NASA Technical Reports Server (NTRS) 19950024443: Kinetic theory model predictions compared with low-thrust axisymmetric nozzle plume data

9 Pages·0.47 MB·English

by  NASA Technical Reports Server (NTRS)

NASA-CR-19893_
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Kinetic Theory Model Predictions
Compared with
Low-Thrust Axisymmetric Nozzle
Plume Data
B.R. Riley" and S.J. Fuhrmant
University ofEvansville, Evansville, Indiana
and
P. F. Penko*
NASA-Lewis Research Center, Cleveland, Ohio
Abstract
Asystem of nonlinear integral equations equivalent to the steady-
state Krook kinetic equation was used to model the flow from a low-thrust
axisymmetric nozzle. The mathematical model was used to numerically
calculate the number density, temperature, andvelocity ofa simple gas as
itexpands into anear vacuum. With these quantities the gas pressure and
flow directions ofthe gas near the exit plane were calculated and compared
with experimental values for alow-thrust nozzle of the same geometry and
mass flow rate.
,,t
,o
o _ rm Introduction
U ur_ A coordinated program tostudygas flowsfrom low-thrustnozzles
0_ _ 0
(,1N) isinprogressatNASA-Lewis Research Center.1Thiseffortincludes
numerical mathematical modeling and experimental measurements to
0 determine plume parameters.
c.,,j
Intwo previous papers, model results were presented for plume
flows from a two-dimensional = nozzle and a three-dimensional
axisymmetric _nozzle. Nocomparisonofthe modelresultswere made with
experimental data. Inthispaper, for the first time, the model results and
r- experimental data are compared. The two earlier papers showed the
U
L feasibility ofutilizingthe Boltzmann•equation withaKrook-type relaxation
lEDI,-.-uj
scatteringterm for binary collisionsto model plume flows for low-thrust
U_
nozzles whentheflowgoesthroughthetransitionregion(continuumtofree-
aO p," molecularflow). Inparticular,it wasdemonstratedthatan iterativeprocess
U ,.,u Z
for solvingthe Boltzmann equation wouldconverge togivea steady-state
solution using a reasonable amount of computer time. The spherical
:¢
Z_ computationalvolumewasquite small(-3 cmradius)andthe nozzlesatthe
t-- .J
exitplane were only3/4 cmindiameter.
The model work described here isfor a larger nozzle (exit plane
,_)
."_Z diameter -3.2 cm) andextendsthecomputational volume (-20 cmradius)
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U '_ CL
C_ Copyright © 1993 by American InstituteofAeronautics andAstronautics,
I ,LP1)-- Inc. All rights reserved.
eC --3 < *Professor and Chair, Department of Physics
! tUndergraduate Student, Department of Physics
I-'CA 0
I :F +_ *Aerospace Engineer, Member AIAA
<_)
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inanattempt to predict parameters inthe near andintermediate regionsof
the plume. The nozzle geometry and propellant flow rate in the
experimental setupare identicaltothoseassumed inthemodel,thusmodel
predictionscanbedirectlycompared withtheexperimental measurements.
Kinetic Theory Model
Mathematical Formulation
The Krookkinetic equation5or iterate equation' for steady-state
flowthat was solved numericallyby integratingover velocityspace along
a characteristic lineis
v.V,f(v,r) =A(n,(r)f.-f) (1)
where f. (shown below) isan equilibrium Maxwellian velocitydistribution
normalized to one at agiven r.
fo= {l/(2_k'/')_Z}exp-{(m/(2kT)(v-u) 2} (2)
The boundary conditions forf (probability distribution function)are
discussed inthe next section of this paper.
The first iteration values for n(r), u(r), and T(r) (number density,
mean velocity, and temperature) at each grid point can then becalculated
using the following definitions
n(r) =.f.t'.ff(r,v) d3v (3)
u(r) = (1/n) .fJ"j vf(r,v) d3v (4)
T(r) ={rn/(3kTn)}___(v-u) 'f(r,v) cPv (5)
The values of n, u, and T are put into Eqs. (1) and (2) and the
process 3is repeated until convergence is obtained. At convergence the
values for n, u, and Tat the grid points do not significantly change from
the values obtained in the previous iteration. Because the nozzle is
axisymmetric, only parameter values in one plane through the axis of
symmetry are necessary. Figure 1shows the plane polar grid system used
in the model calculations.
Nozzle Parameters and Flowfleld Boundary Conditions
As mentioned earlierthe experimental flowconditionsand nozzle
geometry were identical with those in the model. The nozzle was
axisymmetric withahalfangleattheexitplane of20deg. Thenozzle throat
diameter was3.18 mm andthearea ratio(exit/throat) was 100. The plume
gas was N2witha flowrate of6.8 xlOskg/s.
