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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–69

http://d0022-40

n CorrE-m

journal homepage: www.elsevier.com/locate/jqsrt

Flow-radiation coupling for atmospheric entriesusing a Hybrid Statistical Narrow Band model

Laurent Soucasse a,n, James B. Scoggins a,b, Philippe Rivière b, Thierry E. Magin a,Anouar Soufiani b

a von Karman Institute for Fluid Dynamics, B-1640 Sint-Genesius-Rode, Belgiumb Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, F-92295 Châtenay-Malabry Cedex, France

a r t i c l e i n f o

Article history:Received 23 July 2015Received in revised form12 April 2016Accepted 12 April 2016Available online 19 April 2016

Keywords:Atmospheric entryNonequilibrium plasma radiationNarrow-band modelFlow-radiation coupling

x.doi.org/10.1016/j.jqsrt.2016.04.00873/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail address: l.soucasse@imperial.ac.uk (L. Sou

a b s t r a c t

In this study, a Hybrid Statistical Narrow Band (HSNB) model is implemented to make fastand accurate predictions of radiative transfer effects on hypersonic entry flows. The HSNBmodel combines a Statistical Narrow Band (SNB) model for optically thick molecularsystems, a box model for optically thin molecular systems and continua, and a Line-By-Line (LBL) description of atomic radiation. Radiative transfer calculations are coupled to a1D stagnation-line flow model under thermal and chemical nonequilibrium. Earth entryconditions corresponding to the FIRE 2 experiment, as well as Titan entry conditionscorresponding to the Huygens probe, are considered in this work. Thermal none-quilibrium is described by a two temperature model, although non-Boltzmann distribu-tions of electronic levels provided by a Quasi-Steady State model are also considered forradiative transfer. For all the studied configurations, radiative transfer effects on the flow,the plasma chemistry and the total heat flux at the wall are analyzed in detail. The HSNBmodel is shown to reproduce LBL results with an accuracy better than 5% and a speed upof the computational time around two orders of magnitude. Concerning molecularradiation, the HSNB model provides a significant improvement in accuracy compared tothe Smeared-Rotational-Band model, especially for Titan entries dominated by opticallythick CN radiation.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Spacecraft may undergo severe convective and radia-tive heating during atmospheric entry at high velocitiesfrom the surrounding aerothermodynamic environment.Its accurate prediction during the design of such vehicles istherefore paramount for the success and safety of futureplanetary missions. In particular, the radiative heat trans-fer in the shock layer ahead of the vehicle is known tosignificantly alter the aerothermodynamic environment,especially early in the entry when the velocity is high and

casse).

the density of the atmosphere is low. The numericalsimulation of hypersonic reactive plasma flows coupledwith radiative heat transfer is an active research topic forthe design of thermal protection systems of future spacemissions. Past numerical investigations have shown sev-eral major coupling effects including (i) radiative cooling ofthe shock layer due to the strong emission of the plasma[1–3], (ii) the production of precursor chemical com-pounds ahead of the shock [4], and (iii) the promotion ofablation products released by the heat shield which may inturn contribute to increased radiation blockage in theboundary layer [5]. At high altitudes corresponding to lowdensities, the need to consider detailed nonequilibriumradiation appeared since the middle of the 1980s [6] andthermodynamic and chemical nonequilibrium flowfield

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–6956

solvers, coupled to radiative transfer, became common inthe 1990s [7,8].

The numerical simulation of radiative transfer is achallenging problem because of the spatial, angular, andspectral dependence of the radiation field. The referenceapproach for treating the spectral dependence is the Line-By-Line (LBL) method which consists in finely discretizingthe radiative properties over the relevant spectral range.These radiative properties depend on level populationsand on fundamental spectroscopic data gathered in spec-tral databases such as NEQAIR [9,10], SPRADIAN [11],MONSTER [12], SPECAIR [13]. In the present study, we usethe HTGR database (High Temperature Gas Radiation),which has been previously developed [14–18] for O2–N2

and CO2–N2 plasma applications. This database gathersup-to-date atomic spectroscopic data from various sources(such as NIST [19] and TOPbase [20]) together with abinitio calculations of diatomic molecular spectra andatomic line shapes. It includes bound–bound atomic andmolecular transitions, bound-free transitions resultingfrom various mechanisms, and free–free transitions. Thecovered spectral range is [1000–200,000 cm�1] and thetargeted maximum temperature is 30,000 K. The HTGRdatabase has been used in several studies for LBL radiativetransfer calculations in hypersonic entries. In particular,Lamet et al. [21] performed uncoupled radiation simula-tions of the FIRE II flight experiment using a two-temperature approach to model the thermal none-quilibrium. More recently, Lopez et al. [22] carried outcoupled flow-radiation simulations of the relaxationbehind a shock wave in Air with a consistent state-to-statemodeling of the atomic electronic levels.

A full LBL closely coupled flowfield-radiation model hasbeen developed by Feldick et al. [23] for Earth hypersonicreentries. They used the tangent slab approximation andintroduced optimized variable wavelength steps todecrease the computational costs. The full LBL simulationswere successfully compared to a hybrid line-by-line-graymodel where molecular radiation in optically thin systemswas assumed to be gray inside narrow bands. However,although the LBL method is very accurate, the largenumber of radiative transitions that have to be taken intoaccount makes it very computationally expensive andimpractical for coupled simulations in complex geome-tries. The Smeared-Rotational-Band (SRB) model is acommon way to simplify the calculation of molecularradiation but its accuracy is restricted to small opticalthicknesses. It has been used for instance in Ref. [24],together with a LBL treatment of atomic radiation.

More sophisticated approaches for radiative propertymodeling include the k-distribution methods which arebased on the distribution functions of the absorption coef-ficient over the whole spectrum (see e.g. [25]) or overspectral narrow bands [26]. They have been widely used formodeling IR radiation in the field of atmospheric physics orfor combustion applications, but also for modeling visible,UV and VUV radiation of astrophysical (Opacity DistributionFunction model of Ref. [27]) or thermal [28] plasmas.Recently such models have been developed in the frame-work of hypersonic nonequilibrium flows for air mixtures[29,30] and also in carbonaceous atmospheres [31]. They

are based on the Full-Spectrum Correlated-k approach(FSCK) previously developed for IR applications [32] anduse efficient tabulations and rescaling of the variousrequired distribution functions against temperatures,molecular electronic state populations, and a typical Starkwidth of the atoms. The accuracy of these approaches wasdemonstrated by successful comparisons with LBL results.In the case of carbonaceous atmospheres [31], where onlythree non-overlapping molecular band systems are con-sidered, such an approach is very efficient and easy toimplement. Moreover, it retains a description of radiativeproperties in terms of absorption coefficients and is there-fore applicable to any radiation solver. For more arbitrarygas mixtures, including for instance ablation products, alarge number of overlapping, non-weak molecular electro-nic systems, absorbing in the Voigt regime, and whoseinduced emission contribution might not be negligible,have to be accounted for. In this case, the multi-scale MS-FSCK approach may become quite tedious to implement.Moreover, the spectral information is completely lost whenusing such full-spectrum approaches. This is not an intrinsiclimitation if one is only interested in heat transfer with graywalls, but such models do not enable comparisons withexperiments done in limited spectral ranges.

Recently, Lamet et al. [33] have developed a HybridStatistical Narrow Band (HSNB) model, which combines aStatistical Narrow Band (SNB) model for optically thickmolecular systems with a box model for optically thinmolecular systems and continua, and a LBL description ofatomic line radiation. Band parameters have been com-puted using the HTGR database and tabulated againsttranslational–rotational and vibrational temperatures. TheHSNB model can easily include new radiating species andelectronic systems, using the uncorrelation assumptioninside narrow bands, and arbitrary electronic populationsmay be specified. Also, it can be applied to predict theradiative flux in the case of non-gray walls. Nevertheless, itmight be less computationally efficient than k-distributionmethods.

