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Study of rarefaction effects in gas flows with particle approaches

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Study of rarefaction effects in gas flows with particle approaches

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Content STUDY OF RAREFACTION EFFECTS IN GAS FLOWS WITH PARTICLE
APPROACHES
by
Cedrick Goliati Ngalande
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ASTRONAUTICAL ENGINEERING)
May 2007
Copyright 2007 Cedrick Goliati Ngalande
ii
Acknowledgements
Without guidance and support of a lot of people, culmination of this thesis and my
Ph.D. degree would not have been possible.
I would like to thank my guidance committee chair Dr Joseph Kunc and all the
members of the committee, Dr Sergey Gimelshein, Dr Daniel Erwin, Dr Mike Grunt-
man and Dr Darrell Judge, for their wonderful support. Very, very special thanks
goes to my good friend and dissertation advisor Dr Gimelshein without whose ad-
vise and help, this would not have been possible. Thanks to Natalia Gimelshein,
Dr Alina Alexeenko, Dr Mikhail Shneider and the 'upcoming great scientist', Taylor
Lilly, whose help was very valuable in deed.
In a special way, let me acknowledge my great professor, mentor and friend, Dr
Andrew Ketsdever. Andrew has tirelessly fought for me throughout my Ph.D. studies.
I thank him for his friendship and trust.
Finally I would like to thank my mother, brothers and all my friends for their
encouragement and support.
This thesis was supported in part by the Propulsion Directorate of the Air Force
Research Laboratory at Edwards Air Force Base, California.
iii
Table of Contents
Acknowledgements ii
List of Tables vi
List of Figures x
Abstract xi
1 Introduction 1
1.1 Fundamentals of rareed
ow . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Solutions of the Boltzmann equation . . . . . . . . . . . . . . . . . . 6
1.3.1 Numerical integration of the Boltzmann equation . . . . . . . 7
1.3.2 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Direct Simulations Monte Carlo (DSMC) method . . . . . . . 8
1.3.4 Hybrid approaches . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 The DSMC technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Key elements of DSMC . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Application of the DSMC method . . . . . . . . . . . . . . . . . . . . 12
2 Analysis of Rareed Flows in CHAFF-IV 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The cryogenic pumping capabilities of CHAFF-IV . . . . . . . . . . . 15
2.2.1 Facility background pressure requirement . . . . . . . . . . . . 16
2.2.2 Test particle model . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Pressure and velocity distribution in CHAFF-IV . . . . . . . . . . . . 24
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Rareed Flow Impingement on Spacecraft Surfaces 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Nozzle surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . 34
iv
3.5 Axisymmetric plume-surface interaction . . . . . . . . . . . . . . . . . 39
3.6 Interaction of plume with a plate: Numerical modeling . . . . . . . . 50
3.7 Comparing numerical results with data . . . . . . . . . . . . . . . . . 57
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Radiometric Forces in Rareed Gas Flows 62
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Translational temperature and streamline
owelds . . . . . . . . . . 67
4.4 Radiometric forces as a function of pressure . . . . . . . . . . . . . . 68
4.5 Eects of viscosity index, molecular mass and diameter . . . . . . . . 69
4.6 The impact of chamber size on radiometric forces . . . . . . . . . . . 71
4.7 Impact of surface area and edge eects . . . . . . . . . . . . . . . . . 71
4.8 DSMC and experimental measurements . . . . . . . . . . . . . . . . . 73
4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Acceleration of Polarizable Molecules in Low Density Gases 76
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Particle trapping and acceleration in weakly collisional regime . . . . 83
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Optical Lattices in Collisional Gas Regimes 92
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Energy and momentum deposition in collisional regime . . . . . . . . 95
6.4 Acoustic signals induced by optical lattices . . . . . . . . . . . . . . . 100
6.5 Two-step kinetic/continuum approach . . . . . . . . . . . . . . . . . 102
6.6 Gas mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Optical Lattice Operated Micropropulsion Devices 110
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Laser beam propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3 Optical lattices in free molecular and weakly collisional regimes . . . 112
7.4 Low density microthruster . . . . . . . . . . . . . . . . . . . . . . . . 114
7.5 High density microthruster . . . . . . . . . . . . . . . . . . . . . . . . 119
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8 Concluding Remarks and Summary 125
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.1.1 The CHAFF-IV ground testing facility . . . . . . . . . . . . . 125
8.1.2 Eects of surface roughness in plume impingement . . . . . . 126
v
8.1.3 Radiometric Forces . . . . . . . . . . . . . . . . . . . . . . . . 127
8.1.4 Optical lattices in high and low density gases . . . . . . . . . 127
8.1.5 Micropropulsion devices based on pulsed optical lattices . . . . 129
8.2 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Bibliography 131
vi
List of Tables
3.1 Impact of surface roughness on nozzle properties. . . . . . . . . . . . 53
3.2 The axial force on the cylinder for surface conditions and diameters . 53
3.3 Surface forces for P
0
= 405 Pa. . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Surface forces for P
0
= 155 Pa. . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Eects of viscosity index mass and diameter of the molecules on net
radiometric forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Radiometric forces on solid and hollow plates . . . . . . . . . . . . . 73
7.1 Maximum observed thrust and specic impulse for various pressures
and X
c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
vii
List of Figures
2.1 CHAFF-IV cryogenically cooled, radial n total chamber pumping ar-
rangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Fabricated aluminum nned arrays installed in CHAFF-IV . . . . . . 15
2.3 Sticking coecients of some common gases as a function of gas and
surface temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Test particle model results for pumping eciency as a function of stick-
ing coecient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Increasing height of the panel increases the pumping eciency of the
chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Pumping eciency of the chamber as a function of panel thickness . . 22
2.7 Pumping eciency of the chamber as a function of the length between
panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Diusion pump were used to maintain low pressure in the chamber . 24
3.1 Schematic of the geometric setup in the experiment. . . . . . . . . . . 30
3.2 Scanning electron microscope image showing the surface roughness of
the expanding section of the conical nozzle. . . . . . . . . . . . . . . . 31
3.3 Comparison of the mass density elds for a smooth and rough nozzles. 32
3.4 Comparison of the axial velocity elds for a smooth and rough nozzles. 35
3.5 Comparison of the pressure elds for smooth and rough cylinders with
the inner diameter of 0.79 cm. . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Comparison of the axial velocity elds for smooth and rough cylinders
with the inner diameter of 0.79 cm. . . . . . . . . . . . . . . . . . . . 37
viii
3.7 The impact of the roughness shape on pressure elds for the plume-
cylinder interaction. Cylinder diameter is 0.79 cm. . . . . . . . . . . . 40
3.8 Comparison of the pressure elds for smooth and rough cylinders with
the inner diameter of 1.59 cm. . . . . . . . . . . . . . . . . . . . . . . 41
3.9 Comparison of the axial velocity elds for smooth and rough cylinders
with the inner diameter of 1.59 cm. . . . . . . . . . . . . . . . . . . . 42
3.10 Comparison of the pressure elds for 5 mm long rough and smooth
cylinders with the inner diameter of 1.59 cm. . . . . . . . . . . . . . . 43
3.11 Pressure eld (Pa) over a smooth plate at 0 deg. . . . . . . . . . . . . 46
3.12 Pressure eld (Pa) over a rough plate with triangular grooves. . . . . 47
3.13 Pressure eld (Pa) over a rough plate with rectangular grooves. . . . 48
3.14 Pressure eld (Pa) over a rough plate with triangular grooves. Plate
angle is 10 deg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.15 Force in X direction per unit area (N/m
2
) on a smooth plate for the
plate angle of 0 and P
0
= 405 Pa. . . . . . . . . . . . . . . . . . . . . 52
3.16 Force in Y direction per unit area (N/m
2
) on a smooth plate for the
plate angle of 0 and P
0
= 405 Pa. . . . . . . . . . . . . . . . . . . . . 54
3.17 Force in X direction per unit area (N/m
2
) on a rough plate for the
plate angle of 0 and P
0
= 405 Pa. . . . . . . . . . . . . . . . . . . . . 55
3.18 Force in Y direction per unit area (N/m
2
) on a rough plate for the
plate angle of 0 and P
0
= 405 Pa. . . . . . . . . . . . . . . . . . . . . 56
3.19 Total force versus mass
ow for free expansion and smooth and rough
surfaces: comparison of numerical and experimental modeling. . . . . 58
3.20 Measurements of total force versus mass
ow for free expansion and
smooth and rough surfaces. . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Translational temperature and streamlines shown for helium at a cham-
ber pressure of 2Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Net radiometric force as a function of chamber gas pressure . . . . . . 68
4.3 Net radiometric force normalized by the force on the hot plate as func-
tion of the Knudsen number calculated near the hot surface as a func-
tion of chamber gas pressure . . . . . . . . . . . . . . . . . . . . . . . 69
ix
4.4 Temperature proles are shown for dierent chamber sizes for Helium
gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Net radiometric force as a function of chamber size for Helium gas . . 72
4.6 surface pressure distribution over the cold and hot sides of the solid
plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.7 Comparison of the DSMC and experimental predictions of the net force
on the plate in nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 Phase diagrams for two particles in free molecular and collisional regimes. 85
5.2 Velocities of two untrapped molecules in the free molecular and tran-
sitional regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Velocities of two trapped molecules in the free molecular and transi-
tional regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4 Velocities of three dierent atoms in the free molecular and transitional
regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.5 The velocity distribution function for methane gas at two pressures for
an accelerating lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1 Energy and momentum deposition in an initially stagnant gas for dif-
ferent pressures and a long pulse with I
max
= 0:5 10
17
W/m
2
. . . . 97
6.2 Translational temperature and momentum increase in an initially stag-
nant gas after a short pulse with I
max
= 10
18
W/m
2
. . . . . . . . . . 98
6.3 (a) Translational and rotational temperature increases in nitrogen, (b)
Transient evolution of nitrogen temperatures at 1 atm. A 50 ps pulse
with I
max
= 10
18
W/m
2
. . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Energy deposition for dierent laser intensities. . . . . . . . . . . . . 99
6.5 Pressure (a) and number density (b) as a function of time at two radial
locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.6 (a) Pressure at dierent radial locations. (b) Pressure in the center of
the lattice for bounded and unbounded gas cell. . . . . . . . . . . . . 102
6.7 Comparison of DSMC and NS pressure proles for two time moments
at P=0.1 atm (a), and pressure proles for four time moments 0, 0.05,
0.5, and 1s at P=1 atm (b). . . . . . . . . . . . . . . . . . . . . . . 103
x
6.8 The impact of pressure (a) and beam intensity (b) on acoustic signal
propagation. (a) From left to right, pressures of 0.1, 0.333, and 1 atm.
(b) From left to right, intensities of 1:7 10
17
, 2:5 10
17
, and 5
10
17
W/m
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.9 Gas mixing: pressure elds at dierent time moments for a single lat-
tice (a) and two counter lattices (b). From top to bottom: 0.25s,
0.5s, 1.0s, 1.5s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.10 Gas mixing:
ow velocity elds at dierent time moments for a single
lattice (a) and two counter lattices (b). From top to bottom: 0.5s,
2s, 5s, 10s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.11 Gas mixing: nitrogen mole fraction at 10s (a), 30s (b), and 100s
(c). Top, no lattice; middle, single lattice; bottom, two counter lattices. 108
7.1 Distribution function of molecular velocities at dierent time moments
for 3 torr (left) and 10 torr (right). . . . . . . . . . . . . . . . . . . . 114
7.2 Schematic of a low-density micropropulsion device powered by the op-
tical lattice/gas interaction. . . . . . . . . . . . . . . . . . . . . . . . 115
7.3 Axial velocity elds (m/s) in a low-density micropropulsion device at
two dierent pressures and two time moments. . . . . . . . . . . . . 116
7.4 Temporal change in thrust force for dierent pressures and center of
the lattice locations X
c
. . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.5 Schematic of microthruster integrated with an optical lattice, operating
in the continuum regime. . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.6 Temporal evolution of the axial velocity elds (m/s) inside a micronoz-
zle. Left, axial velocity (m/s); right, translational temperature (K). . 121
7.7 Axial velocity proles along the nozzle axis at dierent time moments. 123
xi
Abstract
The objective of this study is the numerical analysis of gas
ow rarefaction phenomena
with application to a number of aerospace-related problems. The understanding and
accurate numerical prediction of rareed
ow regime is important both for aerospace
systems that operate in this regime, and for the development of new generation of gas-
driven nano- and micro-scale devices, for which the gas mean free path is comparable
with the reference
ow scale and rarefaction eects are essential. The main tool for
the present analysis is the direct simulations Monte Carlo (DSMC) method.
The rst topic is the study of rareed
ows in the CHAFF-IV facility. A test parti-
cle method was used to analyse the pumping eciency of CHAFF-IV, and determine
optimum geometrical conguration of the chamber.
The second topic under consideration is the in
uence of the surface roughness
on nozzle plume
ow and plume impingement for dierent
ow regimes from free
molecular to near-continuum. Surface roughness eects in rocket nozzles are found
to be signicant only in very rareed
ows where Reynolds number is about unity.
The third topic is the eect of rarefaction on radiometric forces. This eect
is shown to be an important factor aecting the radiometric forces. The maximum
radiometric forces for all gases under consideration are observed at a Knudsen number
of about 0.1. For a radiometer vane placed in a nite size chamber, the maximum
force was found to be roughly proportional to the surface area of the vane. This is
xii
an indication that the collision-less area force, and not thermal transpiration edge
force, dominates the radiometric phenomena in that regime. The role of molecular
diameter, viscosity and chamber size on radiometric forces have been found to be
signicant.
The forth topic is the numerical study of the interaction between optical lattices
created by two counter-propagating laser beams and initially stagnant gases, in the
entire
ow regime from free molecular to continuum. It has been shown that in a
weekly collisional regime optical lattices can trap and accelerate neutral molecules
from room temperature level to tens of kilometers over a single laser pulse. In the
collisional regime, the optical lattice - gas interaction was found to result in strong
energy and momentum deposition to the gas.
Two types of optical lattice-based micropropulsion devices have been proposed
for low and high density regimes. For the high density microthruster, an optical
lattice is used to deposit energy and momentum to the region near the nozzle throat
with subsequent increase in propulsion eciency. In the low density microthruster,
a multiple orice
ow is considered, and thrust is produced by molecules accelerated
to high velocities by a chirped lattice potential.
1
Chapter 1
Introduction
The Navier-Stokes equations are a well-established means of predicting a variety of

