Numerical investigation of a hypersonic flow around a capsule in CO2–N2 environment

Journal Pre-proof Numerical investigation of a hypersonic flow around a capsule in CO2 − N2 environment Zineddine Bouyahiaoui, Rabah Haoui, Abderrahmane Zidane

PII: DOI: Reference:

S0997-7546(18)30777-5 https://doi.org/10.1016/j.euromechflu.2019.12.009 EJMFLU 103582

To appear in:

European Journal of Mechanics / B Fluids

Received date : 18 December 2018 Revised date : 9 November 2019 Accepted date : 20 December 2019 Please cite this article as: Z. Bouyahiaoui, R. Haoui and A. Zidane, Numerical investigation of a hypersonic flow around a capsule in CO2 − N2 environment, European Journal of Mechanics / B Fluids (2019), doi: https://doi.org/10.1016/j.euromechflu.2019.12.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier Masson SAS.

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Numerical investigation of a hypersonic flow around a capsule in CO2 − N2 environment Zineddine Bouyahiaouia , Rabah Haouia , Abderrahmane Zidanea

a Laboratoire de M´ecanique Energ´ ´ etique et Syst`emes de Conversion, Universit´e des Sciences et de la

Technologie Houari Boumediene, BP 32 El Alia, 16111 Bab Ezzouar, Algeria

Abstract

A numerical investigation is conducted to study a viscous hypersonic flow

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around a capsule in chemical and vibrational nonequilibrium. A 16−reaction

kinetic model was implemented in our two-temperature Hypersonic Reacting Flow (HRF) code. The vibrational relaxation times are evaluated using Millikan & White semi-empiric formula along with Park’s correction for high temperature. The utilized 16−reaction kinetic model demonstrates good modeling of

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the chemical nonequilibrium. The simulations are performed for three wall conditions: noncatalytic, equilibrium catalytic, and fully catalytic wall. Obtained results show a good agreement with the literature. The heat flux was found to vary from 10 to 41 W /cm2 depending on the wall condition. Keywords: Hypersonic flow, Catalytic wall, Nonequilibrium, Viscous flow,

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Finite Volume Method 1. Introduction

The conquest of planet Mars has become a major issue, especially during

the last decade. Due to overcrowding and lack of resources, the need for vital space becomes a priority for the development and survival of humanity. 5

Researchers began to think of a way to conquer a new planet that brings

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together livable conditions, namely water, air ... etc. As a result, several mis∗ Corresponding author

Email address: [email protected] (Zineddine Bouyahiaoui)

Preprint submitted to European Journal of Mechanics - B/Fluids

November 8, 2019

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sions have been programmed to explore nearby planets like Venus and Mars,

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which turned out to be the perfect candidate. To this end, studies relative to the Mars journey had to be carried out. These studies were to focus on several 10

aspects, such as launch, atmospheric reentry, and landing. Several launches

were carried out with different probes such as Viking 1 and 2 (1975-1982) [1], Pathfinder (1996-1997) [2], MER A and B (2003-2010) [3], Phoenix (2008) [4], MSL (2012) [5] and Exomars (2016) [6].

The Mars atmosphere is composed of 95.7 % carbon dioxide, 2.7 % nitro15

gen, and 1.6 % argon. It is characterized by its low-pressure density: near 7 Pa at high altitude and less than 600 Pa on the ground. This makes the braking

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of capsules weak and experimentation difficult and expensive. Therefore we resort to numerical calculations by developing CFD codes. Several numerical studies using different solvers were published in order 20

to study the different aspects of the reentry, such as the study of chemical and vibrational non-equilibrium flow behind a normal shock wave [7], the flow

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around a blunt body for Martian reentry [8–12], aerothermal heating effects on capsules [13, 14] and many other aspects involved in such flows. Regarding the numerical studies of aerodynamic and thermal effects during 25

reentry, we can cite among them:

Gnoffo et al. [15] investigated the influence of sonic-line location on Mars

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Pathfinder probe aerothermodynamics. They used two codes, the first code HALIS provides a perfect inviscid gas solution, and the second is a real gas, laminar and viscous code, LAURA [16]. The variation of angle-of-attack from 30

0°to 11°is studied to provide an understanding of the resulting aerodynamics and heating distribution changes. The work concluded that at small angle-ofattack (α ≤ 5°), the sonic line location shifts from the shoulder to the nose cap and back again to the leeside symmetry plane because of the heat capacity ratio

γ and cone half-angle of 70°. It is also showed that the sonic line movement affects the heating distribution.

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35

Mitcheltree et al. [17] proposed an aerothermal heating prediction for Mars

Microprobe using the CFD code LAURA and experimental measurements. The 2

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heating rate at the stagnation point at 0 angle-of-attack (α = 0°) was predicted

40

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to be 194 W /cm2 associated with a pressure of 0.064 atm. No significant heat-

ing augmentation due to radiation was predicted according to their model, and

the forebody shock layer should remain laminar. The effect of angle-of-attack was also investigated and shows that, on the forebody frustum, the experimental data is as much as 30 percent lower than the CFD prediction.

