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AOO-36826

AIAA 2000-3668

Numerical Modeling of Axisymmetric andThree-Dimensional Flows in MEMS nozzles

A.A. Alexeenko, S.F. Gimelshein, and D.A. LevinGeorge Washington University,Washington, DC 20052

RJ. GollinsUniversity of Minnesota,Minneapolis, MN 55455

36th AIAA/A5ME/SAE/ASEE Joint PropulsionConference and Exhibit

16-19 July 2000 / Huntsville, ALFor permission to copy or republSsh, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 22091


Numerical Modeling of Axisymmetric andThree-Dimensional Flows in MEMS nozzles

A.A. Alexeenko*, S.F. Gimelshein*, and D.A. Levin*George Washington University

Washington, DC 20052

R.J. Coilins*University of Minnesota,Minneapolis, MN 55455

AbstractA numerical study of three-dimensional effects on theperformance of a micronozzle fabricated from flat sili-con wafer is performed by both continuum and kineticapproaches. The nozzle operates in a low Reynoldsnumbers regime and viscous effects dominate the gasexpansion. Thrust losses occur because the shear onthe wall is greater in the nozzle of a flat configurationcompared to an axisymmetric conical nozzle. There-fore, the prediction of the micronozzle performancebased on axisymmetric or two-dimensional modelingcan lead to significant design errors.

I IntroductionUsing the advances in Micro-Electro-Mechanical Sys-tems (MEMS) many micron sized mechanical deviceshave been constructed. Bulk micro-machining in siliconand other materials has been used to produce pumps,small motors, channels, mechanical sensors, and otherdevices.1 MEMS technology has been considered forthe production of micron-sized rocket motors, howeverseveral questions must be addressed before their utilitycan be assessed. One of the most important issue isan estimate of thrust performance at the small scale,possible with this new technology.

* Graduate Student, Student Member AIAA. E-mail [email protected]

* Senior Research Scientist, Member AIAA.t Research Professor, Senior Member AIAA. E-mail

[email protected]* Professor Emeritus, Senior Member AIAA.

Copyright ©2000 by A. A. Alexeenko, S.F. Gimelshein, D.A.Levin, R.J.Collins. Published by the American Institute of Aero-nautics and Astronautics, Inc. with permission

The dramatic change in the linear dimension affectsboth the mechanics of working fluid and geometric de-sign of micro devices. Prom the fluid mechanics point ofview, a model is needed to take into account microflu-idic aspects.2 A typical flow in a cold-gas micron-sizeddevice has low Reynolds number on the order of 102 -103 and viscous effects therefore will be much more sig-nificant than in conventional large nozzles. Other con-sequences of the small linear scale for supersonic nozzleflow are rarefaction effects resulting in the possibilityof velocity slip and temperature jump at the gas solidsurface interface. The surface area to volume ratio inmicro devices is high and the wall effects may dominatethe fluid behavior inside the nozzle, thus requiring anaccurate modeling of fluid-surface interaction.

Often the geometric shape of a mechanical device ischosen to maximize the performance while minimizingthe cost of manufacture. Since entirely different mate-rials and manufacturing technology is used for MEMS,the geometric shape is different from that for largescale nozzles. The conventional rocket nozzles are al-most always of an axisymmetric shape and often havea contoured section to direct the exhaust gas along theaxis. For the microfabrication of nozzle devices thetechnique is well developed for etching a simple-shapeddevice from a plane silicon wafer. Experimental mea-surements by Bayt and Breuer3 of mass flow and thrustlevels of such a flat contoured nozzle showed that forlow Reynolds numbers Re < 500 nozzle performance isstrongly affected by viscous losses and there is a consid-erable deviation from a two-dimensional Navier-Stokessolution because of the three-dimensional end-wall ef-fects.

The main objectives of this work are numerical studyof viscous effects in micronozzle flow and a compari-son of different geometric configurations, axisymmetricconical and flat three-dimensional, in terms of thrust


performance and flow fields. A two-dimensional modelof a micronozzle is also examined and compared withthe full three-dimensional simulation.