ows of engineering interest. The increased capabilities of computer platforms have
allowed engineers and scientists to develop a large number of applications based on
these equations, which permit the solution of complex physical processes. Unfortu-
nately, there are limitations to the applicability of tools that rely on these equations.
The Navier Stokes equations no longer hold as
ow becomes rareed. This is because
the constitutive equations that relate the shear stress and heat transfer to the other
variables break down. More specically, the linear transport term for mass diusion,
viscosity, and thermal conductivity in the partial dierential equations are no longer
valid in the rareed
ow regime. For this regime, the Boltzmann equation needs to
be considered and solved using a kinetic approach.
1.1 Fundamentals of rareed
ow
In gas dynamics, the basic criterion of the
ow regime is the Knudsen numberKn =

L
, where is the mean free path in a gas and L is the reference
ow scale. The
ow
2
regime is continuum when Knudsen number tends to zero. While studying the gas

ow in this regime, one can disregard its microscopic structure and consider only its
macro parameters such as density, velocity, or temperature, etc. For Knudsen number
tending to innity the
ow regime can be considered as free-molecular. In this case,
particle collisions with the body surface play the determining role.
There is a transitional regime between the free-molecular and continuum ones,
where not only gas-surface collisions but also intermolecular collisions are important.
Free-molecular and transitional regimes are the subject of study of rareed gas dy-
namics. In addition to considerable viscous eects including, apart from viscosity,
heat conduction, relaxation, diusion, and irreversible chemical reactions, an impor-
tant feature of rareed hypersonic
ows is strong thermal non-equilibrium of the
ow,
i.e., the velocity distribution function is substantially non-Maxwellian. Thermal and
chemical relaxation lengths in such
ows become comparable or larger than the ref-
erence
ow scale and the dierence in temperatures of translational, rotational and
vibrational molecular modes become determining.
1.2 The Boltzmann equation
When studying rareed gas regimes, one should consider directly the Boltzmann equa-
tion { the basic equation that describes rareed gas
ows, and to operate within the
framework of the kinetic approach. The Boltzmann equation is a nonlinear integral-
dierential equation closed with respect to a one-particle distribution function, which
in the simplest case of mono-atomic gas determines the numerical density of particles
in a six-dimensional phase space of particle coordinates and velocities.
The Boltzmann equation will now be derived as follows.
Suppose for a moment that in a container there is a single gas of completely
3
identical molecules. The molecules behave like elastic sphere when they collide into
each other. Then even if all molecules initially had the same velocity, in course further
collisions it would soon happen that all possible velocities would occur, from zero up
to velocities much larger the original common velocity of all molecules.
In order to nd the velocity distribution law or law of distribution of velocities
among molecules in the nal state thus reached, it will be assumed that the container
described above has two kinds of molecules in it. Each molecule of the rst kind has
mass m , and each molecule of the second has mass m
1
. The velocity distribution
which prevails at any arbitrary timet will be represented by drawing as many straight
lines (starting from the origin of coordinates) as there arem-molecules in unit volume.
Each line will be the same length and direction as the velocity of the corresponding
molecule. It endpoint will be the velocity point of the corresponding molecule.
Now at time t let
f(u;v;w;t)dudvdw =fds (1.1)
be the number of m-molecules whose velocity in the three coordinate direction lie
within the limits, u and u+du, v and v+dv, and w and w+dw. The number of
m-molecules inside any volume is therefore equal to,
number = fds (1.2)
Same for the m
1
-molecules
number
1
= F
1
ds
1
(1.3)
We now construct a sphere of unit radius whose center is that the origin of coor-
dinates, and on it a surface elementd
. The line of centers of the colliding molecules
4
drawn from m to m
1
must, at the moment of collision, be parallel to a line drawn
from the origin to some point of the surface elementd
. The aggregate of these lines
constitutes the cone d
.
If m
1
is assumed at rest while m moves with velocity g, then the moving m-
molecules traces out a cylinder of height, g cos dt, and cross section area
2
d
.
Such an imaginary cylinder has volume
=
2
g cos d
dt (1.4)
where is the angle between the direction of velocity g and the line connecting the
center of masses of the colliding molecules; is the radius of the molecular collision
cross section;g is velocity of them which is also relative velocity of the two molecules
(since m
1
is assumed at rest).
Since there are fds m-molecules in a unit volume, the total amount of volume
transverse by all these molecules is
=fds
2
g cos d
dt (1.5)
All centers of m
1
-molecules inside the volume will touch during interval dt
one surface element ,
2
d
and hence the number dv of collisions which occur in
the volume element during time dt is equal to the number Z

of the centers of m
1
-
molecules that are in the volume at the beginning of dt . According to equation
1.3, this is
dv =Z

= F
1
ds
1
=fdsF
1
ds
1

2
g cos d
dt (1.6)
Whenever an m-molecule collides with an m
1
-molecule, the number of molecules
within the specied velocity region decreases by one. To nd the total decrease
R
dv
5
suered byfds duringdt as a result of all collisions ofm-molecules withm
1
-molecules,
we must consider, and as constant in equation 1.6 and integrate ds
1
and d
over all
possible values -i.e., we must integrate ds
1
over all space, and d
over all surface
elements for which the angle is acute. We denote the result here as
R
dv .
Collisions betweenm-molecules andm
1
-molecules will not be the only phenomenon
that decreases the numberfds. The m-molecules will also be colliding among them-
selves hence causing further decrease in the fds. The method for calculating this
number is the same as for collisions between m-molecules and m
1
-molecules except
that in this case one replaces m
1
by m and the function F by f, and by the real
diameter r of the m-molecules . Thus the decrease due to this kind of collision is
d =ff
1
ds
1
dsds
1
r
2
g cos d
dt (1.7)
Therefore the total decrease of fds during dt is given by
TotalDecrease =
Z
dv +
Z
d (1.8)
Now we shift our attention to inverse collisions. The number of inverse collision
in unit volume during dt is by analogy to equation 1.6,
dv
0
=f
0
F
0
1
ds
1
ds
0
ds
1

2
g cos d
dt (1.9)
An opposite collision increases both fds and F
1
ds
1
by one. The increase of fds
resulting from all collisions ofm-molecules withm
1
-molecules is given by
R
dv
0
. Sim-
ilarly, this quantity increases by
R
d
0
as a result of collisions of m-molecules with
each other, where
d
0
=f
0
f
0
1
ds
1
dsds
1
r
2
g cos d
dt (1.10)
6
If we subtractfds from the increase of the total decrease, we obtain the net change
df
dt
dsdt (1.11)
which the quantity experiences during time dt.
df
dt
dsdt =
Z
dv
0

Z
dv +
Z

0

Z
(1.12)
In the integrals the
R
dv and
R
dv
0
the integration variables are identical, and
likewise in the integrals
R
and
R

0
If we combine these integrals and divide the whole equation by ds:dt , then it
follows from equations 1.6, 1.7, 1.9, and 1.10 that
df
dt
=
Z
(f
0
F
0
1
fF
1
)
2
g cos ds
1
d
+
Z
(f
0
f
0
1
ff
1
)r
2
g cos ds
1
d
(1.13)
wherer is radius ofm-molecules. This integration is to be extended over all possible
ds
1
and d
. Likewise, one obtains for the function the equation
df
dt
=
Z
(f
0
F
0
1
fF
1
)
2
g cos dsd
+
Z
(F
0
F
0
1
FF
1
)r
2
1
g cos dsd
(1.14)
where r
1
is radius of m
1
-molecule. This is the Boltzmann Equation.
1.3 Solutions of the Boltzmann equation
The Boltzmann equation is a complex integro-dierential equation, and its analytic
solution is available only for a small number of simple, mostly homogeneous, problems.
7
The solution of the Boltzmann equation for practical application requires a numerical
approach to be used. The principal approaches currently used for the solution of the
Boltzmann equation are listed below.
1.3.1 Numerical integration of the Boltzmann equation
In this method the Boltzmann equation is solved numerically through two basic stages:
the evaluation of the collision integral and the integration of the dierential portion
of the Boltzmann equation. The strategy of solution of the dierential portion of the
Boltzmann equation is conventional. An important feature of the integral evaluation
is the simulation of molecular collisions accordingly to the Boltzmann formulation and
the verication of statistical errors (if Monte Carlo method for calculation of collision
integral is used). Some important steps have been made recently in this direction.
First, the existing method has been considerably improved by a new technique of
evaluation of collision integral. It has been supposed for a long time that it is impos-
sible to carry out this evaluation with preserving the conservative variables, and an
articial correction of the solution has been used to avoid parasitic accumulation of
the error. New methods are being based on special quadrature integration formulas
for collision integral that preserve the conservative properties. Second, an approach
based on a special projection technique has been developed. The advantages of the
method of direct numerical integration are a uniform accuracy of computing both low
and high density regions, and also an easy and eective parallelization of the compu-
tational code. Some limitations of the method are mainly related to a considerable
dependence of the computational cost on the problem dimension and to complexity
of simulation of chemically reacting gas mixtures.
8
1.3.2 Molecular dynamics
This was the rst ever physical simulation method. It employs probabilistic pro-
cedures to set initial conguration of the molecules. Calculation of the subsequent
molecular motion including collisions and boundary interactions is deterministic. Col-
lisions occur when cross sections overlap. The cost or computational time of the MD
is proportional to the square of the number of simulated particles.
The major dicult of the MD is that for a given molecular size,
ow geometry
and gas density, the number of simulated molecules is not a free parameter; this often
requires a prohibitive number of simulated molecules to be used.
1.3.3 Direct Simulations Monte Carlo (DSMC) method
The basic dierence between the DSMC method and the molecular dynamics method
is that in the former technique molecules are selected for collision on a probabilistic
basis and, while the molecule size appears in the procedures of establishing the col-
lision rate, it does not aect the collision parameters. The fundamental principle of
the DSMC method is the splitting of continuous motion and collisions of molecules
at a time step t into two sequential stages: free-molecular transfer and collision
relaxation. The DSMC method is non stationary in its nature, and the following pro-
cedure is used to solve stationary problems. In the entire computational domain an
arbitrary initial state of the gas is specied and boundary conditions corresponding to
desired
ow are imposed at time zero. The boundary conditions should be such that
a steady
ow is obtained as the large time state of the unsteady
ow. Realization of
the DSMC method implies the dividing of the computational domain into a grid of
cells. The size of these cells should be suciently small so that the change in gas dy-
namic properties across each cell is small. It is usually selected as t =min(

