AIAA-9 1- 1363

Numerical and Experimental Investigations of Rarefied Nozzle and Plume Flows of Nitrogen

I. D. Boyd

Eloret Ins tit u t e Palo Alto, CA 94303.

P. F. Penko NASA Lewis Research Center Cleveland, OH 44135.

D. L. Meissner University of Toledo Toledo, OH 43606.

AlAA 26th Thermophysics Conference June 24-26, 1991 / Honolulu, Hawaii

For permission to copy or republish, contacf the American lnsfltute of Aeronautics and Astronautics 370 L'Enfcmt Promenade, S.W., Washington, D.C. 20024

N U M E R I C A L A N D E X P E R I M E N T A L I N V E S T I G A T I O N S OF R A R E F I E D NOZZLE A N D P L U M E F L O W S O F N I T R O G E N

lain D. Boyd* Eloret Institute, 3788 Fabian Way, Palo Alto, CA 94303.

Paul F Penko** NASA Lewis Research Center, Cleveland, OH 44135.

Dana L. Meissner*** Department of Mechanical Engineering, University of Toledo, Toledo, OH 43606.

Abs t r ac t

Numerical and experimental investigations are per- formed for the rarefied flow of nitrogen through a small nozzle which is expanded into near-vacuum conditions. Two different numerical studies are undertaken: the first employs a continuum approach in solving the Navier-Stokes equations, and the second employs a particle approach through use of the direct simulation Monte Carlo method (DSMC). The experimental in- vestigation concerns the measurement of pressure, us- ing a Pitot tube, in the nozzle exit plane and near-field of the plume. Comparison of the experimental and nu- merical data a t the nozzle exit reveals that the DSMC technique provides the more accurate description of the expanding flow. It is discovered that the DSMC solutions are quite sensitive to the model employed to simulate the interaction between the gas and the noz- zle wall surface. It is concluded that the simplistic fully diffuse model is quite satisfactory for the present application.

j--

Introduction For the control in orbit of satellites and large struc-

tures a number of small propulsive devices are often used. The thrust level of these engines is usually quite small, so that it was thought previously that they pre- sented few problems of integration with the spacecraft. However, the experience gained in orbit has shown that such rockets can cause a number of deleterious effects which can reduce significantly the effective lifetime of a satellite. While the firing times of the thrusters are

* Research Scientist. Mailing address:

** Aerospace Engineer, Member AIAA.

~

NASA Ames Research Center, MS 230-2, CA 94035

*** Graduate Student. This paper ia,declared a "or? of the U:S. Government and is not

s u b l e i to copynght protection in the United States.

usually only a few seconds, it must be remembered that they are fired repeatedly for a number of years. This can lead to the gradual accumulation of contam- ination from the plume on sensitive surfaces such as solar arrays. In addition, the plume can cause heating or charging of the spacecraft, which can alter the ther- mal balance and damage scientific intruments. Thrust loss and disturbance torques are also unwanted ef- fects which can result from the firing of the control thrusters. Unfortunately, the magnitude of these ef- fects is very much dependent on the thruster design, and the satellite configuration

The assessment of the interaction between the spacecraft and the plume requires an accurate descrip- tion of the expanding flow field. Therefore, a research program has been established with the objective of ad- vancing the theoretical predictions of small rocket ex- pansions through comparison with experimental data. The major phenomena to be investigated include the flow field of low Reynolds Number nozzles, the re- sultant plume flowfields and impacts, and the amhi- ent environment effects which arise during testing in ground-based facilities. The approach adopted is to develop and compare the various theoretical and nu- merical techniques used in predictions and to obtain accurate experimental data for code development and verification. Although the program encompasses all low thrust propulsion concepts, including the neutral flows of resistojets, the plasma flows of arcjet thrusters, and small chemical rockets, the initial emphasis is directed toward an understanding of the nozzle and plume flows of resistojets.

2 A I A A 91-1969

In the first stage of this study, the nozzle flow of carbon dioxide in a small thruster was investigated with two different numerical techniques'. The first of these techniques adopts a continuum approach by solv- ing the Navier-Stokes equations. The code employed is called VNAP2 (Viscous Nozzle Analysis Program) and is widely available. The second technique is the direct simulation Monte Carlo method3 (DSMC) which mod- els the gas at the molecular level. The main purpose of Ref. 1 was to show that the numerically expen- sive DSMC technique could be applied to low den- sity flows in nozzles with throat Reynolds numbers of around 1000. This was made possible by structuring the DSMC algorithm t o take advantage of the vector- ized hardware available on modern supercomputers. The solutions obtained with these two different solu- tion techniques showed differences mainly in the region close to the nozzle wall in the nozzle exit plane.

