Submitted to Journal of Thermophysics and Heat Transfer

Spectral Modeling of Radiative Heat Transfer in CarbonaceousAtmospheres Using Newk-Distribution Models and Databases

Ankit Bansal∗

The Pennsylvania State University, University Park, PA 16802M. F. Modest†

University of California, Merced, CA 95343

Multi-scale correlated-k distribution models are presented for the red and violet bands of CN for applica-tions relevant to entry into the atmospheres of Titan, Mars,Venus and flows over an ablating heat shield. Inthe multi-scale model spectral lines are separated into different scales to improve correlation among absorp-tion coefficients. The methodology for scaling absorption coefficients into such groups is presented. A newemission-weighted full-spectrumk-distribution model is proposed as an improvement over the more popularPlanck function-weighted full-spectrum k-distribution model. In the new model, the emission coefficient re-places the Planck-function as weight in the reordering scheme. Accuracies of the two models are compared bysolving severe two-cell problems. To exploit the full potential of the k-distribution method databasing schemesfor k-distributions are discussed and cpu-time studies are presented. The accuracy of the new model and thedatabase is demonstrated by solving the radiative transferequation along the stagnation line flow field of theHuygens spacecraft.

Nomenclature

a nongray stretching function ofk-distributions, dimensionlessc speed of light, 2.9979× 108 m s−1

E energy, cm−1

f k-distribution, cmg cumulativek-distributionh Planck’s constant, 6.6262× 10−34 JsI intensity, (W/m2-sr)Ib Planck function, W/m2-srk, k∗ reordered absorption coefficient, cm−1

k′ reordered absorption cross-section, cm2

kB Boltzmann constant, 1.38× 10−23 JK−1

n number density, cm−3

S line strength, cm−1ÅT translational temperature, KTv vibrational temperature, KTr rotational temperature, KTe electron temperature, KGreekελ spectral emission coefficient, W/cm3-Å-srκλ spectral absorption coefficient, cm−1

κ′λ

spectral absorption cross-section, cm2

λm overlap parameter for scalem, cm−1

∗Graduate student, Department of Mechanical Engineering; azb162@psu.edu†Shaffer and George Professor of Engineering; MModest@eng.ucmerced.edu

1

φ gas state vectorSubscripts0 reference stateD Dopplere electrong at a given value of cumulativek-distributionJ for a given rotational quantum numberk at a given value of reordered absorption coefficient variablev for a given vibrational quantum numberλ at a given wavelength

I. Introduction

Radiation from the shock layer during atmospheric entry plays a significant role in the design of modern spacevehicles. Efficient and accurate modeling of shock layer radiation is absolutely necessary for the optimum designof new generation space vehicles. A large number of studies has been performed for Earth reentry conditions,1–3

analyzing basic mechanisms and processes contributing toward radiative heat fluxes. A number of similar studies hasbeen performed for entry into the atmospheres of Titan, Saturn’s largest moon,4–6 and Mars.7–9

In the above mentioned atmospheres a number of carbonaceousspecies including C,CN, C2, CO, and CO2 emitsignificant amounts of radiation leading to large heat loadson the spacecraft. Titan, for example, has an atmosphereof N2 with a small amount (2-3%) of methane. At high temperatures,dissociation of methane leads to the productionof the cyanogen radical (CN), which is a very strong radiator. Radiative heating from the CN molecule was predictedto be of critical importance in the design of the European Space Agency’s (ESA) Huygens probe. At peak heating theradiative heat flux was predicted to be as high as twice the convective heat flux.?, 10 For NASA’s Titan Aerocapturemission, analyses have predicted that radiative heating from CN can be as high as 300 W/cm2 at the stagnationpoint,4, 5, 12 more than 4 times the convective fluxes. Over 90% of the contribution toward the radiative heat fluxmay come from the CN (violet) band and the rest from the red band. Similarly, in a CO2–N2 atmosphere, typicallyfound on Mars and Venus, shock layer radiation is dominated by the CO (4+) band system, along with the CN (Violet)and CN (Red) band systems.7–9 Thus, in typical carbonaceous atmospheres the main radiating species are CN andCO. Furthermore, these two species are significant when ablation of thermal protection systems (TPS) is considered.The concentration of ablation products is generally higherin the cooler boundary layer region and, therefore, ablationspecies may have relatively larger absorption than emission. This may result in lowering of radiative heat fluxes, but,in turn, may also lead to larger convective fluxes due to a hotter boundary layer. The main ablation products that emitstrongly and lead to an increase in wall heat flux are CN and C, while other species like the continuum of atomic Cand the vacuum ultraviolet band systems of CO, C3, and H2 also lead to some absorption in the boundary layer.13, 14

The fourth positive band system of carbon monoxide, CO (4+), is one of the most important absorbing species amongthe ablation products.15 A reduction of 7-9 % in the wall heat flux was predicted by Johnston et al.13 for the Apollopeak heating case.

The current work focuses on spectral modeling of violet and red bands of CN. The models presented in thiswork can easily be extended to other carbonaceous species and molecular bands, which will be considered in futurework. The objective of this paper is to develop an efficient and accurate spectral model and database for solvingradiation problems for typical entry conditions in carbonaceous atmospheres. A databasing scheme will be developedto store precalculatedk-distributions, from which required spectral data can be interpolated very quickly. The newk-distribution model along with the databasing scheme will provide an efficient yet accurate way of generating gasradiative properties and solving the RTE.

