Computation of Axisymmetric and Ionized Hypersonic FlowsUsing Particle and Continuum Methods
lain D. Boyd*Cornell University, Ithaca, New York 14853
and 'Tahir Gokgent
Eloret Institute, Palo Alto, California 94305
Comparisons between particle and continuum simulations of hypersonic near-continuum flows are presented.The particle approach employs the direct simulation Monte Carlo (DSMC) method, and the continuumapproach solves the appropriate equations of fluid flow. Both simulations have thermochemistry models for airimplemented including ionization. A new axisymmetric DSMC code that is efficiently vectorized is developed forthis study. In this DSMC code, particular attention is paid to matching the relaxation rates employed in thecontinuum approach. This investigation represents a continuation of a previous study that considered thermo-chemical relaxation in one-dimensional shock waves of nitrogen. Comparison of the particle and continuummethods is first made for an axisymmetric blunt-body flow of air at 7 km/s. Very good agreement is obtainedfor the two solutions. The two techniques also compare well for a one-dimensional shock wave in air at 10 km/s.In both applications, the results are found to be sensitive to various aspects of the chemistry models employed.
Introduction
T HE computation of flows around hypersonic vehiclespresents many problems. One of these concerns the range
of flow regimes encountered by a vehicle. A spacecraft re-en-tering the Earth's atmosphere such as the Space Shuttle beginsthe entry in a low-density regime. The flow around the vehicleis characterized by a high Knudsen number where collisionsare infrequent. Under such conditions, the flow is highlyrarefied, and it cannot be well represented as a continuum.Currently, the preferred solution technique for such flows isthe direct simulation Monte Carlo method (DSMC). As thespacecraft descends in altitude, the atmospheric density risesconsiderably. The Knudsen number of the flow falls linearlywith this rise in density, collisions in a given fluid elementincrease, and the flow becomes more like a continuummedium. For Knudsen numbers below about 0.01 the pre-ferred solution techniques are obtained from continuum com-putational fluid dynamics (CFD). In the continuum approach,numerical methods are employed to solve the equations offluid mechanics. In addition to these issues concerned withflow regimes, the problem of modeling thermochemistry is ofconcern.
The relationship between the particle and continuum simu-lations under conditions of thermochemical nonequilibriumhas been investigated recently by Boyd and Gokgen.1 Evalua-tion of the thermochemical models for pure nitrogen (N2) wasmade through the computation of a number of different one-dimensional shock waves. The conditions in the studies werechosen to place the flows in the near-continuum regime, andto allow separate examination of the effects of vibrationalrelaxation, dissociation, and ionization. Excellent agreementwas obtained between the continuum and particle solutionswhen the relaxation and rate constants employed in the twotechniques were made consistent.
* Assistant Professor, Mechanical and Aerospace Engineering.Member AIAA.
tResearch Scientist, 3788 Fabian Way. Member AIAA.
The present paper represents a continuation of the studiesreported in Ref. 1. The first objective is to extend the compari-sons between the particle and continuum simulations toaxisymmetric flows. In the previous study, one-dimensionalshock waves were considered. Unfortunately, this approachdoes not allow evaluation of the shock standoff distance thatis an important attribute of hypersonic blunt-body flows.Therefore, in the present study, a new axisymmetric blunt-body DSMC code has been developed. The second objective isto consider hypersonic flows of air. This requires the system-atic evaluation of chemical rate constants for a much larger setof reactions than was studied previously for N2. The approachadopted in the current work follows that of Ref. 1. Thus, forall the reactions of interest in air, equilibrium constants are tobe obtained for use in DSMC that are consistent with thevalues used in the continuum simulations. Comparison of theparticle and continuum methods is considered for two differ-ent flows of air under near-continuum flow conditions. Thefirst involves flow at 7 km/s over a 1-m-radius sphere: bothadiabatic and isothermal wall conditions are investigated. Thesecond considers ionization processes in a one-dimensionalshock wave at 10 km/s.
Continuum ApproachIn the continuum formulation, the nonequilibrium gas
model for air consists of eleven chemical species, (N2, O2, NO,N, O, N2
+, O2+, N0 + , N + , 0 + , e-), and the thermal state of
the gas is described by three temperatures: translational, rota-tional, and vibrational (vibrational-electronic), The governingEuler/Navier-Stokes equations are augmented with the equa-tions accounting for thermochemical nonequilibrium pro-cesses. The equation set consists of sixteen partial differentialequations: eleven mass conservation equations for species,two momentum equations for two-dimensional flows, andthree energy equations. In this study both the viscous Navier-Stokes and the inviscid Euler solutions are presented using thetwo-dimensional axisymmetric and one-dimensional codes.