The purpose of the present study is to compare re- cently acquired experimental data with solutions ob- tained using these numerical methods. A brief descrip- tion is made of the experimental facility, and the type of measurements which have been undertaken at the exit of the nozzle and in the plume. The two numer- ical techniques employed in the study are also briefly reviewed. Particular attention is given to the physical and computational limitations of these methods based on the assumptions inherent t o their formulation. This investigation provides the opportunity.for verification of the DSMC technique in an expanding gas. In ad- dition, i t will be shown that a portion of the flowfield is particularly sensitive to the model assumed for the interaction between the gas and the surface of the noz- zle wall. The coupling of experimental measurements to the computations offers a unique opportunity to in- vestigate the validity of these models.

Numerical Investigations

1. Continuum ComDutations

A numerical investigation of the nozzle flow has been undertaken with the code VNAP (Viscous Noz- zle Analysis Program)2. The code solves the full set of Navier-Stokes equations by finite differences for a compressible fluid in timedependent, non- conservative form using the explicit, tw-step Mac- Cormack Method. With this method the equations

. I

are marched in time from a specified initial flow field to obtain a steady-state solution. For this partico- lar problem, with its relatively low Reynolds Number, laminar flow was assumed. The model employed at the nozzle wall may be specified as adiabatic or isothermal (for a given temperature distribution): each with no slip. The code produces a body-fitted, non-orthogonal grid in physical coordinates that is transformed to a rectangular grid in the computational plane. For the problem in this study, the grid consisted of 31 radial by 51 axial lines. The grid was clustered near the wall and in the nozzle throat, the regions of highest gradi- ents, to capture adequately the rapidly changing flow conditions. The input includes parameters describing the nozzle geometry, as well as inlet total pressure, tc- tal temperature and zero radial velocity. The subsonic inflow boundary is handled by the method of charac- teristics that allows the axial velocity, static pressure, and temperature to evolve with the solution. The code has been verified in a number of studies for flow in larger thrusters and rockets.

W Since the stream is exhausting to an ambient pres- sure of near zero, the exit-plane boundary condition for the subsonic region of the flow was modified from that originally provided in the code. For a proper state- ment of the problem, a pressure must be specified in the subsonic region of the flow at the exit plane which occurs near the wall. For this problem, however, the exit pressure in the subsonic region is not known a priori because of the restriction of ending the compu- tation at the exit plane. This condition was modeled by extrapolating the exit pressure from interior points (as is normally done in the supersonic region) a t each time step along each grid line in the subsonic region, then using the new values as the specified pressure. In this manner, the exit-plane pressure was continually updated iteratively. This scheme for handling the exit boundary provided a relatively smooth solution for the properties in the vicinity of the exit plane, though it did in effect impress a back pressure on the flow.

I t is well established that the Navier-Stokes equa- tions break down at high Knudsen numbers due to the failure of the linear forms of the constitutive relations for viscous stress and heat transfer. This problem has received detailed investigation in a study relating to the problem of a normal shock wave'. It has been

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AIAA 91-1363 3

I

shown that solutions of shock waves obtained with the Navier-Stokes equations give much poorer correspon- dence to experimental data than computations per- formed with the DSMC technique, or with the Bur- nett equations (a higher order extension of the Navier- Stokes equations). However, no such study has been performed to investigate the effect of the failure of the Navier-Stokes equations under rarefied flow conditions for an expanding gas. This is one of the primary aims of the present study.