This study aims at showing the ability of the HSNBmodel to predict accurately and efficiently coupled radia-tion effects on hypersonic entry flows. For this purpose, a1D stagnation-line flow model for blunt, hypersonic vehi-cles [34] has been coupled with a newly developed radia-tive transport code using the HSNB formulation. Earth entryconditions corresponding to the FIRE 2 experiment, as wellas Titan entry conditions corresponding to the Huygensprobe, are considered. Thermal nonequilibrium is describedby a two temperature model, although non-Boltzmanndistributions of electronic levels provided by a Quasi-Steady State model are also considered for radiative trans-fer. Section 2 details the 1D model for coupled flow andradiation along the stagnation-line of the vehicle. The HSNBmodel [33] is presented in Section 3 with new additionalfeatures. The results are finally discussed in Section 4.

2. Governing equations

We consider a plasma flow constituted of atoms,molecules and free electrons under chemical and thermal

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–69 57

nonequilibrium conditions. The thermal state of theplasma is described according to a two-temperature modelin which the translation of heavy species and rotation ofmolecules are assumed to follow a Boltzmann distributionat the temperature T, and the translation of electrons,vibration of molecules, and electronic excitation of heavyspecies are assumed to follow a Boltzmann distribution atthe temperature Tve. The energy of a species s per unitmass is then defined by

esðT ; TveÞ ¼ etrs ðTÞþeves ðTveÞþefs ; ð1Þwhere etrs , eves , and es

fare the translation–rotational, vibra-

tion-electronic-electron, and formation energies respec-tively, for species s. The rigid-rotor and harmonic-oscillator models are used to describe rotational andvibrational modes when computing species thermo-dynamic quantities, while species electronic energies arecomputed from the electronic energy levels taken fromRef. [35].

The pressure p of the plasma is modeled using theperfect gas law

p¼XsAH

ρsrsTþρereTve; ð2Þ

where ρs is the species mass density, rs the perfect gasconstant per unit mass of species s, H is the set of heavyspecies and the subscript e refers to free electrons.

2.1. Stagnation-line flow modeling

We follow the methodology of Ref. [36] for deriving a 1Dplasma flow model along the stagnation-line of a sphericalbody (see Fig. 1). The problem is considered symmetricaccording to the azimuth angle ϕ and the velocity compo-nent uϕ is set to zero. Mass fractions and temperatures areassumed to be only a function of the radius r

ys ¼ ys rð Þ; T ¼ T rð Þ; Tve ¼ Tve rð Þ; ð3Þwhile velocity and pressure are split into radial and tan-gential components following

ur ¼ ur rð Þ cos θ; uθ ¼ uθ rð Þ sin θ; p�p1 ¼ p rð Þ cos 2θ;ð4Þ

where the pressure is assumed to follow a Newtonianapproximation [37]. Introducing this decomposition intothe 3D Navier–Stokes equations in spherical coordinatesand then taking the limit when θ tends to zero leads to a setof 1D equations for the stagnation-streamline quantities of

Fig. 1. Spherical body of radius R0 subjected to a hypersonic flow at u1 .Azimuth and zenith angles are ϕ and θ, respectively.

the form

∂U∂t

þ∂Fc

∂rþ∂Fd

∂r¼ ScþSdþSkþSrad: ð5Þ

U is the conservative variable vector. Fc and Fd are theconvective and diffusive fluxes. Sc and Sd are convective anddiffusive source terms resulting from the expansion alongthe stagnation-line. Sk is the kinetic and energy transfersource term vector and Srad is the radiative source termvector. Specifically, these vectors are written as

U¼ ρs; ρur ; ρuθ ; ρE; ρeve� �T

; ð6Þ

Fc ¼ ρsur ; ρu2r þp; ρuruθ ; ρurH; ρureve

� �T; ð7Þ

Sc ¼ �ðurþuθÞr

2ρs; 2ρur ; 3ρuθ�2p�p1urþuθ

; 2ρH; 2ρeve� �T

;

ð8Þ

Fd ¼ jrs; �τrr; �τrθ ; qr�τrrur ; qver� �T

; ð9Þ

Sd ¼ �1r2jrs; 2ðτθθ�τrrþτrθÞ; τθθ�3τrθ ;�

2ðqr�τrrur�τrθur�τθθuθÞ; 2qver�T; ð10Þ

Sk ¼ _ωks ; 0; 0; 0; Ωve

h iT; ð11Þ

Srad ¼ _ωrads ; 0; 0; Prad; Prad;ve

h iT; ð12Þ

where the overline symbol introduced in Eqs. (3) and (4) todesignate stagnation-line quantities has been omitted forthe sake of clarity. The total energy and the vibrational-electronic-electron energy per unit volume are defined byρE¼Psρsesþρu2

r =2 (only the radial velocity componentcontributes to the kinetic energy due to the ansatz Eq. (4))and ρeve ¼Psρse

ves . Note that the formation energy is not

included in eve according to the definition given in Eq. (1).This choice is important for the source terms describedbelow. The total enthalpy is defined by H¼ Eþp=ρ.

The radial species diffusion fluxes jrs are obtained bysolving a simplified Stefan–Maxwell equation for heavyspecies [38]. The electron diffusion flux is computed fromthe ambipolar assumption that makes the electric currentequal to zero. The radial heat fluxes are defined by

qr ¼Xsjrshs�λtr

∂T∂r

�λve∂Tve

∂r; ð13Þ

qver ¼Xs

jrshves �λve

∂Tve

∂r; ð14Þ

where λtr and λve are the thermal conductivities of theenergy modes in equilibrium with the temperatures T andTve respectively, hs ¼ esþps=ρs, hves ¼ eves for sae andhvee ¼ evee þpe=ρe. The components of the viscous stresstensor τrr, τrθ and τθθ are given by

τrr ¼43μ

∂ur

∂r�urþuθ

r

� �; τrθ ¼ μ

∂uθ∂r

�urþuθr

� �; τθθ ¼ �1

2τrr ;

ð15Þ

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–6958

where μ is the viscosity.In Eq. (11) _ωk

s are the collisional species mass produc-tion rates. Reaction rate constants are assumed to followan Arrhenius law. Forward reaction constants are takenfrom the literature depending on the mixture considered.Backward reaction constants are computed such thatequilibrium relations are satisfied [39]. The source termΩve is written as

Ωve ¼ �pe∂ur

∂rþ2

urþuθ∂r

� �þXsAM

ρsev0s �evsτVTs

þρeet0e �eteτET

þXseves _ωk

s �XpAR

ΔHp _χ kp: ð16Þ

The first term on the right-hand-side of Eq. (16) representsthe internal work done by the electron pressure. The sec-ond term represents vibration-translation energyexchange where ev0s and evs are vibrational energies ofspecies s at the temperature T and Tve, respectively. Theassociated relaxation time τs

VTis computed from the

experimental data of Ref. [40] taking into account the hightemperature correction from Ref. [41]. The third termcorresponds to electron-heavy translation energyexchange where et0e and ete are translational energies ofelectrons at the temperature T and Tve, respectively. Theexpression of the relaxation time τET is given in Ref. [42].The fourth term represents the coupling between chem-istry and vibrational-electronic energy. Finally, the fifthterm accounts for the energy removed from the electronbath due to the set R of electron impact ionization anddissociation reactions, where _χ k

p is the molar rate of pro-gress and ΔHp ¼

Psνspe

fs=Ms is the chemical heat released

per unit mole (νsp is the stoichiometric coefficient forreaction p and Ms is the molar mass of species s) of reac-tion p. The set of chemical reactions considered will bespecified for each case in Section 4.