Application of modern CFD techniques to modelthese flows is important since it allows one to obtaindetailed information on flow structure and peculiari-ties. There are two general approaches to treat a fluidin different flow regimes - continuum when the scale offlow phenomena is large compared to the fluid micro-scopic structure, and kinetic, for a rarefied flow wherephenomena at the molecular level become important.Since the flow regime varies from near-continuum atthe nozzle throat to rarefied at the nozzle exit, accu-rate modeling is a challenging task for both approaches.Both kinetic and continuum numerical approaches areused in this study.

Several earlier papers presented computational re-sults for axisymmetric micronozzle flows.4"7 Most ofthem employed the direct simulation Monte Carlo(DSMC) method, the most widely used approach formodeling rarefied gas flows. The work by Chung et al5was performed with the goal to make a comparison be-tween numerical modeling and experimental data. Inthe paper by Ivanov et al s both DSMC and contin-uum methods were used to simulate the axisymmetricand two-dimensional flow in a nozzle at low Reynoldsnumbers. The present paper is the first application ofthe DSMC method to modeling three-dimensional mi-crothrusters. The solutions of the Navier-Stokes equa-tions are also obtained to elucidate the area of appli-cability of the continuum approach.

The flow of molecular nitrogen in micronozzles is an-alyzed in terms of flowfields and performance charac-teristics. The outline of the paper is as follows. InSection II the geometric setup and flow conditions areexplained. Section III describes the numerical methodsand computational requirements for the cases underconsideration. Results of numerical modeling are pre-sented in Section IV. A discussion of the flow featuresis also given and the capabilities of numerical methodsare compared. Then, nozzle performance parametersare compared for different geometric configurations.

II Micronozzie configurationsand flow conditions

Two different micronozzle configurations are consid-ered, axisymmetric and three-dimensional. The ax-isymmetric conical nozzle has an expansion angle of15 deg, throat diameter Dt = 300 pro., and an exitto throat area ratio of 100. A schematic of the three-dimensional (hereafter referred as flat) nozzle is shownin Fig. 1. The throat width is equal to the ax-

isymmetric nozzle throat diameter Dt, and the heighth = 300 ftm. The expansion angle is 15 deg, and thearea ratio is 10. The flat nozzle dimensions are derivedfrom the experimental study reported recently8. Theflat nozzle has the same cross section in XY symmetryplane as the axisymmetric nozzle, as well as the samenozzle length of 5.038 mm. The details on nozzle geom-etry and computational domain in XY symmetry planeare given in Fig. 2.

h, height

Figure 1: Schematic of flat micronozzle.

Figure 2: Nozzle geometry and computational domainin XY plane.

For the two geometric nozzle configurations, the flowof molecular nitrogen was calculated at a stagnationpressure pc = 10 kPa and stagnation temperatureTc = 300 K. Stagnation and critical conditions for asonic flow at the throat are given in Table 1. TheKnudsen number at the throat for the both nozzlesis 5 x 1Q~3 and the corresponding Reynolds numberbased on the throat half-width is 200. The tempera-ture at the nozzle wall is assumed to be constant andequal to the stagnation temperature at the chamber.


Table 1: Flow conditions

Test gas N2

Stagnation temperature Tc 300 KStagnation pressure pc 10 kPaCritical pressure pt 5.2 kPaCritical temperature Tt 250 KWall temperature Tw ____300 K

III Numerical methodsIII.l Continuum methodA continuum model was used to describe the gas flow inthe axisymmetric and three-dimensional nozzles. Nu-merical solution of the Navier-Stokes equations for vis-cous fluid flow was obtained with a finite-volume spatialdiscretization on a structured three-dimensional gridimplemented in the General Aerodynamic SimulationProgram (GASP).9

Molecular nitrogen was considered a perfect gas andthe Sutherland model10 was used for the approxima-tion of temperature dependence of the gas viscosity.Viscous derivative terms in the momentum and energyconservation equations are computed with second-orderaccuracy on the interior and gas-solid interface cells.The third-order upwind-biased scheme is applied forspatial reconstruction of volume properties on the cellboundaries. To obtain a steady state solution two fac-tor approximate factorization is used for time stepping.