The thermochemistry model is basically that proposed byPark2 and Park et al.3 The relaxation time for vibrational-translational energy exchange is taken from Millikan andWhite4 with Park's modification that accounts for the limitingcross section at high temperatures. For vibration-dissociationcoupling, the chemical reaction rates are prescribed by Park's
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BOYD AND GOKgEN: IONIZED HYPERSONIC FLOWS 1829
Table 1 Leading constants in chemical rate data (mVmolecule/s)
N2 + e--*N + N + e~OZ + MD-O + O + MD02 + MA^O + O + MANO + MD-N + O + MDNO + MA-N + O + MA
O + NO-N + O2N + O2-*O + NON2 + O-N + NON + NO-~N2 + O
N + O-NO+ + e ~NO + + e--N + OO + O-O2
+ + e~O2+ + e--O + ON + N-*N2+ + e-N2
+ + e--N + NN2 + N+-*N + N2
+
N + N2+-*N2 + N +
O + O2+-O2 + O +
O 2+O+ -*O + O2+
O + NO+-*O2 + N +O2 + N+-O + NO +
N 2 +O + -*O + N2+
O + N2+-~N2 + O +
O2 + NO+-*NO + O2+
NO + O2+-O2 + NO +
N + NO+-*O + N2+
O + N2+-N + NO +N2 + O2
+-O2 + N2+
O2 + N2+-*N2 + O2
+
O + NO+-*N + O2+
N + O2+-*O + NO +
NO + O+-O2 + N +O2 + N+-NO + O +
N + NO+-*N2 + O +N2 + O+-N + NO +
N + e~-*N + + e~ + e~O + e~-» O+ + e~ + e~
Continuum3
i . i6x lo-sr-1-6
4.98 x lO-sr-1-6
1.49 X 10-5^-1.63.32 x lO-^r-1-5
1.66 x 10-sr-1-5
8.30xlO~ 1 5
1.83 x 10~13
1.39 x 10-17
See Eq.(3)1.06 x 10-12J--1.0
See Eq.(3)1.46 X !Q-2iTi.o
See Eq.(3)i . isx io-27r2-7
See Eq.(3)7.31 x lO-^r1-5
See Eq.(3)1.66 x io-18r°-5
See Eq.(3)6.64 x io-18r-°-09
See Eq.(3)1.66 x io-18r°-5
See Eq.(3)LSI x io-18r°-36
See Eq.(3)3.99 x I0-17r°-41
See Eq.(3)1.20 X 10~16
See Eq.(3)1.64 x 10- 17
See Eq.(3)i.20x io-17r°-29
See Eq.(3)2.32 X IQ-2ST1.90
See Eq.(3)5.67 x 10- nr-i-os
See Eq.(3)4.15 X 104r-3-82
6.48 x 103r~3-78
Particle (present)7.97 x io-13r-°-5
7.14 x lO-8^-1^1.49 x io-5r-L6
3.32 x lO-9^-1-5
1.66 x lo-sr-1-5
8.30 x 10- 15
1.83 x 10~13
1.39x 10-17
4.60 x io-15r-°-55
1.06 x lO-12^-1-0
4.06 x 10- ̂ r-1-36
1.46 X lO-21^1-0
2.20 x io-13r-°-19
i . isx io~27r2-7
9.22 x I0-15r°-29
7.31 x lO-23^1-5
1.57 x io-17r°-85
i.66x io-18r°-5
2.34 x 10-14r-°-61
6.64 x lo-18r-°-09
4.99 x 10- 14
1.66 x io-18r°-5
3.04 x io-18r°-29
1.51 x io-18r°-36
1.98 x 10- ̂ r0-11
3.99 x I0-17r°-41
6.20 x lo-16r-°-051.20 x 10- 16
1.74X 10- isr0-30
1.64 x 10-17
4.59 x lO-18r-°-04
i.20x io-17r°-29
8.92 x io-13r-°-97
2.32 x lO-25^1-90
2.44 x I0-26r2-10
5.67 x 10- "r-1-08
3.97 x io-18r-°-71
5.81 X lO-Sj-1-00
1.59 x lO-Sj-1-00
Particle15
6.17 x io-9r-L6
i.85 x io-8r-L6
not included4.58 x lo-nr-1-0
i.38 x lo-^r-1-0
3.83 x 10-13r-°-5
7.66 x 10-13r-°-5
3.60 x lO-22^1-29
5.20 x io-22rL29
5.30 x io-17r°-10
2.02xlO-17r°-10
2.55 x I0-2°r°-37
4.03xlO~9r-1-63
6.42 x I0-22r°-49
3.83xlO~9r-1-5 1
2.98 x I0-20r°-77
8.88X 10-10r-L23
i.67xio-17r-°-18
2.37 x io-18r-°-52
i.89 x io-16r-°-52
1.89 x io-16r-°-52
not includednot included
1.06 x io-16r-°-21
1.77 x io-17r-°-21
1.72 x io-14r-°-17
not included2.83 x I0-17r°-40
4.10 x io-18r°-40
not includednot includednot includednot includednot includednot includednot includednot included1.00 x 10- 14
3.00 x 10- 12
j = diatomic species N2, O2, NO, N2+, O2
+, NO+ . bMA = atomic species N, O, N + , O + .