;
res
),
where

is the mean collision time, and
res
is the mean residence time, so that the
9
molecules did not cross more than one cell during one time step. After the steady
ow
is reached, sampling of molecular states within each cell is performed during the time
long enough to avoid statistical scattering. All macroscopic gas dynamic parameters
are then determined from these time averaging data.
The DSMC method has become de facto the main tool for the study of complex
multidimensional
ows of rareed hypersonic aerothermodynamics. This is primarily
conditioned by a number of its obvious merits: comparative simplicity of transition
from one-dimensional problems to two- and three-dimensional ones; a possibility of
using various models of gas particle interaction, including the models of internal
degrees of freedom and chemical reactions, without substantial complication of the
computational algorithm; a possibility of eective application of the method on up-
to-date parallel computers.
A detailed explanation of this method is given in the next section.
1.3.4 Hybrid approaches
Hybrid approaches are sometimes used in cases where both continuum and kinetic
regime
ows are present. An example of such a case is a shockwave where the
ow
on one side of the wave is continuum and the kinetic on the other.
The coupling between continuum and kinetic methods implies usually the dividing
of the computational domain into subdomains. A solution in one of them is obtained
by solving the Navier-Stokes equations, the DSMC method being used in the other
one. An important question here is setting the interface between these subdomains.
In regions with strong deviations from the equilibrium state the DSMC method is
used, and in other regions a continuum scheme is used based on the solution of Euler
or Navier-Stokes equations.
Of all the above mentioned methods, the DSMC is the most reliable and widely
10
used. Most
ow analysis in this thesis is done with this method.
1.4 The DSMC technique
To circumvent the dicult of a direct solution of the Boltzmann equation, an alter-
native technique known as direct simulation Monte Carlo (DSMC) was proposed by
G. Bird in 1963. The DSMC method maybe regarded as a numerical method for
solving the Boltzmann equation, and has been shown to be equivalent to solving the
Boltzmann equation. The DSMC technique uses model particles that move and col-
lide in physical space to perform a direct simulation of the molecular gas dynamics.
Using DSMC technique, a complete
ow eld can be modeled at the molecular level.
In comparison, the high computational expense associated with direct numerical sim-
ulation of the continuum Navier-Stokes equations has placed a limit on the size of the
problem that can be tackled.
Bird (Bird, 1994) denes DSMC as a computational technique in which a computer
is used to provide a direct physical simulation of
uid
ow rather than a numerical so-
lution of equations that provide the conventional mathematical model of the
ow. The
gas is represented in the computer by some thousands of model molecules. The veloc-
ity components and position coordinates of the simulated molecules are stored in the
computer and are modied with time as molecules are concurrently followed through
representative collisions and boundary interactions in simulated physical space.
1.5 Key elements of DSMC
The process of eecting aerodynamic investigations using DSMC starts from dening
the geometrical model, physical properties of its surfaces (wall temperature, in a
general case being dierent at various parts of a body), gas-surface interaction laws,
11
and the freestream conditions.
Then the main DSMC procedure is used for computations. It is conventionally
regarded as a technique for the computer modeling of real gas by several hundred
thousands of simulation molecules. Each of these molecules represent a very large
number of real gas molecules. The main feature of DSMC is a selection of molecules
for collision on a probabilistic basis and, while while the molecular size appears in
the procedures for establishing the collision rate, it does not aect the collision pa-
rameters.
The main principle of DSMC is the splitting of continuous process of molecular
motion and collisions into two successive stages at the time step t. The compu-
tational domain is divided into cells of size x such that the variation of the
ow
parameters in every cell is small. The time step t should be small as compared
with the mean collision time

