IBRA'I'IONAL. NONEQUILIBRIUM. EFFECTS. IN RAREr_IED ..... 9 Lordi, J.A. and
Mates, R.E., "Rotational. Relaxation in. Nonpolar Diatomic Gases", Physics of ...
https://ntrs.nasa.gov/search.jsp?R=19900003182 2017-10-05T19:27:39+00:00Z
/'/1 ROTATIONAL
AND ,?IBRA'I'IONAL IN RAREr_IED,
ORIIGIINAL
PAGE
{IF
QUALITY
POOR
NONEQUILIBRIUM
HYPERSONIC
NASA
Ames Research
Moffett
for an investigation
transfer
is calculated
tion Monte Carlo method. developed lational
energy
and rotational
modes
brational dence
relaxation
predicted
between
time to follow
and chemical
ionization
and radiation
degree of thermal Larger, obtained are found included
when
= average
amount of chemical
veloped
ulates
g
= relative
velocity
of collision
g*
= characteristic
mr
= reduced
mass of collision
n
= number
density
on a probabilistic
tant aspects
probabilities
thermal
energy
is
ciated
Vehicle
in which
with
the various
to reduce
computational
of energy
is a constant
= reference
temperature
= Variable
Hard Sphere Member
(NASA-CR-I_25s9) NON_,_UILI_RIUM
degrees
HYPEKSO'4IC
FLOW
of the DSMC
to model
the temperatures
modes
asso-
of the gas (transare unequal.
In
such phenommanner
between
for internal
in which
these various mechanism.
If
energy transfer,
p CSCL
exchange
scheme,
significant
have been calculated
one of the most interesting
Orbital
G3/tJ2
surround-
such as the Space Shuttle or
Transfer
Vehicle
NgO-
O}.A
in shock
applications
has been to the flowfield
space vehicle
AND V I aRAI" IONAL RARCFIEI') 11
model z .
nonequilibrium
technique
an Aero-Assisted
AIAA
Corp.)
is the ability
impor-
in plumes 3 .4 and for compressions
waves 5,6. Recently,
for VHS model
(Eloret
col-
exchange
this energy
of thermal
for expansions
for ZR
ROTATIUNAL _FFECT_ IN
level by calculating
overheads,
is accepted
for sim-
in a simplistic
phenomenological
ing a hypersonic Scientist,
technique
for each type of transfer
collision
de-
then post collision values are sampled from the local equilibrium distribution function in accordance with the Larsen-
By employing
VHS
energy
modelled
(DSMC)
One of the most
calculations
Borgnakke
temperature
basis.
of DSMC
the probability
velocity
T,,f
time
The method
lational,rotational,vibrational,electronic)
modes function
numerical
flowfields.
nonequilibrium
a particular
temperature
time
for relaxation time
Monte Carlo method
the flow at the molecular
lisions
in the results
velocity distribution
relaxation
correction relaxation
of low density
ena are usually
velocity
for
cross section
vibrational
by Bird I is an important
the calculation
order
= relative
* Research
excitation
The Direct Simulation
are in-
nonequilibrium.
Orbital Transfer
exchange
Introduction
affect the
of internal
f(g)
= characteristic
re-
in the flowfield.
exchange
energy
vehicle.
coefficient
= translational
= effective
8000K
The introduc-
differences
exchange
VHS model
have been chosen such that
observed
energy
in VHS model collision cross section
representative
= heat-transfer
T*
= parameter = reference
calculations
Cn
T
of vibrational collision rate
rv
effects
molecular
= probability = molecular
into the DSMC
energy
= Aero-assisted
/./
17re f
Nomenclature AOTV
of rotational
numbers
ranging,
in the calculation
= probability
= Park's empirical = total vibrational
nonequilibrium
a significant
_R
Tp
is found to significantly
with the different
Zt
at moderate
space
i
number
infinite temperature collision number = translational collision number
these
nonequilibrium
and more widely
for species
collision
_T
may be neglected.
tion of these new models
ZR
= rotational
depen-
of about
of a hypersonic
cluded while the flow conditions
= mass fraction
w
to include an empirical
by making
streamline
Xi
= Landau-Teller
collision
is assessed
Both thermal
theory
/
Center
the vi-
allows
time. The effect of introducing
dependent
of the stagnation
and vibra-
the temperature
in excess
model is extended
sult for the relaxation
technique
This model
by the Landau-Teller
the vibrational
///
for simulating
the translational
For temperatures
temperature
is made of a recently
A new model
is also explained.
temperatures.