2. DSMC ComDutat ions

The direct simulation Monte Carlo method is now a familiar tool employed in the numerical simulation of flows involving rarefied gases. I t has been applied to expanding flows in a number of studies, although al- most none of these have made direct comparison with experimental data. This is somewhat surprising when one considers the quantity of such data that is avail- able mainly in the volumes which publish results from the international symposia on rarefied gas dynamics. The present study therefore makes an important con- tribution in that numerical and experimental studies of an expanding flow have been performed simulta- neonsly. The DSMC code employed in the present work is effectively structured for optimum use on vec- tor supercomputers, and has been described in detail previously5. A computational grid of 760 by 50 non- uniform cells was required to meet the restriction of a cell length of one local mean free in the flow direction. I t was found that the transitory stage of the simu- lations required 75% of the total computational cost. The total time required to achieve asolution has there- fore been reduced signifcantly by employing only half the usual number of simulated particles in the tran- sitory stage. Then, following McDonald', this set is cloned once before commencing with the sampling of particle quantities during the steady stage of the sim- ulation. The DSMC computations are started at the nozzle throat using the input of macroscopic properties from the continuum solution. Several gas/surface in- teraction models have been considered including spec- ular reflection, diffuse reflection, varying fractions of these two extremes, and the Cetcignani-Lampis model implementedin DSMC by Lord'. For nitrogen, the rc- tational energy exchange model of Boyda is employed

:-

-1' . .~~ ..

ber with temperature. The exchange of vibrational energy is assumed frozen due to the low temperatures encountered in the flowfield. The code has been writ- ten specifically to compute the flow in the nozzle, in the plume forward of the nozzle, and in a small por- tion of the backflow. This region has been designed to include a larger area than that investigated exper- imentally, and to ensure that all flow contributing to that region has been included. It is not, however, the purpose of this study to provide a definitive computa- tion of the backflow region. This would be performed in a more numerically efficient way by starting a fresh computation at the nozzle exit plane, using the macro- scopic results from the present study. This is proposed as a topic worthy of future investigation.

The limitations of the DSMC technique applied un- der the present conditions are more of a computational nature, and quite different from the physical difficul- ties experienced by the Navier-Stokes equations. The numerical cost of DSMC computations is linearly pro- portional to the density of the flow. This is because the size of a computational cell should be scaled with the mean free path, which is, of course, itself scaled with the density. Therefore, a high density flowfield requires a large number of computational cells. This means that more particles must be simulated, and conse- quently more collisions computed. The computations of the nozzle flow are quite expensive for the DSMC code because the conditions are those of a relatively high density, continuurn regime. This computation represents one of the most numerically intensive sim- ulations undertaken with the DSMC technique. Rea- sonable computational execution times have only been achieved through the vectorization of many parts of the algorithm, for efficient performance on a Cray- Y/MP supercomputer.

Exuerimental Investieation

The experimental data was taken at the NASA Lewis Research Center with an apparatus in a large vacuum tank. The apparatus contains a heat ex- changer and a converging-diverging nozzle that heats and expands a gas to simulate a thruster. The appara- tus was designed specifically for making measurements in a rarefied, expanding flow. I t allows measurement in the nozzle exit plane and in a considerable portion

which in esence varies the rotational collision num- of the plume by using traversing mechanisms. Full de-

4 A I A A 91-1969

tails of the experimental procedures are given in Ref. 9. For this phase of the study, nitrogen was used as the working fluid for the purpose of obtaining data with a diatomic gas. Pressure measurements were made with a Pitot tube in the near-field plume of the nozzle. De- tails of the nozzle geometry are listed in Table l and a set of flow conditions for the experimental data are listed in Table 2. A schematic diagram of the thruster and Pitot tube is shown in Fig. 1.

Test Facility

The experimental tests for this study were con- ducted in a spacesimulation vacuum tank. The test apparatus was mounted in a 1 m diameter section about 1 m in length that is attached to a larger tank 4.9 m’in diameter and 19 m long. The facility pumping system consists of twenty oil-diffusion pumps in paral- lel with four blowers, and in series with four roughing pumps. A detailed description of the facility can be found in Ref. 10.

The pumping system was able to maintain a vac- uum during the tests of order lo-’ Pa with a nozzle flow rate of 0.068 g/s. The vacuum pressure was mon- itored with a hot-cathode, ionization gauge mounted on the test section and connected to a digital meter.

Test Apparatus

The test apparatus consisted of a heater and noz- zle that simulated a thruster, and traversing tables that permitted surveying the plume with pressure probes. The nozzle flow was first heated by passing through an annular area comprised of a 12.7 mm di- ameter cartridge-heatiug element inside of a 17.3 mm internal-diameter tube. The pressure and temperature of the flow were measured downstream of the heater in a 22.1 mm diameter section. The measurements were effectively nozzle-inlet stagnation conditions as the ratio of the area of the measurement station to the throat area was about 48:l. The pressure was sensed hy a capacitance manometer having a full-scale range of 1 . 3 3 ~ 1 0 ~ Pa. The temperature was measured with a half-shielded, chromel-alumel thermocouple at the centerline of the measurement station, and con- nected to a digital voltmeter with self-contained, cold- junction compensation. Volumetric flowrate of the ni- trogen was measured with a thermal-type flowmeter.