. Free motion of molecules and their collisions are
considered successively at this time step t:
1. Collisions of particles belonging to the given cell in each cell of are carried
out independently in each cell of physical space, i.e. the collisions of particles in
the neighboring cells are not considered. Since the distribution function variation is
supposed small in the cell, when a pair of particles for collision is chosen the relative
distance between them is not taken into account. The post-collision velocities are
calculated in accordance with the conservation laws of linear momentum and energy.
2. All molecules located in the computational domain are displaced by the distance
determined by their velocities at the moment and by the time step t. If a molecule
leaves the computational domain, then its velocity is recomputed according to the
boundary conditions. At the same step t new particles entering the computational
domain are generated in accordance with the distribution function specied at the
domain boundaries.
12
Thus, the following principals steps are specied in the main DSMC procedure:
entering new molecules
molecular motion
gas/surface collisions
indexing of molecules over collision cells
collisions
sampling of macroparameters
After nishing the main computation, the results of computations are processed
to obtain the
owelds, total and distributed surface quantities and energy distri-
bution funstions. The analysis of results is performed by means of a postprocessing
system that provides for the color visualization of
owelds and distributed surface
characteristics.
1.6 Application of the DSMC method
The DSMC method is the primary computational tool used to support recent and
ongoing space projects for which rareed hypersonic regime is critical to spacecraft
performance is the DSMC method. The recent years have seen a very successful
application of this method as a predictive tool for many complicated problems. These
include modeling of aerodynamics of spacecrafts,
ow around slender and convex
bodies,
ow around concave bodies, materials design and Processing (lm growth
and etching), and many other problems.
13
Chapter 2
Analysis of Rareed Flows in the
Collaborative High altitude Flow
Facility (CHAFF-IV)
2.1 Introduction
With recent major advances in the aerospace industry, the problem of availability of
reliable experimental platforms is becoming very evident. Testing of space instru-
ments and systems require a space environment. Currently, newly designed instru-
ments are taken onboard the spaceshuttle to be tested in space. While space based
investigations have a reliable testing environment, they are very limited in scope.
There is limited space available on board the space shuttle or rockets. Space shut-
tle missions are dependent on weather, rotation of the earth, political decisions and
availability of large amounts of funds needed to operate launches. Even well-funded
space institutions have to wait on a line for a long time to get their time on the space
shuttle.
14
Such problems with space based investigations have made the idea of ground based
testing facilities more appealing. Ground based testing has many advantages. It will
dramatically cut the cost of space instrument investigations. Experimentalists will
nd it easy to change congurations of their experiments. And because samples do
not need to be
own to space, it will be possible to use a large number of samples
in dierent investigations and thereby ensuring more reliable results. The major dis-
advantage of ground-based testing facilities is the challenge of faithfully reproducing
the space environment.
The limitations of ground-based facilities in accurately predicting the eects of
thruster operations on spacecraft systems has always been driven by by the facility's
background pressure (Ketsdever (2001)). In space, thruster plume would escape and
move away from the thruster. In a ground facility, however, the plume hits the
walls of the testing facility as soon as it escapes the thruster. The re
ection of the
plume makes it dicult for accurate testing to be made on thruster parameters. The
challenge is to pump out the plume quickly so that it does not bounce back into the
chamber. Ground facilities therefore require large and ecient pumping systems.
The Collaborative Altitude Flow Facility (CHAFF-IV) was designed in an eort
to obtain meaningful spacecraft-thruster interaction data by maximizing the facility's
high vacuum pumping. See Figure 2.1. CHAFF-IV incorporates a total chamber
pumping (TCP) concept by lining the entire facility with an array of cryogenically
cooled, radial ns as shown in Figures 2.1 and 2.2. The aluminum nned arrays are
contained inside a 3m diameter by 6m long stainless-steel vacuum chamber. There
are 112 ns in total.
In this chapter, rareed
ows in the CHAFF-IV are analyzed. First a test particle
method is used to simulate the CHAFF-IV's pumping capabilities, and then a DSMC
investigation is done to model the shadowing eect of a plenum
ow.
15
Figure 2.1: CHAFF-IV cryogenically cooled, radial n total chamber pumping ar-
rangement
Figure 2.2: Fabricated aluminum nned arrays installed in CHAFF-IV
2.2 The cryogenic pumping capabilities of CHAFF-
IV
A test particle modeling is used to help optimize the radial n arrays by determining
the ideal length, spacing, thickness and orientation of the ns. Comparison is also
16
made between two types of simulation scenarios. One scenario assumes that a parti-
cle's sticking coecient is updated every time the particle hits a surface whereas the
other scenario assumes constant sticking coecient.
2.2.1 Facility background pressure requirement
In typical ground-based facility with a fraction of its inner surface occupied by pump
inlets, the thruster euents are typically stopped and randomized by facilities sur-
faces. The random motion of the scattered propellant molecules ineciently bring
them to a pumping inlet. The background pressure of a propellant gas can be ap-
proximated by
p
b
=
4
_
Mk
p
T
b
Tp
mv
0
f
p
A
s
=
_
MkT
b
m
_
V
(2.1)
where
_
M is the propellant mass
ow, k Boltzmann's constant, T
b
and Tp are back-
ground and propellant gas temperatures respectively,m is the molecular mass of the
propellant gas, v
0
is the average thermal speed of the background gas, f
p
is the frac-
tion of the facility inner surface occupied by pumping inlets or pumping surfaces,A
s
is the inner surface area of the facility, and V is the facility's pumping speed. In
order to minimize the eect of chamber induced charge exchange collisions in xenon
ion thrusters, background pressures on the order of 310
6
Torr are required. For a
typical Hall thruster mass
ow rate of 5mg/s, equation 2.1 indicates that pumping
rates in the excess of 2.510
5
L/s are required. For a cold gas system with a nitrogen
propellant
ow rate of 1g/s, pumping rates on the order of 10
7
L/s are required to
maintain free molecular
ow in the plume back
ow region.
Clearly a critical background number density for the thruster plume interaction
studies is reached when the background mean free path becomes less than or equal
17
to the largest internal dimension of the facility. Therefore, the background number
density should be
n
b

1
p
2
b

c
(2.2)
where
b
is the background gas collision cross section and
c
is the critical mean free
path.
As equation 2.1 indicates, the background pressure can be minimized by having
large available pumping areas (f
p
A
s
). For a given chamber geometry, the pumping
rate is maximized by increasing the fraction of the inner surface area which acts as a
pump. This suggests that high pumping rates can be achieved when the entire inner
surface of the facility is a pumping surface. The TCP concept has driven design of
several interaction facilities.
For the radial n TCP array, the fraction of eux from a thruster impinging on
the array that is able to return to the thruster vicinity is given by
F
r
= (1)

w
2
2h(w +t)
+
t
(w +t)