Edquist et al. [18] presented an aerothermodynamic design environment 45

for a Mars Science Laboratory entry capsule heatshield. The design was based

on Navier-Stokes flowfield simulation and entry trajectory from the 2009 launch. Baldwin-Lomax algebraic turbulence model was used and agreed well with

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fully turbulent experimental heating data. In the design conditions, a correc-

tion bias was included to account for heatshield TPS surface roughness effects 50

on turbulent shear stress and heat flux. At the maximum heat flux location, results showed that the total bias for the heat flux plus uncertainty is 50% more than the baseline heating. The peak design conditions found were: 197W /cm2

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heat flux, 471P a shear stress, 0.371atm pressure and 5477J/cm2 total heat load. McDaniel et al. [19] worked on the aerothermal analysis of the Phoenix 55

Martian entry vehicle using two CFD codes, namely DPLR and LAURA. Both codes solve the reacting Navier-Stokes equation on a structured finite volume grid for non-equilibrium gases. The difference between the two codes is the nu-

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merical method, DPLR [20] uses a modified Steger-Warming flux splitting for Inviscid fluxes with third-order accuracy using a monotone upstream centered 60

scheme for conservation laws (MUSCL) extrapolation with a minmod flux limiter. LAURA uses Roe’s averaging for inviscid fluxes with second-order correction via Yee’s symmetric total variation diminishing scheme. Results showed that the flow over the Phoenix vehicle could be modeled as laminar. Also, the axisymmetric CFD solution was faster and cheaper to generate than the 3D solution with conservative results. Finally, the two codes yielded consistent results within 5% on the forebody, and the maximum heating rates occurred at

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the stagnation point and the shoulder. Hollis and Borrelli [21] made a review article comparing computational 3

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tools, ground and flight tests data in order to illustrate the encountered challenges in the numerical modeling of these phenomena and to provide test

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cases to evaluate the computational fluid dynamics codes predictions. Cap-

sules used in this article were Mars Science Laboratory (MSL), Orion Crew Exploration Vehicle (CEV), and Fire II flight investigation of the reentry environment. This review shows the effectiveness of computations at predicting 75

the effects of physical phenomena such as turbulence, chemical and vibrational non-equilibrium, rarefied flow, and radiation transport. It was found that there is still the need to validate computational models at non-equilibrium condi-

tions for both Earth and Mars atmospheric environments. The accuracy of

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chemical and vibrational models seemed to be not fully validated, especially for the Martian CO2 environment.

Bansal et al. [22] developed a new flow solver to simulate coupled hypersonic flow-radiation over a reentry vehicle using existing solvers and tools available in OpenFOAM. The hypersonicFoam solver developed combines fea-

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tures from two solvers, the first is rhoCentralFoam, and the second is reactingFoam. An excellent agreement was observed in flowfields obtained by the new solver. The heat transfer rates for the coupled case were observed to be significantly reduced compared to the uncoupled rates. Preci and Auweter-Kurtz [23] presented a numerical simulation using the

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URANUS non-equilibrium code. The flow field was calculated for several chemical and thermal models of CO2 -N2 inflow mixture. A sensitivity analysis of the radiative heat flux has also been performed by employing the PRADE radiation database and the HERTA algorithm. Results showed that the calculated flow in front of the vehicle was in good agreement with the results of other authors, contrary to those at the wall, which showed significant discrepancy due 95

to simple wall boundary conditions. Armenise et al. [24] did a comparative study for Mars free stream condi-

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tions for two models: the state to state and multi-temperature models. Calculations concluded that the multi-temperature model achieved a better agreement for the heating predictions than the state-to-state. There were some discrepan4

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cies for the prediction of the species mole fractions, especially close to the wall.

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For a noncatalytic wall, the heating values obtained for Mars Pathfinder with the multi-temperature model were in better agreement with postflight calculations. The multi-temperature approach is much easier to implement and costs less compared to the state-to-state model. 105

There are also recent works concerning Martian reentry flows, among them the works of Yatsukhno et al. [25], [26] and [27], as well as other experimental works allowing the derivation of new thermochemical models such as [28–33]. All the studies cited before used different numerical approaches; however

as stated by Hollis and Borrelli [21], the numerical models still need to be val-

idated, especially for the Martian atmospheric environment. Thus, the present

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work is another contribution to the investigation of aerothermodynamics and thermal heating for Martian reentries of the MER capsule. It is focused on the flow simulation around a space capsule, taking into consideration three wall to during a reentry.

