Real Gas Effects on a Planetary Re-entry Capsule

Real Gas Effects on a Planetary Re-entry Capsule S. Bisceglia* and F. Grasso.† University of Rome “La Sapienza”, Rome, 00184, Italy G.Ranuzzi‡ CIRA, Italian Aerospace Research Center, Capua, 81043, Italy This paper provides a detailed investigation of nonequilibrium real-gas effects of hypersonic speed in the forebody region of a typical re-entry vehicle. A finite rate chemistry model based on the Scott model is implemented here and for an extensive investigation of the catalysis modeling relevance simulations with non-catalytic wall conditions are computed as well as those with full catalytic boundary conditions. Turbulence modeling is considered to investigate its coupling with finite rate chemistry. The method developed is then used to compute the detailed flow features of a hypersonic flow around the forebody of a re-entry vehicle. The computed pressure and shear stress coefficients along the profile are then analyzed and the stagnation heat flux is compared to the experimental data. A comparison with the results obtained by Kurotaki with his finite rate chemistry model is here presented as well.

Nomenclature D E e h kb kw n R S V W X Y γ ∆h0 ∆s η Θ µ τ

ω&

= = = = = = = = = = = = = = = = = = = =

diffusion coefficient total energy internal energy, electron charge enthalpy Boltzmann constant catalytic recombination rate outward unit normal, number density vector of residuals thermal nonequilibrium source term cell volume vector of unknown molar fraction mass fraction specific heat ratio, recombination coefficient enthalpy of formation cell face length thermal conductivity characteristic temperature mixture viscosity relaxation time

= chemical nonequilibrium source term

Subscripts E e

= inviscid contribution = electronic, electron

*

Aeronautical Engineer, Department of Mechanics and Aeronautics, via Eudossiana 18. Professor, Department of Mechanics and Aeronautics, via Eudossiana 18, Associate Yellow AIAA. ‡ Research Engineer, Aerothermodynamics and Space Propulsion Laboratory. †

1 American Institute of Aeronautics and Astronautics

el mol R i rot s, r T tr V v

= = = = = = = = = =

free electron molecular species rotational impinging rotational sth, rth species translational translational, translation-rotation viscous contribution,vibrational-electron-electronic vibrational

Superscripts e s T +

= = = =

electronic sth species transpose ion

I.

L

Introduction

ast two decades have brought important developments for aerothermodynamics with the emergence and maturing of the discrete numerical methods for aerodynamics/ aerothermodynamics complementary to the ground-simulation facilities and the parallel outstanding growth of computer power. The benefits of numerical simulation for flight vehicle design are enormous: much improved aerodynamic shape definition and optimization, provision of accurate and reliable aerodynamic data and highly accurate determination of thermal and mechanical loads. The discrete numerical methods of aerothermodynamics permit now the simulation of high speed flow past real flight vehicle configurations with thermo-chemical and viscous effects, the description of latter being still handicapped by insufficient flow-physics models. Accurate evaluation of heat flux is important for reusable space transportation system design to minimize weight, maintenance and refurbishment while there is an increasing need to model finite rate chemistry effects in Computational Fluid Dynamics (CFD). The same increasing need concerns improvements in the physical understanding of theory and physics since currently available tools are still far to model correctly finite rate chemistry6. The objective of the present work is to provide a detailed investigation of the non-equilibrium real-gas effects in the forebody region of a typical re-entry vehicle, providing comparison between different chemistry models. Table 1: Flight Conditions Mach Number

9,06

14,71 20,09 23,89 25,96

Altitude [Km]

48.4

55.7

63.6

71.7

79.9

Tw [K]