II. Radiation models

NEQAIR9616 has been the most widely used code to perform nonequilibriumradiative calculations, providingline-by-line data of nonequilibrium radiative propertiesof hypersonic shock layer plasmas, along with a primitiveone-dimensional radiative transport algorithm. Other codes that have been used for simulation of CO2/N2 plasma areSPECAIR17 and SPARTAN.18 The HARA code developed by Johnstonet al.13 has been used for solving radiationproblems in ablating atmospheres. A comprehensive validation of different radiation codes for CN has been given byWright et al.,19 who found that the spectroscopic properties given in NEQAIR96 are sufficient to model CN bands,

2

even though there are significant differences between the excitation models employed by different codes. In this work,spectroscopic data for the red and violet bands of CN are mostly taken from NEQAIR96 with some values replacedby more accurate data from Laux’s work.17

Molecular radiation is marked by the presence of a large number of ro-vibrational bands, with each band havinglarge numbers of rotational lines. This, in turn, requires an extremely large number of spectral points to describeemission and absorption from all lines. A snapshot of the absorption spectrum of the CN violet band is given inFig. 1, identifying the five strongest vibrational bands. The violet band contains about 59 vibrational bands, each bandcontaining more than 200 rotational lines. Similarly, the red band contains about 235 vibrational bands. The strongspectral structure of radiative emission requires a line-by-line (LBL) solution of the RTE at several hundred thousandwavelengths, making it extremely expensive and unfeasiblefor solving molecular radiation problems coupled with aflow solver. The violet band is strongly self-absorbed and, therefore, it is important to accurately represent the spectralcoefficients for this band. On the other hand, the red band is relatively optically thin and does not require very accuratespectral representation. It will be shown that a gray calculation for this band provides results accurate within 5%compared with the LBL method.

To reduce the computational effort of the LBL method, a smeared rotational band (SRB) model was developed byPatchet al.,20 which was further improved and extended by Chambers.21 The SRB method is used in the LORAN andthe HARA codes to model molecular band radiation. In the SRB model a number of rotational lines are combined,such that the average shape of each vibrational band is captured. Also, the formulation is done in such a way that themethod returns accurate total radiative flux in the optically thin limit. It was shown that, for optically-intermediateconditions, the smeared-rotational band approach is inadequate. Johnstonet al.22 found that representing the CN violetband by the SRB model leads to over-prediction of the radiative flux by up to 40% relative to the LBL method. Theyproposed a simple modification to the SRB model that improvedits accuracy in optically-intermediate conditions. Inthe new model, called SRBC (smeared-rotational band corrected), the rotational temperature is tuned in such a waythat the band shape conforms well with the actual shape and gives more accurate results than the conventional SRBmodel.

It has been shown in the field of atmospheric radiation23, 24 that, for a small spectral interval in a homogeneousmedium, the absorption coefficient may be reordered into a monotonick-distribution, which yields exact results ata small fraction of the time required for line-by-line calculations. It was found in the field of combustion that suchk-distribution reordering can also be done for the full spectrum and applied to inhomogeneous media.25, 26 It wasrecognized27–29 that, in high temperature combustion applications, at significantly different temperatures differentspectral lines dominate the radiative transfer, and the assumption of a correlated absorption coefficient breaks down.Similarly, in a mixture of gases the correlation breaks downin the presence of strong concentration gradients, asrecognized by Modest and Zhang.25 To overcome some of these difficulties, Modest and Zhang developed two differentapproaches: the multi-scale full-spectrum correlated-k distribution method MSFSCK,30 in which different lines areplaced into separate “scales” based on their temperature dependence, and the multi-group full-spectrum correlated-k-distribution method MGFSCK31 where different spectral positions are placed into different spectral groups accordingto their temperature and pressure dependence. In the multi-scale approach the spectral scales may overlap, and theoverlap between scales is then found in an approximate way, which may lead to additional inaccuracies. In contrast,the multi-group model avoids the problem of overlap betweenthe spectral groups, but it is not trivial to apply to gasmixtures. Here the multi-scale approach will be adopted to group spectral lines.

Most k-distribution models to date have been developed for molecular band radiation in thermodynamic equi-librium and at relatively lower temperatures. Recently, Bansalet al.32 have developed a multi-group FSCK methodfor high-temperature nonequilibrium atomic radiation problems, which has greatly enhanced the efficiency of radiativecalculations in the Earth’s atmosphere. They also developed a database of narrow-bandk-distributions, from which anydesired full-spectrumk-distribution can be calculated on-the-fly, resulting in further reduction of computational time.33

From an FSCK point of view the problem of molecular radiationin hypersonic nonequilibrium flows is somewhat morechallenging than the combustion problems and nonequilibrium atomic radiation problems, because of thermodynamicnonequilibrium and the presence of a large number of species, each with a large number of ro-vibrational bands. Inthermodynamic nonequilibrium the Planck function is not a function of just one temperature; rather it depends on alarge number of parameters in a complex way. This makes the development and implementation of thek-distributionmodel very challenging in such applications.

3

III. Basic Equations

In a simplified form the absorption and emission coefficients of molecular species at any given wavelength can bewritten as the sum of contributions from all overlapping lines as

ελ =∑

ℓ

εcℓnℓ′Φ

(

λ − λℓ

bD(T)

)

(1)

κλ =∑

ℓ

εcℓλ5

2hc2(nℓ′′ − nℓ′)Φ

(

λ − λℓ

bD(T)

)

(2)

wherenℓ′ andnℓ′′ represent populations of upper and lower states associatedwith line ℓ, Φ is the Doppler line shapefunction. Since most molecules are present in regions of lowpressure and low number density of electrons, the effectsof collisional and Stark broadening are likely to be insignificant and Doppler broadening is adequate to model lineshape. The population density of any ro-vibrational level is given by

nℓ =ni

QivJ

exp

[

−hckB

(

EvTv+

EJTr

)]

(3)

whereni is the population andQivJ is the ro-vibrational partition function associated with the i-th electronic level. Atequilibrium the electronic state population is given by theBoltzmann distribution as

ni =nMQivJ

∑

j

g jQjvJ exp

(

−hcEjkBTe

) exp

(

−hcEikBTe

)

(4)

wherenM is the number density of the molecule and the summation in thedenominator is taken over all electroniclevels. Most previous excitation predictions for the CN molecule have been carried out assuming an equilibriumBoltzmann population distribution, which has has been shown to lead to over-prediction of radiative heat fluxes bya factor of two compared to those predicted by nonequilibrium models.34 In case of nonequilibrium, a collisional-radiative approach instead of Eq. (4) is used to calculate the electronic state populations. As the purpose of this studyis to develop spectral models – which should work with both the equilibrium and nonequilibrium electronic statepopulation distributions – here a Boltzmann distribution is assumed to populate the electronic states of CN (X2Σ+,A2

∏

, B2Σ+). Using the above relationships, the emission and absorption coefficients can be written as

ελ =∑

ℓ

εcℓni

QivJ

exp[

−hckB

(Ev′

Tv+

EJ′

Tr

)]