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conditions in order to predict the heat flux range that the heat shield is exposed 115

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2. Mathematical model

Figure 1: MER Entry Capsule Configuration. Dimensions in meter

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The MER geometry is illustrated in fig 1. The outer mold line (OML) con-

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sists of a 70° sphere-cone heatshield and a 43° conic backshell [34]. The Navier-

Stokes equations in a flux-vector formulation in a Cartesian coordinate system 120

for a compressible viscous flow expressed in 2D-axisymmetric form is given as:      ∂Wi,j X  → − → − → mes Ci,j + Fi,j i + Gi,j j . − ηa − H.aire Ci,j = Ω 0 0 a∈{x,x ,y,y } ∂t

(1)

mes(Ci,j ) is the measurement (in m3 ) of a infinitely small volume of center

(i, j) and aire(Ci,j ) is the surface of the symmetry plane passing by the center of elementary volume with ηa the integrated normal. Where vectors W , F, G, H and Ω are given by:

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ρ

ρu ρv ρe

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         W =        

ρs

ρm ev,m

                 

(2)

with ρ the mass density of the fluid, ρs and ρm are the partial densities of species s and molecules m. u and v are the components of the velocity vector.      2 ρu + p − τxx     ρuv − τxy  ,  (ρe + p) u − uτxx − vτxy + qx     ρs u   ρ e u ρu

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         F =        

m v,m

6

(3)

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(4)

     −2τxy     2p − 2τyy  ,  −2uτxy − 2vτyy + 2vτzz + 2qy     0   0

(5)

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         H =        

     ρvu − τxy     ρv 2 + p − τyy  ,  (ρe + p) v − uτxy − vτyy + qy     ρs v   ρ e v ρv

m v,m

0

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         G =        

0 0

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         Ω =        

0 0

ωc,s

ωv,m

                 

(6)

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The total energy per unit of mass e is written as: e=

ns X s=1

cv,s T +

nm X

Ym ev,m +

s=1

ns X

Ys h0f ,s +

s=1

 1 2 u + v2 2

(7)

h0f ,s is the formation enthalpy of the species s in J/kg given in table 1, and cv,s 130

is the specific heat at constant volume for each species. The pressure of the mixture is obtained by the equation of state:

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! ns p X R = T = RT ρ Ms

(8)

s=1

where R is the gas constant, Ms the molar weight for species s, and R the specific gas constant for the mixture.

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lows: ωv,m = ρm

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The source term of vibration energy production ωv,m is calculated as fol135

ev,m (T ) − ev,m (Tv ) + ev,m (Tv ) ωc,s τm

(9)

The first term of the right-hand side of equation (9) represents the LandauTeller equation [35] for the vibrational-translational energy exchange mode.

The second term is the loss/gain of vibrational energy due to chemical effects. This term is significant, but to avoid the use of complex models that are more 140

computationally expensive, we opted for the mentioned above model along with Park’s CVD model for the influence of vibrational nonequilibrium on kinetics.

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ev,m (T ) is the equilibrium energy of vibration at the temperature of translationrotation expressed as:

ev,m (T ) = 145

R Mm θv,m θ  exp Tv,m − 1

(10)

θv,m is the characteristic temperature of vibration for any molecule given in

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table 1. ev,m (Tv ) is the vibrational energy at the temperature of vibration Tv specified for each molecule.

The vibrational characteristic time τm is defined from the characteristic times related to the collisions between the molecule m and the species s:

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X X 1 s = s τm,s τm

(11)

where Xs is the molar fraction of the species s, and τm,s is the characteristic time of vibration-translation exchanges of the molecule m with species s, defined by the semi-empirical formula of Millikan and White [36] as:   1   pτm,s = exp am,s T − 3 − bm,s − 18.42 [atm.s]

(12)

With:

1

4

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2 3 am,s = 0.00116µm,s θv,m

1

4 bm,s = 0.015µm,s

8

(13)

(14)

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Molecule

θv,s , K

CO2

945 (2)

1890 3360 3395

O2

2239

C2

2669

CO

3074

NO

2817

h0f , kJ/mol -393.510 0.0 0.0

830.457

-110.535 91.271

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N2

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Table 1: Vibrational characteristic temperatures and standard formation enthalpies

CN O C

438.684

-

249.175

-

716.680

-

472.680

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N

3074

µm,s being the reduced molar weight in g/mole given as: µm,s =

155

Ms Mm Ms + Mm

(15)

For temperatures superior to 8000 K, the equation gives relaxation times lower

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than those observed in experiments. To overcome this problem, C. Park [37] suggests the following relation to be added to the vibrational relaxation time: p

τm =

1 Nt cm σm

(16)

where cm is the molecular average velocity in m/s defined as follows: r 8RT cm = πMm

(17)

with σm being the effective collision cross-section to vibrational relaxation in m2 computed using the following,

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σm = 10−21 9



50000 T

2

(18)

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and Nt is the density of the number of collision particles of species s, where Na

