Numerical study of 2D/3D micronozzle flows
Gennady N. Markelov and Mikhail S. Ivanov
Computational Aerodynamics Laboratory
Institute of Theoretical and Applied Mechanics
Siberian Division of Russian Academy of Sciences, Novosibirsk, Russia
Abstract. Methods for creating thrusters with very low thrust using micronozzles have been actively
developed recently. The flow in such nozzles is characterized by low Reynolds numbers and, as a consequence,
by high viscous losses. DSMC calculations for flows in plane and three-dimensional micronozzles have been
conducted in the present paper. A comparison of DSMC results with available experimental data of the
nozzle performance is presented. It is proposed to increase the efficiency of micronozzles by helium injection
to the near-wall flow region at the throat. This leads to a significant increase in the gas velocity near the
nozzle wall and in the specific impulse.
INTRODUCTION
Low-thrust propulsion systems are required to provide a very accurate attitude control of small satellites,
micro satellites or even nano satellites, for drag-free motion of larger spacecraft or the outmost accurate pointing
capabilities for deep-space planetary probes. To obtain these low thrust values (0.1 -r 10 mN), a small nozzle
scale and low chamber pressures are usually used. This results in the throat Reynolds number within the range
between 10 and 100. Due to such low levels of Reynolds numbers, the viscous losses are significant in small
scale nozzles.
The technology used in micromachined cold gas thruster opens new possibilities for creating effective low
thrust propulsion systems. This is due to a possibility of manufacturing nozzles with a throat size of 10
microns and using a high chamber pressure (1-10 atm). This permits obtaining a low thrust level for rather
high Reynolds numbers. For instance, for axisymmetric or three-dimensional nozzles with a fixed expansion
area ratio, a tenfold change in the linear dimensions requires a 100-fold increase in the chamber pressure to
preserve a prescribed thrust level. However, the problem of taking into account viscous losses remains because
the throat Reynolds number Re increases only by an order of magnitude.
A number of micronozzles have been developed [1,2], and their performance has been experimentally studied
using mass flow and thrust measurements [1,3]. Preliminary performance tests for micronozzle indicate that
specific impulse efficiencies (measured to ideal) drop rather dramatically for Re < 1500 [3]. Therefore, a
comprehensive study is indispensable to understand the special features of viscous flow within such nozzles
and determine the optimal micronozzle geometry, which ensures high performance. For example, the optimal
value of the half-angle of diverging section is 15 deg for small rocket nozzles [4]. At the same time, for reaching
the maximum value of specific impulse for low-Reynolds-number nozzles, the half-angle should be greater, for
instance, 26 deg [5].
A performance evaluation of micronozzle was conducted using the Navier-Stokes (NS) solver and the DSMC
method (see, for example, [1,6-9]). A detailed comparison of an axisymmetric micronozzle flow and performance
obtained both by the NS solver and the DSMC method were presented in Ref. [8] within a wide range of
Reynolds numbers Re = 100 — 1000. It was also shown [8] that the use of the continuum approach can lead to
overprediction of the specific impulse even for rather high Reynolds numbers Re ~ 1000.
The main objective of the present paper is a detailed DSMC study of viscous flows in 2D/3D micronozzles.
The influence of the throat radius, expansion area ratio and flow three-dimensionality on the micronozzle
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
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TABLE 1. Nozzle performance
case
2D, sharp corner
2D, Rc/Rt = 1.0
2D, sharp corner
3D
3D, injection
3D, sudden
expansion ratio
4:1
4:1
7:1
7:1
7:1
7:1
mxW3,g/sec
1.476
1.472
1.479
1.425
1.217
1.285
F,mN
0.949
0.943
0.981
0.908
0.841
0.671
Isp,sec___CD
65.66
0.925
0.922
65.47
67.67
0.926
64.96
0.893
0.762
70.49
53.26
0.805
0.842
0.837
0.871
0.806
0.746
0.596
0.912
0.909
0.939
0.902
0.979
0.739
performance is examined and compared with experimental data. Special attention is paid to the analysis of
helium injection to the near-wall flow region at the throat on the flow structure and specific impulse.
NOZZLE GEOMETRIES AND PLENUM CONDITIONS
Micronozzles presented in [3] are fabricated by reactive ion etching and are inherently planar. The etches
are 308 microns deep. The nozzles have similar converging sections with a 7:1 inlet area ratio and 45 deg-half
angle. The diverging sections have a 20 deg half-angle. The nozzle with a 34 micron throat width and 7:1
expansion area ratio was chosen for DSMC study in this paper.