To simulatethe 0.02 Paexperimental background pressure inthe
vacuum chamber, theplume inthe model was assumed toexpand intoa
background number density set at 1.2x10'8 N2 molecules/m 3 with a
temperature of300 Katzero mean velocity.Thiswas accomplished inthe
model by assuming experimental vacuum chamber conditions for the
surface locatedat200 mmwithangles fromthecenter linegreater than 50
deg (See Fig. 1). For the surface at 200 mm inthe first50 deg from the
centerline, the flow was assumed to have only an outward velocity
component.
The flowparameters at the nozzle exit plane used for the inflow
boundaryconditionswereobtainedfromBoydL7who utilized a discrete
simulationMonte Carlo (DSMC) computational method to model the flow
insidethenozzle. Boyd'smodelassumes diffusescatteringalongthe inside
ofthe nozzle wall.Thus the boundary layereffecton theflow externaltothe
nozzle willbe inherent through the exit surfaceboundary conditions.The
parameters at exit surface grid points are given in Table 1. Boundary
conditionsfor theexternal wall ofthe nozzle were modeled withf=0.
Convergence Criterion and Computer Time
The convergence to a solution for the iteration process was
assumed whenchanges inthequantities n,u,and Tcompared toprevious
iterationvalues approached zero. A solutionfor the539 gridpointsshown
in Fig. 1took approximately 20 h ofCPU time on the CRAY -XMP at the
NASA-Lewis Research Center. An equivalent solutionwas obtained with
about 1/2 the number of gridpoints and correspondingly less computer
time. Most ofthe computer time was taken inthe relatively high-number
density regionsnearthe nozzle exitplaneand alongthe radiilinesatlarge
angles (angle >50deg) wherethe numberdensityisnearthe background
level.
For convergence at a gridpoint, ctest, defined to be the square
root of the sum of the squares of the fractional changes inthe number
density,the components ofvelocity,and the temperature was requiredto
be small. Typically the convergence criterion for ctest was <0.01. A
correctionfactor that rangedfrom 1.0 to1.1 asr inFig. 1ranged from 46.4
mmto 200 mmwas applied tothe numberdensityto insurethat the mass
flow rate ateach radius was maintained at- 6.8xl0Skg/s. Thiscorrection
was needed when the grid point separations were large.
Experimental Data
The experimental data used inthis paper were taken inavacuum
tank for space-simulated tests at NASA-Lewis Research Center. The
details ofthe experimental setup are described inapaper by Penko set al.
and the correction factor that was applied to the pressure measurements
due to the shock wave caused by the measuring probe is discussed by
Shapiro. oA conical probe was used to measure the flow angles and Pitot
tubes with diameters of 1mm and 6.4 mm were used to measure the total
pressure in appropriate regions of the plume. The 1-mm tube was used
near the exit plane where large pressure gradients exist and the 6.4-mm
tube was used inthe far-field region of the plume. Anoverall measurement
accuracy inthe pressure measurements of+9% was estimated at the exit
plane. This improved to :_..2%inthe far field. The 30-deg conical probe
used to measure the flow angle inthe farfield was 6.4 mm indiameter at the
base and had an estimated accuracy of ±1 deg.
Model Results Compared with Experimental Data
Model Prediction Overview and Parameter Manipulations
The model resultsare depicted incontourplotsfor the gridpoints
as seen in Fig. 1with the exclusion of the grid pointsthat have rvalues
lessthan 46.4 mm. Figures 2-5 are contourplotsfor the numberdensity,
axialcomponent ofvelocity,radialcomponent ofvelocity,andtemperature,
respectively. These plotsgiveacomplete visualperspective of the model
results.The contourplotinFig.6showsthe Pitot pressuresas calculated
from the model whichcan be compared directly withthe data.
Pitot pressures were calculated from the model values of n, u,
and T at the grid points with shock wave and supersonic rarefied flow
effects built into the model results. Since the Pitot probe acts as an
intrusive device around which a shock wave develops in the supersonic
flow, the pressure drop acrossthe shock wave must becalculated inorder
tocompare with theexperimental data. Thus, the Rayleigh supersonic Pitot-
tube formula* was applied to the model prediction for pressure.