Finally, _ωrads is the radiative mass production rate of

species s, Prad is the total radiative energy source term andPrad;ve is the radiative energy source term for the energymodes in equilibrium with Tve.

2.2. Radiative source terms

For computing the radiative source terms of Eq. (12),we make use of the 1D tangent slab approximation[43,44]. The radiative properties of the medium areassumed to vary only along the stagnation-line directionand are assumed to be constant in planes that extend toinfinity, perpendicular to this direction. It is a reasonableapproximation in this work as we will consider shock layerof which the size is small compared to the radius of thevehicle.

The radiative source terms are functions of the spectralradiative intensity Iσ which is governed by the radiativetransfer equation

cos ϑ∂Iσðr;ϑÞ

∂r¼ ησ rð Þ�κσ rð ÞIσ r;ϑ

� ; ð17Þ

for a non-scattering medium of optical index equal to 1,where ησ and κσ are the emission and absorption coeffi-cients at wavenumber σ, r designates the position along

the stagnation-line and ϑ is the angle made between theoptical path and the stagnation-line.

The radiative mass production rate in Eq. (12) corre-sponds to the production or destruction of a species sduring a bound-free radiative process p (photoionizationAþhcσ⇌Aþ þe� or photodissociation ABþhcσ⇌AþB)

_ωrads rð Þ ¼

Xp

νspMs

N A

Z 1

0

2πκpσ rð Þ R π0 Iσ r;ϑ�

sin ϑ dϑ�4πηpσ rð Þhcσ

dσ;

ð18Þwhere νsp is the stoichiometric coefficient for the boundfree process p and species s (positive for product, negativefor reactant), N A is the Avogadro number, h is the Planckconstant and c the speed of light.

The total radiative energy source term Prad is equal tothe opposite of the divergence of the radiative flux qrad,which gives

Prad rð Þ ¼ �∂qrad rð Þ∂r

; qrad rð Þ ¼ 2πZ 1

0

Z π

0Iσ r;ϑ�

cos ϑ sin ϑ dϑ dσ:

ð19ÞFor computing the source term Prad;ve, we first neglect theradiative energy exchanges with translational and rota-tional modes. Then, we take into account the fact thatduring a photoionization process, a part of the photonenergy goes into the formation energy [22]. This leads to

Prad;veðrÞ ¼PradðrÞ�XpAJ

Δhp _ωradp;e ðrÞ; ð20Þ

where J is the set of photoinization processes, _ωradp;e is the

electron mass production rate for the bound-free processp, and Δhp is the ionization energy per unit mass ofelectron.

2.3. Numerical implementation

When radiation is ignored (Srad ¼ 0) we follow theimplementation of Munafò and Magin [34] for thenumerical solution of Eq. (5) of which we recall the mainfeatures. Equations are discretized in space by means ofthe finite volume method. The Roe scheme [45] is used tocompute the convective flux at cell interfaces. Boundaryconditions, which will be specified in Section 4, areimplemented through ghost cells and are imposed interms of primitive variables. The time integration is fullyimplicit and is performed until steady state is reached. Inorder to advance the solution from the time-level n to thetime-level nþ1, the fluxes and source terms are linearizedaround the solution at the time-level n by means of aTaylor-series expansion where the Jacobians are computednumerically.

When radiation coupling is considered the radiativesource terms are added explicitly to the previous algo-rithm. They are not computed at each flow time step butare typically updated every 200 flow time steps. Startingfrom an uncoupled solution, time integration is performeduntil steady state is reached. In order to reduce the com-putational time, radiative calculations are carried out on acoarse mesh in which the convective cells are grouped byfive based on an extensive grid convergence analysis. Theangular integration of the intensity field is achieved by

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–69 59

computing the intensity at each point for 20 regularlyspaced cos ϑ values between �1 and 1. This relativelycoarse angular discretization was found sufficiently accu-rate for the planar geometry associated with the tangentslab approximation. The spectral integration over thewavenumber σ in Eqs. (18) and (19) is achieved using theHybrid Statistical Narrow Band model, which will bedetailed in Section 3. Finally, radiative source terms arelinearly interpolated from the cell centers of the radiationmesh to the cell centers of the flow mesh.

3. The Hybrid Statistical Narrow Band model forradiative transfer

The spectral radiative intensity at an arbitrary point s ofan optical path starting at point s¼0 is given by thesolution of the radiative transfer equation, such that

Iσ sð Þ ¼ Iσ 0ð Þτσ 0; sð ÞþZ s

0ησ s0ð Þτσ s0; sð Þ ds0; ð21Þ

where τσ s0; sð Þ ¼ exp � R ss0 κσ s00ð Þ ds00� is the spectral trans-

missivity between points s0 and s. We search for an

expression of the averaged intensity IσðsÞΔσ over a spectralnarrow band Δσ. First, the radiative mechanisms aregrouped into different contributions: e.g. a molecularelectronic system, a set of atomic lines, or a continuumprocess. These contributions are assumed to be statisticallyuncorrelated, which allows us to write

Iσ sð ÞΔσ ¼ Iσ 0ð ÞΔσ∏kτkσ 0; sð ÞΔσ

þXk

Z s

0ηk s0ð Þτkσ s0; sð ÞΔσ ∏

k0 akτk

0σ s0; sð ÞΔσ ds0; ð22Þ

where the index k refers to a radiative contribution.Numerical tests have shown that the uncorrelationassumption between atomic lines and molecular lines wasvalid with an accuracy of about 1%. Note that we alsoassume in Eq. (22) the mean intensity at the starting pointof the path (s¼0) to be uncorrelated with the totaltransmissivity as we will only consider gray radiationleaving the boundaries of the domain.

For the evaluation of the term ηkσðs0Þτkσðs0; sÞΔσ

in Eq.(22), we use different procedures which are presented inthe following subsections: (i) a SNB model for opticallythick molecular systems; (ii) a box model for optically thinmolecular systems and continua; (iii) a LBL treatment foratomic lines. When all contribution types are included, theresulting method is named the Hybrid Statistical NarrowBand (HSNB) model.

The criterion retained to decide whether a molecularsystem is thick or thin is based on the maximum value ofthe optical depth κσ l for a plasma at thermodynamicequilibrium with T¼8000 K, p¼2 atm and l¼10 cm. If themaximum value of κσ l is greater than 0.1, the molecularsystem is considered as thick.

3.1. Statistical narrow band model for optically thick mole-cular systems

For an optically thick molecular system, the emissioncoefficient and transmissivity are strongly correlated.However, the ratio ησ=κσ can be considered uncorrelatedto the transmissivity τσ , allowing us to write

ηkσðs0Þτkσðs0; sÞΔσ ¼ ηkσðs0Þ

κkσðs0Þ∂τkσðs0; sÞ

∂s0

Δσ

Cηkσðs0Þκkσðs0Þ

Δσ∂τkσðs0; sÞ

∂s0

Δσ

:

ð23ÞThis assumption is valid at thermal equilibrium where theratio ησ=κσ is equal to the Planck function which is nearlyconstant within a narrow band. For thermal none-quilibrium conditions, Lamet et al. [33] checked that thisassumption remains satisfactory for atmospheric entryflow applications. Discretizing the optical path intohomogeneous cells of size Δsi ¼ siþ1�si, the contributionof optically thick molecular systems to the mean intensityis thus written as

Ithickσ ðsjÞΔσ

¼XkAT

Xj�1

i ¼ 0

τkσðsiþ1; sjÞΔσ�τkσðsi; sjÞ

Δσ� �

ηkσκkσ

Δσ

i

∏k0ASk0ak

τk0σ ðs�i ; sjÞ

Δσ;

ð24Þ

where T is the set of optically thick molecular systems andS is the set of all the systems. A mean equivalent point s�i isintroduced to simplify the spatial integration between siand siþ1, such that

τkσðs�i ; sjÞΔσ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτkσðsiþ1; sjÞ

Δστkσðsi; sjÞ

Δσr

: ð25Þ

This pragmatic choice does not cause any significant loss ofaccuracy.