D
0
X

2
(2.3)
where is the sticking coecient, w is the radial n-to-n spacing, h is the length
of the n in the axial direction, t is the n thickness, D
0
is the characteristic thrust
diameter, and X is the distance from the thruster exit plane to the from edge of
the radial n array. Design of the radial n arrays can be optimized through the
minimization of the geometric term in the brackets of equation 2.3 by an appropriate
selection of the n geometry. For CHAFF-IV radial n target array, h = 25.4 cm, t
= 0.32cm, and w varies from 1 to 6 cm due to the radial nature of the array.
18
2.2.2 Test particle model
A test particle model was used to investigate the pumping characteristics of the
radial n array of the CHAFF-IV. Although the CHAFF-IV n array has 112 similar
wedge shaped sections, the computational domain involved only one wedge section for
simplicity. Free molecular
ow was assumed in the volume between the panels which
allowed individual molecule trajectories to be followed. Pumping statistics were built
from individual particle dynamics to represent the physical problem. Molecules were
emitted from a point source on the centerline of the facility a distance of 79.6cm from
the front edge of the n array. The particles were given randomly selected velocity
components in the horizontal and vertical directions, (Ketsdever, et al).
All surface interactions are treated as fully diuse implying that the molecules
accommodate to the surface temperature. The sticking coecient can be updated
based on the gas temperature and the presumed temperature of the wall from a
database on Table 2.3. Initially, the incident molecules can either strike the front
thickness of the radial n or enter the volume between the two ns. The molecules
are followed until they eectively stick to the panel or cross the front panel of the
radial array in which case they are considered backscattered molecules. The sticking
probability model is based on a Monte Carlo acceptance-rejection scheme. The initial
sticking coecient of 300 K carbon dioxide molecules on a liquid nitrogen cooled
cryogenic surface is assumed to be 0.63, (Ketsdever, et al).
The surface behind the radial n array is modeled as a
at plate. This surface can
either be cryogenically cooled to represent a pumping surface or at room temperature
to represent a chamber wall. Results were obtained used both congurations of the
back wall.
The results of the test particle modeling for the fraction of the backscattered
molecules from the radial n array and the
at panel are shown in gure 2.4
19
Figure 2.3: Sticking coecients of some common gases as a function of gas and surface
temperature
Figure 2.4: Test particle model results for pumping eciency as a function of sticking
coecient
The solid radial n data line is for simulations which update the sticking coecient
for each interaction with a surface. If the molecule strikes a cryogenic surface and
is not pumped, the sticking coecient may become unity if the molecule should hit
another cooled surface. If the molecule strikes a chamber wall, the sticking coecient
is reset to the origin orice expansion value.
20
The dashed line is for a constant sticking coecient throughout the simulation.
The radial n arrays outperform the simple
at panel for free molecule
ow for sticking
coecients less than approximately 0.55. As expected, the two radial n results
converge for very large sticking probabilities. Figure 2.4 shows the utility of the
radial n arrays for the pumping of high energy propulsion system
ows where the
initial sticking coecients are expected to be extremely low.
The code was also run for dierent panel height, thickness and width. The fol-
lowing results were obtained;
Panel height
As panel height is increased, the pumping eciency of the CHAFF-IV increases as
shown in gure 2.5. This is because as height is increased the pumping surface area
of the arrays increases.
Panel thickness
At low sticking coecients, increasing thickness has the eect of lowering pumping
eciency. Whereas at high sticking coecients, increasing panel thickness will result
in an increase in pumping eciency. This is shown in Figure 2.6.
Thus, a
at plane would work well with neutral because they have high sticking
coecients. The ions, which have very low sticking coecients, will be pumped best
by the proposed radial panel array.
Panel separation
As distance between individual panels is varied, the eciency of the chamber varies
depending on the sticking coecient. For small sticking coecients, an increase in
separation length increases pumping eciency to a point beyond which a further
21
Panel Height, m
Pumping Efficiency
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Sticking Coefficient = 0.0
Sticking Coefficient = 0.4
Sticking Coefficient = 0.6
Sticking Coefficient = 0.9
Sticking Coefficient = 1.0
Figure 2.5: Increasing height of the panel increases the pumping eciency of the
chamber
22
Thickness, m
Pumping efficiency
0 0.02 0.04 0.06 0.08 0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sticking coefficient = 0.0
Sticking coefficient = 0.4
Stickign coefficient = 0.6
sticking coefficent = 0.75
Sticking coefficient = 0.9
Sticking coefficient = 1.0
Figure 2.6: Pumping eciency of the chamber as a function of panel thickness
23
Separation, m
Pumping efficiency
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.2
0.4
0.6
0.8
1
Sticking coefficient = 0.0
Sticking coefficient = 0.1
Sticking coefficient = 0.4
Sticking coefficient = 0.7
Sticking coefficient = 1.0
Figure 2.7: Pumping eciency of the chamber as a function of the length between
panels
increase in the width results in the decrease of eciency. There is, therefore, a width
that provides optimum eciency as shown in 2.7. For large sticking coecients, an
increase in width results in a decrease of pumping eciency.
24
2.3 Pressure and velocity distribution in CHAFF-
IV
A DSMC study was conducted to investigate pressure and velocity distribution in
the CHAFF-IV. Although CHAFF-IV is a cryogenical, space simulation facility, only
diusion pumps were used for this study as shown in Figure 2.8.
Figure 2.8: Diusion pump were used to maintain low pressure in the chamber
Mass
ow and thrust force measurements were made for nozzle
ow with nitrogen
as a propellant. These measurements were made for dierent values of plenum and
background pressures. The condition for these experiments to be a good representa-
tion of space-based measurements is that there must be no induced velocity in the
chamber. If pressure in the chamber is too high, there will be more collisions which
will lead to an induced velocity. The DSMC method is used to verify these conditions.
25
Experimental background
A companion experiment was conducted in CHAFF-IV, by A. Ketsdever, and com-
prised of a comprised of a nozzle of 8-mm-diameter and 9-mm-length to a plenum of
dimensions 66mm x 35mm x 17mm. The measurements of mass
ux have been made
at the gas inlet with an MKS mass
ow meter. The force measurements were made
by sensing the angular displacement resulting from a torque applied to a damped
rotary system. Angular de
ection was detected by measuring linear displacement at
a known radial distance using a linear variable dierential transformer.
This experiment showed a uniform pressure in the chamber.
Numerical modeling
The DSMC study was performed in order to provide more detail on gas properties to
the above experiment. The computations were conducted for a worst case scenario,
that is the highest plenum and background pressure for which the experiments were
conducted. The highest plenum pressure for these experiments was 38 Torr with a
background pressure of 0.7 mTorr.
The diusion pumps were modeled by specifying the pump orices as having a
sticking coecient greater than zero. The rest of the inner surface of the chamber
was regarded as having zero sticking coecients, i.e. no sticking at all.
The rst step was to identify proper sticking coecient for the pump orices. This
was done by running the code using dierent sticking coecients. For each sticking
coecient a corresponding pressure value was recorded. This is the value of pressure
at the center of the back lid of the chamber.
The sticking coecient, for which the pressure value matched the experimental
value, was taken as proper sticking coecient for the pump orices. An analysis was
then conducted of the pressure and velocity distribution in the chamber. The analysis
26
showed that pressure at the wall of the chamber matched with that at the back of
the plenum. This was an indication that pressure in the chamber was uniform. The
analysis also showed that there was no noticeable induced velocity in the chamber.
2.4 Conclusions
In this chapter, analysis of rareed
ow in the CHAFF-IV chamber has been con-
ducted. The performance of CHAFF-IV as a cryogenic pump was studied using a
test particle method and reasonable results were found. However, obtaining accurate
information on chamber performance is dicult because of many variables involved.
This is especially true when ions are involved. Ions possess much higher kinetic energy
than the neutrals and will most certainly sputter the chamber walls on impact. A
good simulation of the pumping capabilities of the CHAFF-IV must take into account
the sputtered particles because they will most certainly reduce the performance of
the chamber.
A DSMC investigation was carried out to verify that there are no signicant pres-
sure gradients or induced velocities in the chamber. Satisfaction of these condictions
is critical for analysis of experimental data and proper representation of space envi-
ronment.
27
Chapter 3
Rareed Flow Impingement on
Spacecraft Surfaces: Eects of
Surface Roughness
Numerical and companion experimental eorts were undertaken to assess the eects
of a cold gas (T
0
=300 K) nozzle plume impinging on a simulated spacecraft surface.
The nozzle
ow impingement is investigated numerically using the direct simulation
Monte Carlo method. The Reynolds number ranges from approximately 2 to 600 for
nitrogen propellant. The thrust produced by the nozzle was rst assessed for vacuum
expansion. Subsequently,
at and rough plates were added downstream of the nozzle
parallel to the plume
ow. The DSMC results were compared with companion thrust
measurements in CHAFF-IV. Three plates were used in the experimental study, an
electropolished plate with smooth surface, and two rough surface plates with equally
spaced rectangular and triangular grooves. A 15% degradation in thrust was observed
both experimentally and numerically for the plate relative to the free plume expansion
case. The eect of surface roughness on thrust was found to be small due to molecules
28
back scattered from the plate to the plenum wall. Additionally, the in
uence of
surface roughness in the diverging part of the nozzle on nozzle thrust was examined
numerically, and found to be signicant at Reynolds numbers less than 10.
3.1 Introduction
When in orbit, spacecraft require on-board or secondary propulsion systems to per-
form orbit transfer, orbit maintenance, and attitude control maneuvers. An important
issue in the use of any spacecraft propulsion system involves the assessment and re-
duction of eects caused by the interaction between the thruster plume and spacecraft
surfaces (Boyd and Ketsdever (2001)). Direct impingement of a thruster plume on
surfaces can generate unwanted torques, localized surface heating, and surface con-
tamination. Self impingement (i.e. the impingement of a thruster plume on a host
satellite surface) generally occurs for small surface angles with respect to the propul-
sion system's thrust vector or occurs in the thruster back
ow. Cross impingement
(i.e. the impingement of one spacecraft's thruster plume onto another spacecraft) can
occur at essentially any angle and is becoming increasingly important with the advent
of microsatellite constellations. Many studies, both numerical (Lengrad et al (1997),