As was shown by Ivanov et al 6, an extrapolationboundary condition at the exit of a nozzle can suffi-ciently decrease the accuracy of the performance pre-diction at low Reynolds numbers. Therefore, an exte-rior region of nozzle was also included in the computa-tional domain. Two-zone grids resolving gradients nearwall boundaries and along the axis are used in the com-putations. Grid convergence study showed that the so-lution is grid independent for grid dimensions 200 x 40(zone 1) and 100 x 60 (zone 2) and larger for the ax-isymmetric and 2D case and 200 x 40 x 20 (zone 1) and100 x 60 x 20 (zone 2) for the 3D case.

The GASP MPI capabilities allow iterations in dif-ferent regions of the computational domain to be per-formed in parallel. Using this option on a two processorSGI Octane the total computational time for the ax-isymmetric case with the grid 200 x 40 and 100 x 60 isabout 8 hours and about 50 hours for the 3D case.

Table 2: Nozzle performance characteristics for GASPsolution with different inlet conditions for an axisym-metric nozzle.

subsonic inletuniform throat

thrust, mN1.081.07

spi SGC

66.0665.62

A no-slip boundary condition is used in these compu-tations to model gas-surface interaction at a fixed walltemperature. The temperature of the wall is set to thestagnation temperature at the chamber. The inlet con-ditions are obtained from ideal nozzle theory based onstagnation gas properties and an inlet area ratio. Forthe axisymmetric case both subsonic and critical inletconditions are considered. For the subsonic inlet con-ditions the boundary layer is thin at the nozzle throat.This is illustrated in Fig. 3 where the velocity compo-nent in X direction along the nozzle axis is shown. Thedifference between solutions for these two types of inletconditions is therefore small. The comparison of Machnumber fields is given in Fig. 4. There is a very smalldifference in the results. For a quantitative compari-son, the X-component velocity distribution along thenozzle axis is given in Fig. 5. The nozzle performancecharacteristics of the axisymmetric nozzle are also veryclose (see Table 2).

300

250

1200

.-150

100

50

5E-OS 0.0001Y, m

0.00015

Figure 3: Axial velocity profile at the throat of theaxisymmetric nozzle.

The constant critical throat condition is thereforeused to as the boundary conditions for the axisymmet-ric and three-dimensional solutions calculated with the


as

0.001-

•O.001

•O.OD2-0 ' ' ' 0.001 0.002 0.003

X,m

Figure 4: Density fields (kg/m3) for uniform andnonuniform throat conditions.

0.001 0.002 0.003 0.004 0.005X, m

Figure 5: Axial velocity profile along the axis of theaxisymmetric nozzle.

Navier-Stokes and DSMC methods.

III.2 DSMC methodThe Direct simulation Monte Carlo method11 is a sta-tistical computational approach used for solving rar-efied gas dynamics problems. During last several yearsit has been successfully applied to modeling differentflows in the near-continuum regime. In this work, aDSMC-based software SMILE (Statistical Modeling Ina Low-density Environment)12 is used in all DSMCcomputations. The majorant frequency scheme13 of theDSMC method is utilized to model collisions betweenmolecules. For a one-component gas, this scheme re-defines the collision frequency used in the simulation

ZMFS = n(aT(v)v) max'

where v is the relative velocity of colliding particles,<TT is the total collision cross section, n is number ofsimulated particles in the cell. The collision frequencyin a cell is calculated using the number of simulatedmolecules in the cell and the maximum value of theproduct of the total collision cross section and the rela-tive collision velocity. The number of potential collisionpairs is therefore maximized with respect to the prod-uct of the cross section and the relative velocity. Oncea pair is selected the probability that a collision occursis evaluated by an acceptance-rejection test of the ratioof

P =

The intermolecular potential is assumed to be thevariable soft sphere model14. The Larsen-Borgnakkemodel15 with temperature-dependent Zr and Zv anddiscrete rotational and vibrational energies is used forthe energy exchange between translational and internalmodes.