From statistical assumptions concerning the intensityand the position of the lines within a narrow band Δσ, theSNB model [46] provides an expression for the meantransmissivity of a homogeneous column of the form

τσðlÞΔσ ¼ 1

Δσ

ZΔσ

exp �κσ lð Þ dσ ¼ exp �Wδ

!; ð26Þ

where δ is the mean spacing between the line positions

within Δσ and W is the mean black equivalent width ofthese lines. The mean black equivalent width can be set asa function of three band parameters: the mean absorption

coefficient κσΔσ of the absorbing species and two over-

lapping parameters βDΔσ

and βLΔσ

related to Doppler andLorentz broadening. For addressing non-homogeneousoptical paths, both the Curtis–Godson and the Lindquist–Simmons approximations have been considered. TheLindquist–Simmons approximation is known to be moreaccurate but also more computationally expensive com-pared to Curtis–Godson. Numerical tests have shown thatthe precision of the Lindquist–Simmons approximationwas required during coupled calculations because it pre-vents numerical instability in the free-stream region.Details are given in Appendix A.

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–6960

The required band parameters for radiative transfer

calculations (ησ=κσΔσ

, κσΔσ , βLΔσ

, βDΔσ

) have been cal-culated in Ref. [33] for thick molecular systems of air andin Ref. [47] for thick molecular systems in Martian atmo-spheres. They have been tabulated according to two tem-peratures T and Tve for 199 spectral bands of constant sizeΔσ ¼ 1000 cm�1 in the range [1000–200,000 cm�1]. Asexplained by in Ref. [33], the parameters can be convertedfor treating arbitrary electronic level populations as eachelectronic molecular system is treated independently.

3.2. Box model for optically thin molecular systems andcontinua

If a molecular system k is optically thin for allwavenumbers σ (κkσ l⪡1, l being a typical length of theproblem), the mean transmissivity over a spectral narrowband can be simply expressed by

τkσðs0; sÞΔσ ¼ exp � R ss0 κkσðs″ÞΔσds00

� �. In addition, the cor-

relation between the emission coefficient and the trans-missivity is weak such that one can write

ηστσΔσCησ

ΔστσΔσ . These two simplifications also holdfor continua because of their weak spectral dynamics.

Therefore, the contribution to the mean intensity of theset of optically thin molecular systems and continua, B, isthen written as

Iboxσ ðsjÞΔσ

¼XkAB

Xj�1

i ¼ 0

ηkσi

Δσ∏

k0ASτkσ ðs�i ; sjÞ

ΔσΔsi:

ð27Þ

Band parameters (ηkσΔσ

, κkσΔσ

) for optically thin mole-cular systems have been tabulated according to two tem-peratures T and Tve. As for thick molecular systems, theycan be converted for treating arbitrary electronic levelpopulations.

Band parameters for continua have been tabulatedaccording to one temperature (Tve for free–free processes,photoionization and photodetachment, and T for O2 pho-todissociation). For bound-free processes, three para-meters have been tabulated in order to treat chemical non-equilibrium between the species involved in the process:

the spontaneous emission coefficient ηeqσΔσ

, the true

absorption coefficient κabsσΔσ

and the induced emission

coefficient κie;eqσ

Δσ. For example, for a photoionization

process ðAþhcσ⇌Aþ þe� Þ, the spontaneous and inducedemission coefficients have been tabulated according to thepartial pressure of species A, assuming chemical equili-brium between A and Aþ concentrations. Under chemicalnonequilibrium conditions, the actual radiative propertiescan be retrieved according to

ηkσΔσ ¼ ηk;eqσ

Δσχneq; ð28Þ

κkσΔσ ¼ κk;absσ

Δσ�κk;ie;eqσ

Δσχneq; ð29Þ

χneq ¼ nAþ ne�

nA

QA

2QAþ ξexp

EionkBT

ve

� �; ð30Þ

where ns is the number density of species s, Qs is theelectronic partition function of species s, ξ is the volu-metric translational partition function of free electrons,Eion is the ionization energy of species A and kB is theBoltzmann constant. Similar relations can be derived forphotodissociation processes.

For both continua and thin molecular systems, the sizeof the spectral bands and the spectral range are the sameas for thick molecular systems.

3.3. Line-by-line treatment of atomic lines

For atmospheric entry applications, many atomicradiative transitions are optically thick. Attempts havebeen made to derive a SNB model for atoms but the resultswere not sufficiently accurate due to the weak spectraldensity of atomic lines [33]. Therefore, the contribution ofatomic lines to the mean intensity is treated in a LBLmanner,

Iatσ ðsjÞΔσ

¼Xj�1

i ¼ 0

ηatσκatσ ∣i τatσ ðsiþ1; sjÞ�τatσ ðsi; sjÞ

� Δσ

� ∏k0A Sk0 a at

τk0σ ðs�i ; sjÞ

Δσ; ð31Þ

where the first average is computed exactly over thenarrow band.

High resolution atomic radiative properties are com-puted according to

ησ ¼Xul

nuAul

4πf seul σ�σulð Þhcσ; ð32Þ

κσ ¼Xul

nlBlufaulðσ�σulÞ�nuBulf

ieulððσ�σulÞÞ

h ihσ; ð33Þ

where Aul, Bul and Blu are the Einstein coefficients relatedto spontaneous emission, induced emission and absorp-tion of the transition u-l, nu and nl are the number den-sities of the upper and lower levels and σul is the wave-number of the transition, ful

se, ful

aand ful

ieare the line profiles

associated to spontaneous emission, absorption andinduced emission, respectively. The line shapes are relatedto one another to retrieve equilibrium at Tve [46]

f ieul σ�σulð Þ ¼ f seul σ�σulð Þ σul

σ

� 3; ð34Þ

f aul σ�σulð Þ ¼ f seul σ�σulð Þ σul

σ

� 3exp

hcðσ�σulÞkBT

ve

� �: ð35Þ

A Voigt profile is considered for the spontaneous emissionline shape ful

se. Einstein coefficients are taken from the NIST

database and collisional broadening data are taken fromRefs. [15,48].

The LBL treatment of atomic lines is not too penalizingbecause of the small number of atomic lines (of the orderof few thousand) compared to the number of molecularlines (of the order of several million). Furthermore, thespectral grid dedicated to atomic radiation can be muchsmaller as compared to the LBL spectral grid including allradiative contributions. For this work, an adaptive spectralgrid which combines the 11 point stencil per line proposed

Table 1Conditions for the trajectory points t¼1634 s and t¼1642.66 s of the FIRE2 experiment and for the trajectory point t¼191 s of the Huygens probeentry. Radius of the vehicle R0, wall temperature Tw and free streamconditions (temperature T1 , velocity u1 , total mass density ρ1 and mass

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–69 61

in Ref. [49] and a refinement procedure between two linesin order to accurately capture the far wing regions hasbeen implemented. More details regarding this procedureare given in Appendix B.

fractions y1).