1501

1571

1413

1078

808

For our purpose we have chosen as test case the first Japanese orbital experiment vehicle successfully launched in 1994. The vehicle has a blunt-cone shape, which has aerodynamic static stability at re-entry. The OREX vehicle1 was launched and inserted into about 450 km circular orbit; its velocity and flight path angle at re-entry interface (altitude = 120 km) were 7433 m/s and -3.17° respectively, providing no aerodynamic lift at re-entry flight. Space transportation vehicles of this kind enter the atmosphere at a flight Mach number higher than 25, this generates a very strong shock wave in front of the nose19, 20 and, as a result, the air in the shock layer is heated to temperature T>10,000 K so that the characteristic flow time become comparable to the characteristic times for chemical reaction and vibrational relaxation. Hence in the shock layer the flow is not in thermal and chemical equilibrium. From a mathematical point of view the problems we have to deal with in the present context are well known14, 15, 31, 37 and can be formally written down but the detailed inputs and physico-chemical models4, 9, 10, 28, 38, 39 required to describe the constitutive properties and thermochemical behavior of the flow medium are far from being complete or final. Such models and their built-in parameters are essential to close the problem for a detailed analysis and application to 2 American Institute of Aeronautics and Astronautics

any specific configuration. Depending on the location of the fight vehicle on its trajectory, it encounters regimes ranging from free molecular flow down to continuum flow, with a transitional slip-flow regime along the way33. Here we deal with continuum regime problems of fight in the lower altitudes (below ~80 km). The governing equations describing the flow of a multi-species chemically reacting gas mixture are statements of16: global continuity; total mass conservation; total momentum conservation; total energy conservation; individual species conservation, including production and consumption via chemical reactions; energy conservation26 in different internal molecular modes: rotation, vibration, electronic excitation, including their exchanges mutually21, 30 and with translational modes; gas characterization by thermal and caloric equations of state. The associated boundary conditions are statements of14: momentum transfer at gas-solid interface, allowing in general for slip-velocity between the two phases; heat transfer at gas-solid interface, allowing in general for temperature-slip between the two phases; mass transfer at the gas-solid interface, allowing in general for heterogeneous reactions involving production and consumption of species, and imperfect energy accommodation. Clearly, closure of the above systems of equations and conditions setting up a mathematical model for the physical conservation and contiguity laws requires specification (in terms of macroscopic thermodynamic state variables of the multicomponent gas mixture) of: transport coefficients17 (viscosity, thermal conductivity, electrical conductivity, species diffusion coefficients); chemical reaction schemes4, 10, 28, 29 and relevant rate coefficients25, 32 both in gas phase and at gas/solid interface; rate coefficients for relaxation17, 24 from nonequilibrium between chemical states and among internal molecular states; emission and absorption of radiation in gas phase and at walls19; ionization coupled to radiation40; combustion characteristics relevant to high-speed propulsion devices; and the old standbys: turbulence and transition13, modeling of which are not described here. In this study numerical calculations performed with different models are compared with flight data to improve the understanding of the physics involved and to assess the accuracy of the chemical model used.

II.

Governing Equations

The governing equations that describe the flow of a multi-species chemically reacting gas mixture are statements of total mass conservation, total momentum conservation, total energy conservation and individual species conservation, including production/ consumption via chemical reactions. In addition the assumption35 is made that a single (translational) temperature T characterizes the translational and rotational modes, whereas a single (vibrational) temperature Tv characterizes the vibrational and electronic modes of heavy particles and the translational modes of free electrons; accordingly, an additional conservation equation for the vibrational-electronelectronic energy must be solved for, and models for the energy exchanges with the roto-translational energy must be supplied. Neglecting7 externally applied electric and magnetic fields it is assumed that charge separation and conduction currents are negligible the conservation equations may be cast11 as:

∂ WdV + ∫ ( FE − FV ) ⋅ ndS = ∫ HdV S V ∂t ∫V

(1)

where

W = [ρ S , ρu , ρv, ρE , ρeV ]

T

F = (FE − FV , G E − GV )

[ = [ρ v, ρuv, ρv

FE = ρ S u , ρu 2 + p, ρuv, ρuH , ρueV GE

S

2

+ p, ρvH , ρveV

] ]

T

T

(FV , GV ) = [− ρ SVS ,σ ,σ ⋅ u − Q,−QV ]T T H = [ω& S ,0,0,0, SV ] and

3 American Institute of Aeronautics and Astronautics

(2)