FLOW
(ZR)o =
Simula-
model that deals with the trans-
modes.
the transfer of energy tional
Description
exchange
into the methods
in the Direct
" /_ _
Field, CA 94035
Abstract Results are reported
-
EFFECTS
Iain D. Boyd * Eloret Institute
IS
by which energy
_iS
12,,o
oncl,as 0230470
(AOTV)
as de-
scribed inRefs.5and6.Thedegree ofthermal nonequilibriumin suchenergetic flowscanhavea significant impact ontheshockstandofflocation,ontheamount of chemicalactivity,andontheconvective andradiative heatload tothevehicle.It is therefore ofgreatimportance thatthe models employed in thecalculation ofenergytransfer be physically realistic.ForAOTVcalculations theprobabilitiesassumed forrotational andvibrational energy exchange areusuallytakentobe 0.2 and 0.02 respectively. The value
This expression
where en is the probability
of energy
for the transfer
translational
modes,
of rotational
experimental
studies
is an estimate
the probability much
energy
of about
exchange,
of the maximum
of energy
smaller
has been derived
for temperatures
the case of vibrational 0.02
energy
except
transfer
from
1000K.
the probability
possible.
would
be expected
collisions.
ea(Za)_
r(2
27
has been derived
-w)
_'2kT"_
_ tr½
k --77
T
+ rC1 and rotational
collision
of
is the relative
to be
Such a formulation
over
in Ref. 10 and is given as
In
In reality,
at very high temperatures.
all possible
must reduce to Eq.(1) when integrated
number,
m,
velocity.
in which
between
the
mass of collision,
In the derivation
the Variable Hard Sphere collision employed
(2)
Zt is the translational
is the reduced
collision
transfer
+
and g
of Eq.(2)
model of Bird u has been
the molecular
collision
rate is given by
It is the !
purpose
of the present
work to describe
implementing
energy
pendent
the local
upon
than calculating
exchange lisions.
translational
is obtained
incorporated
should
by making
ensure
of energy
the probability
col-
that any nonequilibmodes
The effect
is assessed
along
vehicle.
the stagna-
First,
of internal
are
of introduc-
technique
calculations
of a hypersonic
for calculating described.
expressions
over all simulated
into the DSMC
one-dimensional
tion streamline
continuum
with the translational
into the calculation.
ing such expressions
Rather
at each cell in
time 7 , each probability
associated
for
are de-
temperature.
by integrating
This procedure
rium attributes
which
temperature
and then employing
the relaxation
the procedures
probabilities
the translational
the simulation to evaluate
transfer
the methods
energy transfer
are
=
-\mr
where n is the number section
In a separate
employed
in the calculation obtained
of Rotational
The following collision
number
Energy
approximate
expression
was obtained
by Parker s
to give the correct behavior it has been shown
gives good agreement ness for a range is observed
for the rotational
transition
.__(__)
where T* is the characteristic ular
potential,
Parker's
and
expression
a large number nature
Parker's
is derived
of the intermolecvalue.
from an analysis
In the current
so as to obtain results
rate Parker's an expression be developed
dependent
with the more
of Lordi and Mates 9 wh6 performed
classical
for the probability as a function
of energy
have some is evaluated
equipartition
of the thermal
the fact that the collision having
collision
probability
is therefore
smaller
provides
bet-
with calculamodel based
in Fig.
1, even at
transfer
for niin Cn
impact
energy
on the calculated
at each collision, modes
results.
it is found
is not achieved
probability
is biased
translational
employed,
towards those
energies.
and is obtained
that
due to
The mean
for each cell in the computational
do-
by averaging
Eq.(2) over all collisions.
traProbability
of Vibrational
Energy
To incorpo-
bility of vibrational-translational
energy
technique
transfer
13 it
to 0.2 and the variation
molecules,
collision
in Ref.
sophisticated As shown
In a pure gas of diatomic
into the DSMC
of the relative
will therefore
collisions
obtained
of rotational
different
When Eq.(2)
main
formulation
between
the best correspondence
formula
rigorous
work the value of (Zn)o¢
and those of Lordi and Mates.
continuum
While involving
the temperature
is in agreement
calculations.
is chosen
temperature
+ _.) -fT'
is the limiting
of assumptions,
of Eq.(1)
treatment jectory
(Zn)_
_- + (._
rates.
the probability
trogen is certainly
1+
a more
In
shock wave thick-
Indeed,
than the results
employed
1000K
number. simple
It
formulation
et a113 that Eq.(1)
with experimental
that Parker's
with ¢R--0.2.
up to about 4000K.
by Lumpkin
of Mach
on rotational
(1)
shock wave and
et a112 that Parker's
addition,
was
to offer better correspon-
data than those
tions which
ZR =
of a standing
has been found by Belikov
Transfer
(Z n)oo
m, Eq.(2)
dence to the experimental
appears
cross
the interaction
set of calculations
were found
_' (3)
a collision
T, cf, and w defines
potential.
ter correspondence Probability
density, cr_cf defines
at a temperature
the results
[(2-w)T,,s/T]
expressed
Transfer the mean exchange
probamay be
as
must
velocity.