Ivanov et al (1997) and Hyakutake et al (2001)) and experimental (Legge (1991) and
Deependran et al (1997)), have been performed by various investigators to assess the
impingement of plumes onto surfaces.
In recent years, micropropulsion systems have been developed to address the need
for highly mobile microspacecraft. A wide array of concepts will require the expan-
sion of propellant gases through microscale nozzles. Because many micropropulsion
systems will also operate at relatively low pressures, the investigation of low Reynolds
number
ow has become increasingly important (Ketsdever (2000)). In the present
29
study, an experimental and numerical eort has been developed to assess the eects
of a nozzle plume impinging on a simulated spacecraft surface. Special attention is
paid here to the impact of roughness on surface forces and
oweld structure.
The nozzle
ow impingement is investigated experimentally using a nano-Newton
resolution force balance and numerically using the Direct Simulation Monte Carlo
(DSMC) numerical technique. The purpose of this work is to extend previous noz-
zle plume impingement results (Legge (1991) and Ketedver et al (2005)) to the
low Reynolds number
ow range for application to micropropulsion systems. The
Reynolds number range investigated in this study is from 2 to approximately 600,
based on the nozzle throat.
3.2 Experimental setup
All thrust measurements were performed by T. Lilly and A. Ketsdever on the nano-
Newton thrust stand (nNTS), which has been described in detail by Jamison et al.
(Jamison, Ketedever and Muntz (2002)). The nNTS was installed in Chamber IV of
the Collaborative High Altitude Flow Facilities (CHAFF-IV), which is a 3-m-diam
6-m-long cylindrical, high vacuum chamber. The facility was pumped with a 1-m-
diam diusion pump with a pumping speed of 25,000 L/s for molecular nitrogen. The
ultimate facility pressure was approximately 10
6
torr with all operational pressures
below 10
4
torr. A previous study (Ketsdever (2002)) has shown that at these back-
ground pressures and corresponding thrust levels there is no evidence of background
pressure eects on the thrust measurements in CHAFF-IV.
The conical De Laval nozzle used in this study is shown schematically in Fig. 3.1.
The conical nozzle was scaled from the geometry used by Rothe (Rothe (1971)). The
scaled Rothe geometry has a 30-deg subsonic section, a relatively sharp 1-mm-diam
30
Figure 3.1: Schematic of the geometric setup in the experiment.
throat with radius of curvaturer
c
=d
t
/4, a 20-deg diverging section, and an expansion
ratio of 62.4. This geometry was selected because of the extensive experimental data
that exists, which was previously used to verify the DSMC model's accuracy (Ivanov,
et al (1999)). The nozzle was machined from aluminum and attached to a cylindrical
aluminum plenum and mounted on the nNTS. Figure 3.2 shows a scanning electron
microscope image of the nozzle side wall, where the surface features caused by the
machining process are clearly evident. The eect of the rough diverging section walls
on the nozzle's performance parameters will be investigated in the following sections.
After the free expansion thrust was measured, aluminum engineering surfaces with
dierent surface roughness were attached to the thrust stand in the conguration
shown in Fig. 3.1. The following three surfaces were used: (1) a electropolished
at
surface (called smooth hereafter), (2) a surface with triangular (prism-like) grooves
perpendicular to the plume axis, and (3) a surface with rectangular grooves perpen-
dicular to the plume axis. The grooves are equally spaced, and the spacing is 0.05 cm.
The angle of triangular grooves is 90 deg, and the depth of rectangular grooves is the
same as their thickness of 0.05 cm. The length of the plate (plume direction) is 3 cm,
31
Figure 3.2: Scanning electron microscope image showing the surface roughness of the
expanding section of the conical nozzle.
and the width is 3.81 cm. The grooves are made only in the region downstream from
the nozzle exit plane (2.45 cm long).
The total force measured on the nNTS for this conguration is given by
F
tot
=F
th
F
s
+F
b
(3.1)
where F
th
is the thrust produced by the nozzle in the absence of the plate, F
s
is the
incident shear force on the plate (acting in the opposite direction as the thrust force),
and F
b
is the force exerted on the plenum wall due to gas pressure in the back
ow.
The angle of the plate was varied from 0 to 10 deg. The surface temperature was
32
Figure 3.3: Comparison of the mass density elds for a smooth and rough nozzles.
33
held constant throughout at 300 K.
The propellant was introduced to the plenum through an adjustable needle valve
located downstream of a mass
owmeter. In the experimental conguration, the
mass
owmeters were operated in the continuum regime throughout the pressure
range investigated. The propellant used was molecular nitrogen. In this study, the
stagnation pressures ranged from about 0.1 torr to approximately 17 torr, and the
stagnation temperature was measured to be 300 K. The combination of stagnation
pressure and temperature gave maximum Reynolds numbers of 350.
3.3 Numerical method
Three geometric congurations have been considered in the computations. First,
the nozzle expansion into a vacuum has been modeled. The experimental nozzle
geometry has been used for several stagnation pressures ranging from 18 to 1800 Pa.
Second, the interaction of the nozzle plume with the inner part of a hollow cylinder
aligned along the nozzle axis is modeled. The inner cylinder surface is considered as
macroscopically smooth or rough with surface roughness specied as triangular and
rectangular grooves perpendicular to the nozzle
ow direction. The inner diameter of
the cylinder varied from the nozzle exit diameter d
n;e
= 0:79 cm to 2.79 cm. Finally,
a 3D interaction of the nozzle plume with a plate, smooth or rough, is simulated. The
computational geometry included the nozzle with the external side of the plenum and
the plate which size and location correspond to the experimental setup.
The DSMC-based software system SMILE (Ivanov, et al (1998)) was used in all
DSMC computations. The important features of SMILE that are relevant to this
work are parallel capability, dierent collision and macroparameter grids with man-
ual and automatic adaptations, and spatial weighting for axisymmetric
ows. The
34
majorant frequency scheme was used to calculate intermolecular interactions. The
intermolecular potential was assumed to be a variable hard sphere. Energy redistri-
bution between the rotational and translational modes was performed in accordance
with the Larsen-Borgnakke model. A temperature-dependent rotational relaxation
number was used. The re
ection of molecules on the surface was assumed to be
diuse with complete energy and momentum accommodation.
All walls were assumed to be at a temperature of 300 K, except where specied
otherwise, and the propellant gas is nitrogen at a stagnation temperature of 300 K.
A background pressure of zero was assumed in all calculations. In the rst series
of computations (nozzle plume expansion into a vacuum) the computational domain
included a part of the plenum large enough to avoid the impact of the domain size on
the results, and the total number of collision cells and molecules was about 400,000
and 4 million, respectively. For the axisymmetric interaction of a plume with a hollow
cylinder these numbers were 1.5 and 8 million, respectively. The three-dimensional
plume-surface interaction was modeled using a starting surface at the nozzle exit,
generated using an axisymmetric solution of a nozzle plume expansion. An ellip-
tic distribution function was used for in
ow molecules. The number of simulated
molecules and cells was about 20 million and 3 million, respectively.
3.4 Nozzle surface roughness
A close examination of the surface structure inside the actual nozzle manifested a very
rough, groove-like structure, as illustrated in Fig. 3.2, with micron-size grooves set out
perpendicular to the main
ow direction. The evident surface roughness prompted
the authors to numerically study the eect of roughness inside the nozzle on the
nozzle thrust. To this end, axisymmetric DSMC computations were performed for a
35
Figure 3.4: Comparison of the axial velocity elds for a smooth and rough nozzles.
36
Figure 3.5: Comparison of the pressure elds for smooth and rough cylinders with
the inner diameter of 0.79 cm.
37
Figure 3.6: Comparison of the axial velocity elds for smooth and rough cylinders
with the inner diameter of 0.79 cm.
38
rough surface of the diverging part of the nozzle, assumed to have a regular triangular,
saw-tooth structure with the triangle angle of 90 deg, and the distance between the
triangles of 10m. The diuse model of re
ection with full energy and momentum
accommodation was used on triangle surfaces. A total of more than 1,000 was used,
and the results are compared with those obtained for a
at diusely re
ecting surface
(called \smooth" hereafter).
The comparison of number density elds inside a rough and a smooth nozzles
is presented in Fig. 3.3 for the smallest chamber pressure considered, P
0
= 18 Pa.
The gure also illustrates the geometry of the nozzle and the computational domain.
The results show that the in
uence of the surface roughness in the diverging part
propagates into the plenum, and the density for the rough surface is about 10% lower
than the corresponding values inside the smooth nozzle. This is easily explained
by the fact the the
ow is subsonic in the most part of the nozzle, and becomes
supersonic only near the exit. The dierence between rough and smooth is larger
near the surface than at the centerline, and amounts to almost 20% at the nozzle lip.
The larger density for the rough case is explained by a signicant amount of molecules
re
ected on the windside of the triangles and traveling toward the throat.
The molecules re
ected on the windside of triangles increase density and decrease
axial velocity in the diverging part of the nozzle. This decrease in axial velocity,
however, is compensated by the contribution molecules re
ected on the triangle lee
sides, that on average are re
ected in axial direction. The combined eect of these
two trends results in a small in
uence of the surface roughness on the axial
ow
velocity elds, as shown in Fig. 3.4. Although there is a visible dierence near the
nozzle surface, with the rough case values at the surface being lower by over 50m/s,
the proles at the nozzle exit are close.
The quantitative impact of the surface roughness inside the nozzle on the
ow
39
properties is given in Table 3.1, where the nozzle performance properties are shown
for three plenum pressures. As expected, the eect of roughness is maximum at
the lowest pressure, with the rough case mass
ow being over 12% lower than the
corresponding smooth case. Since the axial velocity at the nozzle exit is weakly
aected by the surface roughness, the thrust force is also about 12% lower, and the
specic impulse does not change with roughness. For a ten times larger pressure,
P
0
= 180 Pa, the surface roughness causes only a 3% decrease in the mass
ow and
practically no change in the specic impulse. At an even higher pressure of 1,800 Pa,
no visible in
uence of the nozzle roughness was found.
The conclusion from these computations is that the surface roughness in the nozzle
impacts mostly the density elds; its eect on the axial velocity is much smaller.
Correspondingly, the mass
ows is signicantly reduced by the surface roughness
only for throat-based Reynolds numbers of about unity or lower, when the subsonic
region occupies large part of the diverging part of the nozzle. The surface roughness
was found to have little eect on the specic impulse. All this also shows that the
experimental data on plume and surface forces shown below as function of the mass