Case FIRE 2(1634 s)

FIRE 2(1642.66 s)

Huygens(191 s)

R0 (m) 0.935 0.805 1.25Tw (K) 615.0 480.0 1000.0T1 (K) 195.0 273.0 183.0u1 (km/s) 11.36 10.56 4.788ρ1 (kg/m3) 3:72� 10�5 7.17 �10�4 3:18� 10�4

y1 (%) 77.0–23.0(N2–O2)

77.0–23.0(N2–O2)

98.84–1.16(N2–CH4)

4. Results

Three hypersonic entry conditions have been studied:two conditions of Earth entry corresponding to the tra-jectory points t¼1634 s and t¼1642.66 s of the FIRE2 experiment and one condition of Titan entry corre-sponding to the trajectory point t¼191 s of the Huygensprobe entry. The FIRE 2 experiment has been the subject ofmany numerical studies because of the availability of flightdata [50,51]. In addition, the study of the first trajectorypoint t¼1634 s allows to investigate strong thermalnonequilibrium effects. The Titan test case has been cho-sen to show strong molecular radiation effects comingfrom the CN-Violet system [39,52].

Boundary conditions for the numerical simulations aregiven in Table 1. The wall of the vehicle is assumed to benon-ablative, non-catalytic and isothermal at T ¼ Tve ¼ Tw

and a no slip condition for the velocity is prescribed. In thefree stream, temperature, velocity and mass densities areimposed. For radiation, we assume that the boundaries ofthe computational domain are black walls at Tw and T1.

For Earth entries, a mixture of 11 species (e� , N, Nþ , O,Oþ , NO, N2, N2

þ, O2, O2

þ, NOþ) is considered. Chemical

reactions and rates are taken from Ref. [53]. The radiativesystems taken into account are listed in Table 2. For pho-todetachment processes, the computation of N� and O�

concentrations are based on a chemical equilibriumassumption with N and O. For Titan entries, a mixture of 13species (N, C, H, N2, C2, H2, CN, NH, CH, CH2, CH3, CH4,HCN) at thermal equilibrium is envisaged. For the trajec-tory point considered, thermal nonequilibrium effects areweak and ionization is insignificant. Chemical reactionsand rates are taken from Ref. [54]. Finally, all radiativesystems taken into account are listed in Table 2.

4.1. Accuracy and efficiency of the HSNB model

In order to assess the accuracy and the efficiency of theHSNB model, a comparison with the rigorous Line-By-Line(LBL) method is carried out. In LBL calculations, radiativeproperties of the plasma are computed from the spectro-scopic HTGR database [18] on a high resolution spectralgrid of 4:4� 106 points in order to capture correctly all theatomic and molecular lines.

It is also interesting to compare the HSNB model withthe Smeared-Rotational-Band model, which is often usedas a simple model to treat molecular radiation but maylead to a strong overestimation of radiative fluxes. For thispurpose, we implemented a model similar to theSmeared-Rotational-Band that will be called hereafterHSNB-Weak. It consists in computing the mean transmis-sivity of thick molecular systems according to τkσ s0; sð ÞΔσ

¼ exp � R ss0 κkσ s″ð ÞΔσ ds00� �

instead of Eq. (26).For the three entry conditions described previously

(FIRE 2 (1634 s, 1642.66 s) and Huygens (191 s)),

calculations have been performed with the LBL, HSNB-Weak and HSNB models from the same flowfield corre-sponding to the coupled result obtained with the HSNBmodel. Table 3 gives the incident radiative flux at the wall,together with the total computational time for one radia-tion calculation, for the different combinations of modelsand test cases. Compared to the reference LBL solutions,the HSNB model provides an accurate prediction of theincident radiative flux, with an error between 3% and 5%and a speed up factor around 80 for the computationaltime. Most of the computational gain comes from thecalculation of LBL molecular spectra which is very expen-sive due to the large number of molecular lines.

From Table 3, the HSNB-Weak model provides reason-ably accurate results for Earth entry with a difference of3.5% and 4.5% for the two trajectory points. However, forthe Titan entry case, the incident radiative flux is over-predicted by 26%. These are expected results based on aprevious assessment of Smeared-Rotational-Band modelsin Ref. [24]. Concerning the computational times, theHSNB-Weak model is 5 times faster than the HSNB modelfor Earth entry cases. The Lindquist–Simmons approx-imation used for computing mean transmissivities overnon-homogeneous paths for thick molecular systems (seeSection 3.1 and Appendix A) is responsible for the lowercomputational efficiency of the HSNB model.

The spectral and cumulated incident radiative fluxes atthe wall are displayed in Fig. 2 for the early trajectory point(1634 s) of the FIRE 2 experiment. It can be seen that thecomplex structure of the LBL spectral flux is correctlycaptured by both HSNB and HSNB-Weak models, with agood agreement on the total cumulated flux (see Table 3for numerical values). The incoming radiation mostlyarises from molecular and atomic transitions in theVacuum Ultraviolet. The accuracy of the HSNB modelshould also be assessed regarding the total radiativeenergy source term along the stagnation line. Fig. 2 alsoshows this distribution together with the difference withLBL calculations normalized by the absolute maximumvalue of the total radiative source term. The differences donot exceed 5% for both HSNB and HSNB-Weak models. Thehighest discrepancies are located near the shock position,where the radiation emission is at a maximum.

Table 2Radiative systems considered for Earth and Titan entries.

Earth entriesAtomic lines N, Nþ , O, Oþ

Thick molecularsystems

N2 (Birge–Hopfield 1 and 2, Worley–Jenkins,Worley, Caroll–Yoshino), O2 (Schumann–Runge), NO (β, β0 , γ, γ0 , δ, ε)

Thin molecularsystems

N2 (first and second positive), NO (11,000 Å,infrared), N2

þ(first and second negative,

Meinel)Bound-free processes Photoionization (N, O, N2, O2, NO), photo-

dissociation (O2), photodetachment(N� , O�)

Free-free processes N, O, Nþ , Oþ , N2, O2

Titan entriesThick molecularsystems

N2 (Birge–Hopfield 1 and 2, Worley–Jenkins, Worley, Caroll–Yoshino), CN violet,C2 Swan

Thin molecularsystems

N2 (first and second positive), CN (red,LeBlanc), C2 (Philips, Mulliken, Deslandres–D'Azambuja, Ballik and Ramsay, Fox–Herzberg)

Table 3Comparison between LBL, HSNB-Weak and HSNB models for FIRE 2(1634 s), FIRE 2 (1642.66 s) and Huygens (191 s) cases. Incident radiativefluxes and computational times for one radiation calculation.

FIRE 2 (1634 s) LBL HSNB-Weak HSNB

qrad;iw (W/cm2) 146.78 151.94 150.85

tCPU (s) 20,480 41 242

FIRE 2 (1642.66 s) LBL HSNB-Weak HSNB

qrad;iw (W/cm2) 553.78 578.81 581.97

tCPU (s) 19,140 51 250

Huygens (191 s) LBL HSNB-Weak HSNB

qrad;iw (W/cm2) 82.68 104.26 86.24

tCPU (s) 13,158 5 105

Fig. 2. FIRE 2 (1634 s). Comparison between LBL, HSNB-Weak and HSNB modradiative source term along the stagnation line and differences with LBL calcula

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–6962

The spectral and cumulated incident radiative fluxes atthe wall and the total radiative source term along thestagnation line are shown for the Titan test case in Fig. 3.While the HSNB model reproduces with a good accuracythe LBL calculation, both the spectral flux and the totalradiative source term are strongly over-predicted by theHSNB-Weak model. This failure comes from an incorrecttreatment of the CN-Violet molecular system in the spec-tral range [25,000–29,000 cm�1]. For this case, the valueof the HSNB model is clearly realized.