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is the Avogadro number,

ρNa Nt = Pns s=1 Xs Ms

(19)

Combining the two relations, the following expression for the vibrational relaxation time is obtained:

p

MW τm = τm + τm 165

(20)

The source term of the conservation equation of chemical species ωc,s is given by: ωcs = Ms

r  X 00 0 υs − υs Jr

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r=1

(21)

Where Jr is the elementary sheet balance for a chemical reaction written as : ns X s=1

0

kf

νs As ⇐⇒ kb

ns X

00

νs A s

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 0 00  Y ρ !νs   Y ρ !νs   s s Jr = kf − kb    Ms Ms s s

0

(22)

s=1

(23)

νs and νs” are, respectively, the stoichiometric mole numbers of the reactants 170

and products of species s for each chemical reaction r. kf and kb being the forward and the backward (or reverse) reaction rates.

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Θf Ta

(24)

Θ a

(25)

Bf

−

B

− Tb

kf = Af Ta exp

kb = Ab Ta b exp

The kinetic model used in the present work is a reduced version of the

model found in [38], which is based on the Park Model [7]. It was obtained by 175

eliminating the reactions that involve ionization and NCO. Constants Af , Ab , Bf , Bb , Θf and Θb depend on the considered reaction and are summarized in

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tables 2 and 3.

The reaction rates of CO2 dissociation and O2 taken from different refer-

ences [39, 40] are compared to those included in the present reduced model 10

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(figure 2). It is observed that all the reaction rates have relatively the same variations in a function of temperature.

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In order to assess the coherence of this model, it was tested for the predic-

tion of equilibrium composition for a temperature range of [1000 K − 4000 K] at two considered pressure values 1500 P a and 15000 P a. The obtained results 185

were compared to those shown by Palmer and Cruden [41] and are represented

in figures 3 and 4. The initial gas mixture used is composed of [96%CO2 − 4%N2 ].

We can see that the CO2 dissociates more quickly at low-pressure than at high pressure when the temperature increases. For temperatures below 2500 K, CO2 is the dominant species. Above this temperature, CO and O

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become the dominant species, this is the case for both values of pressure. The differences observed between our results and those of Palmer and Cruden are due to the fact that the latter use chemical equilibrium, whereas in our case, we use the nonequilibrium relaxation until the equilibrium is reached. In the case of hypersonic flows behind shock-waves, it is customary to ex-

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press the reaction rate of dissociation reactions as a function of both Ttr and Tv,m [42]. We use the vibration-dissociation coupling because of the molecular vibration influence on the dissociation. Indeed, the more a molecule is excited in a vibrational plan, the more it breaks up easily. The coupling allows holding into account this influence. According to this, the average temperature of

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the dissociation reactions is equal to the geometric mean of the temperature of translation rotation and the vibrational temperature of each molecule: p Ta = T Tv

(26)

The heat flux vector q is calculated for each direction x, y, and z. The heat flux for a direction j is constituted of three components: the first component is given by the Fourier law, the second term corresponds to the vibrational heat flux, and the third one corresponds to the species diffusion. The heat flux in a

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2a 2b 3a 3b 4 5a 5b 6a 6b 7a 7b 8 9 10 11 12

Bf

Θf

C2 + M ⇐⇒ C + C + M

3.7 × 10+14

0.00

69900

7.0 × 10+21

−1.60

113200

N2 + Mol ⇐⇒ N + N + Mol

3.0 × 10+22 1.0 × 10+22

−1.50

59750

N2 + A ⇐⇒ N + N + A O2 + A ⇐⇒ O + O + A

14 15 16

2.0 × 10+21

O2 + Mol ⇐⇒ O + O + Mol

2.5 × 10+14

CN + M ⇐⇒ C + N + M CO + A ⇐⇒ C + O + A

CO + Mol ⇐⇒ C + O + Mol N O + X ⇐⇒ N + O + X

CO2 + A ⇐⇒ CO + O + A

CO2 + Mol ⇐⇒ CO + O + Mol N O + O ⇐⇒ N + O2

3.4 × 10+20 2.3 × 10+20 1.1 × 10+17 5.0 × 10+15

N O + D ⇐⇒ N + O + D

N2 + O ⇐⇒ N O + N CO + O ⇐⇒ C + O2 CO + C ⇐⇒ C2 + O

CO + N ⇐⇒ CN + O N2 + C ⇐⇒ CN + N

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Af

−1.60

113200

−1.50

59750

0.00

71000

−1.00

129000

−1.00

129000

0.00

75500

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1

Reaction

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Table 2: Forward coefficients.