To examine the effect of flow three-dimensionality and to study in more detail the special features of the
viscous flow inside the micronozzle in a simple formulation, calculations for a 2D nozzle were performed. This
nozzle replicates the geometry of the micronozzle in the vertical plane of symmetry. To analyze the effect of
the throat radius of curvature, additional calculations were performed for 2D nozzles with a shorter length of
the diverging section and a 4:1 expansion area ratio.
As in [3], nitrogen propellant had the total temperature TO = 300 K, and the Reynolds number determined
from the flow parameters in the nozzle throat and the throat width was Re = 350.
NUMERICAL APPROACH
The calculations were performed using the SMILE software system [10] based on the majorant frequency
scheme [11] of the DSMC method [12]. For modeling particle collisions, the variable hard sphere (VHS) model
and the Larsen-Borgnakke model with a temperature-dependent rotational collision number were used. It
was assumed that the nozzle wall is adiabatic, and gas/surface interaction was simulated using diffuse-elastic
reflection. This means that the magnitude of the molecular velocity remains constant during reflection, but
the angular distribution of the velocity vector follows the cosine-law of diffuse reflection. The temperature of
the nozzle wall was not measured in [3], but it was demonstrated in [4,8] that variation in the wall temperature
within wide limits (from adiabatic to TW/T$ = 0.6) has no noticeable effect on the nozzle performance.
A uniform flow with parameters determined using one-dimensional isentropic relations is set at the entrance
of the converging section. The computational domain includes the vicinity of the nozzle exit plane and extends
slightly downstream and the vacuum conditions are imposed on outflow boundaries. The 2D (3D) nozzles
considered have one (two) plane(s) of symmetry; therefore, to decrease the computer resources, the calculations
were performed for 1/2 (1/4) part of the computational domain.
RESULTS AND DISCUSSIONS
Effect of throat radius of curvature
It is well known that the throat radius of curvature, Rc, may affect the nozzle performance. It is shown in [4]
that the change in Rc/Rt (Rt is the throat radius) from 1/8 to 3/4 increases the mass flow, ra, and thrust, F,
by 3-4%. For small rocket nozzles, the optimal value of the radius of curvature is Rc/Rt — 0.625. Therefore,
first the effect of the parameter Rc/Rt is studied, where Rt is the throat width in our case.
Figure 1 shows the density contours for 2D nozzles having Rc/Rt = 1-0 and a sharp corner at the throat.
For Rc/Rt = 1.0, a greater increase in density near the wall of the diverging section is observed. At the same
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time, the values of density in the core flow in the nozzle exit plane are roughly identical for both nozzles. The
Mach numbers at the nozzle exit are also close (Fig. 2), but the behavior of Mach contours in the vicinity of
the throat are qualitatively different. For example, for a nozzle with a smooth variation of the wall surface, the
line M = 1 touches the nozzle wall only near the nozzle lip. For a nozzle with a sharp corner, the line M = 1
touches the nozzle wall at two points: sharp corner at the throat and nozzle lip.
0.08
0.73
0.1
0.7
0.6 0.5 0.35 0.25
0.5 0.35 0.25
a) sharp corner
a) smooth throat
FIGURE 1. Density contours for 2D nozzles with expansion area ratio 4:1 (values in kd/m3)
0.2
2.2
U-6
J2.4
a) smooth throat
a) sharp corner
FIGURE 2. Mach number contours for 2D nozzles with expansion area ratio 4:1
The difference in the throat shape also leads to significant differences in friction and pressure distributions
along the nozzle wall in the vicinity of the nozzle throat (Fig. 3). The flow inside the nozzle with Rc/Rt = 1.0
is characterized by a smooth variation of friction and pressure. The presence of a sharp corner leads to a sharp
peak of pressure and rather complex behavior of pressure near the nozzle throat. First, the pressure rapidly
decreases and then increases to a value close to the pressure in the nozzle with Rc/Rt = 1-0. Figure 3 also
shows the gas velocity immediately near the wall. The change in the gas velocity near the throat corresponds to
the behavior of pressure. Along the diverging section, the velocity is almost constant (50 m/sec). It increases
dramatically only in the vicinity of the nozzle lip because of the fast expansion of the gas into vacuum in this
region. The effect of vacuum conditions extends to 20 microns inside the nozzle, which corresponds to 10-15
local mean free paths (Fig. 4). This also leads to an increase in friction and a decrease in pressure in the
vicinity of the nozzle lip.