An additional assessment of the accuracy of the HSNBmodel is proposed in Appendix C where we have repro-duced the LBL results of Ref. [31] concerning the trajectorypoint t¼189 s of the Huygens probe entry.

4.2. Analysis of radiation effects on flow and heat transfer

In this section, the results of coupled simulationsobtained with the HSNB model are compared to the resultsof uncoupled simulations to show how radiative transferaffects the aerothermodynamic fields and the heat fluxesat the wall of the vehicle.

4.2.1. FIRE 2 (1634 s)The FIRE 2 flight conditions of the early trajectory point

t¼1634 s correspond to a high velocity entry into a lowdensity atmosphere (see Table 1). The plasma flow around thevehicle is then in strong thermal nonequilibrium that may notbe correctly described by multi-temperature models [42].

In order to investigate non-Boltzmann effects, weimplemented for this particular case the Quasi-Steady State(QSS) model proposed by Johnston [55]. For each radiationcalculation, non-Boltzmann populations of electronic levelsof N and O as well as the first electronic levels (X, A, B, C) ofN2 and N2

þare determined from simple correlations

depending on the macroscopic state of the flow (electrontemperature, total number densities). An additionalassumption is made concerning N2 VUV systems (Birge–

els. Left: spectral and cumulated incident fluxes at the wall. Right: totaltions normalized by the maximum absolute value.

Fig. 3. Huygens (191 s). Comparison between LBL, HSNB-Weak and HSNB models. Left: spectral and cumulated incident fluxes at the wall. Right: totalradiative source term along the stagnation line and differences with LBL calculations normalized by the maximum absolute value.

Table 4FIRE 2. Standoff distance δ, conductive flux at the wall qw, radiative flux at thewall qw

radand incoming radiative intensity at the wall Iw over spectral intervals

Δσ1 ¼ 16;667�33;333 cm�1� �

and Δσ2 ¼ 2500�50;000 cm�1� �

.

FIRE 2 (1634 s) Uncoupled Coupled Coupled QSS Flightdata

δ (cm) 5.36 5.05 5.21 –

qw (W/cm2) 94.9 76.3 76.5 –

qwrad

(W/cm2) 203.6 150.0 74.7 –

Iw(Δσ1) (W/cm2/sr) 2.05 2.12 1.28 0.1Iw(Δσ2) (W/cm2/sr) 8.57 7.16 4.71 1.3FIRE 2 (1642.66 s) Uncoupled Coupled Flight data

δ (cm) 4.06 4.01 –

qw (W/cm2) 635.6 617.0 –

qwrad

(W/cm2) 791.3 581.7 –

Iw(Δσ1) (W/cm2/sr) 11.65 9.28 10.5Iw(Δσ2) (W/cm2/sr) 71.01 53.63 63

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–69 63

Hopfield 1 and 2, Worley–Jenkins, Worley, Caroll–Yoshino):the population of the upper energy levels of these transi-tions, which are above the dissociation limit, are computedaccording to a chemical equilibrium assumption with atomicNitrogen. The non-Boltzmann populations are taken intoaccount in the HSNB model using Eqs. (32) and (33) foratomic radiation and using expressions given in Ref. [33] forband parameters of molecular systems. It should be under-lined that this QSS model used for radiation is not consistentwith the flow modeling, though it provides a first approx-imation of thermal non-equilibrium effects. The full con-sistent state-to-state coupling between flow and radiationhas been recently achieved for instance in Ref. [22] foratomic electronic states.

Table 4 compares the uncoupled, coupled, and coupledQSS results concerning the shock standoff distance, the con-ductive flux, and the radiative flux at the wall. The couplingeffect is to decrease all of these quantities due to the radiativecooling associated with the strong radiative emission in theshock layer. The shock layer thickness decreases because the

plasma density increases and the heat fluxes decrease becausethe temperature levels are lower. In the coupled QSS case, thesame trends are obtained but to a lesser extent. Table 4 alsoshows the incoming radiative intensity at the wall over twospecific spectral ranges corresponding to the experimentalflight data, given with an uncertainty of 20%. All uncoupled,coupled, and coupled QSS results are far from the flight data,although the QSS case is much closer than the two others.

Fig. 4 displays temperatures along the stagnation line forthe uncoupled, coupled, and coupled QSS cases. The uncou-pled temperature profile can be split into four regions fromright to left: the free-stream, the shock, the equilibriumplateau and the boundary layer close to the wall. Because ofthe low density, the shock region is wide and in strongthermal nonequilibrium. In the boundary layer, Tve is slightlygreater than T, because of atomic recombination which cre-ates vibrational energy. When radiation is considered (cou-pled case), the temperature distributions are significantlyaffected. The shock layer spreads out and the equilibriumzone is shortened. Radiative cooling lowers the peak andplateau temperatures. In particular, the maximum of Tve

decreases from 14,670 K (uncoupled) to 13,470 K (coupled).Another interesting feature is that the free-stream region isno longer at thermal equilibrium because the radiationabsorption from the shock increases electronic and vibra-tional energy.

Fig. 4 also shows the species molar fractions along thestagnation line. For the uncoupled case, the main chemicalmechanisms are the dissociation of molecular nitrogenand oxygen and the ionization of atomic nitrogen andoxygen through the shock. A significant amount of Nitro-gen monoxide is also produced in the shock region. Theionization level is quite important in the plateau as theelectron molar fraction reaches 0.15. In the boundary layer,the ionization level drops down and atomic nitrogen startsrecombining. For the coupled case, the fall of the twotemperatures slows down the ionization reactions and theelectron molar fraction reaches a maximum of 0.08. Thefree-stream region ahead of the shock becomes chemicallyreacting under the effect of radiation: atomic oxygen is

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–6964

produced by photodissociation and electrons are producedmainly by photoionization of molecular oxygen.

These coupling effects on temperature and compo-sition are also noticeable in Fig. 4 for the coupled QSScase, however they are much weaker. In order tounderstand this behavior, the radiative source termalong the stagnation line is plotted in Fig. 5 and splitaccording to the atomic, molecular, and continua emis-sion contributions. From uncoupled to coupled calcula-tions, the peak of the radiative source term is decreasedby a factor two due to radiative cooling. From coupled tocoupled QSS calculations, the peak of the radiativesource term is further reduced and atomic contributionalmost vanishes at the shock location. The reason is thatthe Tve Boltzmann distribution leads to an over-estimation of the population of the highest electronicenergy levels of atomic N and O, as well as N2 and N2

þ. In

particular, for the N2 VUV systems, the dissociationequilibrium assumption makes the population of theupper electronic energy level associated with thesesystems close to zero and thus cancels their contributionto radiation. The remaining molecular emission peak inFig. 5 comes mostly from NO radiation, of which elec-tronic energy levels are assumed to be populatedaccording to the temperature Tve. An incorrect treatmentof the thermal state of NO might be responsible of theremaining discrepancies between the coupled QSSresults and the flight data.

4.2.2. FIRE 2 (1642.66 s)The FIRE 2 flight conditions of the trajectory point

t¼1642.66 s correspond to a quasi-thermal equilibriumsituation. Due to the higher density, the kinetic energytransfer between the internal energy modes is much fasterthan for the trajectory point t¼1634 s. Thus, the QSSmodel has not been considered in this case.