--1L ¢_
v
_°r_¢_(g) f(g)dg
(4)
where¢_is theprobability ofenergy exchange forthecollisionwithrelativevelocityg,fig) is therelativevelocity distribution function,andr_ is thevibrational relaxation time.In thefollowing,wewishto choose a formfor¢_ suchthatthevibrational relaxation timeisgivenbytheexpression obtained byMillikanandWhite14whocorrelated anumber ofexperimental results. Theprobability ofenergy exchange is assumed tobeoftheform -o"
Cl
¢_ = _og
exp(--:--) g
where
c_and Zo are constants,
locity.
The form of Eq.(5)
sumed
by Landau-Teller
tegral obtained
(5)
and g* is a characteristic
is identical
by substituting
Eq.(5)
into Eq.(4)
be evaluated approximately using the method descent 15. The result obtained with this method the temperature sion number
dependent
is obtained
nature
as-
The incan only
of steepest reveals that
of the vibrational
colli-
when
collision
(6)
that employed
description
of the preceding
with the expressions characteristic hard sphere
for the constant
velocity
analysis
interaction
in Ref.
expression
Cv = 1.20
energy
exchange
For a
An expression which over all collisions is
-
_P
is
43300
× lO-ng3exp(
)
(7)
velocity.
obtained
with the DSMC
are plotted
result.
Due to the approximation
the results
in Fig.
are not identical,
lying less than a factor The vibrational generally 8000K.
troduced
used in evaluating
with the DSMC
of hypersonic
To better simulate
temperature
always
time is implemented
gration
is being performed
empirical
flow surrounding may be as high
temperatures
Park 16 in-
relaxation
Having
derived
(9)
for to=0.25
3 for the modiIt is seen that the
which allow
collision
agreement.
the effect of introducing DSMC
lem chosen
calculations
follows
through stream
streamline
enter
conditions.
molecules
is assumed
accomodation.
reflected
the wall
achieved
back
from
moves
velocity to conserve
As the molecules
molecules
wave
forms
A steady
investigated
try and allows
por-
This procedure
the atmosphere
try. The exit phase is significantly
which
to the square
and momentum
represent rather
has
when the
between
the
the exit of the than the reen-
less energetic
us to neglect ionization
are
flow is
from a downstream proportional
component.
of
to be a diffuse reflec-
a shock
energy
is open
of the up-
molecules are removed in the region lying shock standoff location and the surface.
through
The
to one
end is the surface
away from the surface.
by removing
of the normal
domain
representative
At the opposite which
vehicle.
flow may be reduced
tor with full energy
gradually
space
that of Bird 17 who has shown
One end of the computational which
into
The prob-
is the flow along the stagna-
in front of a hypersonic
that the stagnation dimension.
these developments
is now assessed.
for consideration
procedure
both the rota-
numbers to vary with tem-
energetic
space vehicle 1 rp = r_r_2
expressions
perature,
The conditions
time,
by av-
Calculations
been shown
(8)
the results in Fig.
re-
that the inte-
are in excellent
tion of the flow with probability
correction
To = rLr + "rp where TLT is the Landau-Teller
of about
it is of the
method
To ensure
time of Eq.(8).
results
a temperaEq.(ll)
is independent
correctly,
relaxation
integrated
(ll)
From
in the DSMC
are shown
and continuum
(10)
to the vibrational
over all collisions.
molecules
when
to include
modification
Eq.(11)
'_
l
the result
eraging
fied vibrational
then
-- to) kTrey J
for cry if desired.
Park's
the space vehicle,
and White is
the vibrational-translational
at such elevated
the following
Eq.(4),
results
time of Millikan
the translational
process
employing
with the continuum
to be valid up to temperatures
In the consideration
as 40,000K.
technique
of 2 above the exact solution.
relaxation
recognized
space vehicles, exchange
2 together
[2(2
laxation
solution Results
to Eq.(10)
that it is possible
form
tion streamline
Eq.(7)
energy exchange.
to)T,,f]
reduces
seen that for hard spheres, collision
to
is obtained
Vl20"ref
be noted
ture dependent
manner
is taken to be constant,
V,_crTefF(2_to)[T/(2-
tional and vibrational
probability for nitrogen
to the vibrational
in a similar
1
together
15.
(to---0), the instantaneous
of vibrational-translational
cross-section
Zo in Eq.(5) and the
g* are included
This correction
cry
It should
and _ is the av-
for rotational-translational
the following
Cp=
cross-section
may be evaluated
If the excitation
DSMC A detailed
velocity.
number
and nitrogen c_ = 3 - 2to
excitation
erage molecular
ve-
to that originally
with c_ set equal to zero.
a_ is the effective
than reen-
and radiation
effects
inthecalculations. An and the free stream The wall surface non-catalytic. 02,
altitude
of 90km has been selected
flow conditions
is assumed
are listed
to be constant
A five species Ref.
recombination
18).
ysis. Calculations cases.
have been completed
The first of these follows
dependent
ergy transfer
expression
energy
as a function
exchange
of temperature
Eq.(2) and Eqs.(6,11).