Pressure, kN/rn2
Friction, N/m2
Velocity, m/sec
100-1—————
100
3000
2000
10-
100
X, micron
200
300
1.0
0
100
X, micron
200
300
100
X, micron
200
300
FIGURE 3. Parameters along the nozzle wall
For fixed area ratios, the nozzle with Rc/Rt = 1.0 has a slightly greater length, which leads to an increase
in the boundary layer thickness (sf. Mach number contours at the throat in Fig. 2) and friction forces. This
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results in slightly lower values of the gas flow rate through the nozzle, thrust, and even vacuum specific impulse.
Table 1 lists both dimensional quantities and dimensionless quantities normalized to the values obtained using
one-dimensional isentropic relations for a 7:1 expansion area ratio. Thus, the use of throat curvature does not
offer a better micronozzle performance for Re = 350.
The influence of an expansion area ratio
The values of thrust along the nozzle are plotted in Fig. 4 for the nozzles under consideration. Note that
the thrust monotonically increases, and there is no maximum like that for Rot he nozzles at Re = 270 [2].
Therefore, an increase in the expansion area ratio allows one to reach a greater efficiency of the nozzle, which
is demonstrated by the curve for a 2D nozzle with a 7:1 expansion area (see Fig. 4). Note that the thrust curve
does not have a maximum even for this degree of expansion.
F/F ideal
0.90i———————————————————————————
0.85
—- R /R =1.0,4:1
— - • sharp corner, 4:1
0.80
——— sharp corner, 7:1
0.75
0.70
0
100
200
300
X, micron
FIGURE 4. Mean free path contours (values in micron) and Thrust
The distributions of friction, pressure, and velocity along the nozzle wall with a 7:1 expansion area ratio are
similar to those for the nozzle with a sharp corner considered above (see Fig. 3). A large value of the expansion
area ratio leads to a further decrease in friction and pressure along the nozzle wall. However, the slip velocity
remains constant and equal to the velocity for a nozzle with a lower expansion area ratio. The density and
Mach number contours are shown in Fig. 5. The Mach number at the nozzle exit is 3.14, which is rather close
to 3.54 obtained using one-dimensional isentropic relations. The performance of this nozzle (thrust and specific
impulse) is 3% higher than the corresponding values for a shorter nozzle with a 4:1 expansion area ratio (see
Table 1).
0.6
0.2
FIGURE 5. Density (left) and Mach number (right) contours for 2D nozzle with expansion area ratio 7:1
Effects of flow three-dimensionality
To analyze three-dimensional effects, DSMC calculations of this micronozzle with side walls were performed.
A boundary layer is formed at the side wall. The boundary-layer thickness at the nozzle exit is 20% of the
nozzle half-depth. A weak shock wave emanates from the throat and reaches the vertical plane of symmetry
near the nozzle exit. As a result, the flow in the vertical plane of symmetry of such a 3D nozzle experiences
almost no effect of the side wall and is in good agreement with 2D simulation (cf. Figs 5 and 6). The influence
of the side-wall boundary layer exerts a stronger effect on the surface streamlines along the diverging section
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(Fig. 7). Under the action of the side-wall boundary layer, the streamlines are deflected toward the vertical
plane of symmetry.
FIGURE 6. Density (left) and Mach number (right) contours for 3D nozzle (top - vertical plane of symmetry, bottom
- horizontal plane)
Figure 7 shows a comparison of velocity profiles obtained by 2D and 3D simulations in the vertical and
horizontal planes of symmetry. Velocity profiles in the vertical plane of symmetry are slightly different for 2D
and 3D flows. This difference is caused by the fact that an accurate 3D simulation requires an enormous total
number of model particles (more than 108). Only 107 model particles were used in the simulation. However,
the study of the effects of the total number of model particles on the nozzle performance showed that mass
flow and thrust remain almost unchanged.
The boundary layers on the side wall and along the nozzle wall in the vertical plane of symmetry experience
the action of an identical pressure gradient. However, the thickness of the side-wall boundary layer increases
more slowly (approximately by a factor of 2) than the boundary layer on the nozzle wall. The reason is that
the gas velocity near the side wall is much greater. Figure 8 shows the gas velocity in a cell adjoining the side
wall. In the converging section, the velocity decreases first from a prescribed velocity value at the inlet plane,
then smoothly increases to 150 m/sec in the cross section downstream from the throat, and then decreases
again. On the major part of the diverging section, the value of velocity is almost constant and two times higher
than the gas velocity near the wall in the vertical plane of symmetry.
