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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;
[email protected]†Shaffer and George Professor of Engineering;
[email protected]
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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,
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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.
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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
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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)
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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)
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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)
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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.
References1Park, C., “Calculation of Stagnation-Point Heating
Rates Associated with Stardust Vehicle,”Journal of Spacecraft and
Rockets, Vol. 44,
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
capsule,”AIAA Paper No. 2008-1213, 2008, 46th AIAA Aerospace
Sciences Meeting and Exhibit, Reno, Nevada.3Bose, D., McCorkle, E.,
Thompson, E., Bogdanoff, D., Prabhu, D. K., Allen, G. A., and
Grinstead, J., “Analysis and Model Validation of
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
for Aerocapture at Titan,”AIAA Paper No. 2003-4953, 2003, 39th
AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit,
Huntsville,Alabama.
5Wright, M. J., Bose, D., and Olejniczak, J., “The Impact of
Flowfield-Radiation Coupling on Aeroheating for Titan Aerocapture,”
Journalof Thermophysics and Heat Transfer, Vol. 19, No. 1, 2005,
pp. 17–27.
6Olejniczak, J., Prabhu, D., Bose, D., and Wright, M. J.,
“Aeroheating Analysis for the Afterbody of a Titan Probe,”AIAA
Paper No. 2004-0486, 2004, 42nd AIAA Aerospace Sciences Meeting and
Exhibit, Reno, Nevada.
7Park, C., Howe, J., Jaffe, R., and Candler, G. V., “Review of
Chemical-Kinetic Problems of Future NASA Missions, II: Mars
Entries,”Journal of Thermophysics and Heat Transfer, Vol. 8, 1994,
pp. 9–23.
8Kudiyavtsev, N. N., Kuznetsova, L., and Surzhikov, S.,
“Kinetics and Nonequilibrium Radiation of CO2/N2 Shock
Waves,”Journal ofThermophysics and Heat Transfer, Vol. 19,
2001.
9Grinstead, J. H., Wright, M., Bogdanoff, D., and Allen, G.,
“Shock Radiation Measurements for Mars Aerocapture Radiative
HeatingAnalysis,”Journal of Thermophysics and Heat Transfer, Vol.
23, 2009, pp. 249–255.
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.
11
-
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.
16Whiting, E., Park, C., Liu, Y., Arnold, J., and Paterson,
J.,“NEQAIR96, Nonequilibrium and Equilibrium Radiative Transport
and SpectraProgram: User’s Manual,” Nasa reference publication
1389,NASA/Ames Research Center, Moffett Field, CA 94035-1000,
December 1996.
17Laux, C. O.,Optical Diagnostics and Radiative Emission of Air
Plasmas, Ph.D. thesis, Stanford University, Stanford, CA, 1993.18da
Silva, M. L., “An adaptive line-by-line–statistical model for fast
and accurate spectral simulations in low-pressure plasmas,”Journal
of
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
Aeroheating,”AIAA Paper No. 2006-382, 2006, 44th AIAA Aerospace
Sciences Meeting and Exhibit, Reno, Nevada.20Patch, R. W.,
Shackleford, W. L., and Penner, S. S., “Approximate Spectral
Absorption Coefficient Calculations for Electronic Band Systems
Belonging to Diatomic Molecules,”Journal of Quantitative
Spectroscopy and Radiative Transfer, Vol. 2, 1962, pp.
263–271.21Hartung-Chambers, L., “Predicting Radiative Heat Transfer
in Thermochemical Nonequilibrium Flow Fields,”NASA Technical
Memoran-
dum 4564, 1994.22Johnston, C. O., Hollis, B. R., and Sutton, K.,
“Radiative Heating Methodology for the Huygens Probe,”AIAA Paper
No. 2006-3426, 2006,
9th AIAA /ASME Joint Thermophysics and Heat Transfer Conference,
SanFrancisco, California.23Lacis, A. A. and Oinas, V., “A
Description of the Correlated-k Distribution Method for Modeling
Nongray Gaseous Absorption, Thermal
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-
tures,”Journal of Heat Transfer, Vol. 124, No. 1, 2002, pp.
30–38.26Modest, M. F., “Narrow-band and
full-spectrumk-distributions for radiative heat
transfer—correlated-k vs. scaling approximation,”Journal
of Quantitative Spectroscopy and Radiative Transfer, Vol. 76,
No. 1, 2003, pp. 69–83.27Rivière, P., Soufiani, A., and Taine, J.,
“Correlated-k and Fictitious Gas Methods for H2O near 2.7µm,”
Journal of Quantitative Spectroscopy
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
Flows,” Journal of Heat Transfer, 2010, in print.34Caillault,
L., Walpot, L., Magin, T. E., Bourdan, A., and Laux, C. O.,
“Radiative Heating Predictions for Huygens Entry,” Journal of
Geophysical Research, Vol. 111, 2006.35Modest, M. F.,Radiative
Heat Transfer, Academic Press, New York, 2nd ed., 2003.36Pal, G.,
Modest, M. F., and Wang, L., “Hybrid Full-Spectrum
Correlatedk-Distribution Method for Radiative Transfer in Strongly
Nonho-
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,”
Proceedings of the ICHMT 4th International Symposium on
Radiative Transfer, edited by M. P. Mengüç and N. Selçuk,
Istanbul, Turkey, 2004.38Johnston, C. O.,Nonequilibrium Shock-Layer
Radiative Heating For Earth and Titan Entry, Ph.D. thesis, Virginia
Polytechnic Institute and
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