Φ

(

λ − λℓ

bD(T)

)

(5)

and

κλ =∑

ℓ

εcℓλ5

2hc2n j

Q jvJ

{

exp[

−hckB

(Ev′′

Tv+

EJ′′

Tr

)]

−nin j

Q jvJ

QivJ

exp[

−hckB

(Ev′

Tv+

EJ′

Tr

)]

}

Φ

(

λ − λℓ

bD(T)

)

(6)

where indicesi and j refer to the upper and lower electronic state of the transition, respectively. Under nonequilibriumthe Planck functionIneb is defined as the ratio of emission and absorption coefficient,

Inebλ =ελ

κλ=

∑

ℓ

εcℓni

QivJ

exp[

−hckB

(Ev′

Tv+

EJ′

Tr

)]

Φ

(

λ − λl

bD(T)

)

∑

ℓ

εcℓλ5

2hc2n j

Q jvJ

{

exp[

−hckB

(Ev′′

Tv+

EJ′′

Tr

)]

−nin j

Q jvJ

QivJ

exp[

−hckB

(Ev′

Tv+

EJ′

Tr

)]

}

Φ

(

λ − λl

bD(T)

)

(7)

which no longer is a smooth function and varies erratically similar to the absorption coefficient as is shown in Fig. 2.

IV. Multi-scale Planck function-weighted full-spectrum k-distribution: MSFSCK

The biggest challenge with the full-spectrumk-distribution method is that it can be relatively inaccurate for inho-mogeneous problems with strong gradients in temperatures and concentrations. In the multi-scale method the accuracy

4

of the correlated-k method is improved by separating the absorption coefficients from a single gas or a gas mixtureinto a number of scales. Due to nonequilibrium effects, the multi-scale full-spectrumk-distribution model is ideallysuited for gas mixtures in nonequilibrium plasmas. Some lines tend to become stronger with increasing temperaturewhile others tend to become weaker. The spectrum can be broken into scales to separate lines depending on theirtemperature dependence, in order to improve the accuracy ofthe correlated-k distribution method. In this section thebasic methodology of the multi-scalek-distribution method is presented. Following the work of Zhang and Modest,30

the gas mixture is split intoM different scales. The total absorption coefficient for the gas mixture can then be writtenas

κλ(φ) =M∑

m=1

κmλ(φ) (8)

whereφ is the gas state vector which, in general, includes number densities of different species and up to four temper-aturesT, Te, Tv andTr . Following the convention to treat different electronic bands of the same molecule as seperatespecies, summation in Eq. (8) is over all scales of all bands.The RTE for the nonequilibrium gas mixture can bewritten similar to the equilibrium case as35

dIλds=

M∑

m=1

κmλ(φ)Inebmλ(φ) −( M∑

m=1

κmλ(φ))

Iλ (9)

whereInebmλ is the nonequilibrium Planck function andκmλ(φ) is the absorption coefficient for scalem. Now, splittingthe total spectral intensity intoM terms, one for each scale, i.e.,

Iλ =M∑

m=1

Imλ (10)

substituting Eq. (10) into Eq. (9) and separating the equation for them-th scale,

dImλds= κmλ(φ)Inebmλ(φ) −

( M∑

n=1

κnλ(φ))

Imλ, m= 1, 2...M (11)

Each of theseM equations is an RTE for the spectral intensity emitted by onegas scale, but attenuated by the entiregas mixture. Next, Eq. (11) is reordered by multiplying withδ(km− κmλ(φ0)) and integrating over the entire spectrum.Hereκmλ(φ0) is the absorption coefficient of them-th scale evaluated at some reference stateφ0. This leads to

dImkds= k∗m(φ, km)I

nebm(φ) fm(φ, φ0, km) −

∫ ∞

0

( M∑

n=1

κnλ(φ))

δ(km− κmλ(φ0))Imλdλ (12)

provided that at every wavelength across the entire spectrum, whereκmλ(φ0) = km, we also have a unique value forκmλ(φ) = k∗m(φ, km) everywhere within the medium. HereImk and fm are defined as

Imk =∫ ∞

0Imλδ(km− κmλ(φ0))dλ (13)

fm(φ, φ0, km) =1

Inebm(φ)

∫ ∞

0Inebmλ(φ)δ(km− κmλ(φ0))dλ (14)

wherefm(φ, φ0, k) is the Planck function weighted full spectrumk-distribution for scalem, which depends on refer-ence state conditionsφ

0through the absorption coefficient and local conditionsφ through the nonequilibrium Planck

function. The above equation can be rewritten as

dImkds= k∗m(φ, km)I

nebm(φ) fm(φ, φ0, km) − λm(φ, km)Imk (15)

whereλm is defined as the overlap factor of them−th scale with the entire gas. In general,

λmImk =∫ ∞

0

M∑

n=1

κnλ(φ)δ(km− κmλ(φ0))Imλdλ

=

M∑

n=1

∫ ∞

0κnλδ(km− κmλ(φ0))Imλdλ = Imk

M∑

n=1

λnm (16)

5

Hereλnm is the spectral overlap of scalem with scalen. Note thatImλ = 0 whereverκmλ = 0, asImλ is related toemission from them-th scale. Also,λmm= k∗m and if there is no overlap,λm = k

∗m.

Now, dividing Eq. (15) by thek-distribution at the reference statefm(φ0, φ0, km), the RTE is transformed into themuch smootherg-space

dImgds= k∗m(φ0, φ, gm)am(φ, φ0, gm)I

nebm(φ) − λm(φ0, φ, gm)Img (17)

with

Img = Imk/ fm(φ0, φ0, km) =∫ ∞

0Imλδ(km− κmλ(φ0))dλ

/

fm(φ0, φ0, km) (18)

wheregm is the cumulativek-distribution andam(φ, φ0, gm) is a weight or nongray stretching function given by

gm(φ0, φ0, km) =∫ km,max

km

fm(φ0, φ0, km)dkm (19)

am(φ, φ0, gm) =fm(φ, φ0, km)

fm(φ0, φ0, km)=

dgm(φ, φ0, km)

dgm(φ0, φ0, km), (20)

where in Eq. (19) a monotonicallydecreasingcumulativek-distribution function has been defined for numerical pre-cision reasons. Also, in numerical calculations it is difficult to evaluate the ratio of thek-distributionsfm, due to theirerratic behavior (having singularities at each minimum andmaximum of the absorption coefficient26); it is much moreconvenient to evaluate the derivativedgm/dkm, as indicated in Eq. (20).