The test apparatus was designed specifically for

making measurements in an expanding flow by use of a traversing mechanism. The mechanism consisted of a rotary table mounted atop a pair of linear tables. The range of table travel was 24 mm in the radial di- rection, 36 mm in the axial direction, and 360 degree in rotation. The tables were manually positioned via a mechanical link between the rotary handles 011 each table and handles outside the vacuum chamber.

A Pitot tubes of 1.0 mm diameter was used to mea- sure pressure of the flow in the exit plane and plume. A capacitance manometer having a full-scale range of 1 . 3 3 ~ 1 0 ~ Pa, and accuracy of 0.1% of full scale, was used to measure the Pitot tube pressure. The manometer was mounted directly on the rotary table. Nozzle wall temperatures were measured by chromel- alumel thermocouples tack-welded to the outer wall surface and evenly spaced between the nozzle throat and exit plane.

Calculation of Pitot Pressure

To compare computed and experimental results, Pitot pressures were first calculated from the numer- ical results using the computed state variables. The calculated Pitot pressure is the pressure that would be measured if a Pitot tube were inserted in the com- puted stream. Since the flow was both supersonic and rarefied, the calculation involved a twestep process:

1. The total pressure behind a normal shock was calculated via the normal-shock relationship

w

where P, and M are the static pressure and Mach number ahead of the shock, 7 the ratio of specific heats, and Po the total pressure behind the shock.

2. Po is then corrected for rarefaction effects by

where Po,,, is the corrected total pressure behind the shock, Re,, the probe Reynolds number, and To, the stream total temperature.

The equation in Step 1 is the customary pressure ra- tio across a normal-shock from the Rankine-Hugoniot relations. Here, the values of P, and M are outputs of the numerical codes. The relationship in Step 2 is a correction of Po, calculated in Step 1, to account for probe rarefaction effects, where Po, represents the

-L’

A l A A 91-1363 5

pressure that would actually be measured with a Pitot tube. The relation in Step 2 was obtained from Ref. 11, a report on rarefaction effects of Pitot tubes in rarefied wind-tunnel flow. In Ref. 11, the factor P,,/P,, which is always greater than or equal to unity, was plotted as a function of Re,, with total stream temperature, To, as parameter. The probe Reynolds number was defined as

Rep = ~ m U m D p l ~ 2 (3)

where pm and U- are the free-stream density and ve- locity preceding the shock, D, is the probe diameter, and p2 is the gas viscosity a t the stream temperature behind the shock.

In the probe Reynolds number, the probe diam- eter was that of the actual experimental apparatus, and the freestream density and velocity were taken directly from the output of the codes, except for the DSMC results, where the mass density was calculated from number density, the quantity actually computed by the code. The viscosity, j12, was obtained from the numerical results by first computing the temperature, Tz, downstream of a normal shock, using the computed values of upstream Mach number and static tempera- ture from the numerical schemes in the normal-shock relation. The viscosity was then found from the tem- perature power law,

w

(4)

where prej is the value for viscosity of nitrogen at T,,,. Having the probe Reynolds number, the correction

factor for Step 2 was obtained by entering the paramet- ric curve in Ref. 11 for T, = 700K at the appropriate value of Re,, with P,,/P, read from the ordinate. Fi- nally, the Pitot pressure, derived from numerical code results, was obtained by,

(5)

The comparisons of analytical and experimental re- sults in the following section are of Po,, derived from the computed results, and the actual data reading from the capacitance manometer on the test apparatus. No corrections, as such, were made to the values measured experimentally.

4

Using the input of Tables 1 and 2, the continuum code was run using the adiabatic wall model to ob- tain a solution of the nozzle flow. The first test of the continuum code was to compare the wall temperatures computed against those measured experimentally us- ing thermocouples. This comparison is shown in Fig. 2 where it is observed that the computed temperatures are higher than the measured values. The contin- uum code was therefore run again with an isother- mal wall condition together with a wall temperature distribution based on the experimental data shown in Fig. 2. The solutions of this continuum run at the nozzle throat and along the nozzle wall were then em- ployed to obtain a solution of the nozzle flow and the near field expansion region with the vectorized DSMC code. The DSMC computation required 3 CPU hours on the Cray YMP while the execution time for VNAP was 0.81 CPU hours.