model where the dissociation rate is governed by the geometricaverage of translational and vibrational temperatures. Therotational relaxation times are calculated assuming a constantcollision number of 5. For dissociation-vibration coupling, thestandard approach adopted is to specify the average vibra-tional energy lost or gained due to dissociation and recombi-nation as 30% of the dissociation energy.2 To show howsensitive the flowfield is to the dissociation-vibration cou-pling, some computations are also made taking this energyequal to twice the local vibrational energy. For the viscouscontinuum computations, transport properties such as speciesviscosity, heat conductivity, and species diffusion coefficientsare specified using the curve fits given by Gupta et al.5
The numerical approach to solve the governing equations isfully implicit for fluid dynamics and chemistry. It uses fluxvector splitting for convective fluxes, and shock capturingwith an adaptive grid strategy. The details of the numericalmethod can be found in Refs. 6-8.
Particle ApproachAxisymmetric Direct Simulation Monte Carlo Code
The new DSMC code is based on a one-dimensional stagna-tion streamline code for air described in Ref. 9, and a nonre-acting axisymmetric code for nozzle and plume flows describedin Ref. 10. The DSMC algorithms employed in the code areefficiently vectorized as described in Ref. 11. Nonuniform,body-fitted computational grids are employed. Weighting fac-tors are used in the radial direction to reduce the number of
particles required in the simulation. The code simulates trans-lational, rotational, vibrational, and electron kinetic energydistributions. Dissociation and recombination reactions arecomputed using the vibrationally favored dissociation (VFD)model.12 All other reactions are simulated using the model ofBird.13 For dissociation-vibration coupling, the activation en-ergy is removed statistically from the energy modes. On aver-age, the fraction of the dissociation energy removed from thevibrational mode during reaction is:
fv
+ fr + fv(1)
where ft, ft, and ft, are the total numbers of translational,rotational, and vibrational energy modes, respectively, thatparticipate in the collision. This fraction varies between 0.2and 0.4 depending on the nature of the collision partners andthe value assumed for ft. The chemical rate coefficients em-ployed in the code are discussed in the next section.
The code executes with a performance of 1 jus per particleper time step on a Cray Y-MP vector computer. Thus itrequires one hour to process 3600 steps in a simulation con-taining one million particles. In the transient stage of thesimulation, a reduced number of particles is employed. Oncethe steady state of the simulation is reached, cloning of theparticles increases the total number of particles per cell to amore acceptable level. In this way, a significant amount ofcomputational resources may be saved.
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1830 BOYD AND GOK£EN: IONIZED HYPERSONIC FLOWS
New Discrete Simulation Monte Carlo Chemistry ModelThe rate coefficients employed in the reactions of interest in
the present study are given in Table 1. These are described inthe usual Arrhenius form:
discrepancies were discovered. Reactions 38 and 39 of Ref. 15are written:
k(T) = aTb Qxp(-Ea/kT) (2)
where a and b are empirically determined constants, Ea is theactivation energy, and Tis the controlling temperature. Threedifferent sets of coefficients are given corresponding to thoseused: one in the continuum code, one in the present DSMCcode, and one in previous DSMC investigations. The values ofthe activation energy used in the three sets of rate data areunchanged for each separate reaction. Therefore, the expo-nential term in the Arrhenius form has been omitted fromTable 1.
The rate expressions employed in the continuum code arethose recommended in the review by Park et al.,3 Generally,only the forward rate coefficients kf are specified. In thedissociation reactions, numbers 1-3, the controlling tempera-ture in the continuum two-temperature approach is given byTa = (7TV)1/2. For nitrogen dissociation, the particle code em-ploys the rates shown with a parameter of 4> = 2 in the VFDmodel. In a recent study by Haas and Boyd14 the parameterfor oxygen dissociation was determined to be </> = 0.5. Fornitric oxide dissociation, no vibration-dissociation couplingis included.