As it can be seen in Table 4, the standoff distance andthe conductive flux slightly decrease and the radiative fluxdiminishes by around 25% when radiation is coupled to theflow. In addition, coupled results are in fair agreementwith flight data which are given with an uncertainty of

Fig. 4. FIRE 2 (1634 s). Temperatures and co

20%. Temperatures and composition along the stagnationline are shown in Fig. 6. For the uncoupled case, the flow isin thermal equilibrium everywhere except in the shockregion. A chemical equilibrium zone with flat molar frac-tion profiles is clearly distinguishable between theboundary layer and the shock. The electron molar fractionis around 0.07 in this zone. When radiation is considered,as for the previous trajectory point, the vibration-electronic-electron temperature Tve increases and a slightfraction of O, e� and O2

þare produced in the free stream

by photodissociation and photoionization of O2. The tem-perature is slightly decreased in the shock layer, leading toa lower ionization level.

4.2.3. Huygens (191 s)This Titan entry case has been selected to show the

ability of the HSNB model to handle strong optically thickmolecular radiation. For early trajectory points, a state-to-state electronic specific model of the CN molecule isrequired [39,55]. As thermal nonequilibrium modeling hasbeen already discussed for the FIRE 2 t¼1634 s case, wewill focus here on an quasi-equilibrium case at the tra-jectory point t¼191 s.

When coupled radiation effects are taken into account,the standoff distance decreases from 10.77 (uncoupled) to10.33 cm (coupled), the conductive heat flux decreases from24.2 to 22.0W/cm2 and the radiative flux decreases from95.6 to 80.57W/cm2. The radiative flux obtained in theuncoupled case is in agreement with Ref. [52]. They found aradiative flux of 75W/cm2 but they applied to their results acorrection coefficient of 0.75 to model the 3D effects.

The temperature and CN molar fraction along the stag-nation line are plotted in Fig. 7. In the shock layer, we cannotice a lower temperature due to radiative cooling and ahigher CN molar fraction because of lower dissociation rates.

The role of CN radiation, especially the CN-Violetsystem, has been highlighted when discussing the accu-racy of the HSNB model in Section 4.1. Indeed, it can beseen in Fig. 8 that the dominant contributors toradiation are the CN-Violet, followed by the CN-Redmolecular systems. N2 radiation also contributes

mposition along the stagnation line.

Fig. 5. FIRE 2 (1634 s). Total radiative source term along the stagnation line.

Fig. 6. FIRE 2 (1642.66 s). Temperatures and composition along the stagnation line.

Fig. 7. Huygens (191 s). Temperature and CN molar fraction along the stagnation line.

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–69 65

significantly at the emission peak. However, C2 radiationis weak. When radiation is coupled to the flow, the emis-sion is reduced at the peak because of the lowertemperature.

5. Conclusion

We have shown in this paper the ability of the HSNBmodel to accurately and efficiently predict radiation effects

on hypersonic entry flows. The HSNB model reproducesthe LBL results with an accuracy better than 5% and aspeed up of the computational time around two orders ofmagnitude. Concerning molecular radiation, the HSNBmodel provides a significant improvement compared tothe Smeared-Rotational-Band model in the case of Titanentry dominated by optically thick CN radiation. Takinginto account the coupling with radiation, both convectiveand radiative heat fluxes at the wall decrease. The standoffdistance is reduced due to radiative cooling of the shock

Fig. 8. Huygens (191 s). Total radiative source term along the stagnation line.

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–6966

layer. For Earth entry cases, we observed slower ionizationlevels as well as the appearance of O, e� and O2

þin the free

stream produced by the photodissociation and the pho-toionization of O2.

This study was focused on a 1D flow-radiation model inorder to demonstrate the feasibility of coupled simulationswith the HSNB model. However, 2D or 3D simulations maybe required to determine the spatial distribution of theheat flux on the wall. For these geometries, coupled cal-culations are not practical with a LBL description ofradiative properties. Though through the use of paralleli-zation, the HSNB model may be used to compute radiativesource terms accurately in a reasonable amount of time.The Monte Carlo method would be probably the bestalgorithm to solve the radiative transfer equation inthis case.

The analysis of the first trajectory point of the FIRE2 experiment has shown that the two temperatureapproach was not satisfactory to model the thermalnonequilibrium state of the plasma. The results obtainedusing non-Boltzmann electronic populations provided bythe QSS model of Johnston [55] were closer to the flightdata even if some discrepancies still remain. For such entryconditions (high velocity, low density), a self-consistentelectronic specific collisional-radiative model should bedeveloped. The HSNB model could be used since it iscompatible with arbitrary populations of electronic states.

Finally, it is worth mentioning the usefulness of thepresented HSNB model for studying coupled ablation-radiation phenomena in the boundary layer of futureentry vehicles. Light-weight carbon-phenolic ablators havealready been extensively used for several recent missionsand are likely to dominate future space missions as welook forward to evermore challenging entry environments.One notable effect that ablators may have on the radiationfield is the ability of blown ablation products to absorbradiant energy from the shock layer and carry this energydown stream of the stagnation region. As several of therelevant carbonaceous species have already been includedin the HTGR database and their HSNB band parametershave already been presented, the HSNB model could beused to significantly reduce the cost of ablation-radiationstudies in the future.

Acknowledgments

This research was sponsored by the European ResearchCouncil Starting Grant no. 259354 and by the EuropeanCommission Project ABLAMOD (FP7-SPACE No. 312987).The authors gratefully acknowledge Dr. AlessandroMunafò for his previous developments on the stagnation-line flow solver.

Appendix A. Curtis–Godson and Lindquist–Simmonsapproximations for SNB model

For a homogeneous optical path, analytical expressionscan be derived for the mean black equivalent width W=δintroduced in Eq. (26) for mean transmissivity calculations.These expressions depend on the broadening mechanismand a prescribed distribution law for line intensities withinthe narrow band and are functions of two parameters: amean absorption coefficient k per unit partial pressure ofthe absorbing species and an overlapping parameter β .

For a non-homogeneous optical path, the Curtis–God-son approximation consists in using the expressionsderived for homogeneous media with averaged para-meters k

�and β

�that we define according to

u¼Z s

s0pað s00Þ ds00; ðA:1Þ

k� ¼ 1

u

Z s

s0pa s00ð Þkðs00Þ ds00; ðA:2Þ

β� ¼ 1

uk�

Z s

s0βðs00Þpa s00ð Þkðs00Þ ds00; ðA:3Þ

where u is the mean pressure path length and pa thepartial pressure of the absorbing species. Expressions ofthe mean black equivalent width WLðs0; sÞ=δ andWDðs0; sÞ=δ for both Lorentz and Doppler broadening aregiven in Table 5. In order to obtain the mean blackequivalent width in the Voigt broadening regime, we usethe expression proposed of Ref. [56]

WV ðs0; sÞδ

¼ uk�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�Ω�1=2

q; ðA:4Þ

Table 5Expressions of mean black equivalent width for Curtis–Godson approximation and its derivative for Lindquist–Simmons approximation. The first columnindicates the broadening type and the distribution function considered for line intensities. For Doppler broadening, the tailed-inverse exponential dis-tribution is used for air systems (N2, O2, NO) and the exponential distribution is used for carbonaceous systems (CN, C2).