A comparison of the two solutions is made in Fig. 3 in which Mach number contours are shown. The con- tinuum solution of the nozzle flow occupies the upper portion of the figure, and the DSMC solution of the nozzle and the near field expansion are shown in the lower portion. The vertical line from the nozzle lip in the DSMC computation represents the plane at which the computed data in the backflow region have been cut off for the purposes of this study. This has been performed due to the amount of statistical scatter ex- hibited by this data. As stated earlier, this region of the flow will be computed in the future with a separate simulation to be begun at the nozzle exit plane

In a similar way to the flow of carbon dioxide in a small nozzle, computed in Ref. 1, significant differ- ences were observed in the two numerical solutions. In the DSMC solution, for the flow near the nozzle lip, it is found that the sonic line intersects the lip, and the flow is turned towards the lip, whereas in the continuum solution the flow is parallel with the nozzle wall. Also, while the solutions are in agreement in the near-continuum flow regime near the nozzle throat, the Mach number computed with DSMC becomes grad- ually greater than the corresponding continuum val- ues. Further differences are revealed by examining the macroscopic flow quantities in the nozzle exit plane. In Fig. 4, the number densities computed with the

6 A I A A 91-1363

two different numerical techniques are compared. It is found that the density computed with DSMC is gen- erally less than that obtained with VNAP. A trend consistent with this finding is shown in Fig. 5 where the velocity profiles for the two methods are compared. It is found that the velocity provided by DSMC is al- ways greater than that given by VNAP. It is of par- ticular importance to note that the velocity given by DSMC at the nozzle wall is non-zero. This indicates that the nc-slip boundary conditions assumed in the continuum computations are invalid. This failing in the continuum formulation is particularly important in the present study due to its significant effect on the boundary layer thickness. The temperatures com- puted at the nozzle exit with the two numerical meth- ods are‘given in Fig. 6. Both translational and rota- tional temperatures are obtained from DSMC, whereas the continuum solution assumes thermal equilibrium giving just one temperature. The continuum temper- ature is always larger than both DSMC temperatures. These latter values show only a small degree of non- equilibrium in the low density region close to the nozzle wall. The nonequilibrinm between the translational and rotational energy modes is more accentuated as the plume expands away from the nozzle. This phe- nomenon is observed clearly in Fig. 7 where the trans- lational temperature, shown in the lower portion of the figure, is compared with the rotational temperature, shown in the upper portion. At the nozzle lip, the translational temperature is turned into the lip, and seems t o curve around the blunt end of the wall. By comparison, the rotational temperature freezes quite quickly in this region, due to the relatively low den- sity.

In Fig. 8, the Pitot pressure profiles in the noz- zle exit plane are shown. Four separate profiles are shown: the experimental data, the continuum solu- tion, and two DSMC solutions employing two differ- ent galsurface interaction models. It is found that the DSMC computation simulating the wall diffusely gives the best correspondence t o the experimental data. The continuum solution overpredicts the Pitot pressure, in the isentropic core of the expansion, and then un- derpredicts the results in the thick laminar boundary layer. The DSMC solution employing specular wall reflection essentially provides a non-viscous Euler so-

lution with no boundary layer. The sensitivity of the exit plane profile to the surface model is readily seen by comparison of the two DSMC solutions. Further DSMC computations employing varying combinations of diffuse and specular reflection, and the Cercignani- Lampis model7, all gave profiles lying between the two extremes indicated in Fig. 8 . The fact that the diffuse model agrees so well with the experimental data is a strong indication that this model is well suited to the subsonic flow conditions along the nozzle wall. This study provides one of the very rare occasions in which gas/surface interaction models have been assessed di- rectly through comparison with experimental data.

It is difficult to ascertain whether the differences between the continuum and particle simulation re- sults are due to the failure of thk Navier-Stokes equa- tions in a rarefied expanding flow, or instead to the no-slipp nozzle wall boundary condition employed in VNAP and the outflow boundary condition. In Ref. 3 (page 188), it is shown that the translational tem- perature is smaller than the continuum temperature, and the rotational temperature larger, for a one- dimensional steady spherical expansion at a similar Knudsen number as the nozzle flow. The exact diver- gence of the translational and rotational temperatures is very much dependent upon the rotational energy exchange model employed in the simulation. The two models used in Ref. 3 are different from that employed in the present study. Also, the present application is complicated by two-dimensional and boundary layer effects. However, it is reasonable to expect similar nonequilibrium behavior as that which occurs in the 1-d, steady expansion. Therefore, it is concluded that the continuum computation does not predict the flow correctly because of its failure to capture thermal non- equilibrium effects due to the physical limitations of the Navier-Stokes equations, and the use of a no-slip boundary condition.