The reverse rates kr for each reaction are obtained by usingthe principle of detailed balance:
kr(T) = Ke(T) (3)
The following temperature-dependent form proposed byPark2 is employed for the equilibrium constant:
A5/z2 (4)
where the A-t are constants and z = 10000/7". Unfortunately,this form for the equilibrium constant is not mathematicallyconvenient for direct implementation in the DSMC chemistrymodels. However, a set of reverse reaction rates for use inDSMC has been proposed by Bird.15 These have been used ina number of studies and are reviewed in the following subsec-tion. To limit the number of factors involved in the compari-sons made in the present study, it is the aim of the researchersto maintain consistency between the relaxation rates employedin the particle and continuum solution techniques. Therefore,a form for the equilibrium constant that takes the traditionalArrhenius form is fit as a function of temperature to Park'sexpression. This form for the equilibrium constant may beused in the DSMC chemistry models. The resulting rate con-stants for the reverse reactions are listed in Table 1. Generally,good agreement is obtained between the new DSMC expres-sions and Park's expressions, particularly over the tempera-ture range of interest, i.e., from 10,000 to 20,000 K.
For reactions 19 and 20 in Table 1, the temperature-depen-dent form proposed in Ref. 3 is not convenient for use in theDSMC chemistry models. Once again, a fit is made to Park'sexpression in an Arrhenius form that may be employed in theparticle chemistry models. The new form, which is given inTable 1, gives fair correspondence to Park's results over thetemperature range of interest. These reactions are probablymore accurately simulated as a two-step mechanism as dis-cussed previously by Carlson and Hassan.16
Old Discrete Simulation Monte Carlo Chemistry ModelIn the previous study for nitrogen dissociation, it was dis-
covered that the temperature exponent for reaction 9b shouldbe -0.18 and had been mistakenly reported in Ref. 15 as-0.52. In reviewing the rates of Ref. 15 for air, further
andN+-N + NO
Actually, the second reaction should represent the reverse ofthe first. There is probably a typographical error in reaction 30of Ref. 15 that is written:
Clearly, this statement does not conserve mass, and by match-ing activation energies with reaction 27 it is presumed thatreaction 30 should read:
NO + O+-O + NO+
In comparing the reaction sets in Refs. 3 and 15 several of thecharge exchange reactions are different. In the present workthe aim is to make a comparison with the continuum solution,so the set proposed in Ref. 3 is employed. Experience showsthat these charge-exchange reactions generally play a relativelyinsignificant role in air thermochemistry up to 10 km/s.
Further anomalies occur in the associative ionization reac-tions 6-8 in Table 1. In Ref. 15 the forward rates are given as:
N + 0-NO+ + kf = 2.55 x 10-20r°-37
O + 0-O2+ + e~ : kf = 6.42 x I0~22r°-49
N + N-N2+ + e- : kf = 2.98 x I0-20r°-77
where the units are identical to those used in Table 1 . In theoriginal work of Park and Menees17 these rates are:
N + O-NO + + e~ : kf = 2.55 X I0-21r°-37
O + O-
N + N^
e~ :kf = 6.42 x I0~21r°-49
e~ : kf = 2.98 x I0-21r°-77
The leading constant in each of the three reactions is incorrectin Ref. 15 by an order of magnitude, and presumably thereverse reaction rates are also in error by this same amount.The inconsistencies of the rates used in Ref. 15 for the associa-tive ionization reactions do not affect the present study as theupdated continuum rates of Ref. 3 are being employed. Never-theless, the magnitude of these errors clearly has some signifi-cance for previous calculations that employed this data set.
1.25
0.75 -
0.50 -
0.25 -
-0.25 0 0.25 0.50 0.75 1
Fig. 1 Computational grid employed in the continuum technique:dimensions are in meters.
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BOYD AND GOK£EN: IONIZED HYPERSONIC FLOWS 1831
1.6
cT 1-2_E
^ 0.8
c0Q 0.4
0.0
ContinuumO DSMC
i i i
-0.08 -0.06 -0.04 -0.02Distance (m)
0.00
Fig. 2 Comparison of continuum and particle solutions of densityalong the stagnation streamline for axisymmetric dissociated flow.
oubx
Tem
pera
ture
(K-•>
ho
o
o
n
^
V
V v^
7 VV "fT^^r
\ i
Tt (DSMC)TV (DSMC)
S*̂ 3-4&6=S
1 1 I
-0.08 -0.06 -0.04 -0.02 0.00Distance (m)
Fig. 3 Comparison of continuum and particle solutions of transla-tional and vibrational temperature along the stagnation streamline foraxisymmetric dissociated flow.