ρ sVs = − ρDs∇X s 2 3 Q = −(ηtr + ηrot )∇T − (ηv + η E + ηel )∇TV − ∑ ρhs Ds∇X s

σ = µ (∇u + ∇u T ) − µ (∇ ⋅ u ) I

s

(3)

Q = −(ηv + ηE + ηel )∇TV − ∑ ρhV Ds∇X s S

smol

E = ∑ Ys es + s

ρ = ∑ ρs s

u 2 + v2 2

p

H =E+

ρ

p = ∑ ρ s RsT + pe

pe = ρ e ReTV

s≠e

where, for simplicity, a gradient law has been assumed to hold for the diffusion and conduction terms. Note that s

e

s

for heavy particles, hV = eV while for electrons hel = hV = eel + p e

ρ e and that the vibrational-electron-

electronic energy source term is:

SV = − p e ∇ ⋅ u + S T −V + S T − E + S E − R

(4)

where the term − p e ∇ ⋅ u is the electron pressure work, S T −V

ST − E are respectively the translation-vibration energy exchange and the heavy-particle electron-electronic energy exchange while S E − R represents the energy contribution due to coupling between chemistry and vibration and to electron-electronic excitation.

A. Turbulence model The turbulence modeling methodology relies on a linear eddy viscosity two-equation (k-ε) model that accounts for the coupling of turbulence with chemistry and vibration through properly defined turbulent Prandtl numbers as described in Ref 13. B. Thermodynamic models The internal energy of species s is obtained assuming energy mode separability. For atomic species one has34

3 Rs T + evs + ∆hs0 2

(5)

5 Rs T + evS + e ES + ∆hs0 2

(6)

3 ReTV 2

(7)

es = while for diatomic species

es = The electron translational energy is given by

eel =

and the vibrational and electronic energy contribution are obtained assuming Boltzmann distribution at the vibrational-electronic temperature Tv, thus yielding

eVs = Rsθ sV

(

exp θ

1 V s

)

TV − 1

4 American Institute of Aeronautics and Astronautics

(8)

∑ g ⋅θ exp(− θ T ) ∑ g exp(− θ T ) N Se

s E

e = Rs

i =1

e s ,i

s ,i

e s ,i

N Se

i =1

e s ,i

s ,i

V

(9)

V

where g S ,i represents the ith state degeneracy. Because of vibration-dissociation coupling, dissociation occurs mainly at the higher vibrational levels. For harmonic oscillator behavior, only a finite number of such levels can be taken into account in evaluating the vibrational energy. The number of vibrational levels before the occurrence of dissociation depends on the dissociation energy of each molecule

C. Transport properties The mixture viscosity is determined according to Chapman-Enskog34 theory N −1

µ=∑ s =1

where

integrals22

X

s

= Ys W

s

ms X s mX + N e (e2 ) ( 2) (2) ∆ + ∆ X ( T ) X ( T ) ∑r ≠ e r s,r e s ,e V ∑ r =1 X r ∆ e, r (TV )

(10)

is the molar concentration of the s’th species. Approximations for the collisions

∆(sk,)r , k = 1,2 are obtained by means of Yos formulas19. For a mixture of gases in thermal

nonequilibrium the heat conduction includes the contributions of different energy modes. For the vibrational and electronic thermal conductivity we use a simplified expression corresponding to partial excitation of the two modes, thus obtaining

(

)

(

)

X s cυSV Rs ηV = k ∑smol ∑r ≠e X r ∆(s2,)r (T ) + X e ∆(s2,)e (TV ) X s cυSE Rs N −1 η Ec = k ∑s =1 ∑r ≠e X r ∆(s2,)r (T ) + X e ∆(s2,)e (TV )

(11)

The free-electron translational thermal conductivity is expressed as

η el =

15 k ∑ smol 4

∑r ≠ e 1.45 X r ∆

Xe (2) e,r

(TV ) + X e ∆(e2,)e (TV )

(12)

The diffusion coefficients of species s in the mixture are determined according to Ref. 11

kT X t2Ws (1 − Ws γ s ) Ds = p ∑r ≠ s γ r ∆(s1,)r where X t

(13)

= ∑s X s . In a weakly ionized gas7, with the assumption of zero electric current and charge

neutrality, the induced electric field affects the diffusion of the charged species. The effective diffusion coefficient of the electrons (De) is then proportional to the ambipolar diffusion coefficient of the ions, i.e.