W” J

In Fig. 9, the Pitot pressure profiles at an axial dis- tance of 12 mm from the nozzle exit plane are shown. It is to be noted that VNAP provides a solution. for the nozzle flow only, so no continuum data has been obtained. The two profiles obtained with the DSMC technique show the effect of applying the correction of Eq. (2) to the data. Once again, the DSMC solu- tions give excellent correspondence to the experimen-

L9

A I A A 91-1363. 7

tal data. These trends are observed further down- stream at distances of 24 mm and 36 mm from the nozzle exit plane in Figs. 10 and 11. It is most en- couraging to find that the excellent agreement between the DSMC computations and the experimental data is continued into the plume expansion. The correction factors become larger as the density, and the Reynolds number, decrease. The curve from Ref. 11 used for the corrections is not ideal as it is was constructed from just afew experimental cases. It is the intention to im- prove this situation in the future by a thorough inves- tigation of this phenomenon with the DSMC method.

The measurement of Pitot pressure at locations away from the plume axis involves rotation of the probe. The experimental data is recorded as the maxi- mum pressure obtained during this rotation. Following Bailey12, this point also provides the angle of the flow at each location in the flowfield. An example of this data is shown in Fig. 12 at the location of 12 mm from the nozzle exit plane. It is found that the DSMC com- putations offer very good agreement with the probe measurements, thus lending further support in favor of the simulation results.

I,

Concludine R e m a r k s

By comparison with new experimental data, it has been shown that the direct simulation Monte Carlo method provides an accurate description of a rarefied flow in the nozzle and the near-field expansion of a small rocket. This study thereby provides verification of the DSMC method in an expanding flow for the first time. For a number of wall models it was found that diffuse reflection gave the best agreement with experi- ment. A solution of the Navier-Stokes equations under the same conditions failed to predict the flow correctly. It is well established that, for rarefied flows, the Knud- sen number becomes so great that the Navier-Stokes equations are invalidated. In previous studies of shock waves, ii has been shown that the DSMC technique of- fers better correspondence to experimental data under conditions where the Navier-Stokes equations break down. It is therefore concluded that the failure of the continuum solution is effected by the physical limi- tations of the Navier-Stokes equations, although the modeling of the gas-surface interaction along the noz- zle wall also plays an important role.

w-

Acknowledgement

Support (for I.D.B.) by NASA (Grant NCC2-582) is gratefully acknowledged.

References

Boyd, I.D., Penko, P.F., and Carney, L.M., “Eff- cient Monte Carlo Simulation of Rarefied Flow in a Small Nozzle”, AIAA Paper-90-1693, AIAA/ASME 5th Joint Thermophysics and Heat Transfer Coufer- ence, Seattle, WA, June 1990.

Cline, M.C., “VNAP: A Program for Computation of Two-Dimensional, Time-Dependent Compressible, Viscous Internal Flow”, Los Alamos Laboratory Re- port LA-7326, Nov. 1978.

Bird, G.A., Molecular Gas Dynamics, Clarendon Press, Oxford, 1976.

Lumpkin, F.E., Development and Evaluation of Con- iinuum Models for Translaiional-Roiational Ponequi- librium, Ph. D. Thesis, Stanford University, March 1990.

Boyd, I.D., “Vectorization of a Monte Carlo Method For Nonequilibrium Gas Dynamics”, Journal of Comp- utational Phvsics (to appear, 1991).

McDonald, J.D., A Computationally Efficieni Parti- cle Simulation Meihod Suited to Vector Computer Ar- chitectures, Pb. D. Thesis, Stanford University, Jan- uary 1990.

Lord, R.G., “Application of the Cercignani-Lampis Scattering Kernel to Direct Simulation Monte Carlo Calculations”, in Rarefied Gas Dynamics, edited by A.E. Beylich, VCH Press, Weinheim, Germany, 1991, pp. 1427-1433.

Boyd, I.D., “Analysis of Rotational Nonequilibrium in Standing Shock Waves of Nitrogen”, AIAA Journal, Vol. 28, No. 11, 1990, pp. 1997-1999.