ow rate is not aected by the nozzle surface roughness.
3.5 Axisymmetric plume-surface interaction
The study of the plume-surface interaction with dierent roughness structure has been
started with an axisymmetric nozzle-hollow cylinder geometry. The axisymmetric
geometry was selected due to the relative simplicity of the
ow as well as numerical
eciency compared to a 3D plume-surface interaction. A plume from the nozzle is
directed into a cylinder with the axis coinciding with the plume axis, and a diameter
varying from d
n;e
to d
n;e
+ 8 mm, where d
n;e
is the nozzle exit diameter. A plenum
40
Figure 3.7: The impact of the roughness shape on pressure elds for the plume-
cylinder interaction. Cylinder diameter is 0.79 cm.
41
Figure 3.8: Comparison of the pressure elds for smooth and rough cylinders with
the inner diameter of 1.59 cm.
42
Figure 3.9: Comparison of the axial velocity elds for smooth and rough cylinders
with the inner diameter of 1.59 cm.
43
Figure 3.10: Comparison of the pressure elds for 5 mm long rough and smooth
cylinders with the inner diameter of 1.59 cm.
44
pressure of 1800 Pa was used in all calculations presented in this section.
The rst conguration considered is shown in Fig. 3.5, where the pressure elds
are given for a smooth and rough (triangular roughness shape) cylinders. The larger
distance between the triangles of 0.001 mm was used for the roughness characteristic
size to be on the order of or larger than the gas mean free path. Note that the
starting surface was used in these computations, with the in
ow boundary located in
the middle of the diverging part of the nozzle, and the in
ow parameters taken from
the full nozzle modeling with a smooth surface.
The most important conclusion here is that the surface roughness does not have
a pronounced eect on gas pressure. Pressure is one to three percent higher for the
rough case, with qualitatively the same
ow structure. Initially, the pressure drops
in the nozzle. Then, there is a pressure increase in the center of the cylinder due to
the formation of the viscous layer at the cylinder surface. Even a smaller dierence
between the rough and smooth cases is observed for the axial
ow velocity elds
presented in Fig. 3.6. Although there is a minor dierence near the surface, with
the velocity somewhat lower near the rough cylinder, the velocity near the axis is
practically not aected by the surface roughness.
The described behavior is similar to that inside the nozzle shown in the previous
section. The molecular explanation of this eect given for the low-pressure gas
ow
inside the nozzle is however not applicable for this case of a relatively high pressure.
The gas mean free path inside the cylinder is about 10
4
m, which corresponds to
the Knudsen number of about 0.1 based on the roughness size, and 0.01 based on the
cylinder diameter. The
ow is therefore near-continuum; the gas stagnates inside the
triangular cavities of the rough surface, and acts as a pseudosurface in terms of
ow
development. As a result, the cavities do not signicantly aect the
ow.
This is also illustrated in Fig. 3.7 where the pressure elds are shown for rect-
45
angular and triangular roughness shapes. The roughness shape does not impact the