5 American Institute of Aeronautics and Astronautics

De =

me me amb 2 D NO + D NO + = m NO + m NO +

(14)

D. Internal energy transfer (relaxation) Referring to the vibrational-electron-electronic equation in Eq.(4), the source term SV includes the translationalvibrational energy exchanges S T −V , the heavy-particle-electron translational energy exchanges S T − E and the vibrational-electron-electronic energy lost (or gained) in chemical reactions modeled according to Landau-Teller34 theory:

S T −V =

∑ ρs

S E − R . The T-V energy exchanges are

eVs (T ) − eVs (TV )

smol

τs

(15)

where the vibrational relaxation time is defined as the sum of the molar averaged Millikan-White relaxation time

τ

M −W s

and the collision-limited time

τ sP τ s = τ sM −W + τ sP

(16)

The heavy-particle-electron translational energy exchanges are derived from classical kinetic theory36, under the assumption of a displaced Maxwellian distribution function of molecular velocity. Neglecting5 frictional heating of the electrons due to differences between the electron and heavy-particle velocities, one obtains

S T − E = 2m e n e ∑ r

3 kb (T − TV )ν er* 2 mr

(17)

with ν er is the collision frequency for electrons and species r. The energy removal contribution due to coupling8, *

21, 30

between chemistry and vibration and to electron-electronic excitations is assumed to be related to the average vibrational and electron-electronic energies

SV − R = ω& e eel + ∑ ω& s eVs

(18)

smol

III.

Finite Rate Chemistry Model

Finite rate chemistry is modeled by assuming that seven species are present [O, N, NO, O2, N2, NO+, e-] and using the 17 reaction mechanism described in Ref. 11. The forward and backward reaction rate constants are evaluated by means of the Arrhenius formulas, and the reaction-rate-controlling temperature for the dissociation reactions is set to Td

= (TTV )

1/ 2

.

A. Catalysis model and boundary condition Compared to the homogeneous reaction kinetics, the mechanisms and rate processes determining the gas/solid chemical processes taking place between the hot gas as it approaches a hypersonic vehicle wall are even less firmly founded and quantified23. The understanding and control of these processes is of decisive importance for their development and fabrication of thermal protection systems for planetary atmospheric re-entry vehicles. The recombination rate defined as the ratio between the flux of atoms that recombine at the surface with the total flux of atoms impinging the surface. The latter flux is given by kinetic theory2 of gas (Hertz-Knudsen relation):

6 American Institute of Aeronautics and Astronautics

Z i = [s]

k b Tw 2πms

(19)

Where [s] and mS are the concentration and the mass of the atomic specie s, Tw is the wall temperature and kb is the Boltzmann constant. The wall production rate of monoatomic species is:

ω s = −k w ρ s

(20)

With

kw = γ s

k bTw 2πms

(21)

Catalytic speed and where γs is the recombination coefficient of specie s. The wall conditions for monoatomic species are written as follows:

r r

ρ sVs ⋅ n = − k w ρ s

(22)

Where n is the vector normal to the wall. For the reaction rate of the chemical reaction at the wall is possible to assume a behavior following the Arrhenius empirical law:

γ s = Pe

−

Ea kbTw

(23)

Where P is a steric factor accounting for the directional effects, that are difficult to predict, and Ea is the activation energy. Eq. 23 has been achieved by Scott 32, 34 and is implemented in our model. One characteristic of such models is that usually activation energies cannot really be considered as unique over the considered wide range of temperatures. Moreover such a relation obtained by empirical setting is limited to conditions very similar to that of the process and extrapolation is a risky procedure2, 10, 34, for high temperatures these models are limited to a constant maximum value. The species wall boundary conditions (b.c.) were determined by using (a) the noncatalytic wall condition, (b) the fully catalytic wall condition or (c) the finite rate chemistry boundary condition.