0.5
0.4
0.3
0.2
0.1
0.0
N (continuum)——— O (continuum)
D N (DSMC)O O(DSMC)
-0.08 -0.06 -0.04 -0.02Distance (m)
0.00
Fig. 4 Comparison of continuum and particle solutions of atomicmole fractions along the stagnation streamline for axisymmetric disso-ciated flow.
Presentation of ResultsComputations are performed in air for two different config-
urations: 1) an axisymmetric blunt-body flow at an enthalpywhere dissociation effects are dominant, and 2) a one-dimen-sional shock-tube flow in which a higher enthalpy producessignificant ionization. The results for these studies are de-scribed in the following sections.
Dissociated Blunt-Body FlowThe first comparison considers Earth entry conditions for
the forebody flow over a 1-m-radius sphere at an altitude of 70km. The freestream conditions are listed in Table 2 and give aKnudsen number of 8 x 10 ~4 based on the freestream meanfree path and the sphere radius. The freestream enthalpy issuch that the flow is dominated by dissociation reactions.Initially, a continuum solution is generated in which the vis-
cous terms are neglected. The wall boundary condition isadiabatic in the continuum code and specular in the DSMCapproach. These conditions are chosen to reduce computa-tional overhead in the DSMC calculation associated with thesignificant rise in density that occurs next to a cold wall.
The continuum calculation uses a 100 x 60 grid. The geome-try and computational grid is shown in Fig. 1 where only afraction of the cells are plotted. Solutions are obtained inabout 90 min on a Cray Y-MP when 1500 steps are employed.Grid adaptation is implemented for improved resolution ofthe shock. The DSMC calculation employs a similar computa-tional grid of 286 x 136 cells. The variation of cell size alongthe stagnation streamline maintains a length of one mean freepath through the shock front, and is relaxed to three mean freepaths behind the shock. Over 1 million particles are employedin the steady stage of the simulation. Sampling of flow proper-ties is performed over 5000 time steps, and the total executiontime for the simulation is 5 h.
In Fig. 2 the computed profiles for density along the stagna-tion streamline obtained with the two solution techniques areshown. The shock standoff distances predicted by the particleand continuum methods are in remarkably good agreement.The gradual rise in density behind the shock moving towardsthe wall is also computed in a consistent manner by the twotechniques. It is observed that the shock thickness is slightlygreater in the DSMC computation. This is the expected trend,and may be attributed to viscosity effects that are neglected inthe continuum solution.
The translational and vibrational temperatures along thestagnation streamline computed using the DSMC and CFDtechniques are compared in Fig. 3. In general, the comparison
1.25
0.75
0.50
0.25
y 0 -
-0.25 0 0.25 0.50 0.75 1
-0.25 -
-0.50 -
-0.75 -
-1.25
Fig. 5 Comparison of continuum (upper) and particle (lower) flow-field contours of the local-to-freestream density ratio for axisymmet-ric dissociated flow.
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1832 BOYD AND GOKCEN: IONIZED HYPERSONIC FLOWS
1.25
0.75
0.50 -
0.25 -
y 0 -
-0.25 -
-0.50 -
-0.75 -
-0.25 0 0.25 0.50 0.75 1-1.25
Fig. 6 Comparison of continuum (upper) and particle (lower) flow-field contours of the translational temperature for axisymmetric disso-ciated flow: temperature is scaled by 0.001.
is again very favorable. As with density, the DSMC methodpredicts a thicker structure for the shock front. However, thepeak value and postshock relaxation of the translational tem-peratures show excellent agreement. The comparison for vi-brational temperature shows some disagreement. The DSMCtechnique predicts an earlier rise and a higher peak value incomparison with the continuum solution. Comparison ofthese solutions are interpreted in the following way: 1) Thegood agreement for density indicates that the particle andcontinuum methods model similar rates of vibrational andchemical relaxation. In particular, the effect of the vibrationalstate of the gas on the rate of dissociation appears to be ingood agreement in the two simulations. 2) The differences invibrational temperature indicate that the simulation of dissoci-ation collision mechanics is simulated differently in the parti-cle and continuum methods. In particular, the removal fromthe vibrational modes of the activation energy required fordissociation is not the same in the two techniques.
In summary, it appears that the coupling of vibration todissociation is consistent whereas the coupling of dissociationto vibration is not. The continuum and DSMC rotationaltemperature profiles were found to exhibit similar agreementto that found for the translational mode.
