NASA Technical Reports Server (NTRS) 20040095951: Rarefied Transitional Bridging of Blunt Body Aerodynamics

8 Pages·0.17 MB·English

by  NASA Technical Reports Server (NTRS)

Rare(cid:12)ed Transitional Bridging of Blunt Body
Aerodynamics
R. G. Wilmoth, R. C. Blanchard, J. N. Moss
NASA Langley Research Center, Hampton, VA, USA
1 Introduction
Bridging relations are frequently used to provide engineering aerodynamic
predictions in the transitional regime between free-molecular and contin-
uum(cid:13)ows. Suchpredictionsareimportantformissionstudiesofhigh-speed
planetaryentryvehiclestoprovideaccurateanalysisoftrajectorydynamics,
vehicle stability, and landing footprint. However, empirical parameters in
thesebridgingrelationsmustbederivedfromavailableexperimental,(cid:13)ight,
and computational data, where the latter is often generated using Direct
Simulation Monte Carlo (DSMC) methods. Therefore, bridging relations
have been sought that are applicable to a wide range of vehicles over a
wide range of conditions and use a minimum number of empirical parame-
ters [1, 2]. In the absence of a \universal" bridging relation, it is desirable
toreducetheexperimentalandcomputationale(cid:11)ortrequiredto\calibrate"
the bridging function for a particular class of vehicles.
The increasein planetaryentryandEarthsamplereturnmissionsin recent
years has increased the need for rare(cid:12)ed transitional (cid:13)ow analysis. Mission
studies haverevealeda number of instances wheretransitional aerodynam-
ics plays a signi(cid:12)cant role in trajectory and attitude control design, and
DSMC analyses have been performed to investigate these issues [3, 4, 5].
These analyses have produced a rather extensive database of transitional
aerodynamics for spherically-blunted cone geometries, and therefore it is
appropriate to use this information to develop improved bridging relations
for this class of vehicles. Furthermore, the good agreement recently shown
betweenDSMC analysesand(cid:13)ightdata[6,7]providesincreasedcon(cid:12)dence
in the accuracy of the DSMC predictions.
The analysis in this paper uses rare(cid:12)ed aerodynamic predictions (Knudsen
number, Kn1 0:001) for Viking, Path(cid:12)nder, Microprobe, Mars’01 Or-
(cid:21)
biter, and Stardust vehicles under the nominal entry conditions shown in
Fig.1. Thesevehicleshavespherically-bluntedconeforebodieswithconicor
biconic afterbodies. Viking, Path(cid:12)nder, and Mars’01 Orbiter have 70-deg
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21st International Symposium on Rare(cid:12)ed Gas Dynamics
half-angle cone forebodies, while Stardust and Microprobe have forebody
conehalf-anglesof60and45deg, respectively. Thesemissionsaredesigned
for direct entry at nominal velocities ranging from 4.5 to 12.6 km/s except
forMars’01Orbiterwhichcurrentlyplanstoperformanaerocapturetrajec-
tory entering the Marsatmosphereat about 6.6 km/s and exiting at about
3.5 km/s. All (cid:13)ights are Mars missions except Stardust which is a comet
sample return to Earth. Viking results are of particular interest because
(cid:13)ight data are available for comparison.
Several bridging approaches were explored, and a standard sine-squared
relation[8]andanerf-log relation[7]werechosentodescribethevariations
inaerodynamicquantitiesfromthefree-moleculartothecontinuumregime.
The purpose of this paper is to show the accuracy to which these relations
describe the transitional aerodynamics for this particular class of blunt-
body (cid:13)ows and to provide a nominal set of empirical parameters for use in
engineering predictions of blunt-body aerodynamics.
2 Transitional Bridging Relations
Transitional bridging methods generally fall into the class of either \local"
or \global" techniques. The functional relation used by both methods can
be expressed in the form
C = Pb Cfm+(1 Pb) Ccont (1)
(cid:1) (cid:0) (cid:1)
where C is a local (pressure, friction, etc.) or global (drag, lift, etc.) coe(cid:14)-
cient, andthesubscriptsfmandcontrefertothevalueofthecoe(cid:14)cientin
the free-molecular and continuum limits, respectively. Pb in Eq. (1) is the
bridging function that de(cid:12)nes the variation in the coe(cid:14)cient between the
free-molecularandcontinuumlimitsusingKnudsennumber,Machnumber,
Reynolds number, or various combinations as independent variables.