Approximation Curtis–Godson W ðs0 ;sÞδ Lindquist–Simmons �1

δ∂W ðs″ ;sÞ

∂s″

Lorentz, tailed-inverse exponential2βL

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ uk

�

βL�

r�1

� �pa s″�

kðs″Þ2xrþð1� r2 Þffiffiffiffiffiffiffiffiffiffi1þ2x

p

ð1� r2 þ2xÞffiffiffiffiffiffiffiffiffiffi1þ2x

p

x¼ uk�

2βL� ; r¼ βL ðs″ Þ

βL�

Doppler, exponential βD�ffiffiffiπ

p R þ1�1

xexpð� ξ2 Þ1þxexpð� ξ2 Þ dξ

pa s″ð Þk s″ð Þffiffiπ

pR þ1�1

exp �ξ2ð Þ1þxexp �r2ξ2ð Þð Þ dξ

x¼ uk�

βD� x¼ uk

�

βD� ; r¼ βD ðs″ Þ

βD�

Doppler, tailed-inverse exponential βD�ffiffiπ

pR þ1�1 ln 1þxexp �ξ2

� � dξ pa s″ð Þk s″ð Þffiffi

πp

R þ1�1

exp �ξ2ð Þ1þxexp �r2ξ2ð Þð Þ dξ

x¼ uk�

βD� x¼ uk

�

βD� ; r ¼ βD ðs″ Þ

βD�

Fig. 9. FIRE 2 (1634 s). Comparison between Curtis–Godson and Lind-quist–Simmons approximations.

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–69 67

Ω¼ 1� 1

uk�WL ðs0; sÞ

δ

!224

35

�2

þ 1� 1

uk�WD ðs0; sÞ

δ

!224

35

�2

�1:

ðA:5ÞAn alternative approach for treating non-homogeneous

optical path is the Lindquist–Simmons approximation [57].It consists in finding expressions for the space derivative ofthe mean black equivalent widths in Lorentz ∂WLðs00; sÞ=∂s00and Doppler ∂WDðs″; sÞ=∂s00 broadening regimes. Theexpressions used is in this work are given in Table 5. Theyinvolve both local parameters kðs″Þ, βðs″Þ and averagedparameters k

�, β

�. The non-uniform mean black equivalent

widths in each broadening regime are then obtained byspatial integration, according to

WL=Dðs0; sÞδ

¼Z s

s0

1δ∂WL=Dðs″; sÞ

∂s″ds00; ðA:6Þ

and Eqs. (A.1) and (A.2) are used again to get the meanblack equivalent width in the Voigt broadening regime.

A comparison between the HSNB Curtis–Godson andthe HSNB Lindquist–Simmons is given in Fig. 9 for the FIRE

2 (1634 s) test case. The difference between the twoapproximations is very tiny and for most of the points ismuch lower than the difference between HSNB Lindquist–Simmons and LBL calculations. However, we can see amuch larger discrepancy in the free stream: the radiativesource term predicted by the HSNB Curtis–Godson modelbecomes negative while it remains positive for the HSNBLindquist–Simmons and LBL models. A negative energysource term in this cold region is a major computationalissue for coupling because negative temperatures can bepredicted after few iterations. This is why the Lindquist–Simmons approximation has been chosen for the coupledsimulations even though it requires around three timesmore computational time.

Appendix B. Adaptive spectral grid for atomic radiation

This appendix describes the method for generating anadaptive spectral grid for the LBL treatment of atomicradiation in the HSNB model (Section 3.3). We start froman 11 point stencil [49] defined by σul, σul7Δ, σul beingthe line center and Δ the 5 point half-stencil

Δ¼ 18γV ;

12γV ;ΔσW ;ΔσFW ;

252γV

� �: ðB:1Þ

The estimated distances from the line center to the linewing ΔσW , and to the far wing ΔσFW are computed byΔσW ;ΔσFW� �¼ 2

π 1þζ�

γLþαγD, where γL and γD are thehalf widths at half maximum of the line related to Lorentzand Doppler broadening respectively. The values of the ζ,α constants are taken to be f1;1:8g forΔσW , while they are

chosen as f2:6;5:8g for ΔσFW . γV ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2Dþγ2L

qis the esti-

mated Voigt half width at half maximum. It was shownthat such stencil provides a reasonably accurate resolutionof line intensities for low pressure atmospheric entryconditions due to the low degree of line broadening in thatregime [49].

This fixed point method is likely to work well in spectralregions with a high number of electronic transitions andwith a large degree of line overlap because the majority of

Fig. 10. Huygens (189 s). Comparison between LBL, HSNB and LBL resultsfrom Bansal and Modest [31]. Contribution of the CN red and violetsystems to the radiative source term along the stagnation line.

L. Soucasse et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 180 (2016) 55–6968

the points are distributed around the line centers. For areasin which there are large distances between neighboring linecenters, the method is likely to provide poor estimates ofspectral quantities due to the large error in interpolating thespectral values in the far line wing regions.

For this reason, we implemented an adaptive refinementprocedure in order to accurately compute the far line wingregions for atomic spectra which have a relatively weakspectral density. To begin, the complete line list for atomsconsidered is first ordered by ascending line center values.Then, each region between two consecutive lines is con-sidered. For each consecutive line pair, an adaptive mesh iscreated based on the two corresponding line shapes. We willdenote the left line shape properties with the superscript L,and the right properties with an R. First, the approximatecenter point between each line is defined simply asσLR ¼ ðσL

ulþσRulÞ=2. Next, the following set of points are

added to the mesh based on the 11 point stencil, butensuring that points added for each line do not overlap oneanother:

σLulþΔL

; 8ΔLoσLR�σLul ðB:2Þ

σLul�ΔR

; 8ΔRoσRul�σLR ðB:3Þ

For lines which are sufficiently close, the above proce-dure will prevent unnecessary points from being added tothe spectral grid. For lines which are very far apart incomparison to their line widths, the above set of points areaugmented by adding points recursively to the centerregion by successively bisecting the two intervals closestto the last points added by the stencil above until thespacing between the outermost two stencil points for eachline is at least half the size of the spacing between theoutermost stencil point and the next point. In other words,two bisection fronts are propagated towards the linecenters until the spacing between points matches that ofthe two outermost stencil points of each line.

The accuracy of this adaptive procedure has been suc-cessfully compared with the fine LBL spectral grid of 4:4�106 spectral points that we use for full LBL calculations [58].The differences we obtained were negligible compared to theaccuracy of the full HSNB model which is around a fewpercent (see Section 4.1). The spectral size of the adaptivemesh for atomic radiation was around 4� 104 points for theapplications considered in this paper, which allows thecomputational time to be two order of magnitude less. Thusthis adaptive spectral mesh procedure constitutes a decisivedevelopment for the practical use of the HSNB model.

Appendix C. Accuracy of the HSNB model against lit-erature results

The accuracy of the HSNB model has been assessed inSection 4.1 by comparison with LBL results, where bothHSNB parameters and LBL radiative properties wereobtained from the HTGR spectroscopic database. The pur-pose of this appendix is to compare the HSNB model withreference results obtained by other researchers.

We have considered the LBL radiation simulation ofBansal and Modest [31] of the trajectory point t¼189 s of

the Huygens probe entry from uncoupled flowfield takenfrom Johnston [55]. The spectroscopic constants weretaken from Laux [59] and nonequilibrium populations ofthe CN electronic states were considered.

In order to reproduce their results we have computedthe radiative source term along the stagnation line withboth the HSNB model and the LBL approach. For the CNelectronic states, both Boltzmann populations and none-quilibrium populations based on the QSS model of Boseet al. [60] have been taken into account. The comparison isshown in Fig. 10 for the CN red and violet systems. First ofall, we can see that for a given population assumption, ourLBL and HSNB results give similar results. When the QSSmodel is used, we obtain a good agreement with the LBLresults of Bansal and Modest [31] for both LBL and HSNB.Note that nonequilibrium effects are not negligible in theconsidered simulation and lead to about 16% difference atthe peak value of the radiative source term.

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