work by set-
the temperature
in Eq.(2)
Finally,
and
number
by Fritzsche
_,
and Cukrowski
While the primary tigation
of thermal
of chemical
shown in Fig. of chemical
are calculated species
of the parameters
using
used in the
tational energy
ious reactions purpose
energy
to include
Unfortunately,
Initially,
available
to participate
reactions
internal
chemical
nonequilibrium.
when
be
steric factor
that the use of the rotational
standoff
distance,
local density
Ref.
which is defined
temperature
important. oxygen
are observed
the vehicle.
have been performed
oxide,
that the shock
in the simulation.
value,
is located
at
equilibrium
of about
by the fact
molecules
temperature result
that the various at the upstream
location
is also found to be the case for the oxygen
the standoff
amination
at the upstream
identical
conditions
of highly
such that the steady
state is introduced
This allows
investiga-
that the large translational
molecules.
boundary
energetic,
energy
is caused
However, and this
molecules.
Ex-
for each species revealed
temperature
backscattered
ex-
boundary
are in fact in
for nitrogen,
The present
obtained
any
then it might be
modes
surface.
to be 0.11m for the case where the exchange probare constant. The other cases are then run under
to produce
at this location. thermal
prob-
0.2 from
in Fig. 4a would lie
by forcing
of the results
for
that, up to this
quantities
As the rotational
and oxygen
closer to the trarlslational
become
than those for
is quite large at the upstream
that the rotational
Fig. 7 reveals
surface
a value for the exchange
collisions
for both the nitrogen
It should increases
probabilities
greater
is de-
0.29 at the free-stream
until a distance
of
distance abilities
at the same point in the simulation.
of about
This is explained
probability
transfer
increases.
of the vehicle
in the calculation
is 6 times the free stream
of 0.11m from the vehicle
For nitric
energy
probability
to be everywhere
is not obtained
low free-stream
of the exchange
with a maximum
boundary. ability
effects
values
point, NO gas exists in insufficient
as the point at which the
calculations
The
of
16 shows
a distance
of rotational
that the energy
again as the cooling
the flow, so let us
value of 0.2 and gradually
exchange
change
of 90km,
the probability to the nominal
are shown
It may be seen from
At the relatively
be noted
expected At an altitude
nitrogen
for
This effect
in the calculations
of the importance
and Discussion
probability.
creases as the translational
nitrogen
of such a pro-
Bird's
energy contributions
Results
of course
from the vibrational
will at least give an indication
including
energy
into the steric
it would
contributions
It is proposed
very close
reduced
of Fig. 6 in which the ro-
dominates
temperature,
all of the rotational
obtained
mod-
that the maximum
Fig. 4b that molecular
The frac-
on the flow-
exchange
by examination
on this data.
exchange
5 the rotational
is significantly
species.
concentrate
a tem-
energy
It is found
for each of the three molecular
in the var-
by Bird 2° in his general
to be incorporated
expensive
for re-
of introducing
In Fig.
attained
probabilities
amount
with the two different
are shown.
may be explained
The
along the stagnation
rotational
obtained
temperature
tational
ro-
basis, and where re-
the implementation
is numerically
is employed. energy
by allowing
For the dissociation
modes. cedure
allowed
in each collision
more realistic
the importance
The effect of this assumption
field is also analysed factor.
to the energy
on a restricted
are those employed
code.
available
effects,
cannot be ignored.
steric factor for the VHS model.
tions of rotational
methods
rotational
25,000K.
with the profiles re-
A significant
the effect
dependent
profiles
a signif-
in the flowfield.
into the calculations.
temperature elling
consider
the case of the variable
nonequilibrium
contributions
by Bird's
is that proposed
focus of the present work is the inves-
action are only included quired
in Eq.(2),
is observed
and species
en-
of each of the five species is
4b for the same case.
activity
probability
of about
5 and 6. The variation
Let us now perature
As expected,
agreement
of the mass fraction
en-
19.
nonequilibrium
in Refs.
streamline
and
probabilities
required
ported
for the case
exists, with the translational
a peak temperature
shown offer excellent
for rotational
energy exchange models are given in Table 2 for rotation, and in Table 3 for vibration. The value for the translational relaxation
reaching
pro-
and vibrational
are both kept constant.
results
in the various
are plotted
of rotational
icant degree of nonequilibrium mode
distance
4a, the temperature
streamline
the probabilities
both the rotational
and molecular
Details
ergy exchange
from the anal-
In Fig.
files along the stagnation
(see
involved,
in the standoff
undertaken.
in which
reactions
the previous
calculations
(N2,
for three different
Secondly,
is introduced.
the vibrational
chemical
have been omitted
ting _bR=0.2 and _bv--0.02. species
is employed
Due to the low densities
reactions
tion of any movement 1.