A commonly used global bridging function is the sine-squared relation
2
Pb =sin (cid:30) (2)
where
(cid:30)=(cid:25) (a1+a2log10Kn1) (3)
(cid:1)
and a1 and a2 are constants related to the Knudsen numbers, Knfm and
Kncont that de(cid:12)ne the free-molecular and continuum limits. Eqs. (2)-(3)
have been used for many of the planetary vehicles covered in this paper.
However, de(cid:12)ciencies in this bridging relation were noted based on Shuttle
(cid:13)ight data [8]. Also, the function is intended to be applied only within the
(cid:25)
limits 0 (cid:30) 2.
(cid:20) (cid:20)
2
Marseille, France, July 26-31,1998
The bridging function
1 p(cid:25) Kn1
Pb = 1+erf log (4)
2(cid:18) (cid:26)(cid:1)Kn (cid:20)Knm(cid:21)(cid:27)(cid:19)
was used by Ivanov [7] to describe Soyuz aerodynamics. In Eq. (4), Knm
1
is the center of the transitional regime de(cid:12)ned at Pb = 2 and (cid:1)Kn is the
logarithmic width of the transitional regime. This relation, which Ivanov
refers to as erf - log Kn bridging, was applied in [7] as a local bridging
relation but is applied in this paper as a global bridging relation.
3 Computational Approach
Axisymmetric and full three-dimensional free-molecular, DSMC, and con-
tinuum methods wereused to compute the (cid:13)ow(cid:12)elds and aerodynamics for
allvehicles. AxisymmetricDSMCcalculationswereperformedusingtheG2
code of Bird[9] while three-dimensionalfree-molecular,DSMC andNewto-
niancalculationswereperformedusingthe DSMCAnalysisCode(DAC) of
LeBeau [10] and companion free-molecular/Newtonian codes described in
[11]. A small number of Navier-Stokes results are included to complement
the Newtonian predictions in the continuum limit.
Computations were performed for each vehicle over a range of (cid:13)ight condi-
tionsandattitudesthatcorrespondtoanominaltrajectoryforthemission.
Detailed (cid:13)ow conditions for the varioustrajectoriesare givenin [3, 4, 5, 6].
4 Results
4.1 Viking Flight Comparisons
DSMC predictions of the ratio of normal-to-axialforce are shown in Fig. 2
for the Viking 1 lander. Included are the measured (cid:13)ight data and re-
sultsforbothsine-squaredanderf-logbridgingrelations. Theoverallagree-
mentbetween(cid:13)ightdata,DSMCpredictionsandbridgingrelationsisexcel-
lent. However,the sensitivityof the accelerometersused for the (cid:13)ight mea-
surements was not designed for the small forces experienced at the higher
Knudsen numbers, and the (cid:13)ight data exhibit considerablescatter at these
Knudsen numbers [12]. The erf-log function is evaluated outside the free-
molecular and continuum limits imposed on the sine-squared function to
illustrate the asymptotic behavior of the erf-log function.
4.2 Bridging Function Correlations
Drag predictions are shown in Fig. 3 for zero angle of attack, (cid:11), for all ex-
cept the Vikingdata whichcorrespond tothe nominal (cid:11)=11deg achieved
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21st International Symposium on Rare(cid:12)ed Gas Dynamics
during Mars entry. The data demonstrate the typical behavior of vehicles
in transitional (cid:13)ow, and the drag variations are very similar. Notable ex-
ceptionsoccurnearthefree-molecularlimit wherethelow-velocityMars’01
exit trajectory yields higher drag coe(cid:14)cients than the other vehicles and
at the continuum limit where the dragcoe(cid:14)cients arelowerfor the smaller
cone-angle vehicles. Additional small di(cid:11)erences in vehicle drag are caused
by di(cid:11)erences in trajectory (cid:13)ight conditions.
Although the complete set of aerodynamic predictions is too extensive to
include here, results were also computed at various angles of attack, (cid:11), to
yield an aerodynamic database for use in mission studies. The most com-
plete set of calculationsavailablearefor the various70-degbodies (Viking,
Path(cid:12)nder, and Mars’01 Orbiter). A subset of these results was selected
for detailed correlations with both bridging functions. The functions were
(cid:12)t in a least-squares sense to DSMC predictions of the aerodynamic coe(cid:14)-
cients, and the \best-(cid:12)t" parametersfor Eqs. (2) and (4) were determined.
The root-mean-squared(RMS) errorfor eachfunction wascalculatedas an
overall measure of the \goodness"of (cid:12)t.
Typical bridging parameters for various aerodynamic coe(cid:14)cients for the
Mars’01 case are given in Table 1. In general, the two bridging relations
give about the same accuracy in axial and drag forces while the erf-log
relation gives a somewhat better (cid:12)t to other aerodynamic coe(cid:14)cients.