Obviously, the formation of the boundary layer on the side wall decreases the effective expansion area ratio.
A comparison of the performance of 2D and 3D nozzles shows that flow three-dimensionality slightly decreases
the discharge coefficient from 92.6% to 89.3%. The thrust is determined by the velocity at the nozzle exit
squared; therefore, the thrust efficiency decreases stronger and amounts to 80.6% instead of 87.1% for the 2D
micronozzle. Figure 8 shows the variation in the vacuum specific impulse Isp along the nozzle. Beginning from
the distance of 20 microns from the nozzle throat, the 3D simulation predicts a lower value of Isp than the 2D
one. Note that the curve Isp has a maximum in the vicinity of the nozzle exit plane. Hence, a further increase
in the expansion area ratio for a fixed angle of the diverging section will degrade the micronozzle performance
due to growing of side-wall boundary layer.
Light gas injection
The main reason for the decrease in micronozzle efficiency is the presence of a thick boundary layer inside the
nozzle. One of possible techniques for decreasing the boundary layer thickness is to increase the slip velocity
of the gas on the wall. For an axisymmetric sharp-throat micronozzle, the injection of a light gas (helium) to
the near-wall flow region behind the nozzle throat significant increased the slip velocity and Isp due to helium
injection was from 65 to 73 sec, and the nozzle efficiency reached 92% [9]. This technique can be interpreted
as a special kind of lubrication.
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Uelocity, m/sec
600 r^
400
200-
25
75
micron
FIGURE 7. Surface stream lines and profiles of velocity normal to the wall
Velocity, m/sec
50
100
125
Isp, sec
701———
100
65
10
60
1.0
55
300
100
200
100
X, micron
x, micron
FIGURE 8. Velocity and vacuum specific impulse
200
300
Though the 3D nozzle has a rather high specific impulse (90% of the value predicted by one-dimensional
isentropic relations), it is interesting to see whether it is possible to increase the specific impulse by helium
injection. The simplest change in the geometry of the 3D micronozzle for the injection is a decrease in the
depth of the subsonic part of the nozzle. A sudden expansion in the transverse direction is formed in the nozzle
throat, wherein helium is injected. Helium pressure is equal to nitrogen pressure in the throat, and the total
temperature of helium is 300 K. The velocity of helium injection is chosen in such a way that the impulse of
the helium jet is equal to the decrease in the impulse of the nozzle caused by the smaller depth of the inlet.
Figure 9 shows the mole fraction of helium and the gas velocity in the horizontal plane of symmetry. In the
figure, the difference in mole fraction between adjacent contours is 0.1. Mixing of the gases results in a 10%
content of helium observed in the middle between the side wall and the vertical plane of symmetry. During
injection, helium is first accelerated, then its velocity decreases both from the side wall (the boundary layer is
formed) and from the side of the main flow (the mixing layer is developed). As a result, a velocity maximum
(1000 m/sec) is observed at a distance of 40 microns from the nozzle throat. Though helium is injected at
a pressure equal to nitrogen pressure, a strong interaction between two flows is observed. This leads to the
formation of a weak shock wave, which is reflected from the vertical plane of symmetry, and to the growth in
pressure by 40% in the vicinity of helium injection (Fig. 10). In addition, the flow rate of nitrogen decreases
not by 10% in accordance with the decrease in the inlet depth but by 20%. As a result, the nozzle thrust with
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account of helium injection is 92% of the micronozzle thrust. The specific impulse increases to 70.6 sec (see
Table 1). Thus, an increase in the plenum pressure of nitrogen and helium to compensate for the decrease in
thrust caused by helium injection allows one to enhance the nozzle performance.
To demonstrate clearly that exactly the viscous interaction of two flows led to the decrease in nitrogen flow
rate, the flow inside this nozzle without helium injection was calculated. In this case, the mass flow is 90%,
which is in perfect agreement with the decrease in the inlet width. Note that the flow inside such a nozzle is
characterized by the formation of a rather strong shock wave (see Fig. 10) caused by a sudden expansion of
geometry. As a result the thrust and specific impulse dramatically decreased (see Table 1).
a) helium mole fraction contours
b) x-velocity contours (values in m/sec)
FIGURE 9. The influence of helium injection for 3D nozzle ( horizontal plane of symmetry)
a) with helium injection
b) without helium injection
FIGURE 10. Pressure contours for 3D nozzle with sudden expansion of geometry (horizontal plane of symmetry,
values in kN/m2)
Comparison with NS results and experimental data
Figure 11 shows the discharge coefficient and thrust efficiency (ratios to isentropic values) measured and
calculated by the 2D NS solver for a large interval of Reynolds number [3]. The performance of the 2D sharp
throat micronozzle predicted by the DSMC method is in good agreement with the NS results. The NS and
2D/3D DSMC predictions of the discharge coefficient lie within the measurement error (see Fig. 11). This
means that a 2D simulation is sufficient for reliable prediction of the mass flow.