A. Evaluation of overlap factor

To predict the effect of overlap among various gas scales is not a straightforward task. Ask-distributions do not retainany information of the spectral location of absorption lines in various band systems, it is impossible to combine theindividualk-distributions into a single mixturek-distribution exactly. The overlap factor in Eq. (17) is evaluated in anapproximate way, such that the multi-scale FSCK model obtains the exact result for emitted intensity emanating froma homogeneous layer.30 First, Eq. (17) is integrated over a homogeneous slab of thicknessX. For a homogeneous slabthe weight functionam = 1 andk∗m(φ0, φ, gm) = km(φ, gm). This gives

Img =km(φ, gm)Inebm(φ)

λm(φ, gm)

(

1− e−λm(φ,gm)X)

(21)

The total intensity for them-th scale can then be determined from

Im =∫ 1

0

km(φ, gm)Inebm(φ)

λm(φ, gm)

(

1− e−λm(φ,gm)X)

dgm =∫ ∞

0

km fm(φ, km)Inebm(φ)

λm(φ, km)

(

1− e−λm(φ,km)X)

dkm (22)

Equation (11) can also be reordered by multiplying withδ(k − κλ(φ)), i.e., the total absorption coefficient. For ahomogeneous cell this gives

dImkds= km(φ, k)I

nebm(φ) − kImk (23)

where

km(φ, k) =1

Inebm(φ)

∫ ∞

0κmλ(φ)I

nebmλ(φ)δ(k− κλ)dλ (24)

Integrating Eq. (23) leads to

Imk =km(φ, k)Inebm(φ)

k

(

1− e−k(φ)X)

(25)

The total intensity is obtained by integrating overk-space as

Im =∫ ∞

0

km(φ, k)Inemb(φ)

k

(

1− e−k(φ)X)

dk, (26)

6

which is exact for the homogeneous case. The total intensityobtained by the two methods for a homogeneous layerfrom Eq. (22) and Eq. (26) must be identical. Comparing the results leads to the following two requirements:

λm(φ, km) = k(φ) (27)

andkm(φ) fm(φ, km)dkm = km(φ, k)dk (28)

A computationally convenient way to solve the above, as suggested by Zhang and Modest,30 is∫ km

0km(φ) fm(φ, km)dkm =

∫ k=λm

0km(φ, k)dk (29)

This is an implicit relation from whichλm-values need to be evaluated for a set ofkm-values.

B. Scaling of absorption coefficients

Since the assumption of a correlated absorption coefficient may not be valid for the entire spectrum, a criterion needsto be set to create scales of absorption coefficients. In this section a methodology is presented to separate absorptionlines into scales based on their temperature behavior.30

First, the dependence of absorption line strengths,S, on temperature is analyzed. In Fig. 3 normalized absorptionline strengths are plotted as a function of temperature for anumber of selected lines. Since the absorption coefficientprimarily depends on the lower state population (n′′

ℓ≫ n′

ℓ, see Eq. (6)), different lines were picked based on quantum

numbers of the lower statev′′, J′′. All line strengths were normalized by line strengths at a temperature of 12,000K.The figure on the right is a zoomed (stretchedy-axis) view of the figure on the left. The behavior of lines with smallvalues ofv′′, J′′ is very different, qualitatively as well as quantitatively, from thoselines which have larger values foreither of the quantum numbers. Lines with small quantum numbers become very strong at lower temperatures as isshown by the ratios of line strengths in Fig. 3. Based on this visual observation, an unoptimized scaling scheme is firstchosen, which is summarized in Table 1.

Next, the scaling model is optimized by following the optimization scheme proposed by Palet al.36 According tothis scheme, an averageS-profile is first calculated as a function of temperature for each unoptimized scale using thefollowing expression.

Si(T j) =

∑

ℓ(i) Sℓ(T j)Ibλℓ (Tref)∑

ℓ(i) Sℓ(Tref)Ibλℓ (Tref)(30)

wherei is scale index,j is temperature index, and summation in the above expressionis taken over all rotational linesthat are assigned to scalei; Sℓ is the line strength,Ibλℓ is the equilibrium Planck function at the center of lineℓ andTrefis the reference temperature taken equal to 12,000 K. The resulting S(T)-profiles are plotted in Fig. 4.

Next, for each line normalized departures of its relative temperature dependence from the averageS-profiles areevaluated and the minimum value is found as

ǫℓi =∑

j

(Si(T j) −CℓiSℓ(T j))2

S2i j(31)

where

Cℓi =

∑

j Si(T j)Sℓ(T j)∑

j S2ℓ(T j)

(32)

Thus, if ǫℓ1 < ǫℓ2 the line goes into Scale 1 and vice-versa. This scaling modelwas applied to the violet band, and itwas found that the scaling model summarized in Table 1 is adequate. The absorption spectrum for the two scales isgiven in Fig. 5. Since the red band is optically thin, no scaling is required and a single scale is sufficient.

V. Multi-scale emission-weighted full-spectrumk-distribution: EMSFSCK

In this section a new FSCK model is proposed, which was found to perform better than the Planck-functionweighted MSFSCK model under severely nonhomogeneous conditions. Equation. (11) for Scalem is rewritten as

dImλds= εmλ(φ) −

(M∑

n=1

κnλ(φ))

Imλ (33)

7

As before, reordering Eq. (33) by multiplying withδ(km− κmλ(φ0)) and integrating over the entire spectrum leads to

dImkds= εm(φ) fm(φ, φ0, km) −

∫ ∞

0

(M∑

n=1

κnλ(φ))

δ(km− κmλ(φ0))Imλdλ (34)

wherefm(φ, φ0, k) is the emission-weighted full spectrumk-distribution for scalem, and is defined by

fm(φ, φ0, km) =1

εm(φ)

∫ ∞

0εmλ(φ)δ(km− κmλ(φ0))dλ (35)