Under flow and geometry conditions examined in thiswork, the wall can be expected to significantly affectthe gas flow due to the gas-surface interaction. TheDSMC technique allows a detailed treatment of this in-teraction. Gas molecules can lose or gain a fraction ofimpulse and energy upon a collision with a surface, andthe outcome of the collision depends on physical andthermal properties of the surface and gas. An ideallysmooth surface would reflect particles specularly withthe reflection angle equal to the incident one. Realsurfaces have a significant degree of roughness at themolecular scale that results in inelastic diffusive colli-sion of molecule with the surface.

One of the most widely used gas-surface interactionmodels, the Maxwell model, assumes that a fraction(1—a,j) of incident particles is reflected specularly whilethe remaining fraction a^ experiences a diffuse reflec-tion that means particle velocities are distributed ac-cording to the Maxwellian distribution with the surfacetemperature. This parameter a<i is equal to tangentialmomentum accommodation coefficient:

ad = Pir ~

Pir

where PT is the tangential momentum, and indices i,rrefer to incident and reflected particles. Experimentaldata16 for silicon interacting with nitrogen flow sug-gests the accommodation coefficient ad = 0.8. TheMaxwell model with different o^ between 0 and 1 andthe surface temperature Tw = 300 K is used in theDSMC computations.


DSHC,N=2107

DSMC, H = 10'

DSMC, N = 5 10*

DSMC, N =1.25 10°

Figure 6: Translational temperature profiles (K) in 3Dmicronozzle. SMILE solution for different number ofparticles.

DSMC,N = 210'

DSMC, N = 10'

DSMC, N = 5 10*

DSMC, N = 1.25 10*

0.0025' 0.0075

Figure 7: Velocity component Ux profiles (m/sec) in3D micronozzle. SMILE solution for different numberof particles.

The majorant frequency scheme used in SMILE wasstrictly derived from the Leontovich master kineticequation for the AT-particle distribution function. Inany system of a finite number of particles, N, there arestatistical correlations between particles17 that ariseeven in case particles where initially independent. Themaster kinetic equation differs from the Boltzmannequation, the principal equation that describes rarefiedgas flows, by the presence of a source term dependenton a pair correlation function, g = /2 — fifi- Here, f\and /2 are one- and two-particle distribution functions,respectively. The correlation function g oc N~l, andtherefore the correlation term vanishes when N -> oo.

The statistical dependence is inherent hi any sys-

tem of N particles. The number of simulated particlesis therefore a crucial parameter for any DSMC study.There should be enough particles in the simulation sothat the statistical correlations do not affect the re-sult of the simulation and it can be considered a so-lution of the Boltzmann equation. It was suggestedby Gimelshein et al18 to use the number of moleculesin a cube with the linear dimension of the local meanfree path, A, as an estimate for statistical correlations.Usually, there should a few molecules in A3 for thecorrelations to be negligible. The number of particlesrequired is especially severe in case when the DSMCmethod is applied to model the three-dimensional near-continuum fiows. These requirements are connectedwith both flow dimensionality and the high density ofthe gas (L e., small A).

It is therefore important to analyze the influence ofthe number of particles in order to have results inde-pendent of particle correlations. Convergence studies ofthe DSMC solution in terms of the number of particleswas conducted for the three-dimensional micronozzle.The translational temperature and velocity componentUx profiles along the nozzle centerline are plotted hiFig. 6 and 7 for different number of simulated particlesin the computational domain: N = 1.3 • 106, 5 • 106,107, 2 • 107. The simulations show that there is a sig-nificant dependence of the results on N in the region ofhigh density. As it is seen from the profiles, the DSMCsolution for a smaller number of particles is shifted.

IV Results and discussionIV. 1 Axisymmetric conical nozzleLet us consider first the flow in an axisymmetric con-ical micronozzle that was studied with the continuumand DSMC methods. A comparison of density fieldsobtained by the two techniques is given in Fig. 8. Thedensity is normalized by its value at the nozzle throat.