Penko, P.F., Boyd, I.D., Meissner, D.L., and De- Witt, K.J., “Pressure Measurements in a Low-Density, Nozzle Plume For Code Verification”, AIAA Paper-91- 2110, AIAA/SAE/ASME/ASEE 27th Joint Propul- sion Conference, Sacramento, CA, June 1991.

lo Finke, R.C., Holmes, A.D. and Keller, T.A., “Space Environment Facility for Electric Propulsion Systems Research”, NASA TN-D-2774, 1965.

l 1 Stephenson, W.B., “Use of the Pitot Tube in Very Low Density Flows”, AEDCTR-81-11, Oct. 1981.

8 AZAA 91-1969

\

l 2 Bailey, A.B., “Flow-Angle Measurements in a Rar-

efied Nozzle Plume”, AIAA Journal, Val. 25, 1987, Test Gas Nz pp 1301-1304. Stagnation Temperature 699K

Table 2. Nozzle Flow Conditions.

Stagnation Pressure 6400Pa Throat Reynolds Number 851 Throat Knudsen Number 2 . 2 5 ~ 1 0 - ~ Table 1. Nozzle Geometry.

Throat Diameter 3 . 1 7 5 ~ 1 0 - ~ m Exit to Throat Area Ratio Exit Angle 200

100

E. 550

Nitrogen In

1

\ --- . e-- - *-----*-----*

I I I

PO

+ VOll tage

/ I /

Heater

Rotation (Shielded

Thermocouple)

Fig. 1. Schematic diagram of thruster and probe con- figuration.

Pliot Pressure, P

x10 21 7

6

L o 5 x

c c ._

z 4

$ 3 - 5 2 z

1

u.50000 1 . 00000 1.50000 2.00000 2 .50000 3.00000 3.50000 4.00000 4.50000 5.00000 5.50000 6 . 0 0 0 0 0 6.50000 7 .00000 1 . 5 0 0 0 0

- - - -

0 I I I I 1 I I

0.0 0.4 0.8 1.2 1.6 Radial Distance (cm)

Fig. 4. Comparison of computed number density p r e files in the nozzle exit plane.

I I I I I 1 \

0.0 0.4 0.8 1.2 1.6 .Radial Distance (cm)

Fig. 5. Comparison of computed velocity profiles in the nozzle exit plane.

AIAA 91-1369 9

Fig. 3. Comparison of Mach number contours com- puted with particle (lower) and continuum (upper) methods

Continuum DSMC (Trans) DSMC (Rot)

400

200

0 0.0 0.4 0.8 1.2 1.6

Radial Distance (cm)

Fig. 6. Comparison of computed temperature profiles in the nozzle exi’t plane.

10 A I A A 91-1369

Fig. 7. Comparison of temperature contours com- puted with the particle method for the translational (lower) and rotational (upper) energy modes.

Experiment Continuum DSMC (diffuse)

----_ - DSMC (specular)

300

200

100 I-+ '. -- .- '.

0

Experiment ................ DSMC, Eq. 1

a DSMC. Eq. 2 Y

150 v) v)

a 100 4- E 0 50

n

w

- 0.0 0.4 0.8 1.2 1.6 0.0 0.5 1 .o 1.5 2.0

Radial Distance (crn) Radial Distance (cm)

Fig. 8. Comparison of computed (particle and con-

nozzle exit plane.

Fig. 9. Comparison of computed (particle) and mea-

from the the nozzle exit. tinuum) and measured Pitot pressure profiles in the sured Pitot pressure profiles at a distance of 12 mm i-'

AIAA 91.1963 11

Experiment ............ ... DSMC (Eq. (1)) - DSMC (Eq. (2))

- -

0.0 0.5 1.0 1.5 2.0 2.5 Radial Distance (Cm)

Fig. 10. Comparison of computed (particle) and mea- sured Pitot pressure profiles a t a distance of 24 mm from the nozzle exit.

h m a 2

2

h

3 u) u)

Q 0 c 4-

140

120

100

80

60

40

20

0 ., 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Radial Distance (cm)

Fig. 11. Comparison of computed (particle) and mea- sured Pitot pressure profiles a t a distance of 36 mm from the nozzle exit plane.

80

- DSMC

-

0 1 2 3 4 Radial Distance (Cm)

Fig. 12. Comparison of computed (particle) and mea- sured flow angle a t a distance of 12 mm from the nozzle exit.