at 1000K and is
gas mixture
NO, N, O) with 19 different
Table 2(a),
in Table
obtained
in Fig. 4a
by a very small fraction atoms
and nitric
oxide
InFig.8,thevibrational temperature profilesareshown icalnonequilbirium hasbeenincorporated intocontinuum forthetwoenergy exchange probability models. Asinthe calculations21.22 itis unsatisfactory forthediscrete particle caseofrotational relaxation, it isfoundthatthemaximum solution methods tolagbehind in thisarea.Thereis areal vibrational temperature is significantly reduced.Thein- requirement forimprovement inthemodelling ofchemical dividualexchange probabilities foreachmolecular species reactions in theDSMCtechnique. Inanattempt to assess areplottedin Fig.9. Carefulattention should begivento theeffectof theparticular chemical modelemployed, the thelabelling oftheaxes.Onlyaportionofthedomain mod- same flowconditions havebeenrecalculated withthetotal elledalongthestangation streamline is included.Beyond rotational energynowbeingavailable forreaction. distances ofabout 0.25mfromthevehicle, thecollision rate In Fig. 11,themassfractionof molecular nitrogen is is toolowto allowanyvibrational energyexchange for shownforthecases whereconstant exchange probabilities thevariable probability model.It istoberemembered that employed forbothrotational andvibrational energy. theDSMCtechnique employs a discrete number of simu- were It isseenthatareduced amount ofdissociation isfoundfor latedmolecules inthesimulation whichundergo a discrete theincreased internalenergy contribution. Thiswouldbe numberof collisions.Forcalculated exhange probabiliexpected as Bird'ssteric factors a ssume perfect equilibrium tieswhichareverysmall,e.g. 10-4, it is therefore diffiforthevarious thermal modes. Therefore, whentheinternal culttoaccurately resolve theexchange phenomenon. Inditemperatures liebelowthetranslational result,whichisthe rectcontrast totheresultsfoundfortherotational exchange casefortheflowalongmostofthestagnation streamline, a probabilities, themaximum translational temperature gives smaller rateofreaction will occur. risetothemaximum exchange probabilities. ForbothniThethermal nonequilibrium effects discussed abovefor trogenandoxygen thesemaximum valuesfallwellbelow 0.02whichisthevaluepreviously assumed. It issatisfying therestricted rotational energy contribution tochemical retonotethatthevaluesfornitricoxidelieweUabove those actions wererepeated in thenewsetofcalculations. Howoftheothermolecules. Thisisthetrendexpected fromex- ever,largerdifferences forotherflowquantities weredisInFig.12areshownthemass fractions foratomic periment, andshows thatsuchbehavior cannowbeincor- cernible. porated intotheDSMCcalculations through application of andmolecular oxygenforthetwocases whereconstant and thevariable exchange probability model. variable exchange probabilities wereemployed. Thereisa amount of molecular oxygen present inthemixture It istherefore foundthattheintroduction ofthevariable smaller when thevariable m odels a reintroduced. Similar behavior exchange probabilities leadstoagreater degree of thermal fornitrogen, asshown inFig.13,buttoalesser nonequilibrium in theflowdueto thefactthatthecalcu- isobserved extent. T heexact e xplanation for thesetrendsiscomplilatedprobabilities arealwaysmuchsmaller thanthevalues bythepresence ofthenumber ofcompeting reactions of0.2and0.02normally assumed forenergy exchange with cated which areaccounted forinthe calculations. The differences therotational andvibrational modes respectively. However, thesolutions obtained withthevariable exchange atthemacroscopic level,onlya smalleffectis observed between probabilities arenowlarge enough t obenoticed in such through theintroduction ofthetemperature dependent modquantities asdensityandvelocitywhichare els.Oneexample isshowninFig.10wherethemassfrac- macroscopic plotted i n Figs.14and15respectively. It is foundthatthe tionof molecular andatomicnitrogen is compared. It is is greaterforthevariable probability calculations seenthatagreater amount of nitrogen atomsareproduced density in theregion lyingbetween t hefreestream andtheshock in thecasewherevariable probabilities areemployed. In location.Thereafter, thevariable resultis slightly Table 4 various results obtained fromthese simulations are standoff withtheconstant probability. In Fig. shown. It isfoundthatboththeshockstandoff distance and belowthatobtained 15,it is seen thatthevelocity solutions obtained withthe theheatcoefficient increase slightlywiththeintroduction of methods areclearlydifferent withthevariable refirstthevariable rotational exchange probability, andthen different sultlaggingbehind.Thesefindings areof a similarmagadditionally forthecorresponding modelforvibration. nitude to thedifferences noted by Moss etal in Ref. 18 It isperhaps surprising thattheintroduction ofthenew fordifferent sets of chemical reaction r ate constants. It is models hashadsuchlittleeffectontheimportant flowquanfoundthatfurtheruncertainty insuchcalculations tities.However, thisis almost certainly duetothefactthat therefore arises f romthetechniques employed in thesimulation of theinternal energymodes areonlyincluded in thechemiandchemical nonequilibrium effects. calnonequilibrium phenomenon onarestricted basis.Ata thermal timewhenthesignificant roleofinternalenergy in chemVarious globalresultsobtained withthesecalculations
arecompared in Table5 withthatpreviously obtained for tions previously undertaken at NASA Langley for the flow theconstant exchange probabilities. Theassumption of a along the stagnation streamline of the Space Shuttle Orbiter. fullrotational energy contribution forchemical reactions reReferences suitsinanincreased effectthrough theinclusion ofthevariableenergy exchange probability models. Theheat-transfer 1 Bird, G.A., Molecular Gas Dynamics, Clarendon Press, Oxford, 1976. coefficient, themassfractionof molecular nitrogen atthe surface, andtheshockstandoff distance areallaffected for 2 Borgnakke, C. and Larsen, P.S., "Statistical Collision Model for Monte Carlo Simulation of Polyatomic Gas Mixthedifferentmodelling methods. tures", Concluding The
calculations
of temperature change nificant
reported
and species
probabilities impact
in the simulation.