The flowfields for the axisymmetric micronozzle arethose typical for cold gas thrusters. The gas experi-ences about two orders of magnitude decrease in den-sity along the nozzle axis (see Fig. 9). In the radialdirection the density decreases near the wall becausethe temperature of the wall is higher than that of gas.Contours of the velocity component in the axial direc-tion obtained by both models are plotted in Fig. 10.It is seen that the numerical solution of Navier-Stokesequations agrees well with the DSMC results inside thenozzle and at the core flow outside the nozzle. There isa significant difference between the two solutions onlyin the region of the nozzle lip. The reason for that is arapid expansion of gas and a high flow rarefaction thatis difficult to capture by continuum methods.


0.002

0.001

-0.001

-0.002

1.DODD03313

0.001 0.002 0.003 0.004 0.005 0.008X, m

0.002

0.001

0

-0.001

-0.002 -

10

0.001 0.002 0.003X, m

0.004 0.005 0.006

Figure 8: Density contours in axisymmetric micronoz-zle. SMILE - upper part, GASP - bottom.

Figure 10: Velocity component in X direction (m/sec)in axisymmetric micronozzle obtained by SMILE (top)and GASP (bottom).

10 V

Figure 9: Comparison of the density profiles along thenozzle axis, axisymmetric case.

Velocity contours qualitatively illustrate how thethickness of the boundary layer grows downstream fromthe nozzle throat. To study the boundary layer growthin more detail, the distribution of the X-component ofvelocity at the nozzle exit is given in Fig. 11. The veloc-ity gradient is large close to the axis and the boundarylayer occupies most of the exit area. The difference be-tween the velocities at the wall for the two approachesis due to the difference in their boundary conditions.However, a comparison of velocity profiles along thenozzle axis presented in Fig. 12 shows a small differ-ence between the two solutions.

Translational temperature contours are shown inFig. 13 for two different approaches. The agreement issatisfactory inside the nozzle except in the vicinity of

600

500

I 400

=M300

zoo

100

0.0005V, m

0.001 0.0015

Figure 11: The X-component of velocity at the nozzleexit plane in axisymmetric case.

the lip where the impact of flow rarefaction is again sig-nificant. The GASP solver assumes there is an equilib-rium between translational and internal modes. Whilethe vibrational mode is essentially frozen at the lowtemperatures under consideration, and vibrational ex-citation is not an important factor, the rotational tem-perature may be significantly different from the trans-lational one. Figure 14 shows the DSMC rotational andtranslational temperature profiles along the nozzle axiscompared with those obtained by GASP. Differencesbetween the translational and rotational temperaturesbeyond the nozzle exit can be observed. The differ-ence between the translationai temperatures obtainedby the continuum and kinetic approaches caused both


250

0.002 0.004 O.OOBX, m

Figure 12: The X-component of velocity along the noz-zle axis in axisymmetric case.

Figure 14: Comparison of temperature profiles alongthe nozzle axis for axisymmetric case.

by the rarefaction and wall effects increases startingfrom 1 mm downstream from the nozzle exit.

0.002

o.ooi

E. a

-0.002 -

11

0.001 0.002 0.003X,ro

0.004 0.005 0.008

Figure 13: Translational temperature contours in K foraxisymmetric micronozzle computed with SMILE (top)and GASP (bottom).

IV.2 Flow in a flat micronozzle com-puted using 2D and 3D models

A two-dimensional model can be used to describe theflow in a nozzle of a flat geometric configuration if theinfluence of the end-walls is negligible. However, as it isshown in the previous section for an axisymmetric flow,the entire area of the nozzle exit is affected by the wallboundary layer. In the flat nozzle case with the sameflow conditions at the throat an even larger impact ofviscous effects can be anticipated because of a greater

surface area to volume ratio. A full three-dimensionalmodeling is therefore required to simulate the gas flowand accurately predict the performance characteristicsof a three-dimensional high aspect ratio micronozzle.