0.00

75500

1.4 × 10+22

−1.50

63275

2.8 × 10+09

6.9 × 10+21 2.0 × 10+12

CN + C ⇐⇒ C2 + N

1.00

20000

0.50

38000

−0.18

69200

1.0 × 10+14

2.0 × 10+17

−1.00

58000

0.00

38600

1.1 × 10+14

−0.11

23200

0.10

14600

5.0 × 10+13

0.00

13000

0.00

27800

2.1 × 10+13

CO2 + O ⇐⇒ O2 + CO

63275

3.9 × 10+13

1.6 × 10+13

CN + O ⇐⇒ N O + C

−1.50

direction j is computed as follows: qj = −keq

X ∂Y ∂T −ρ hs Ds s s ∂xj ∂xj

(27)

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keq denotes the coefficient of total thermal conductivity, i.e., it includes translationalrotational and vibrational thermal conductivities, which is a function of Prandtl

210

number P r and dynamic viscosity µ. Ds is the diffusion coefficient given in 12

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2a 2b 3a 3b 4 5a 5b 6a 6b 7a 7b 8 9 10 11 12

Bb

Θb

C2 + M ⇐⇒ C + C + M

3.886 × 10+11

0.62

0.0

1.015 × 10+19

−1.24

0.0

N2 + Mol ⇐⇒ N + N + Mol

4.351 × 10+19 5.856 × 10+19

−1.19

0.0

−0.50

0.0

1.543 × 10+13 1.129 × 10+14 1.155 × 10+14

N2 + A ⇐⇒ N + N + A O2 + A ⇐⇒ O + O + A

1.171 × 10+19

O2 + Mol ⇐⇒ O + O + Mol

3.083 × 10+16

CN + M ⇐⇒ C + N + M

2.281 × 10+13

CO + A ⇐⇒ C + O + A

CO + Mol ⇐⇒ C + O + Mol N O + X ⇐⇒ N + O + X

N O + D ⇐⇒ N + O + D

CO2 + A ⇐⇒ CO + O + A

CO2 + Mol ⇐⇒ CO + O + Mol N O + O ⇐⇒ N + O2 N2 + O ⇐⇒ N O + N CO + O ⇐⇒ C + O2

CO + N ⇐⇒ CN + O N2 + C ⇐⇒ CN + N

14 15 16

CN + C ⇐⇒ C2 + N

D : CO2 , N O, C, N , O

X : CO, O2 , C2 , N2 , CN

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A : C, N , O

13

0.37

0.0 0.0

0.27

0.0

2.343 × 10+14

0.02

0.0

0.02

0.0

1.100 × 10+10

1.00

4000.0

0.50

0.0

3.399 × 10+13

−0.18

14280.0

2.044 × 10+13

−1.00

7351.0

0.00

5814.3

2.921 × 10+15

−0.63

0.0

0.38

1327.1

1.582 × 10+16

−0.59

0.0

M : CO2 , CO, O2 , C2 , N2 , N O, CN , C, N , O Mol : CO2 , CO, O2 , C2 , N2 , N O, CN

0.0

0.27

3.281 × 10+09

CO2 + O ⇐⇒ O2 + CO

−1.19

0.0

1.031 × 10+12

CN + O ⇐⇒ N O + C

0.0

0.37

3.043 × 10+16

CO + C ⇐⇒ C2 + O

−1.24

2.485 × 10+15

4.400 × 10+15

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13

Ab

re-

1

Reaction

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N°

pro of

Table 3: Backward coefficients.

0.78

23464.0

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pro of

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(b)

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(a)

Figure 2: comparison of dissociation rates for CO2 (a) and O2 (b)

function of the Lewis number Le as:

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Ds = ks

Le ρcp s

(28)

Lewis number characterizes mass diffusion (ratio of energy transport due to diffusion and thermal conduction). In our case, it is fixed at 1.0 for Martian applications[43]. cp is the specific heat at constant volume for all energy 215

modes.

For a pure gas, the viscosity is given by the interpolation of Blottner et al.

[44]:

h µ µ µi µs = 0.1 exp As ln (T ) + Bs ln (T ) + Cs

µ

µ

(29)

µ

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Coefficients As , Bs and Cs depend on the species considered and are given in table 4. For air species, coefficients are valid for a temperature range from

220

1000K to 30000K as proposed by Gupta et al. [45]. For carbon species William 14

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pro of

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Figure 3: Equilibrium molar fractions comparison for p = 1500 P a: solid lines is our model and

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dashed for Palmer and Cruden [41]

Figure 4: Equilibrium molar fractions comparison for p = 15000 P a: solid lines is our model and

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dashed for Palmer and Cruden [41]

[46] has built a viscosity law in the form of Bl¨ottner’s model from the gases kinetic theory of Neufeld et al. [47] for a temperature range from 100K to 15

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pro of

20000K. For multicomponent gas, such as a chemically reacting mixture, the Table 4: Blottner’s viscosity law coefficients µ