In Fig. 4, the mole fractions of atomic nitrogen and oxygencomputed with the particle and continuum methods along thestagnation streamline are compared. Excellent agreement isobtained for each of the profiles. Again, this illustrates thatvibration-dissociation processes are simulated in a consistentmanner for the thermochemistry of air. The comparisonsshown in Figs. 2-4 are very similar to those obtained in theprevious study for one-dimensional shocks of nitrogen.1
To give a perception of the overall comparison between theparticle and continuum computations for the forebody flowover the sphere, contours of density are shown in Fig 5. Theagreement for the solution techniques prevails throughout theflowfield. This correspondence is also observed in Fig. 6 wherecontours of translational temperature are shown. It is particu-larly satisfying to observe the good agreement obtained forshock stand-off distance in the present two-dimensional com-putations. This was identified as one of the primary aims ofthe investigation. Having obtained confidence in the agree-ment of the DSMC and continuum methods in the near-con-tinuum flow regime, comparisons under rarefied flow condi-tions can now be made.
Sensitivity of the flowfield to the dissociation-vibration cou-pling in the continuum computations is demonstrated in Fig.7. Two different continuum solutions are presented for thetranslational and vibrational temperatures. The first run isthat shown in Figs. 2-4 in which 30% of the dissociationenergy is removed from the vibrational mode following reac-tion. The second run is made by removing twice the localvibrational energy. As the peak vibrational temperature isabout 10,000 K, this second approach then removes at maxi-mum about 12% and 17% of the dissociation energy for N2and O2, respectively. Hence, this second run removes muchless vibrational energy after dissociation. This is reflected inthe higher peak vibrational temperature, and the movement ofthe shock front into the body surface.
Similarly, the sensitivity of the flowfield to the chemistrymodel employed in the particle computations is demonstratedin Fig. 8. Two different particle solutions are presented for thetranslational and vibrational temperatures. The run with thenew rates is that shown in Figs. 2-4 in which the new reactionrates, variable rotational and vibrational relaxation rates, andthe VFD model are employed. The second run is made with
30
^0
0)
20
10
Tt (0.3Ed)TV (O.SEd)Tt (2Ev)TV (2Ev)
-0.08 -0.06 -0.04 -0.02 0.00Distance (m)
Fig. 7 Comparison of continuum solutions of translational and vi-brational temperature along the stagnation streamline for axisymmet-ric dissociated flow using different dissociation-vibration models.
30
20
10
Tt (new model)TV (new model)Tt (old model)TV (old model)
-0.08 -0.06 -0.04 -0.02Distance (m)
0.00
Fig. 8 Comparison of particle solutions of translational and vibra-tional temperature along the stagnation streamline for axisymmetricdissociated flow using different chemistry models.
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BOYD AND GOKgEN: IONIZED HYPERSONIC FLOWS 1833
1.6
oT- 1.2x
0.8Continuum
o DSMC
0-0.08 -0.06 -0.04 -0.02
Distance (m)
Fig. 9 Comparison of continuum and particle solutions of densityalong the stagnation streamline for axisymmetric dissociated flow:assessment of viscous effects.
30
? 25o
i 20^CD 15
CD0.ECD
10
—— Tt (continuum)- - - - - T v (continuum)n Tt (DSMC)o TV (DSMC)
0-0.08 -0.06 -0.04 -0.02
Distance (m)
Fig. 10 Comparison of continuum and particle solutions of transla-tional and vibrational temperature along the stagnation streamline foraxisymmetric dissociated flow: assessment of viscous effects.
the uncorrected reaction rates of Ref. 15, fixed probabilities ofrotational and vibrational relaxation of 0.2 and 0.02, and noVFD coupling. The solutions are surprisingly insensitive to thechemistry models employed. However, the peak vibrationaltemperature obtained with the old model is significantly higherthan that obtained with the new model. Also, it is preferableto perform the particle and continuum computations withcorresponding reaction rates so that changes in one set may beintroduced in a straightforward manner into the other.