(b)

⎛ ∂Ys ⎞ ⎜⎜ ⎟⎟ = 0 ⎝ ∂y ⎠ w Ysw = Ys∞

(c )

(k s ρYs )w = ⎢ ρDs ⎛⎜⎜ ∂Ys ⎞⎟⎟⎥

(a)

⎡ ⎣

(24)

⎤

⎝ ∂y ⎠⎦ w

where ks, is the recombination rate and Ys the mass fraction of the specie s, ρ the over-all density, Ds the diffusion coefficient.

7 American Institute of Aeronautics and Astronautics

Figure 1.

The OREX configuration (Unit: mm)

IV.

Numerical Solution

The solution of the governing equations for high-speed flows requires the use of robust and accurate schemes. Time integration is then performed by a three-stage Runge-Kutta algorithm with a point-implicit treatment of the source terms. Space and time discretizations are separated by exploiting a method-of-lines approach, and a system of ordinary differential equations is then obtained for every computational cell. A cell-centered finite volume formulation is employed. By approximating surface and boundary integrals by means of the mean-value theorem and midpoint rule, the governing equations are cast 11, 12in the following discretized form:

S i, j

dWi , j dt

4

+ ∑ (Fnum ⋅ n∆s )β = S i , j H i , j

(25)

β =1

The inviscid discretization is based on an upwind biased second-order TVD scheme that includes for nonequilibrium and ionization. The scheme has good properties of monotonicity and conservativity in the presence of discontinuities, and it yields second-order accuracy and oscillation-free solutions. By enforcing consistency at cell face i + 1 / 2, j , the numerical inviscid-flux discretization is cast in the following form:

(

)

(F

n + GE , num n y )i + 1 , j =

E , num x

2

⎤ 1⎡ ⎢(FE nx + GE n y )i , j + (FE nx + GE n y )i +1, j + Ri + 1 , j Φ i + 1 , j ⎥ 2⎣ 2 2 ⎦

(26)

The expression for the elements of the vector Φ i +1 / 2 , j is obtained by characteristic decomposition in the direction normal to the cell face:

ϕl

1 i+ , j 2

⎞ ⎞ ⎛ ⎞⎛ 1 ⎛ = ψ ⎜⎜ a l 1 ⎟⎟⎜⎜ g l 1 + gil, j ⎟⎟ − ψ ⎜⎜ a l 1 + γ l 1 ⎟⎟α l 1 i j i j i j i j i+ , j + , + , + , + , 2 ⎝ 2 ⎠⎝ 2 2 ⎠ 2 ⎠ ⎝ 2

Where

8 American Institute of Aeronautics and Astronautics

(27)

αl γl

1 i+ , j 2

(

= R −11 Wi +1, j − Wi , j

1 i+ , j 2

i+

2

)

(

)

⎞ g l − g il, j 1 ⎛ = ψ ⎜⎜ a l 1 ⎟⎟ i +1, j αl 1 2 ⎝ i+ 2, j ⎠ i− , j

(28)

2

⎛ ⎞ g il, j = min mod⎜⎜α l 1 ,α l 1 ⎟⎟ i+ , j i− , j 2 ⎠ ⎝ 2 min mod(x, y ) = sgn ( x )max{0, min[ x , y ⋅ sgn (x )]}

Here ψ

(z ) is an entropy correction to z . The min-mod limiter is used11 for the second-order antidiffusive flux

contribution g, for its computational efficiency and speed of convergence. The values at the interfaces are calculated by using a generalization of Roe’s averaging to allow for thermal and chemical nonequilibrium. In presence of nonequilibrium ionizing flows, stiffness arises for the disparity between the characteristic time scale of the relaxation process and the fluid-dynamic one. The stiffness can be reduced by preconditioning the system of discretized (ordinary differential) equation. Time integration is performed by means of a three stage Runge-Kutta algorithm, where the source terms are treated semi-implicitly, and the preconditioning matrix that is related to the partial Jacobian of the source term. 2 1.75 1.5

Y/L

1.25 1

0.75 0.5 0.25 0

0

Figure 2.

V.