µ

µ

As

Bs

Cs

CO2

-0.026654

1.107305

-14.291274

N2

0.0203

0.4329

-11.8153

O2

0.0484

-0.1455

-8.9231

C2

-0.011809

0.8486858

-13.182071

CO

-0.014044

0.887198

-13.269815

NO

0.0452

-0.0609

-9.4596

CN

-0.007770

0.779370

-12.825216

O

0.0205

0.4257

-11.5803

C

-0.007140

0.768602

-12.956246

N

0.012

0.5930

-12.3805

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Molecule

225

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mixture value of µmix must be calculated from the values of µs of each chemical species s by means of mixture rules. A common viscosity expression is Wilke rule, which states that:

µmix =

X s

1 M = √ 1+ s M 8 s0

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With:

φs,s0

P s0

Xs µs Xs0 φss0

 ! −1 !1 !1  2  µs 2 Ms0 2    1 + µs0 Ms 

(30)

(31)

Ms , Ms0 Xs and Xs0 are respectively the molar weights and the molar fractions of species s and s0 . 230

For the thermal conductivity kmix of mixture, the same equation (30) can

be used, replacing µ by k. 3. Numerical code

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The numerical study presented in this paper was performed using our in-

house code HRF (Hypersonic Reacting Flow). This code was specifically de-

235

signed for nonequilibrium hypersonic compressible flow studies and is based 16

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on the finite volume method in order to solve nonequilibrium Euler equations

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using a two-temperature model. The disturbance of domain technique [48] is

used to transform the 3D model into a 2D-axisymmetric model, and the time integration is performed using an explicit iterative approach (Euler forward 240

method). It was first used for the numerical simulation of nonequilibrium

nozzle flow [42, 49], and later, it was upgraded with Van-Leer’s flux vector splitting numerical scheme [50] in a first-order version, in order to be used for the simulation of hypersonic nonequilibrium compressible flows around a

blunt body [51, 52] or for atmospheric reentry of a space vehicle [12]. An al245

ternative of this in-house code was written in order to solve nonequilibrium

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Navier-Stokes equations for the simulations of viscous hypersonic compressible flows. Details of the axisymmetric formulation, discretization, and Van-

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Leer’s flux vector splitting can be found in ref. [53].

Figure 5: Computational domain and boundary conditions

The boundary conditions used in simulations are illustrated by figure (5)

and detailed in the following:

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250

17

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3.1. Freestream Conditions

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Inlet velocity, pressure, and temperature are fixed since the flow is supersonic at the freestream. 3.2. Body Surface 255

Since the flow is considered viscous, the no-slip wall condition must be

applied. Therefore, the velocity at the surface is set equal to zero (u = v = 0).

Regarding temperature, it is maintained at a value of Tw = 1500K. In this study, the wall shear stress is calculated by: ! V ∂Vt =µ t τw = µ ∂n wall ∆n

re-

where:

→ − → − Vt = V . t

(33)

− ∆n = (∆x, ∆y) .→ n

(34)

and:

with:

and:

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260

(32)

∆x = x (i, j + 1) − x (i, j)

(35)

∆y = y (i, j + 1) − y (i, j)

(36)

265

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→ − Here we assume that the coordinate of the unit vector t is in the direction → − − of the shear force at the wall and the unit vector → n is normal to t figure 6. The total friction drag is caused by shear stress acting on the top and bottom surfaces.

Concerning the catalysis model, thee conditions applied at the wall must

be specified in order to take into consideration these phenomena. Several definitions can be found in the literature. In this work, the catalysis modeling is based on the definition given by Anderson Jr [54]. The non-catalytic wall condition implies that there are no variations in the

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270

chemical composition near the wall, and thus we have:

18

∂Ys ∂n

=0

pro of

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Figure 6: Grid near the wall

For a fully catalytic wall, all atoms recombine in order to form molecules. This condition is applied as follows: Yatoms = 0. 275

Equilibrium catalytic wall as indicated by its designation supposes that lo-

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cal chemical equilibrium is reached near the wall at the given temperature and pressure, i.e. at Tw = 1500 and a pressure of pw ≈ 9000 P a. For these condi-

tions, the composition at equilibrium is the same as the freestream. Therefore the condition will be: Ys = Ys (f reestream). 280

3.3. Axis of Symmetry

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The conditions at the axis of symmetry boundary are no flow and no scalar

flux across the boundary.

3.4. Outlet Boundary Conditions

At the exit of the computational domain, the values of the flow parameters

285

are extrapolated from the interior values, including in the boundary layer. 4. Results and Interpretations

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4.1. Inviscid perfect gas simulation of the whole domain (front and wake) A first simulation of the flow over the whole MER capsule is conducted

using our inviscid solver with non-reacting gas. The free stream conditions are 19

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290

listed in table 5. The numerical grid used for our calculations is illustrated

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in figure 7, with 60 cells normal to the wall and 197 streamwise. The chosen

convergence criteria for our simulations is a maximum residue of 10−5 . The calculations converged after 131098 iterations equivalent to an execution time of 7328 s.