probabilities
obtained
tude smaller
than the constant
nificant amount cal reaction. energy
is included change
modes
models
value previously
in other flow quantities are found
quantities
a sig-
for each chemionly the rotational and is found
the variable
as the shock standoff
of the energy It is probable
calculations
more energetic
flowfields
entry of an AOTV
in detail
or Mars return
in expanding
vehicles.
and to
nonequilibrium
the role of the variable
temperature of radiation
4 Boyd,
the
In such
plays a sigeffects.
Acknowledgement
supplied
author
gratefully
by DrJ.N.Moss
acknowledges
Lip
Bird,
Flow
Journal
G.A.,
by Direct
of Spacecraft
I.D. and Stark,
Effects
Thruster
Plume",
of Mechanical
Aerospace
Engineering.
5 Moss,
J.P.W.,
in the Isentropic
Institution
and Brock,
FJ.,
Simulation
Monte
and Rockets,
Vol. 23,
mal
"Assessment Core
to appear
Design
tics and Aeronautics, 6 Dogra,
Part
70lynick,
in Ther-
Transfer
Vehicles,
J.N.,
and Simmonds,
Streamline
s Parker,
J.G.,
Diatomic
Gases",
in Astronau-
Flows",
A.L.,
"Direct
Flow for Hypersonic
Rent,
D., Moss, J.N. and Hassan, on AOTV
of Transi-
Conditions",
Orbital
Paper 8743405,
terbodies 1989.
of
AIAA, New York, 1985, pp. 113-139.
of Stagnation AIAA
Simulation
Reentry
of the
G: Journal
Volume 96 of Progress
V.K., Moss,
Reentry",
Satellite
in Proceedings
Engineers,
of Aeroassisted
edited by H.E Nelson,
of Impinge-
of a Small
J.N. and Bird, G.A., "Direct
January
1987.
H., "Influence
AIAA Paper 89-0311,
of AfJanuary
9 Lordi,
on the one-dimensional
and Vibrational
Physics
of Fluids,
Relaxation
Vol.
in
2, No. 4, 1959,
J.A.
and Mates,
Diatomic
R.E.,
Gases",
"Rotational
Physics
Relaxation
of Fluids,
in
Vol. 13, No.
2, 1970, pp. 291-308. 10 Boyd,
I.D.,
in Rarefied tion. 11 Bird,
"Rotational-Translational
Nonequilibrium
G.A.,
ing Context", Fisher,
information calcula-
Flows",
Energy submitted
Simulation
Gas Dynamics,
Transfer
for publica-
Relaxation
in Proceedings
on Rarefied
and Aeronautics,
in an Engineeredited by Sam S.
in Astronautics
and Aero-
New York, 1981, pp. 239-255.
A.E., Sharafutdinov, Rotational
to appear
Symposium
Carlo
74 of Progress
Part I, AIAA,
12 Belikov,
nautics
"Monte in Rarefied
Volume
nautics,
Jet", the
"Rotational
pp. 449-462.
"Nitrogen The
L.T.,
in
exchange
such as that around
of the vibrational
role in the determination
18, 1975,
re-
It is a require-
such conditions
flows,
Vol.
No. 4, 1986, pp. 363-367.
Nonpolar
for the
skirt of an AOTV, should be considered.
flow, the freezing
to de-
with atmospheric
the role of chemical
In addition,
be given
probabilities
will be more pronounced associated
Physics,
distance and
that the effects observed
ment for the future to investigate
probabilities
ex-
effect on
should
transfer
Nozzle
Simulation to
energy
energy
to have a pronounced
that consideration
scribed in this paper.
nificant
to be largest
coefficient.
It is concluded
in the present
energy,
Melfi,
tional Flow for Hypersonic
for the
to contribute
has been considered,
the implementation
aerobrake
assumed.
J.E.,
"Rocket
Control
en-
of magni-
investigation
are found
the heat-transfer
such flows.
in the amount
In the case where the total rotational
such important
consider
nonequilibrium
to that available
in the collision
of Computational
Carlo Method",
ment
are an order
are allowed
In the current
be important.
en-
exchange
models
of energy
contribution
and vibrational
the vibrational
for nitrogen
exchange
when the internal
exval-
of thermal
Indeed,
observed
energy
energy
can have sig-
The constant
due to a reduction
performed.
different
internal
calculations
3 Hueser,
larger than the values calculated
ergy transfer
The differences
the introduction
obtained.