To examine a possible contribution of the third di-mension at given conditions, the modeling of 2D and3D flows in a flat micronozzle was conducted using thecontinuum and kinetic approaches. The density con-tours are shown in Fig. 15 for both the 2D and 3Dflow models. Whereas the density decreases graduallyin the 2D case, in the 3D nozzle the flow experiencestwo successive expansions, at the nozzle throat and theexit. This is due to the contribution of two differentprocesses, the viscous dissipation and the flow expan-sion. This is also seen in Fig. 16 where the pressurefields are shown for the two flow models under consid-eration. Again, there is a significant difference betweenthe two cases. The pressure is higher inside the nozzlefor the 3D case due to the wall effect, with the coreflow values approximately seven times larger. In theexpansion region, downstream of the nozzle exit, thepressure is lower for the 3D case (the flow expands inthree dimensions in this case).

A comparison of velocity fields for the two cases isgiven in Fig. 17. As expected, the 2D model predictsthe values of the velocity component in X direction tobe larger inside the nozzle. For the 3D case, the velocityincreases at first 1 mm downstream from the throat(the flow expansion dominates there), and then slightlydecreases towards the exit since the wall effects becomemore important. The velocity has a local maximum of450 m/s at ~1 mm from the throat. The flow expandsrapidly after the exit, and the velocity at 2 mm fromthe exit is even greater than in the 2D case.


K.m

Figure 15: Density contours (kg/m3) in a fiat micronoz-zle computed for a 3D (top) and 2D (bottom) cases bythe DSMC method.

Figure 17: Contours of velocity component in X direc-tion (m/s) for a flat micronozzle calculated using the3D (top) and 2D (bottom) models.

LI., mil

(J..

Figure 16: Pressure contours (Pa) in a flat micronozzlecomputed for the 3D (top) and 2D (bottom) cases bythe DSMC method.

The translational temperature fields are also quali-tatively different for the two cases(see FigurelS). Thetemperature decreases downstream in the direction ofthe nozzle axis and increases at the wall (the gas iscooler than the surface) for the 2D case. There is a lo-cal minimum of temperature at 1 mm from the throatin the 3D flow, and temperature values are generallyhigher inside the nozzle and lower downstream fromthe exit plane for this case. The influence of the end-walls is therefore very important for the height to widthratio considered. For larger ratios, a smaller effect isexpected, and likewise there should be a smaller differ-ence between 2D and 3D models since the difference inthe expansion process would be reduced.

The difference in the expansion process is illustrated

Figure 18: Translational temperature contours (K) fora flat micronozzle calculated using the 3D (top) and2D (bottom) models.

in Fig. 19 where the molecular local mean free pathis plotted. The mean free path increases gradually in-side the 3D nozzle. It changes by a factor of threefrom the throat to the exit. For the 2D case it growsmore rapidly in X direction and there is also a stronggrowth across the nozzle near walls. Note that the lo-cal characteristic length for the 3D case is equal to thenozzle height h=300 (an. The local Knudsen numberat the exit plane is therefore about 0.1 which falls intoa regime where the continuum model fails.


umi>aa_• -.H

Figure 19: Mean free path contours (m) for a flat mi-cronozzle calculated using the 3D (top) and 2D (bot-tom) models.

IV.3 Wall effects in axisymmetric and3D nozzles

The surface area to volume ratio represents the relativeimpact of surface and volume forces on the flow. Forthe MEMS scale devices the ratio is very high and thewall effects can dominate the expansion process in theflow through a micronozzle.

To choose reasonable geometric parameters ensuringan expanding flow through a micronozzle, an estimateof the boundary layer thickness at the exit has to bemade, aimed at a decreasing its thickness. A possibleapproach to nozzle design could be to utilize a flow overa plate for an assessment of the boundary layer thick-ness growth. In a nozzle, though, the boundary layerthickness grows much more rapidly than that over overa plate due to the gas expansion. A full simulation istherefore required to examine the boundary layer growsfor a specified nozzle geometry.