4.3 Single-Point Bridging Procedure
o
The bridging parameters given in Table 1 for (cid:11) = 11 are typical of those
obtained for other angles of incidence and vehicles. The parameters for
eachbridgingfunctionvarysigni(cid:12)cantlyforthevariousvehicleattitudesand
shapes. However,theerf-logmethodhasanadvantageoverthesine-squared
methodinthatitissensitivemainlytoonlyoneparameter,Knm,whilethe
sine-squared method hasabout the same sensitivityto both parameters, a1
and a2. Therefore, a procedure can be constructed for the erf-log method
wherein a value for (cid:1)Kn is assumed, and the second parameter, Knm, is
obtainedfromasinglevalueofthecoe(cid:14)cientatsomeintermediateKnudsen
number in the transitionalregion. Such a procedureis illustrated in Fig. 4.
By computing the aerodynamic coe(cid:14)cient, C, at the free-molecular and
continuum limits and at some intermediate value of Kn1, the bridging
functionvalue,Pb,isobtained. GivenPbandfurtherassuminga\universal"
value for (cid:1)Kn in Eq. (4), the second parameter, Knm, can be extracted.
Basedon analysisof all the blunt-body DSMC results available,avalue for
(cid:1)Kn of log 500=6:2146 was chosen. The single-point bridging procedure
wasappliedtothedatawhereKnm wasevaluatedbasedonasingleDSMC
result that typically fell in the range 0:05 Kn1 0:5. A comparison of
(cid:20) (cid:20)
thebridgingfunctionthusobtainedisshowninFig.5fortheMars’01dataat
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Marseille, France, July 26-31,1998
(cid:11)=11deg. The\goodness"of(cid:12)t asdeterminedbytheRMS errorwasnot
signi(cid:12)cantlydegradedbythisprocedure,andalthoughthe\universal"value
of (cid:1)Kn was chosen mainly based on the Mars’01 predictions, this value
applies reasonably well to the other vehicles. Table 2 shows the bridging
parameters and RMS errors for axial force coe(cid:14)cient for all the data sets
available, and the errors are generally less than 5 percent.
5 Concluding Remarks
The bridgingproceduresdiscussed provideanaccurateengineeringmethod
forpredictingrare(cid:12)edtransitionalaerodynamicsofspherically-bluntedcone
entryvehicles. Thesingle-pointprocedureo(cid:11)ersawaytoimprovethebridg-
ing procedures while minimizing the computational e(cid:11)ort. However, the
accuracy of these procedures ultimately depends on accurate knowledge of
the aerodynamics in the free-molecular and continuum limits. The excel-
lent agreement shown for DSMC predictions and bridging relations with
theViking(cid:13)ightdataintransitionalregimeenhancethecon(cid:12)denceinthese
procedures.
References
[1] Kotov, V. M., Lychkin, E. N., Reshetin, A. J. and Schelkonogov, A.
N., An Approximate Method of Aerodynamics Calculation of Complex
Shape Bodies in a Transition Region, Rare(cid:12)ed Gas Dynamics, Vol. 1,
ed. O. M. Belotserkovskii,Plenum Press, 1985.
[2] Potter, J. L, and Peterson, S. W., Local and Overall Aerodynamics
Coe(cid:14)cients for Bodies in Hypersonic Rare(cid:12)ed Flow, AIAA Paper 91-
0336, Jan. 1991.
[3] Moss, J. N., Wilmoth, R. G., and Price, J. M., DSMC Simulations
of Blunt Body Flows for Mars Entries: Mars Path(cid:12)nder and Mars
Microprobe Capsules, AIAA Paper 97-2508,June 1997.
[4] Moss,J.N.,Blanchard,R.C.,Wilmoth,R.G.,andBraun,R.D.,Mars
Path(cid:12)nder Rare(cid:12)ed Aerodynamics: Computations and Measurements,
AIAA Paper 98-0298,Jan. 1998.
[5] Wilmoth, R. G., Mitcheltree, R. A., and Moss, J. N., Low-Density
Aerodynamics of the Stardust Sample Return Capsule, AIAA Paper
97-2510, June 1997.
[6] Blanchard,R.C.,Wilmoth,R.G.,andMoss,J.N.,AerodynamicFlight
Measurements and Rare(cid:12)ed-Flow Simulations of Mars Entry Vehicles,
J. of Spacecraft and Rockets, Vol. 34, No. 5, 1997, pp. 687-690.
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21st International Symposium on Rare(cid:12)ed Gas Dynamics
[7] Ivanov, M. S., Markelov, G. N., Gimelshein, S. F., Mishina, L. V.,
Krylov,A. N., andGrechko,N. V., High-Altitude Capsule Aerodynam-
ics with Real Gas E(cid:11)ects, J. of Spacecraftand Rockets, Vol. 35,No. 1,
1998, pp. 16-22.