The only difference between the new and the old scheme is the weight function used in the definition of the full-spectrumk-distribution. In the old approach the absorption coefficient is taken out of the integral in Eq. (35) byinvoking the correlated-k assumption; however in the new approach no such approximation is made at this point. Nowthe above equation can be rewritten as

dImkds= εm(φ) fm(φ, φ0, km) − λm(φ, km)Imk (36)

where, as before,λm is the overlap factor of them−th scale with the entire gas. Finally, dividing Eq. (36) by thek-distribution at the reference statefm(φ0, φ0, km), the RTE is transformed into the much smootherg-space

dImgds= εm(φ)am(φ, φ0, gm) − λm(φ0, φ, gm)Img (37)

The overlap factor is calculated using the same approach as adopted for the MSFSCK method. This leads to thefollowing two requirements

λm(φ, km) = k(φ)

which is same as for the MSFSCK method and∫ km

0fm(φ, km)dkm =

∫ k=λm

0f (φ, k)dk

orgm(φ, km) = g(φ, λm) (38)

which, as before, leads to an implicit relation from whichλm-values need to be evaluated for a set ofkm-values. Boththe old and the new approaches are based on the same argumentsand, therefore, the overlap factor calculated fromthe two approaches should be the same; however, the new approach is neater and easier to implement. It requiresgenerating two distributions,gm(φ, km) andg(φ, k); and then interpolating ing(φ, k) to calculateλm = k for everygm(φ, km) value. In contrast, Eq. (29) requires evaluation of the twointegrals in addition to a interpolation for findingthe value ofλm

VI. Database of emission-weighted full-spectrumk-distributions

Full-spectrumk-distributions and overlap factorsλ are rather cumbersome to generate. Therefore, to make use ofthe full potential of the FSK method, the requiredk-distributions must be evaluated from a precalculated database. Thedatabasing methodology used here closely follows the work of Wang and Modest37 and Bansal and Modest.33 FromEq. (20) it can be inferred that the quantities required to bestored in the database aregm(φ, φ0, km) andgm(φ0, φ, km).Sincegm values always vary between 0 and 1, it is convenient to fix values ofgm and store values ofkm(φ, φ0, gm)

andkm(φ0, φ, gm), respectively.37 The parameter space for the database is defined by two separate gas state vectors,φ

andφ0. Each of these vectors depends on three temperatures,T, Tr , Tv and on electronic state populationni . Under

nonequilibrium the electronic state population is calculated using the Quasi Steady State (QSS) model,34 which maydepend on a large number of parameters. Any database containing all these dependencies on gas states is impracticalas it would require huge storage space in addition to multi-order interpolation. To restrict the size of the database tomanageable limits, it is imperative to factor the QSS dependency out from the expressions for emission and absorptioncoefficients. From the expression for emission coefficient, Eq. (5), the electronic state population is factoredout andan emission cross-section is defined as

ε′λ(ψ) =ελ(φ)

ni(φ)=

∑

ℓ

εcℓ

QivJ

exp[

−hckB

(Ev′

Tv+

EJ′

Tr

)

]

Φ

(

λ − λℓ

bD(T)

)

(39)

8

whereψ(T,Tr ,Tv) is a reduced gas state vector. The absorption coefficient, Eq. (6), is approximated by assumingthat the upper state populationnℓ′ is related to the lower state populationnℓ′′ with a Boltzmann distribution at thevibrational temperatureTv. This leads to

κλ(φ) =∑

ℓ

εcℓλ5

2hc2nℓ′′ (φ)

[

1− exp

(

−hc

λkBTv

)

]

φ

(

λ − λℓ

bD(T)

)

=∑

ℓ

εcℓλ5

2hc2n j(φ)

Q jvJ

exp

[

−hckB

(

Ev′′

Tv+

EJ′′

Tr

)] [

1− exp

(

−hc

λkBTv

)]

φ

(

λ − λℓ

bD(T)

)

(40)

and an absorption cross-section is defined as

κ′

λ(ψ) =∑

ℓ

εcℓλ5

2hc21

Q jvJ

exp

[

−hckB

(

Ev′′

Tv+

EJ′′

Tr

)] [

1− exp

(

−hc

λkBTv

)]

Φ

(

λ − λℓ

bD(T)

)

(41)

From Eq. (35), the emission-weighted full-spectrumk-distribution can then be rewritten as

fm(ψ, φ0, km) =1

ε′

m(ψ)

∫ ∞

0ε′

mλ(ψ)δ(km− κmλ(φ0))dλ

=1

ε′

m(ψ)n j(φ0)

∫ ∞

0ε′

mλ(ψ)δ(k′

m− κ′

mλ(φ0))dλ (42)

and the cumulative emission-weighted full-spectrumk-distribution is defined as

gm(ψ, φ0, km) =∫ kmax

km

fm(ψ, φ0, km)dkm

=

∫ kmax

km

1ε′

m(ψ)n j(φ0)

∫ ∞

0ε′

mλ(ψ)δ(k′

m− κ′

mλ(φ0))dλdk′

mn j(φ0)

= gm(ψ, φ0, k′

m) (43)

Sincekm(φ) andk′

m(ψ) differ by only a constant for each electronic band, the cumulative k-distribution remains thesame. Furthermore, the reference gas stateφ

0is assumed to depend on a single temperature (i.e., thermodynamic

equilibrium state). With this simplification the parameterspace is reduced toT, Tv, Tr andφ0. This will require atriple linear interpolation for the local gas stateφ, assuming no interpolation is done inφ

0. The gas state conditions

used to generate thek-distributions are given in Table 2. The database is generated for a number of fixed referencestates. In an actual application one of these reference state is selected and no interpolation is performed. For the localgas state, it was found that more points are required for smaller values of all the parameters, for which the line shape isnarrower. To allow for efficient search-free retrieval, this is achieved by selectingequally-spaced points along a powerlaw distribution. Following Wang and Modest37 and Bansal and Modest,33 a compact and accurate emission-weightedfull-spectrumk-distribution database was generated for the red and violetbands of CN, and also for a large number ofmolecular bands of other relevant species. The total size ofthe database is about 10 MB per molecular band.