To understand how the boundary layer grows in a3D nozzle under chosen conditions, Fig. 20 shows thetranslational temperature contours at different crosssections perpendicular to the nozzle axis. The viscouslayer is developed very rapidly, and at a distance of sev-eral throat width, it occupies the entire cross-sectionalarea. There is therefore no inviscid core in the gas flowinside the nozzle at the Reynolds number and aspectratio modelled in this work.

The wall effects and flow expansion strongly impactthe flow in a 3D nozzle as compared to that in an ax-isymmetric nozzle. Figure 21 shows the translationaltemperature profiles along the nozzle axis for the 3Dand axisymmetric cases. After an initial decrease atthe first 1 mm due to the gas expansion, the temper-

Figure 20: Viscous layer growth in a flat micronozzleobtained by 3D SMILE.

ature increases in the 3D nozzle. Such an increase iscaused by the viscous dissipation of the flow kinetic en-ergy of the flow due to the shear on the walls. Beyondthe nozzle exit where gas experiences a free expansioninto a vacuum the velocities and temperatures coincidefor the two cases, since the mass flow rates are equal.

A qualitative difference between the two solutionsshown for the temperature profiles is also observed forthe velocity fields. The profiles of the velocity in theX direction are presented in Fig. 22. The velocity in-creases monotonously downstream from the nozzle exitin the 2D case, while in the 3D flow there is a veloc-ity minimum located at X = 0.004 m. The increaseof temperature (see Fig. 21) and decrease in velocityin the three-dimensional nozzle is a consequence of theshear on the walls. For a hypersonic nozzle flow ex-panding into vacuum one would expect the extremumto be at the exit plane. However, the extremum islocated upstream of the nozzle exit because of the sub-sonic region at the walls.These results show that impactof the walls is very pronounced in the 3D case.

The model used to simulate the gas-surface interac-tion is therefore important. Since all results presentedabove were obtained using the tangential momentumaccommodation coefficient a<j — 1, different values ofad were also used to to examine the possible influ-ence of the surface model. The DSMC computationswere performed both for the axisymmetric and three-dimensional nozzles for different values of a<j.

Translational temperature profiles along the nozzleaxis for different ad are shown in Fig. 23.The tempera-ture increases with o^. There is a qualitative differencebetween the solution for ay = 0 (ideally smooth adia-batic surface) and any non-zero ad for which the profilehas a kink after the nozzle exit. There is also a visible


300

DSMC,ilpt»-d»DJ

DSMC. >pw-d - OS

O3HC,H>h»-[|«OJi

D3MC,llph*-d=t.

Figure 21: Translational temperature profile along thenozzle axis for the axisymmetric and 3D nozzles.

Figure 23: Translational temperature profiles along thenozzle axis for different ay hi an axisymmetric mi-cronozzle.

700 -

600

;500

400

300

Figure 22: X-component velocity profile along the noz-zle axis for the axisymmetric and 3D nozzles.

difference between ay = 0, ay = 0.3, and ad = 0.5 pro-files. Experimental work by Arkilis16 suggests that avalue of 04 = 0.8 is recommended for a nitrogen flow ina silicon microchannel. The result for an axisymmetricnozzle flow with a<j = 0.8 is very close to the solutionwith ay = 1.

A greater influence of the gas surface interactionmodel on the solution can be anticipated in 3D casewhere, as it was shown above, the wall effects dominatethe flow inside the nozzle. Two cases were considered,ad — 0.8 and ad = 1. The DSMC results for thesetwo cases are shown in Fig. 24 where the translationaltemperature contours are plotted. There is a subtle dif-ference in the core flow, but generally the temperaturesare very close for these two cases. The difference be-tween the density fields is also small (see Fig. 25). The

specific impulse for these two cases was calculated andthe difference was found to be less than one percent.Note, a similar weak dependence of thrust performancefor non-zero accommodation coefficients was shown formicro-resistojets in a recent paper19.