0.5

1

X/L

1.5

The mesh used.

Results and Discussion

The geometry of the OREX experiment, the capsule-type vehicle successfully launched41 in Japan in1994, is constituted by a spherical nose region with a 1.35 m radius, connected at an angle of 40° with a cone with a half cone angle of 50° (Fig.1) (for further details see Ref. 1). As remembered in Ref. 28 the basic equations the axysimmetric full Navier-Stokes equations. For the propose of our study a grid sensitivity analysis was considered: the adequacy of grid resolution has been assessed by means of preliminary simulations. In particular we have built and tested the following meshes: [80x80], [160x80], [160x160], [320x160] cells obtained by halving the mesh spacing of the base grid. The computed results, not reported because of space limitations, indicate that the [160x160]-cells mesh (Fig.2) represents the optimum choice in terms of grid independence and computational time required. The convergence of flow properties has been checked (in O(200kcyc)). Symmetry boundary conditions have been imposed at the symmetry axis while extrapolation boundary conditions at the inflow as well as at the outflow have been implemented; the flight path here considered allowed to impose the no-slip velocity conditions at the wall, zero normal pressure gradient is imposed and fixed translational wall temperature are set as in Table 1.

9 American Institute of Aeronautics and Astronautics

5

7x10

5

6x10

5

Q 5x10 2 [W/m ]

5

4x10

5

3x10

5

2x10

5

1x10

0

7

10

13

16

19

22

25

28

Mach

Figure 3. Heat flux evaluation. Kurotaki: × fully catalytic, + non catalytic, ο partially catalytic (alpha=1000) ;Present work: ⊕ fully catalytic, û non catalytic, · finite rate chemistry;ı Fay-Riddell, Ê Flight data.

The presence of a wide range of static pressure p∞ and the surface temperature at the stagnation point Tw make this test-case very suitable for the investigation of finite rate catalysis model effectiveness. The vehicle structure consist of two parts (Fig.1): an aeroshell and an equipment section. The aeroshell, which is subjected to aerodynamic heating during re-entry, is made up of a carbon/carbon (C/C) nose cap and an aeroskirt. The aeroskirt is an aluminum alloy honeycomb sandwich panel covered with C/C TPS (thermal Protection System) at the upper part and ceramic tiles at the lower part. As mentioned by Kurotaki28 the coating in the stagnation point indicates that our chemistry models can be applied. The wall temperature Tw and the free stream flow properties are shown in Table 1. The value of the recombination coefficients for both nitrogen and oxygen was set to γ O = γ N = 0.01 which represents10 the optimal choice in the temperature range here considered. Heat flux In Fig. 3 the computed heat flux along the flight path is compared to flight data. Very good agreement between acquired real flight data is observed for values of Mach number up to ~20, and an underestimation of the heat flux for higher values of Mach. To remark the importance of the chemistry model in Fig. 3 are also represented: the results obtained with non catalytic and fully catalytic boundary conditions; the heat flux obtained by means of the Fay-Riddell formula; Kurotaki computations with non-catalytic, fully catalytic and partially catalytic boundary conditions. Among all the computations performed by Kurotaki28 we have selected the results which approximated best the flight data, corresponding to a value of the parameter α (representing the degree of NO production in the Langmuir-Hinshelwood recombination compared with corresponding N2 or O2 production) >100. It is possible to observe that flight data are between the extreme non-catalytic and fully catalytic cases, so that finite rate chemistry modeling in real flight appears to be very important. Differences appear between the present results and Kurotaki model are evident at very high Mach number; these results indicate the need for a further understanding of the model dependence on their parameters. In Fig. 4-6 we report the variation of the Stanton number St = q w / ρ ∞ u ∞ h0 ,∞ − hw with M for non-

[

(

)]

catalytic, fully catalytic and finite rate chemistry boundary conditions along the capsule’s wall; the heat flux increases with M and shows similar trends in the three cases with some differences in the region of the shoulder. In Fig. 7-11 we report the variation of St with M for the different points of the trajectory reported in Table 1. It is possible to observe that the partially catalytic curves are always between the extremes and approximate the fullycatalytic or the non-catalytic wall conditions computations depending on the Mach number.