[8] Blanchard, R. C., and Buck, G. M., Rare(cid:12)ed-Flow Aerodynamics and
ThermosphereStructurefromShuttleFlightMeasurements,J.ofSpace-
craft, Vol. 23, No. 1, 1986,pp. 18-24.
[9] Bird, G. A., The G2/A3 Program Users Manual, G.A.B. Consulting
Pty Ltd, Killara, N.S.W., Australia, Mar. 1992.
[10] LeBeau, G. J., A Parallel Implementation of the Direct Simulation
MonteCarlo Method,tobepublishedinComputerMethodsinApplied
Mechanics and Engineering, 1998.
[11] Wilmoth, R. G., LeBeau, G. J., and Carlson, A. B., DSMC Grid
Methodologies for Computing Low-Density Hypersonic Flows About
Reusable Launch Vehicles, AIAA Paper 96-1812,June 1996.
[12] Blanchard, R. C., and Walberg, G. D., Determination of the
Hypersonic-Continuum/Rare(cid:12)ed-Flow Drag Coe(cid:14)cient of the Viking
Lander Capsule I Aeroshell from Flight Data, NASA TP 1793, Dec.
1980.
sine-squared Parameters erf-log Parameters
Coe(cid:14)cient a1 a2 RMS Error, % (cid:1)Kn Knm RMS Error,%
CA 0.2836 0.1000 3.9 6.5 0.3576 4.1
CN 0.3107 0.0926 17.3 6.3 0.1804 7.0
Cm 0.3367 0.0921 19.8 5.9 0.0825 12.1
Table 1: Bridging relation parameters for various Mars’01 aerodynamic
coe(cid:14)cients. Angle of attack = 11 deg.
Vehicle (cid:1)Kn Knm RMS Error, %
Viking 6.2146 0.1805 4.3
Path(cid:12)nder 6.2146 0.1804 4.6
Mars’01 6.2146 0.4894 4.8
Stardust 6.2146 0.0756 4.8
Microprobe 6.2146 0.0475 5.8
Table2: erf-logBridgingparametersforCA forvariousvehiclesusingsingle
DSMC point to determine Knm.
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Marseille, France, July 26-31,1998
103 VikingI Mars
(Reconstructed Microprobe
102 from
FlightData)
101
Mars
100 Mars’01 Pathfinder
Kn Orbiter
¥
10-1
Stardust Transitional
10-2 Flow
10-3
10-4
10-5
0 5000 10000 15000
V ,m/s
¥
Figure 1: Nominal planetary entry trajectories.
0.20
DSMC
Free-Molecular
Newtonian
0.16 sin2f BridgingRelation
erf-logBridgingRelation
FlightData Kn =25
fm
0.12
C /C
N A
0.08
Kn =0.002
cont
0.04
0.00
10-4 10-3 10-2 10-1 100 101 102
Kn
¥
Figure 2: DSMC predictions and bridging relations
compared to Viking I (cid:13)ight data. Ratio of normal-
to-axial force coe(cid:14)cient, CN=CA.
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21st International Symposium on Rare(cid:12)ed Gas Dynamics
Mars’01OrbiterEntry
2.4 Mars’01OrbiterExit
MarsPathfinder
MarsMicroprobe
Stardust
Viking(a =11o)
2.0
C
D
1.6
1.2
10-3 10-2 10-1 100 101 102 103
Kn
¥
Figure 3: Drag predictions for various blunt
bodies. Zero incidence for all except Viking
for which (cid:11) = 11 deg.
2.2
C
A
1.8
C 1.410-3 10-2 10-1 100 101 102 103
fm 0.6
C
SingleDSMC CN
Computation 0.3
forBridging,
Pb=D CDSMC/D Cmax D Cmax 0.010-3 10-2 10-1 100 101 102 103
D CDSMC Cm,0o.00
-0.04
C
cont
log(Kn¥) -0.0810-3 10-2 10-1 100Kn¥ 101 102 103
Figure 5: Aerodynamics for
Figure 4: Procedure to deter-
Mars’01 Orbiter from DSMC and
mine bridging constant using sin-
erf-log bridging relation based on
gle DSMC computation.
single-point (cid:12)t. Symbols are
DSMC, line is bridging relation.
(cid:11)=11 deg.
8