The degree
increased
that
for rotational
are generally
Journal
pp. 405-420.
dependent
into DSMC
employed
is therefore
show
on the results
ues normally ergy transfer
Remarks
R.G. and Sukhinin, Time of the
Gas Dynamics, AIAA,
Measured
G.I., in Free
16th International Progress
Washington,
in Astro-
1989.
z3Lumpkin,EE.,Chapman, D.R.,andPark,C.,"A New Rotational Relaxation ModelforUsein Hypersonic Computational FluidDynamics", AIAA Paper89-1737, Buffalo,June1989. 14Millikan,R.C.andWhite,D.R.,"Systematics ofVibrationalRelaxation", Journal of Chemical Physics, Vol. 39,
rium Thermal
Radiation
ment Vehicle",
AIAA
19 Fritzsche,
S.
No.
1988, pp. 811-819.
12, 1963, pp. 3209-3213.
15 Boyd, change
I.D.,
Using
"Calculation Discrete
of Vlbraational
Particle
Simulation
Energy Methods",
Regimes
of Aeroassisted
of Rate
Volume AIAA,
17 Bird, G.A.,
"Direct
Flows",
Paper 86-1310,
zs Moss,
AIAA J.N.,
Chemistry
Orbital
of Aeroassisted
tics and Aeronautics,
Simulations",
sub-
Acta
20 Bird, G.A., Killara,
C., "Problems
edited by H.F. Nelson,
Energy
Transfer
Orbital
Vehicles",
in
Vehicles,
in Astronau-
New York, 1985, pp. 511-537.
Simulation
of Typical
June
AOTV Entry
1986.
Bird, G.A. and Dogra,
A.S.,
in Perpendicular
"Relaxation
Directions
of Analytical Physica
Experi-
1988.
Results
Polonica,
Vol.
of
for Hard
by Computer A74,
No.
6,
"General
Programs
Gas Flows",
Australia,
for Numerical
Simula-
Consulting
Pry. Ltd.,
G.A.B.
1988.
in the Flight
Transfer
96 of Progress
Cukrowski,
A verification
tion of Rarefied
16 Park,
Design
Sphere-
and
Flight
January
Ex-
mitted for publication.
Thermal
Translational
for an Aeroassist
Paper 88-0081,
V.K., "Nonequilib-
2z Park, Model Journal
C., "Assessment for Dissociating
of a Two Temperature and Weakly
of Thermophysics
Ionizing
and Heat Transfer,
Kinetic Nitrogen",
Vol. 2, No.
1, 1988, pp. 8-16. 22 Candler,
G., "On the Computation
Nonequilibrium January
1989.
Hypersonic
Flows",
of Shock
Shapes
in
AIAA Paper 89-0312,
Table1.Freeslxeam conditions. Uoo(kin/s) Density (kg/m3) Too(K)
Altitude(km) 90
7.5
Species
3.418x 10-6
188.0
Xo2
Xt¢_
0.23
0.77
Table2.Parameters forRotational Energy Exchange. (ZR)oo T" (K)
Zt
Na
23.3
91.5
1.5
02
16.5
113.5
1.5
NO
19.5
100.0
1.5
Table 3. Parameters Species
for Vibrational
Zo
Energy
Exchange.
g* (m/s)
cr_ (m 2)
N2
8.30x101°
43300
10 -20
02
1.50x10 tl
18200
10 -2°
NO
1.51x10 tl
6400
10 -2°
Table 4. Results q_R
4_
0.2
For Restricted Shock
Rotational
Standoff
Contributions
(m)
To Chemical
CH
Reactions.
Surface
Mass Fraction
0.02
0.110
0.177
0.561
Eq.(2)
0.02
0.113
0.178
0.547
Eq.(2)
Eq.(8)
0.115
0.180
0.540
Table 5. Results Rotational
Contribution
For Different
Rotational
Contributions
To Chemical
Reactions.
fir
qS,
0.2
0.02
0.110
0.177
0.561
Full
0.2
0.02
0.110
0.177
0.552
Full
Eq.(2)
0.02
0.116
0.182
0.530
Full
Eq.(2)
Eq.(8)
0.118
0.188
0.506
Restricted
Shock
Standoff
(m)
Cn
of N2
Surface
Mass Fraction
of
N2
1.0
xl0
0.8
4
3.0
',e" v
2.0
II)
Continuum, DSMC, Eq.(2)Eq.(1 )
Zk
0.6
Trans Rot ......
£
0.4
1.0
E
13_
o.} I--
0.2 i__
,5.
,5,
A
0
1000
3000
tion
Rotational energy
exchange
^
1
0.0
I ....... -4...... .2 .3
(K)
Distance
probability
as a func-
of temperature.