IV.4 Micronozzle performance

The calculated thrust levels and specific impulses fordifferent micronozzle configurations are summarized inin Table 3. For the cases considered the GASP solutionslightly (several percent) overpredicts the thrust valuesobtained by the DSMC method. Comparing axisym-metric and three-dimensional results, the thrust as wellas the specific impulse are lower for a flat micronozzle.The wall effects in the 3D case also cause an about20 percent reduction in thrust as compared to the 2Dmodel. Note that the 2D model gives the highest thrustvalues for the three cases under consideration. The spe-cific impulse is also highest for the 2D model (5 percentgreater than for the axisymmetric nozzle and 20 per-cent greater than the three-dimensional nozzle).

The total impulse flux at different locations down-stream of the throat of the 3D nozzle obtained by theDSMC method is plotted in Fig. 26. There is an un-desirable reduction in thrust in the second half of thenozzle caused by the viscous losses. The way to in-crease the performance would be to make the micronoz-zle shorter or to increase its height.

10


•HIIP:ti I

1.2 -

0.8C

0.6

0.4

0.2

0.001 0.002 0.003X, m

0.004 0.005

Figure 24: Translational temperature contours in thesymmetry plane of a 3D micronozzle for a<j = 0.8 (top)and ad = 1 (bottom).

Figure 25: Density contours in the symmetry planeof a 3D micronozzle for ad = 0.8 (top) and ay = 1(bottom).

Table 3: Nozzle performance characteristics.

CASEAS GASPAS SMILE

2D GASP

2D SMILE

3D SMILE

Thrust, mN

1.07

1.031.171.100.93

i-sp) S6C

65.62

65.50

69.4568.74

56.61

Figure 26: Total impulse flux at different axial loca-tions inside a 3D micronozzle.

V ConclusionsA numerical study of different geometric configurationsof micronozzles - axisymmetric and three-dimensional,has been conducted for a low throat Reynolds numberof 200 using the DSMC method and the solution ofNavier-Stokes equations.

The subsonic inflow conditions as well as criticalthroat conditions were considered in continuum com-putations. The results of the computations were shownto be insensitive to the type of inflow conditions bothfor thrust performance and flow fields.

The DSMC simulation of a three-dimensional flowat a Reynolds number of 200 is very computationallyintensive. It is necessary to take at least 20 million sim-ulated particles to obtain a particle-independent solu-tion.

The DSMC and Navier-Stokes solutions are in asatisfactory agreement for the flow inside the nozzle.There is a significant difference between them in theregion near the lip where the flow expands rapidly. Theuse of an external zone in the continuum approach, thatstarts at the nozzle exit and expands downstream, al-lows one to eliminate the possible impact of the extrap-olation outflow boundary condition at the nozzle exit.This results in thrust values that are in agreement withthose obtained by the DSMC method.

The effect of the wall accommodation coefficient wasinvestigated by the DSMC method. The flow was foundto be weakly dependent on the tangential momentumaccommodation coefficient when it changes from 0.8 to1 both for axisymmetric and three-dimensional cases.The flow changes significantly for accommodation co-efficients smaller than 0.5.

The impact of wall effects on thrust level was ex-

11


amined for axisymmetric and three-dimensional mi-cronozzles. A two-dimensional model was also usedfor the comparison. The flow in a flat nozzle has athree-dimensional structure and is strongly influencedby the end-walls. That causes a significant (about 20percent) reduction in thrust as compared to the two-dimensional model and an axisymmetric nozzle. At-tempts to predict the performance characteristics of a3D microthruster using a 2D model may therefore re-sult in significant design errors.

VI AcknowledgmentsThe work at the George Washington Universitywas supported by the Army Research Office GrantDAAG55-98-1-009, SPAWARSYSCEN San DiegoGrant No. N66001-98-1-8909, and the Ballistic Mis-sile Defense Organization. We would like to thankDr. Clifton Phillips of SPAWARSYSCEN who providedus with computer time on an HP V2500 computer. Theauthors are thankful to Prof. M. S. Ivanov for a valu-able discussion and his attention to this work.

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