A second modification to the model prediction for pressure was
necessary due to the rarefied nature of the flow. The following
equation '.,° was applied to the model results
Iog(P=/Po,) = .089 -.120 Iog(Rep) for Re,<5.6 (6)
P_ is the corrected model predicted Pitot pressure which is to be
compared directly withthe experimental data. ,Doa,nd Re.are, respectively,
the model calculated values for the pressure behind the shock wave and
the Reynoldsnumber. The Reynolds number, Re., was calculated utilizing
the following relation
Re=,=p.U.Dp_ (7)
where p. and U. are the gas density and velocity of the freestream
preceding the shock, whereas Dp and IJ are, respectively, the probe
diameter and the viscosity at stream temperature, T, behind the shock
wave. The temperature, T_was computed from the normal-shock relation
for static temperature (See Ref. 9, p. 118). Once, T_,was known, then the
viscosity, I_,was determined from the power-law relating viscosity of gases
(See Ref. 1, p. 6)by the following formula
= _..(T,/T,.,). (8)
where I_._is the value of the viscosity at the reference temperature, T,_,
and n=0.75.
Direct Comparison of Flow Angle and Pressure with Experimental
Data
To compare the model predictionswiththeexperimental data, the
pressures and flow angles at the grid points in Fig. 1 were linearly
interpolated to pressures and flow angles along the three radial lines at
axialpositionsof 12 mm, 36 mm,and 120 mm asshown inFig. 1.
Results of the comparisons at these three axial positions for
pressureas afunctionofradialdistance isshowninFig.7. Results forthe
flowangles are given in Fig.8 for the 12-ram and 120-ram axialpositions
sincethere was no reliableexperimental data atthe36-mm axial position.
Ingeneralthere isexcellentagreement between themodelpredictionsand
theexperimental data.The biggestvariationbetweenthemodelpredictions
and the experimental data is at the 12-ram axial position for both the
pressure and the flow angle where the experimental data are not as
precise.
Concludlng Remarks
A kinetic theory model was used to calculate flow parameters
external to a low-thrust axisymmetric nozzle. The model predictions for flow
angle and pressure were compared for the firsttime with experimental data
taken for identical flow conditions. Agreement between the model
predictions and the experimental data was very good, which gives one
confidence that the model and numerical techniques used to calculate the
flowfield parameters may be useful in calculating flows from other low-
thrust nozzles. In particular, the model could be used to calculate the
flowfield parameters under space conditions that would be prevalent around
the space station being built by the United States. Preliminary model
calculations using space conditions look promising.
Acknowledgments
This work was performed withthe support of NASA-Lewis research
Center under Grant NAG 3-746.
References
'Penko, P .F., Boyd,I. D., Meissner, D.L, and DeWitt, K.J., "Pressure
Measurements ina LowDensity Nozzle PlumeforCode Verification," AIAA Paper
91-2110, AIAA/SAF_./ASMF_/ASEE27thJointPropulsionConference, Sacramento,
CA., June 1991.
2Riley, B.R., and Scheller, K.W., "KineticTheory Model forthe Flow of
a SimpleGas from aTwo-Dimensional Nozzle," Rarefied Gas Dynamics. Space
Related Studies, edited by E.P. Mutz, D.P. Weaver andD.H.Campbell, Vol. 116,
Progress inAstronauticsand Aeronautics, AIAA, Washington D.C., 1989, pp.352-
362.
3Riley, B.R., "KineticTheory Model forthe Flowofa Simple Gas from a
Three-Dimensional AxisymmetricNozzle," Rarefied Gas Dynamics Proceedings
ofthe17thInternationa/Symposium onRarefied GasDynamics,Aachen, Germany,
1990, edited by A.E. Beylic, VCH Publishers, New York and Germany, 1990, pp.
291-298.
'Patterson, D. G., Introduction to the Kinetic Theory of Gas Flows,
University ofToronto Press, Toronto, Canada, 1971.
SAnderson, D.G., "On theSteady KrookKineticEquation: Part 1,"Journa/
ofF/uidMechanics, Vol. 26, Pt. 1,Sept. 1966, pp. 17-35.
6Boyd, I. D., Private Communication, Eloret Institute, NASA Ames
Research Center, Moffett Field, CA., June 1991.
7Boyd, I. D., Penko, P. F., and Carney, L. M., "Efficient Monte Carlo
SimulationofRarefied FlowinaSmall Nozzle," AIAA Paper90-1693, AIANASME
5thJointThermophysics and Heat Transfer Conference, Seattle, WA., June 1990.
8Penko, P.F., Boyd, I.D., Meissner, D.L.,andDeWitt, K.J., "Measurement
andanalysis ofaSmall Nozzle Plume inVacuum," tobepublished inJ.Propu/sion
andPower.
=Shapiro, A. H., The Dynamics and Thermodynamics of Compressib/e
FluidFIow, Ronald Press, New York, 1953, pp.153-154.
l°Stephenson, W. B., "UseofthePitotTube inVery LowDensity Flows,"
AEDC-81-11, Amold Engineering Development Center, Oct.1981.
