P. A. Gnoffo, R. N. Gupta, J. L. Shinn, Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium, NASA TPí2867, 1989. [15] J. Olejniczak, Computational and experimental study of nonequilibrium chemistry in hypersonic flows, University of Minnesota, Minneapolis, 1997. [16] P. Hammerling, J. Teare, B. Kivel, Theory of radiation from luminous shock waves in nitrogen, Physics of Fluids, 2(4) (1959) 422í426. [17] W. G. Vincenti, C. H. Kruger, Introduction to physical gas dynamics, Krieger, Malabar, 1965. [18] R. C. Millikan, D. R. White, Systematics of vibrational relaxation, J. Chemical Physics, 39(12) (1963) 3209í3213. [19] D. A. Andrienko, K. Neitzel, I. D. Boyd, Vibrational relaxation and dissociation in O2íO mixtures, AIAA 2016í4021, 2016. [20] M. Lino da Silva, J. Loureiro, V. Guerra, A multiquantum dataset for vibrational excitation and dissociation in high-temperature O2íO2 collisions, Chemical Physics Letters, 531 (2012) 28í33. [21] D. A. Andrienko, K. Neitzel, I. D. Boyd, Vibrational relaxation and dissociation of oxygen in molecule-atom mixtures, AIAA 2015í3251, 2015.