The DSMC profiles of the shock front shown in Figs. 2 and3 suggest that viscous effects need to be included in the contin-uum approach even at this relatively low Knudsen number. Toaddress this issue, further simulations are performed in whichthe continuum code is extended in this manner. The contin-uum computation is then repeated with the new code using anisothermal condition at the surface of the sphere with a walltemperature of 6000 K. A further DSMC simulation is alsoperformed with this boundary condition. In Fig. 9 the densityprofiles obtained along the stagnation streamline are com-pared. Note the effect in the continuum approach of includingthe viscous terms. The shock front has a finite thickness thatis in better agreement with the DSMC results than the datashown in Fig. 2. A similar improvement is obtained for thecomparisons of temperature shown in Fig. 10. The initial risein translational temperature predicted by the two methods is ingood accord with the DSMC result. The difference in the peakpostshock translational temperature is due only to a require-ment for finer grid resolution in the continuum calculation.Note also that the poor agreement for vibrational temperatureobserved in Fig. 3 persists in this comparison. The inclusion ofthe viscous terms in the continuum solution improves the
comparison with DSMC. However, it is clear that some of thestructure of the shock front cannot be simulated using theNavier-Stokes equations. This finding has been demonstratedpreviously at lower Mach numbers.18
Ionized Shock-Tube FlowThe flow conditions investigated are listed in Table 2. The
continuum code simulates ionized flow using the methodsdescribed in Refs. 7 and 8. The viscous terms in the Navier-Stokes equations are included in all results shown here. Thegrid employs 400 points with clustering near the shock ob-tained through a refinement study. The solution time is within10 CPU min when 3000 steps are employed.
The manner in which electrons are handled in the one-di-mensional and two-dimensional DSMC codes developed inRef. 8 and in this study is numerically expensive. As electronshave very low masses compared to the heavy atomic andmolecular species in air, they tend to have relatively highthermal velocities and collision rates. The higher thermalspeeds may be estimated as the square root of the ratio of themass of the heavy particle to that of the electron. This ishandled in the present implementation by reducing the compu-tational time step by two orders of magnitude. In addition,charge neutrality is enforced in each computational cell by
Table 2 Flow conditions
Case12
L/oo, m/s700010000
Poo, kg/m3
8.75 x 10-5
1.54 x 10~4
Too, K220300
2.4
°°ox 1-8
CO
)̂ 1 pV^ I . £-
CDQ
—— Continuumo DSMC
0.02 0.04Distance (m)
0.06
Fig. 11 Comparison of continuum and particle solutions of densityin a one-dimensional ionized flow in a shock tube.
60
50
40
30
20
oo ——— Tt (continuum)
- - - TV (continuum)o Tt (DSMC)A TV (DSMC)
0.06
Fig. 12 Comparison of continuum and particle solutions of transla-tional and vibrational temperature in a one-dimensional ionized flowin a shock tube.
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1834 BOYD AND GOKgEN: IONIZED HYPERSONIC FLOWS
0.8
c °'6go2it 0.4
"o
0.2
HHB
D -
- - - O (continuum)n N (DSMC)o O (DSMC)
>— -O — O- -O_ O _<>_ _O_ O _
0.02Distance
0.04(m)
0.06
Fig. 13 Comparison of continuum and particle solutions of atomicmole fractions in a one-dimensional ionized flow in a shock tube.
0.08
0.02 0.04Distance (m)
0.06
Fig. 14 Comparison of continuum and particle solutions of electronmole fraction in a one-dimensional ionized flow in a shock tube.
2.4
oX
1-2
§0 .6Q
Remove (O.SEd)- - - Remove (2Ev)
0.02 0.04Distance (m)
0.06
tion in the DSMC codes. The DSMC simulation employs 1500cells with a total of 100,000 particles. Very long transient timesmust be traversed before reaching a steady state. The totalexecution time for the DSMC code is 3 h.
Density profiles for this case are compared in Fig. 11. Thestandard continuum model for dissociation-vibration coupling,and the new DSMC chemistry model are employed. The shocksare aligned at the point where the density ratio, p/px = 6. It isfound that the DSMC simulation gives reasonable agreementwith the continuum profile. The shock front computed usingDSMC is substantially thicker than the viscous continuumsolution. Also, the rise in density immediately behind theshock (where the profiles are aligned) is slower in DSMC. Thismay be indicative of some kind of induction phenomenon.
The source of the induction behavior may be identified inFig. 12 where the translational and vibrational temperaturesolutions are compared. Consider first the translational mode.It is again apparent that the DSMC method predicts a signifi-cantly thicker shock front than is predicted by the viscouscontinuum calculations. Also note that viscous dissipation inthe continuum method significantly reduces the peak transla-tional temperature in comparison with the DSMC result.However, in the postshock region, the DSMC profile agreesquite well with the continuum solution. For the vibrationalmode, the DSMC solution rises at a slower rate than thecontinuum profile. There is, however, good agreement in therelaxation zone downstream of the shock. The slower rise invibrational temperature predicted by DSMC explains the in-duction behavior observed in the density profile. It is believedthat this behavior is caused by the dissociation-vibration cou-pling algorithm employed in DSMC that removes as much as40% of the dissociation energy from the vibrational modes ofthe particles that are most vibrationally excited (this is a conse-quence of the VFD model). Thus, the DSMC scheme removesmore energy from the vibrational mode than the standardmethod employed in the continuum code. At this point, with-out detailed experimental measurements of vibrational tem-peratures in such flows, it is unclear whether either one of thecontinuum and DSMC simulations is physically accurate.