CN¢rdfnmte Syltem
26.5
o
)1
.0
19,4
0_10.6
Z3
120 mm
-'-"--------'-- 12mm
I 0mm 7 46.4mm 13 82.1ram 19 123.7mm 25 165.3mm
2 4.6ram 8 47.4mm 14 89.0ram 20 130.7mm 26 ]72.3mm
3 14.8mm 9 5.4.4mm 15 96.Omm 21 137,6mm 27 179.2mm
4 25.0mm tO 61.3mm 16 103.0mm 22 144.5mm 28 186.|ram
5 39,gram II 68,2mm 17 II0.Omm 23 151.5mm 29 193.1mm
6 45,4mm 12 75.2mm 18 ll6,Smm 24 15&4ram 30 200.Omm
Fig. 1 Grld system. Grid points are at Intersections of the radii and circular
lines. Allgrld polnts are Inthe y=0plane. The llnes marked as 12 mm, 36 mm,
and 120mm arethe radlal llnes along whlch experlmentsl data was compared
to the predlctlons ofthe model. The nozzle llp Isat radlus 46.4 mm and angle
of 20°.
Number Density Contour Plot
172.3
68.2
47.4
46.4
70 31.0 19.4 8.8 0
Nozzle Lip
ANGLE FROM CENTER LINE(DEG)
Fig. 2 Number density contour plot for flowfleld shown In Flg.1.
AxialVelocityComourPlot RadialVelocl_ ContourPlot
172.3
137.6 ! 1_7.6
103.0 103.0
C_ e,*
(4L2 61.2
47.4 47.4
46.4 46.4
70 31.0 19.4 U 0 70 ,3|,0 $.i 0
_e,U_ Lq,
ANGLE FROM CENTER LINE (DEG) ANGLE FROM CENTER UNE (DEG)
Fig. 3 Axial velocity contour plot for Fig. 4 Radial velocity contour plot
flowfleld shown In Fig. I. for flowfleld shown in Fig. 1.
TemperatureContourPlot Pressure Contour Plot
__!_]'_"' '__i
177,._ 172.3
_ 137.6 i 137.6
Z_]o_.o Z5_o:_.o
m
(dl,2 6_.2
47.4 47,4
46.4 46.4
70 31.0 19.4 8.B 0 70 3L0 19.4 8.8 0
lq_s_ LJ f_ _p
ANGLE FROM CENTER LINE (DEG) ANGLE FROM CENTER LINE (DEG)
Fig. 5 Temperature contour plot for Pig. 6 Pressure contour plot for
flowfleld shown In Fig. 1. flowfleld shown in Fig. I.
Comparision of Pitot Pressure
with Model Prediction
200
" " "_ _ Predk:tlon at 12 mm
6 __ PPrreeddiiccttiioonn aatt 12306 mmmm
re" 150 '_ •• DDeettaalartl 1326mmmm
"_ _ Detll at 120 mm
0 0 e 1_ ------
0 10 20 30 40 50 60
RADIAL DISTANCE (mm)
Flg. 7 Comparlson of experlrnental Pltot pressures and model predlctlon at
three axlal positions. The last radial data polnt at each axial position
corresponds to the end of the approprlate llne In Flg. 1.
Comparison of Flow Angle
with Model Prediction
e _ Prediction at 12 mm
40 /e _ Prediction at 120 mm
,,-. / • Data It 12mm
UJ_ /e _ Deta It 120 mm
P. 30
UJ O
3:<20 • e • * _ *
0
.u.J. 10 • • *
0 o 1'o.... 2'o 3o 40 .... 5'o 6'o
RADIAL DISTANCE (mm)
Flg. 8 Comparlson of experlmental flow angles and model predictlons attwo
axlal posltlone. The last radial polnt ateach axial posltlon corresponds tothe
end of the appropriate line InFig. 1.
8
TABLE 1 BOUNDARY CONDITIONS ATEXIT SURFACE GRID POINTS
Angle Rmdlal Axlel Temperature Number Pltot
Veloclly Velocity Denslty Preuure
(Degrees) (m/I) (m/s) (K) (Molecules/m 3) (Pa)
0.0 0.0 1180 99 4.46E+21 272.0
2.3 35.0 1170 99 4.51E+21 269,0
4.6 66.8 1140 102 4.62E+21 262.0
6.8 99.6 1120 115 4.30;:+21 237,0
8.8 140.0 1070 148 3.361:+21 173,0
10.6 174.0 997 201 2.42E+21 116.0
13.0 215.0 859 278 1.53E+21 61.2
15.0 233.0 732 338 1.13E+21 36.9
17.9 237.0 525 421 7.43E+20 16.4
19.4 216.0 415 487 6.30E+20 11.0
20.0 135.0 339 536 6.01E+20 9.2