VII. Sample Calculations

A. Two cell problem

The accuracy of the new model is demonstrated by solving a simple two-cell problem with strong temperature gradi-ents. The example considers two 1 cm thick cells adjacent to each other and bounded by cold black walls. Concentra-tion of CN in both the cells is 1.0×1016 cm−3. The temperature in Cell 1 is 2000 K while Cell 2 is at 10000 K. It isassumed that all four temperatures (T,Tv,Tr ,Te) are equal and populations of various energy modes are governed bythe Boltzmann distribution. These conditions have been appropriately chosen from actual shock layer conditions. Suchtwo-cell problems with typical conditions serve as an acid test for the method because of their abrupt step-changes inconditions. In actual applications gradients are much morebenign and the accuracy of the model can be expected tobe better.

9

First, results for the single scale model are presented for both the MSFSCK and EMSFSCK approaches. In Fig. 6results for cold wall heat flux from the two FSCK methods are compared with the LBL results for a range of referencestate temperatures. There is only weak dependence on reference temperature for both FSCK models, except at very lowtemperatures; however, the emission-weighted FSCK model performs much better than the Planck function-weightedFSCK model. For the EMSFSCK method the single scale model performs well while for the MSFSCK the absorptionspectrum must be broken into two or more scales to improve itsaccuracy.

The same two cell problem is solved using the two scale model described before and the MSFSCK and EMFSCKapproaches. In the two scale model the first task is to determine the overlap between gas scales. A plot of overlapfactor for Scale 2 is shown in Fig. 7. This is the spectral overlap for emission from Scale 2 and absorption by theentire spectrum.λm is the sum ofkm and the contribution from overlap with all other scales (in this case only Scale 1),implying λm ≥ km always. As mentioned earlier, the two FSCK approaches should predict the same overlap, which isverified from the plot in Fig. 7.

Heat transfer results for the two cell problem are summarized in Table 3 for different spectral models. The overallaccuracy of thek-distribution method is very good even for these extreme temperature gradients. Thek-distributionmethod gives exact results for a homogeneous medium and/or a truly correlated absorption coefficient. Under extremetemperature gradients,k-distributions become uncorrelated, and the accuracy of the method deteriorates. The mainfactor contributing to this uncorrelatedness is the rise ofweak rotational lines at higher temperatures. For the singlescale model, heat flux from the EMSFSCK approach agrees to within 1% with that obtained from the LBL method,while the MSFSCK approach incurs about 8% error. When the effect of overlap is neglected, the two scale EMSFSCKmodel gives good agreement with the LBL method, and the accuracy of the MSFSCK approach improves to 3.5%as compared to 8% for the single scale. However, no-overlap fluxes are considerable higher than the actual heat fluxvalues, indicating that there is significant overlap between the scales. When the approximate overlap model is appliedto the EMSFSCK model the agreement with the LBL method is not very good, indicating that the present approximatetreatment of the overlap is insufficient to model the overlap between scales accurately. Interestingly, the MSFSCKmodel–with the same approximate overlap as for the EMSFSCK model–gives better agreement with the LBL method,apparently due to compensating errors. It can be concluded from this exercise that the one-scale EMSFSCK modelcan be applied with great accuracy to model fluxes from the CN violet band.

B. Stagnation line flow field of Huygens spacecraft

Next, the multi-scale model was applied to the stagnation line flow field of the Huygens spacecraft regenerated fromthe work of Johnston.38 The flow field for the stagnation line is shown in Fig. 8. Wall heat flux and divergence ofheat fluxes along the stagnation line were calculated using the LBL, one-scale EMSFSCK and the gray model. Forthis case the MSFSCK model performs as well as the EMSFSCK model. Very good agreement was found betweenthe EMSFSCK method and the LBL method for both the red and violet bands with a maximum error of 0.3% for theviolet band and 1.3% for the red band in wall heat flux. The EMFSCK method, employing a 16-point Gauss quadraturescheme for spectral integration ing-space, provides excellent agreement for the divergence ofthe heat flux along thestagnation line, as shown in Figs. 9 and 10. For the gray gas approach, significant errors were found for the violetband, indicating that self-absorption is important for this band. However, for the red band, gray results are in verygood agreement with the LBL results.

The entire spectrum combining the red and violet bands can betreated in two different ways. The overlap betweenthe violet and the red band is negligible and one can simply solve RTEs separately for each band and add the resultstogether [FSCK (sum)]; or a singlek-distribution can be generated for the entire spectrum followed by solving theRTE for the single scale [FSCK (tot)]. It can be observed fromFig. 11 that the FSCK (sum) approach performsslightly better than the FSCK (tot) approach. Treating individual bands separately provides better correlation amongspectral absorption coefficients and, hence, better accuracy. However, this will require solving twice as many RTEs:32 when both the bands are treated using a 16-point Gauss quadrature scheme and 17 RTEs if the red band is treatedas gray. Even though all three schemes are very accurate, themost attractive choice is to treat the violet band withthek-distribution model and the red band with the gray gas model,which provides an accurate and efficient methodto model radiative heat transfer. Finally, EMSFSCK resultsobtained from direct on-the-fly evaluation of spectralproperties are compared with those evaluated using the database. The agreement between the two calculated results isexcellent, as shown in Fig. 12. This demonstrates the accuracy of the database in evaluating the spectral properties ofthe red and violet bands through interpolation and look-upappeared to be from central and south-east Asia, Africa orthe mainly Muslim Caucasus region, and brutally attacked them.

10

VIII. Computational E fficiency

To quantify the efficiency of thek-distribution method together with the databasing scheme,computational timeswere measured for heat transfer calculation for the Huygensstagnation line problem discussed above. The LBLmethod requires calculating emission and absorption coefficients at hundreds of thousands of wavelengths and solvingthe RTE for each of the wavelengths. The measured CPU-times for the LBL generation of spectral data and solutionof the RTE are given in Table 4. Since time required in the LBL method directly depends on the spectral resolutionat which spectral data are generated, results from two different resolutions are shown for comparison. The minimumwavelength resolution required for good accuracy is 0.05Å. The FSK method requires calculation ofk-distributionsfrom the database, calculating thea functions and solving the RTE at a number of quadrature points.