A comparison is made in Fig. 13 of the atomic mole frac-tions of nitrogen and oxygen. Consistent with the previouslydiscussed induction behavior, the DSMC profiles lag a littlebehind the continuum solutions near the shock front, but givegood agreement in the postshock relaxation zone. In Fig. 14,the mole fractions of the electrons are shown for the twosimulations. The agreement here is only qualitative with theparticle method predicting about a factor of two fewer elec-trons. Once again, this is partially because of the inductionphenomenon predicted by the DSMC computation.
The sensitivity of the continuum and particle computationsare again investigated in Figs. 15 and 16. In Fig. 15, it is shownthat removing less vibrational energy after dissociation leadsto a significantly higher postshock density. In Fig. 16 it isfound that use of the old DSMC chemistry models leads to a
Fig. 15 Comparison of continuum solutions of density in a one-di-mensional ionized flow in a shock tube using different dissociation-vi-bration models.
associating each electron with the charged particle with whichit was formed.15 This approach is reasonable from a physicalstand point for degrees of ionization of a few percent. How-ever, a great disadvantage of this procedure is that it is numer-ically intensive. Typically, it takes 3 h on a Cray Y-MP toperform a one-dimensional shock-tube flow that includes theeffects of ionization. Therefore, the cost for a two-dimensionalaxisymmetric computation is viewed as being prohibitive. Itwas therefore decided to compare the ionized aerothermo-chemistry models employed in the DSMC and continuumapproaches in a shock-tube flow. An area of future researchwill consider more efficient algorithms for including ioniza-
E 2-°o) 1.5
g 1.0CDQ
0.5
0.00.00 0.02 0.04
Distance (m)0.06
Fig. 16 Comparison of particle solutions of density in a one-dimen-sional ionized flow in a shock tube using different chemistry models.
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BOYD AND GOKgEN: IONIZED HYPERSONIC FLOWS 1835
sharper rise in density at the shock front and no inductionbehavior is observed. The sensitivity of the temperature pro-files to the continuum and particle models follows similartrends to those observed in Figs. 7 and 8, except that thedifferences are more pronounced in the ionized flow. Detailedexperimental measurements of vibrational temperatures andelectron number densities under the type of flow conditionsinvestigated in this study are essential for the development ofaccurate thermochemistry models for particle and continuummethods.
Concluding RemarksAn axisymmetric, vectorized DSMC code with aerothermo-
chemistry has been applied to a blunt-body flow under near-continuum conditions. In a comparison of the DSMC resultswith numerical solutions of the Euler and Navier-Stokes equa-tions for the same flow, excellent agreement was obtained. Animportant aspect of obtaining such good agreement was therequirement of employing consistent chemical rate data in theparticle and continuum techniques. Thus, a new set of back-ward rate constants was developed for the DSMC method.The agreement of DSMC and continuum methods in the near-continuum flow regime is extremely important. It allowsmeaningful comparisons of these methods to be made underrarefied flow conditions. Previously, a comparison of particleand continuum techniques in the transition flow regime wouldhave been difficult to interpret as it had not been establishedthat the methods agreed in any overlapping region.
Because of constraints on available computational re-sources, it was decided to make a comparison of particle andcontinuum methods for ionized air in a one-dimensionalshock-tube flow. Some differences in the solutions computedwith the two techniques were noted, particularly in the shockfront. These were attributed to strong viscous effects notcaptured by the continuum formulation, and to differentmethods employed in the simulations for modeling the effectof dissociation on the vibrational temperature of the gas. Thisis a significant effect for high-enthalpy flows, where the ioniz-ing reactions in the continuum formulation are solely gov-erned by the vibrational temperature. It is concluded thatdetailed experimental measurements of vibrational tempera-ture and electron density are a strong requirement for furtherrefinement of the aerothermochemistry models employed inparticle and continuum simulations of hypersonic flow.
AcknowledgmentsSupport for lain D. Boyd was provided by NASA Univer-
sity Consortium Agreement NCA2-820 and for Tahir G6kc.enby NASA Grant NCC2-420.
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