Table 5: MER free stream conditions

Symbol & unit

Value

Inlet velocity

V [m/s]

5223

Pressure

p [P a]

7.87

Temperature

T [K]

140

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Name

Tw [K]

1500

CO2 Mass fraction

YCO2

0.97

N2 Mass fraction

YN2

0.03

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Wall temperature

Figure 7: Grid and computational domain (60 × 197)

295

From figures 8 and 9, we observe that the detached shock wave is located

at approximately 0.25 m in front of the body. For the after-body area, the gas cools down at the shoulder. Though, hot

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gases are raised to the payload by the recirculation zone developed in the afterbody region (fig. 10). The temperature distribution (fig. 10) for the fore-body

300

is characterized by a sudden increase in temperature to a value of 20715 K after 20

pro of

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Figure 8: Mach number over MER body

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Figure 9: Flow Streamlines

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Figure 10: Temperature over MER body

21

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the shock due to the flow compression. As the flow goes around the body sur-

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face, it accelerates, this causes a temperature diminution. For the after-body, flow velocity decreases with the formation of a recirculation region, which will cause an increase in temperature to a maximum value of 24457 K. 305

4.2. Viscous real gas simulations of the front domain

In order to reduce the computation time, a reduced computational domain representing the fore-body is used for the viscous non-equilibrium case. Sim-

ulations are run using the same freestream conditions as the frozen inviscid simulation.

Figures 11 and 12 represent the tangential velocity profile and tangential

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310

stress normal to the wall at x = 0.2 m respectively. Velocity goes from zero at the wall to a value of around 1000 m/s in the boundary layer with a thickness of 20 mm. Then, it goes from 1000 m/s to 1800 m/s in the flow outside of the boundary layer until reaching the shock at 80 mm from the wall. From figure 11, we also notice that the (80 × 808) grid (80 in the normal direction

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315

and 808 in the tangential direction) gives an almost straight shock which is

expected from the use of a first-order FVS type numerical scheme. Hence, this mesh will be retained to conduct the simulations. Tangential stress shows the exact positions of the shock, the flow behind, and the boundary layer. Note that the stress decreases in the boundary layer because of the fast decrease in

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320

temperature, which is a consequence of the cold wall temperature Tw = 1500 K. Mach number and temperature distributions are illustrated in figures 13

and 14 respectively. The sonic line represented by a discontinued line, shifts from the shoulder to the nose cap and back again on the leeside as reported by 325

Gnoffo et al. [15]. We also notice that the shock position is much closer to the body than in the frozen flow case. This is due to the chemical and vibrational processes. On the other hand, the temperature reaches a peak value of almost

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16000 K. A sudden decrease after the peak is caused by the dissociation reactions which are endothermic and the vibrational relaxation. After a short

330

distance, the temperature reaches the wall value. 22

pro of

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0,30

(80x808) (60x606)

0,25

Normal Position [m]

(50x506) (40x404)

0,20

(30x304) (20x202)

0,15

(10x98) 0,10

0,05

Shock

0,00

Boundary layer

500

1000

1500

2000

re-

0

Tangential velocity [m/s]

Figure 11: Tangential velocity profile in boundary layer for chemically reacting viscous flow

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0,30

Shear stress parallel to the wall at x=0,2 m

0,20

Shock

Free stream

0,15

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Normal position [m]

0,25

0,10

0,05

BL

0,00

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

-2

Shear stress [N.m ]

Figure 12: Tangential stress

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Figure 15 represents kinetic and vibrational temperatures of each molecule

along the stagnation line. As stated before, the temperature reaches a peak

23

pro of

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1,6 1,4

y [m]

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

0

0

re-

0,2 0,2 0,4

x [m]

Figure 14: Temperature contour for viscous

flow

flow

17500

12500 10000

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Temperature [K]

15000

lP

Figure 13: Mach number contour for viscous

Ttr Tv CO2 (1) Tv CO2 (2) Tv CO2 (3) Tv CO Tv O2 Tv N2 Tv NO Tv CN Tv C2

7500 5000 2500

0 −0,03

−0,025

−0,02

−0,015 x [m]

−0,01

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Figure 15: Temperature profiles along the stagnation line

24

−0,005

0

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pro of

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Figure 16: Comparison between temperature variations along the stagnation line for the same

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freestream condition

Figure 17: Mass fraction for a noncatalytic wall along the stagnation line: present work (solid lines), CEA (dashed lines)

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behind the shock front, which induces CO2 -N2 dissociation. Other species

start appearing namely CO, O2 , N O, CN , C2 , C, N and O as illustrated in 25

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pro of

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Figure 18: Mass fraction for an equilibrium catalytic wall along the stagnation line: present work

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(solid lines), CEA (dashed lines)