10
I .1
.0
5000
Temperature l.
^
!
0.0
Fig.
Fig.
4a.
line
for constant
Thermal
nonequilibrium exchange
e-o
J:3 O
(m)
along
Ii co to t_
10 -10
t33 cO
stagnation
stream-
probabilities.
N2
0.6
N ......
o t_
1015
x LtJ
"_........... .4 .5
0.8
0
I0". s -
13..
Vib
,/_
zx
O
0.4 i--
0.2 I
DSMC
02
NO
.,.......
/ 10"20
I
I
! i 11111
2
10
10
I
2. Vibrational
tion
of temperature.
energy
i
i Iii1
exchange
probability
as a func-
Fig.
xl0
l
......
/x
DSMC,
.2
Ref.
.3
Distance 4b.
Species
for constant
Continuum,
.1
(K)
.25
.10
.0
4
10
Temperature Fig.
i
3
12
mass fractions
exchange
_
.5
(m)
along
stagnation
streamline
Constant
Probability
Variable
Probability
I
probabilities.
4
..-,
Eqs.(6,11)
.4
_.
1.5
_= ,_
1.0.
(D 0..
£
E I-.-
13_
-.05
I 1.
0.
I 2.
Temperature Fig.
3. Vibrational
empirical
correction.
energy
exchange
probability
,,
0.0
I 3. (K)
0.5
.0
4. xl0
4
with
Parks
.1
.2 Distance
Fig.
5.
different
Comparison rotational
of rotational energy
exchange
, .3
i .4
.5
(m) temperature models.
profiles
for
010
03
--A,., m
o_ ._
dD
ID
02
£
N2
Et --e--
.008
IQ" !I
02 NO !
.ID
£
•
i 11
006
n
13
03 004 c
03 ¢
01
cO X
......
O
NO
x
LLI
LLI 0.0
.0
I
I
I
I
.4
.2
.3
.4
Distance Fig. 6. Variation ties.
/
.tfi_ I
.0 .05
.0
.5
.10 Distance
(m)
energy
_'_
exchange
probabili-
Fig. 9. Variation ties.
of vibrational
I
I
.15
.20
[] .25
(m)
energy
exchange
probabili-
0.8
4
xl0
of rotadonal
t
.002
30
/
e v
20
¢O
0.6 ¢7¢
....
O
Trans ii /
Rot
N2 Constant N Constant N2 Variable
0.4 ......
tO 03
Q
E
02
t--
10
f
"'"'.
/
N, Variable '
00 0.0
"1_--'---4----_
'
0
1
2
J ......
3
Distance
4
0
5
xl0
4
3
Distance
of molecular
nitrogen.
atomic
4
5
of molecular
and
(m)
I0.
Comparison
nitrogen
of mass
for different
fractions
energy
exchange
models.
0.8
16
12 t
v
nonequilibrium
2
I
(m) Fig.
Fig. 7. Thermal
1
a
! ......
Variable Constant Probability Probability
I
tO
06
O I-'
L1O
E
04
03 03
08
G)
Restricted Full Rot
Rot
I
02
b "M
0
1
2
3
Distance Fig.
8. Comparison
different
vibrational
of vibrational energy
4
00
.....................
'
I.........
.........
I
_
--,,.,_
O
.
O
0.4
5
0
I
l
I
I
1
2
3
4
5
of molecular
nitrogen
(m) temperature
exchange
models.
Distance profiles
for
Fig. 11. Comparison for different
of mass fraction
steric factors.
(m)
02,
0.4
Constant
Constant
O, Constant
g
02,
"_
3
10
0.5 O3 ¢-
Variable
Probabilities
_10 ch
O, Variable
0.3
Probabilities
Variable
(33 U
Lt..
_o 0.2 EIO
:s
L.
o Z
0.1 0.0 .0
.1
; .3
.2 Distance
Fig.
12.
atomic full
Comparison
oxygen
rotational
of mass
for different
.0
.5
of molecular
exchange
to the steric
and
models
Fig.
with
Comparison
models
with
of
density
full rotational
.4
.5
(m)
for different
energy
contribution
ex-
to the steric
factor.
_" E
N2, Constant
6.0-
N, Constant
0.4
4.0
N2, Variable
.-o oo
N, Variable
t_
0.2
>
0.0 .0
.1
.2
13.
Comparison
nitrogen
full rotational
of mass
for different contribution
//
2.0
Constant Variable
.0
.1
(m)
fractions
energy
f
;"
0.0
.5
Distance
atomic
.3
8.0
¢,..)
Fig.
14.
change
factor.
0.6
ii ¢.o
.2 Distance
0.8
to
.1
(m)
fractions
energy
contribution
10
i .4
Distance of molecular
exchange
to the steric
.2
factor.
models
and with
Fig. change factor.
15.
Comparison
models
with
of velocity full rotational
1
1
.3
.4
.5
(m) for different contribution
energy
ex-
to the steric