All calculations were performed on a single Pentium-Xeon 3.0 GHz processor with cache size of 4 MB. In thek-distribution method the database reading time is not included, as this is done only once and does not need to berepeated for each flow condition. Thek-distribution method shows considerable savings in computational time for thegeneration of spectral data, as compared to the LBL data for both bands and for spectral resolutions. The time savingis even more significant in the evaluation of the RTE, as the RTE is evaluated at only few quadrature points, comparedto hundreds of thousands evaluations in the LBL method. Since the red band contains a larger number of rotationallines than the violet band, the savings for the red band are more significant. Other higher order RTE solution methodsfor two- or three-dimensional geometries, e.g., the spherical harmonics and the discrete ordinate methods, are muchmore expensive than the tangent slab method, and one can expect enormous time savings when usingk-distributions.

IX. Conclusions

A multi-scalek-distribution model was presented for the radiating bands of CN in thermodynamic nonequilibrium.State-of-the-art spectroscopic data for the ro-vibrational spectrum were collected from the literature. The red bandofCN was found to be optically thin while self-absorption was observed to be important for the violet band. The overlapbetween the two bands was found to be negligible. To model self-absorption for the violet band accurately using theMSFSCK model, a methodology for scaling the spectral lines was presented. A new emission-weighted MSFSCKwas developed that provides better agreement with LBL results. An efficient and compact database of precalculatedk-distribution was generated. Heat transfer results for a 2 cell problem and the stagnation line flow field of the Huygensspacecraft show that the new model provides very accurate heat transfer results for gas mixtures at a small fraction ofthe cost of LBL calculations.

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2007, pp. 24–32.2Liu, Y., Prabhu, D., Trumble, K. A., Saunders, D., and Jenniskens, P., “Radiation modelling for rhe reentry of the Stardust sample return

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Shock Layer Radiation in Air,”AIAA Paper No. 2008-1246, 2008, 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada.4Olejniczak, J., Wright, M., Prabhu, D., Takashima, N., Hollis, B., and Zoby, E. V., “An Analysis of the Radiative HeatingEnvironment

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10Walpot, L., Caillaut, L., Molina, R., Laux, C., and Blanquaert, T., “Convective and Radiative Heat Flux Prediction of Huygenss Entry onTitan,” Journal of Thermophysics and Heat Transfer, Vol. 20, No. 4, 2006, pp. 663–671.

11Hollis, B. R., Striepe, S. A., Wright, M. J., Bose, D., Sutton, K., and Takashima, N., “Prediction of the Aerothermodynamic Evnironment ofthe Huygens Probe,”AIAA Paper No. 2005-4816, 2005, 38th AIAA Thermophysics Conference, Toronto, Ontario.

12Bose, D., Wright, M. J., Bogdanoff, D., Raiche, G., and Allen, G., “Modeling and Experiment Assessment of CN Radiation Behind a StrongShock Wave,”Journal of Thermophysics and Heat Transfer, Vol. 20, No. 2, 2006, pp. 220–230.

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13Johnston, C. O., Gnoffo, P. A., and Sutton, K. A., “Influence of Ablation on Radiative Heating for Earth Entry,”Journal of Spacecraft andRockets, Vol. 46, No. 3, May-June 2009, pp. 481–491.

14Laux, C., Winter, M., Merrifield, J., Smith, A., and Tran, P.,“Influence of Ablation Products on the Radiation at the Surface of a BluntHypersonic Vehicle at 10 km/s,” AIAA Paper No. 2009-3925, 2009, 41st AIAA Thermophysics Conference, San Antonio, Texas.

15Silva, M. L., Sobbia, R., and Witasse, O., “Radiative Trail of The Phonenix Entry,”AIAA Paper No. 2009-1032, 2009, 47th AIAA AerospaceSciences Meeting Including The New Horizons Forum and Aerospace Exposition, Orlando, Florida.

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Quantitative Spectroscopy and Radiative Transfer, Vol. 108, 2007, pp. 106–125.19Wright, M., Olejniczak, J., Walpot, L., Raynaud, E., Magin,T., Caillaut, L., and Hollis, B., “A Code Calibration Study for Huygens Entry

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Emission, and Multiple Scattering in Vertically Inhomogeneous Atmospheres,”Journal of Geophysical Research, Vol. 96, No. D5, 1991, pp. 9027–9063.

24Goody, R. M. and Yung, Y. L.,Atmospheric Radiation – Theoretical Basis, Oxford University Press, New York, 2nd ed., 1989.25Modest, M. F. and Zhang, H., “The Full-Spectrum Correlated-k Distribution For Thermal Radiation from Molecular Gas–Particulate Mix-

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and Radiative Transfer, Vol. 48, 1992, pp. 187–203.28Rivière, P., Soufiani, A., and Taine, J., “Correlated-k and Fictitious Gas Model for H2O Infrared Radiation in the Voigt Regime,”Journal of

Quantitative Spectroscopy and Radiative Transfer, Vol. 53, 1995, pp. 335–346.29Rivière, P., Scutaru, D., Soufiani, A., and Taine, J., “A Newc− k Data Base Suitable from 300 to 2500 K for Spectrally Correlated Radiative

Transfer in CO2–H2O Transparent Gas Mixtures,”Tenth International Heat Transfer Conference, Taylor & Francis, 1994, pp. 129–134.30Zhang, H. and Modest, M. F., “A Multi-Scale Full-Spectrum Correlated-k Distribution For Radiative Heat Transfer in InhomogeneousGas

Mixtures,” Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 73, No. 2–5, 2002, pp. 349–360.31Zhang, H. and Modest, M. F., “Scalable Multi-Group Full-Spectrum Correlated-k Distributions For Radiative Heat Transfer,”Journal of

Heat Transfer, Vol. 125, No. 3, 2003, pp. 454–461.32Bansal, A., Modest, M. F., and Levin, D. A., “Multigroup Correlated-k Distribution Method for Nonequilibrium Atomic Radiation,” Journal

of Thermophysics and Heat Transfer, Vol. 24, No. 3, 2010, pp. 638–646, DOI: 10.2514/1.46641.33Bansal, A., Modest, M. F., and Levin, D. A., “Narrow-band k-distribution Database for Atomic Radiation in Hypersonic Nonequilibrium

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mogeneous Gas Mixtures,”Journal of Heat Transfer, Vol. 130, 2008, pp. 082701–1–082701–8.37Wang, A. and Modest, M. F., “High-Accuracy, Compact Database of Narrow-Bandk-Distributions for Water Vapor and Carbon Dioxide,”

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State University, Department of Aerospace Engineering, 2006.