Figure 19: Mass fraction for a fully catalytic wall along the stagnation line

figures 17, 18 and 19 for the noncatalytic, equilibrium catalytic and fully cat-

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335

alytic wall respectively. Vibrational temperatures rise just behind the shock due to the excitation of the vibrational modes, then the vibrational tempera26

10000 9000 8000

Pressure [Pa]

7000 6000 5000 4000 3000 2000

Equilibrium catalytic wall Fully catalytic wall

1000

Non-catalytic wall

0 0,1

0,2

0,3

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0,0

pro of

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0,4

0,5

x [m]

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Figure 20: Pressure over the body surface

Figure 21: Heat flux over the body surface

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tures meet the trans-rotational temperature, showing that thermal equilibrium is reached. Thus, the region from x ≈ −0.022 m to x ≈ −0.005 m represents the

340

thermal equilibrium region. Beyond this region, all temperatures decrease to

27

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pro of

6

Equilibrium catalytic wall 5

Fully catalytic wall

Shear stress [N.m

-2

]

Non-catalytic wall 4

3

2

1

0 0,1

0,2

0,3

re-

0,0

0,4

0,5

x [m]

Figure 22: Shear stress over the body surface

12500 10000 7500

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Temperature [K]

15000

lP

17500

Equilibrium catalytic wall Fully catalytic wall Non-catalytic wall

5000 2500

0 −0,03

−0,025

−0,02

−0,015 x [m]

−0,01

−0,005

0

Figure 23: Comparison of Temperature distributions at the stagnation line for the different wall

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conditions

reach wall temperature. Figure 16 illustrates a comparison of temperature profiles along the stag28

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nation line for the same freestream conditions. The first thing to be noticed

345

pro of

is the differences in the temperature peak after the shock, i.e. for our results and those of Preci and Auweter-Kurtz [23] the peak is about 16000 K, while for

Rouzaud et al. [55],Druguet [56] and Potter et al. [57] it is about 7500 K. This is due to the consideration of the vibrational nonequilibrium in our calculations

an those of Preci and Auweter-Kurtz. This is reasonable since, in the case of nonequilibrium, hundred of collisions are required to allow the translational 350

degree of freedom to pass energy to the vibrational degree. After that our temperature decrease rate is different from others, we suspect that this is due to the difference in the employed chemical and vibrational nonequilibrium models.

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Depending on the wall condition (figures 17 to 19), the mass fraction on

the boundary layer differs between the three figures. For the noncatalytic wall, 355

there is no variation of the mass fraction at the wall. In contrary equilibrium and fully catalytic wall depend on the composition at the wall which will induce the temperature distribution in the boundary layer as seen in figure 23.

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For figures 17 and 18, the dashed line represent CEA [58] values for the same conditions at the wall. 360

Pressure, heat flux and shear stress over the body are illustrated in figures 20, 21 and 22 respectively for the three wall conditions. Three regions can be distinguished for all figures. The first region is around the nose cap, where

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the pressure reaches a maximum of almost 9000 P a and diminishes to reach a constant value. For heat flux, the maximum value is around 41 W /cm2 for an 365

equilibrium catalytic wall, 17 W /cm2 for a fully catalytic wall and 10 W /cm2 for a noncatalytic wall. For the latter, the heat flux maximum value is on the same order of magnitude as with those found in literature, such as Rouzaud et al. [55] 12 W /cm2 and Druguet [56] 14 W /cm2 . The differences noticed for

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the heat flux are due to species conditions imposed at the wall

29

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5. Conclusion

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370

A numerical model was developed to investigate the viscous hypersonic flow around a capsule. Chemical and vibrational nonequilibrium were consid-

ered. Therefore, a 16-reactions kinetic model based on the Park model was implemented in our in-house code HRF. Vibrational relaxation times were eval375

uated using Millikan and White formula with Park correction. The flow was considered laminar, and Blottner’s model was used for viscosity computation.

In order to reduce the computation time, an inviscid axisymmetric simulation of the hypersonic flow around the whole capsule was first simulated. Re-

sults from this first simulation were used to locate the shock position in order to reduce the computational domain for the simulation of the viscous hyper-

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380

sonic flow in chemical and vibrational nonequilibrium. The modeling of the chemical and vibrational nonequilibrium was applied to a 2D axisymmetric simulation of the ballistic entry of the MER front part capsule. Three wall

385

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conditions were considered: a noncatalytic wall, an equilibrium catalytic wall, and a fully catalytic wall. We can summarise the significant conclusion of this study in two points: The obtained results showed a good agreement with literature for the aerothermodynamics parameters and the heat flux estimation should be between 10 W /cm2 and 41 W /cm2 which are the bounds to be used for heatshield design to ensure optimum and secure design of the capsule. The accuracy of these results can be improved by using a higher-order numerical

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390

scheme and the inclusion of turbulence phenomena. References

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Declaration of interests

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Zineddine Bouyahiaoui (Corresponding Author).

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☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.