10 American Institute of Aeronautics and Astronautics

0.05

0.08

0.04 0.06

St

St

0.03

0.04

M 0.02

M 0.02

0.01

0

0

0.25

0.50

0.75

0

1.00

0.25

0

x/φ

0.50

0.75

1.00

x/φ

Figure 4. Variation of Stanton number with Mach. Partially catalytic model used; - - M= 9.06, __ - - __ M=14.71, _____ M=20.09, ___ - ___ M=23.89, __ __ M=25.89.

Figure 5. Variation of Stanton number with Mach. Fully catalytic model used; - - M= 9.06, __ - - __ M=14.71, ___ ___ M=20.09, - M=23.89, __ __ M=25.89.

0.04

_____

0.010

Non catalytic Finite rate Fully catalytic

0.008 0.03

St

St

0.006

0.02

M

0.004

0.01

0.002

0

0

0.25

0.75

0.50

0

1.00

0

0.25

x/φ

0.75

1.00

x/φ

Figure 6. Variation of Stanton number with Mach. Non catalytic model used; - - M= 9.06, __ - - __ M=14.71, ___ ___ M=20.09, - M=23.89, __ __ M=25.89.

Figure 7. Stanton number computed along the wall. M= 9.06, altitude h= 48.40 km, V∞ = 3000 m/s

_____

0.015

0.025

Non catalytic

Non catalytic Finite rate Fully catalytic St

0.50

0.010

Finite rate Fully catalytic

0.020

St

0.015

0.010 0.005

0.005

0

0

0.25

0.50

0.75

1.00

0

0

0.25

x/φ

Figure 8. Stanton number computed along the wall. M= 14.71, altitude h= 55.74 km, V∞ = 4759 m/s

0.50

0.75

x/φ

Figure 9. Stanton number computed along the wall. M= 20.09, altitude h= 63.60 km, V∞ = 6223 m/s

11 American Institute of Aeronautics and Astronautics

1.00

0.04

0.08

Non catalytic Finite rate Fully catalytic

0.03

St

Non catalytic Finite rate Fully catalytic

0.06

St 0.02

0.04

0.01

0.02

0

0

0.25

0.50

0.75

0

1.00

0

0.25

x/φ

0.50

0.75

1.00

x/φ

Figure 10. Stanton number computed along the wall. M= 23.89, altitude h= 71.73 km, V∞ = 7049 m/s

Figure 11. Stanton number computed along the wall. M= 25.96, altitude h= 79.90 km, V∞ = 7360 m/s

Skin friction coefficient The skin friction coefficient

C f = τ w 0.5 ρ e u e2 computed along the profile of the capsule is reported in Fig.

12-16; the results indicates a dependency on the finite chemistry model with Mach number. 0.020

0.015

Non catalytic Finite rate Fully catalytic Cf

Non catalytic Finite rate Fully catalytic

0.015

0.010

Cf 0.010

0.005

0.005

0

0

0.25

0.50

0.75

0

1.00

0

0.25

0.75

1.00

Figure 13. Friction coefficient computed along the wall. M= 14.71, altitude h= 55.74 km, V∞ = 4759 m/s

Figure 12. Friction coefficient computed along the wall. M= 9.06, altitude h= 48.40 km, V∞ = 3000 m/s

0.04

0.025

Non catalytic Finite rate Fully catalytic

0.020

Cf

0.50

x/φ

x/φ

Non catalytic Finite rate Fully catalytic

0.03

Cf

0.015

0.02 0.010

0.01 0.005

0

0

0.25

0.50

0.75

1.00

0

0

0.25

Figure 14 . Friction coefficient computed along the wall. M= 20.09, altitude h= 63.60 km, V∞ = 6223 m/s

0.50

0.75

1.00

x/φ

x/φ

Figure 15. Friction coefficient computed along the wall. M= 23.89, altitude h= 71.73 km, V∞ = 7049 m/s

12 American Institute of Aeronautics and Astronautics

The variation of the Cf with Mach number for the finite chemistry computed case is reported in Fig. 17; it is

0.075

0.075

Non catalytic Finite rate Fully catalytic Cf

Cf

0.050

0.050

M 0.025

0

0.025

0

0.25

0.50

0.75

0

1.00

0.25

0

0.50

0.75

1.00

x/φ

x/φ

Figure 17. Variation of skin friction coefficient with Mach number. Catalytic model used: finite rate chemistry; — —M= 9.06, - - M=14.71,______ M=20.09, __ - - __ M=23.89, - - M=25.89.