List of Figures

1 Spectrum of CN-violet band; nCN=1.0x1015 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Nonequilibrium Planck function; Tr=15000 K, Tv=10000 K . . . . . . . . . . . . . . . . . . . . . . 153 Normalized absorption line strengths for selected rotational lines . . . . . . . . . . . . . . . . . . . . 164 AverageS-profiles for the two scales . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 165 Spectrum of absorption coefficient for the two scales . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Percentage error in cold wall heat flux with varying reference temperature . . . . . . . . . . . . . . . 177 Overlap factor for emission from Scale 2 and absorption by entire gas . . . . . . . . . . . . . . . . . 178 Flow field of the Huygens spacecraft . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 179 Heat source along the stagnation line for the CN (violet) band . . . . . . . . . . . . . . . . . . . . . 1710 Heat source along the stagnation line for the CN (red) band. . . . . . . . . . . . . . . . . . . . . . 18

12

11 Heat source along stagnation line for a combination of violet and red band . . . . . . . . . . . . . . . 1812 Comparison of direct FSK results with database results . .. . . . . . . . . . . . . . . . . . . . . . . 18

Scale 1 Scale 2

v′′ J′′ v′′ J′′

0 < 50 1 > 50

1 < 50 1 > 50

2 < 30 2 > 30

>2 all

Table 1. Scales of absorption coefficient for CN-violet band

Table 2. Datapoints for the Narrow-band Database

Parameter Range Number

T(K) 2000− 20000∗ 32

Tv(K) 2000− 20000(Tv > TV) 32

Tr (K) 2000− 20000(Tr > TV) 32

φ0

5000< T

Model Single Scale Two Scale Two Scale

Model No Overlap (λm = km) With Overlap (λm > km)

LBL 289.55 305.53 289.55

MSFSCK 267.94 294.81 286.93

EMFSCK 291.02 308.20 300.9

Table 3. Cold wall heat flux

Table 4. Computational Time Comparison (s)

LBL Spectral LBL LBL FSK FSK

resolution data RTE data RTE

CN (Violet) 0.01Å 4.54 184.28

0.05Å 2.05 36.88 .02 0.05

CN (Red) 0.01Å 28.48 1198.5

0.05Å 10.86 241.50 .02 0.05

14

λ (A)

κ λ(c

m-1

)

3000 3500 4000 4500 500010-6

10-5

10-4

10-3

10-2

10-1

100

101

T=10,000 KT=2,000 K

∆v=-1

∆v=2

∆v=1

∆v=0

∆v=-2

Figure 1. Spectrum of CN-violet band; nCN=1.0x1015 cm−3

λ (A)

I bλne

W/c

m2-µ

-sr

3000 3500 4000 450025000

30000

35000

40000

45000

Figure 2. Nonequilibrium Planck function; T r=15000 K, Tv=10000 K

15

T (K)

S(T

)/S

(12

000)

4000 6000 8000 100000

5

10

15

20

25

v=7, J=2

v=0, J=2

v=1, J=2

v=3, J=2

T (K)

S(T

)/S

(12

000)

4000 6000 8000 100000

0.5

1

1.5

v=7, J=2

v=7, J=100

v=0, J=100

v=2, J=100

v=1, J=100

Figure 3. Normalized absorption line strengths for selected rotational lines

T (K)

Sij

2000 4000 6000 8000 100000

2

4

6

8

10

Scale 1Scale 2

Figure 4. AverageS-profiles for the two scales

λ (A)

κ λ(c

m-1

)

3000 3500 4000 4500 500010-6

10-5

10-4

10-3

10-2

10-1

100

101

Scale 2Scale 1

Figure 5. Spectrum of absorption coefficient for the two scales

16

T (K)

%er

ror

4000 6000 8000 10000

-10

-8

-6

-4

-2

0

2

MFSCKEMSFSCK

Figure 6. Percentage error in cold wall heat flux with varying refer-ence temperature

k2 (cm-1)

λ 2(c

m-1

)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

MSFSCKEMSFSCK

Figure 7. Overlap factor for emission from Scale 2 and absorption byentire gas

Distance from the body (cm)

T(K

)

n CN

(m-3

)

-12 -10 -8 -6 -4 -2 00

2000

4000

6000

8000

10000

12000

0

2E+20

4E+20

6E+20

8E+20

1E+21

T=TrTV=TenCN

Figure 8. Flow field of the Huygens spacecraft

Distance from the body (cm)

∇.q

(W/c

m3 )

-10 -8 -6 -4 -2 0-10

0

10

20

30

40LBLEFSCKGray

erro

r(W

/cm

3 )

-2

0

2

4

6

8

10

Figure 9. Heat source along the stagnation line for the CN (violet)band

17

Distance from the body (cm)

∇.q

(W/c

m3 )

-10 -8 -6 -4 -2 00

2

4

6LBLEFSCKGray

erro

r(W

/cm

3 )

-0.1

0

0.1

0.2

Figure 10. Heat source along the stagnation line for the CN (red) band

Distance from the body (cm)

∇.q

(W/c

m3 )

-10 -8 -6 -4 -2 0-10

0

10

20

30

40LBLEFSCK (tot, 16 RTEs)EFSCK (sum, 32 RTEs)EFSCK (mix, 17 RTEs)

erro

r(W

/cm

3 )

-0.5

0

0.5

1

1.5

Figure 11. Heat source along stagnation line for a combination ofviolet and red band

Distance from the body (cm)

∇.q

(W/c

m3)

-10 -8 -6 -4 -2 0-5

0

5

10

15

20

25

30

35

40

CN (V) DirectCN (V) DatabaseCN (R) DirectCN (R) Database

Figure 12. Comparison of direct FSK results with database results

18

IntroductionRadiation modelsBasic EquationsMulti-scale Planck function-weighted full-spectrum k-distribution: MSFSCKEvaluation of overlap factorScaling of absorption coefficients

Multi-scale emission-weighted full-spectrum k-distribution: EMSFSCKDatabase of emission-weighted full-spectrum k-distributionsSample CalculationsTwo cell problemStagnation line flow field of Huygens spacecraft

Computational EfficiencyConclusions