The 3D DSMC simulation predicts a significantly greater value of thrust efficiency than the measured one.
The measured values of thrust efficiency have a considerable error (about 100%) for Re < 500. However, the
DSMC prediction lies even above the error bars. This difference can be caused by rather high back pressure
and/or wall roughness (of the order of 1 micron) effects. Both effects can lead to a decrease in the nozzle
performance. Therefore, further numerical studies with accounting for these effects are necessary.
CONCLUSIONS
DSMC computations for flows in plane and three-dimensional micronozzles have been conducted for a
Reynolds number Re = 350.
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CD
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
500
1000
1500
2000
2500
3000
3500
500
Re
1000
1500
2000
2500
3000
3500
Re
b) thrust efficiency
a) discharge coefficient
FIGURE 11. Comparison of DMSC results with NS prediction and experimental data
The effect of the nozzle throat shape on the 2D flow structure and nozzle performance was studied. The use
of a throat radius of curvature equal to a half of the throat width ensures a more uniform flow in the nozzle
throat as compared to a sharp corner nozzle. However, the increase in viscous losses due to the larger nozzle
length did not allow improvement of the nozzle performance.
Simulation of a 3D flow showed that the boundary layer thickness on the side wall is approximately half the
thickness of the boundary layer formed on the nozzle wall in the vertical plane of symmetry. The side-wall
boundary layer does not affect the flow in the vertical plane of symmetry for a micronozzle with a 7:1 expansion
area ratio. The effect of three-dimensionality decreases the discharge coefficient from 92.6% to 89.3%, and the
thrust efficiency amounts to 80.6% instead of 87.1% for the 2D micronozzle.
A comparison of 2D/3D DSMC results with experimental data showed that the calculated discharge coefficient is in good agreement with the measured value. Therefore, a 2D simulation is sufficient for reliable
prediction of the mass flow. At the same time, a 3D simulation predicts a significantly higher value of thrust
than the experimental studies. A possible reason is the neglect of rather high back pressure and/or wall
roughness in modeling.
It is proposed to increase the efficiency of micronozzles by helium injection to the near- wall flow region at the
throat. This leads to a significant increase in the gas velocity near the nozzle wall and in the specific impulse.
Acknowledgement This work was supported by the NATO Collaborative Linkage Grant SA(PST.CLG
975431) 5066. This support is gratefully acknowledged.
REFERENCES
1. Bayt, R., Ayon, A., Breuer, K., AIAA Paper No. 97-3169, July 1997.
2. Ketsdever, A.D., Wadsworth, D.C., Wapner, P.G., Ivanov, M.S., Markelov, G.N., AIAA Paper No. 99-2724,
1999.
3. Bayt, R.L., Breuer, K.S., Proc. of the 3rd ASME Microfluids Symp., (1998).
4. Liang, S.M., Huang, K.L., Chen, K.C., J. of Spacecraft and Rockets, 33:5, 693 (1996).
5. Shebalin, J.-R, Tiwari, S.N., AIAA Paper No. 99-3794, June-July 1999.
6. Kirn, S.C., J. of Sapcecraft and Rockects, 31:2, 259 (1994).
7. Boyd, L, Jafry, Y., Beukel, J.,J. of Spacecraft and Rockets, 31:2, 217 (1994).
8. Ivanov, M.S., Markelov, G.N., Ketsdever, A.D., Wadsworth, D.C., AIAA Paper No. 99-0166, Jan. 1999.
9. Ivanov M.S., Markelov G.N., AIAA Paper No. 2000-0468, Jan. 2000.
10. Ivanov, M.S. , Markelov, G.N., Gimelshein, S.F., AIAA Paper 98-2669, June 1998.
11. Ivanov, M.S., Rogasinsky, S.V., Proc. of 18 Int. Symp. on Rarefied Gas Dynamics, VCH, 629 (1991).
12. Bird, G.A., Molecular gas dynamics and the Direct Simulation of gas flows, Oxford: Clarendon Press, 1994.
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