Figure 16. Friction coefficient computed along the wall. M= 25.96, altitude h= 79.90 km, V∞ = 7360 m/s

observed that the skin friction coefficient increases with M.

Pressure coefficients The computed distributions values of the pressure coefficient

C p = ( p − p∞ ) 0.5 ρ e u e2 are shown in Fig. 18-

22; unlike the values of Cf the pressure coefficient Cp is relatively unaffected by chemistry.

2.0

2.0

Non catalytic Finite rate Fully catalytic

1.5

Non catalytic Finite rate Fully catalytic

1.5

Cp

Cp 1.0

1.0

0.5

0.5

0

0

0.25

0.50

0.75

1.00

0

0

0.25

X/φ

0.50

0.75

x/φ

Figure 18. Pressure coefficient along the wall. M= 9.06, altitude h= 48.40 km, V∞ = 3000 m/s

Figure19. Pressure coefficient along the wall. M= 14.71, altitude h= 55.74 km, V∞ = 4759 m/s

13 American Institute of Aeronautics and Astronautics

1.00

2.0

2.0

Non catalytic Finite rate Fully catalytic

1.5

Non catalytic Finite rate Fully catalytic

1.5

Cp

Cp 1.0

1.0

0.5

0.5

0

0

0.25

0.50

0.75

0

1.00

0

0.25

0.50

0.75

1.00

X/φ

X/φ

Figure 21. Pressure coefficient along the wall. M= 23.89, altitude h= 71.73 km, V∞ = 7049 m/s

Figure 20. Pressure coefficient along the wall. M= 20.09, altitude h= 63.60 km, V∞ = 6223 m/s

2.0

2.0

Non catalytic Finite rate Fully catalytic

1.5

Cp

1.5

Cp 1.0

1.0

0.5

0.5

0

0

0

0.25

0.50

0.75

0

0.25

0.50

0.75

1.00

1.00

X/φ

X/φ

Figure 22. Pressure coefficient along the wall. M= 25.96, altitude h= 79.90 km, V∞ = 7360 m/s

Figure 23. Variation of pressure coefficient with Mach number. Catalytic model used: finite rate chemistry; — —M= 9.06, - - M=14.71, _____ M=20.09, __ - - __ M=23.89, - - - M=25.89.

Fig. 23 indicates that the Cp is also very weakly affected by Mach number.

VI.

Conclusions

A detailed investigation of nonequilibrium real-gas effects induced by a hypersonic flow in the forebody region of a typical re-entry vehicle has been carried out. A finite rate chemistry model based on Scott model has been implemented and, in order to investigate the relevance of the catalysis model, simulations with non-catalytic as well as those with full catalytic wall conditions have been carried out. The method here developed has been used to compute the detailed features of the hypersonic flow around the forebody of a re-entry vehicle; for our purposes the Orbital Re-entry Experiment (OREX) vehicle, launched from Japan in 1994, has been chosen as test case. Numerical simulations have been performed to reproduce the forebody heating environment of this capsule covering a wide portion of the flight path and the computed flow field features have been compared to experimental results. The computed shear stress coefficients show a growing dependence from the catalysis model with increasing Mach number, while the shear stress coefficients calculated along the profile seem to indicate independence on the catalysis model and/or Mach number. Good agreement has been found between flight data and computed stagnation heat flux; however differences in the estimates of different finite rate chemistry models with increasing Mach number indicate the need for further understanding of the chemical models used by means of parametric studies, of the implementation of different catalysis model and perhaps the possibility to account for radiation effects.

14 American Institute of Aeronautics and Astronautics

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