ow, and the determining factor in this case is the minimum cylinder diameter and
not the roughness type. The results do no change when deeper cavities are used for
the rectangular roughness shape.
The next conguration considered is that with a larger diameter of the cylinder,
d
n;e
+ 8 mm. In this case, the density inside the cylinder is signicantly smaller, and
the Knudsen number is about 0.1 based on the cylinder diameter. The pressure and
axial velocity elds for the larger cylinder diameter case are shown in Figs. 3.8 and
3.9. There is a local pressure maximum inside the cylinder, although less pronounced
than for the smaller diameter of the cylinder. The
ow inside the nozzle is weakly
aected by the cylinder. Most important, the gas pressure is not in
uenced noticeably
by the surface roughness. The axial velocity is also not aected by the roughness.
In order to understand the eect of the surface roughness that might be obtained in
the experimental setup used in this work, it is necessary to compute axial forces on the
cylinder, and compare them to the thrust forces from the nozzle. The computed values
of the axial force for three dierent diameters of the cylinder are listed in Table 3.2.
Here, d is the dierence between the cylinder and nozzle exit diameters. The total
axial force on the cylinder is comparable with the thrust force of 0:178 10
2
N.
The dierence between the smooth and rough surfaces is small, though. It is within
the statistical error of the computations for d = 0, and slightly increases for larger
cylinders. However, even for d = 8 mm it amounts to about percent of the total
nozzle thrust, making it practically impossible to characterize experimentally.
The computations were performed up to d = 20 mm, when the gas mean free
path near the surface is signicantly larger than the roughness size; no considerable
eect of roughness on the axial force was found. This is somewhat counter-intuitive:
although the incident plume particles have same contribution to the axial force both
46
Figure 3.11: Pressure eld (Pa) over a smooth plate at 0 deg.
for smooth and rough surfaces, the re
ected molecule contribution on rough surface
is expected to be a nite number, compared to zero for the smooth surface. The
small dierence between rough and smooth surfaces in the near-free molecular regime
is because for the cylindrical geometry many molecules that re
ect from the surface
collide with the opposite side of the cylinder. Among all particles that collide with the
surface, the number of particles that experience such multiple re
ections is greater
than those that come directly from the plume, and therefore the eect of surface
roughness is minimized.
47
Figure 3.12: Pressure eld (Pa) over a rough plate with triangular grooves.
48
Figure 3.13: Pressure eld (Pa) over a rough plate with rectangular grooves.
The eect of multiple re
ections of particles on the surface decreases with the
decrease of the cylinder length. The axial force values for a plume impinging on a
short, 5 mm long, hollow cylinder given in Table 3.2 show that there is a 12% dierence
between the smooth and rough cylinders. The absolute magnitude of this dierence
is small compared to the nozzle thrust force, though. This makes the shorter cylinder
case dicult to examine experimentally. The pressure eld for the shorter cylinder is
presented in Fig. 3.10. Pressure inside the cylinder is noticeably higher for the rough
cylinder. The dierence in
ow velocities is smaller than that in pressures, though.
The computations were also performed for an elevated surface temperature inside
49
Figure 3.14: Pressure eld (Pa) over a rough plate with triangular grooves. Plate
angle is 10 deg.
the cylinder, and showed that the smooth/rough surface dierence increases with
temperature. For a surface temperature of 600 K the axial force on the rough cylinder
is about 5% larger than that for a smooth one. Generally, the computations of a plume
interacting with the inner surface of a cylinder allow us to conclude that the surface
roughness has relatively small eect on
ow elds and surface forces, and this eect
is too small to be studied experimentally. The shorter cylinder computations show
that the eect should be signicant in a 3D case of a plume interacting with a
at
surface. This case is considered in the following section.
50
3.6 Interaction of plume with a plate: Numerical
modeling
Consider now the 3D interaction of a rareed plume with a plate. The pressure
ow
eld in the plane perpendicular to the plate surface and coming through the nozzle
axis is given in Fig. 3.11 for a smooth plate and the stagnation pressure of 405 Pa.
The interaction region between the plume and the plate is clearly seen, with the local
pressure maximum located near the plate surface about 6 mm downstream from the
nozzle exit plane. The pressure values in that region are over an order of magnitude
larger than those at the corresponding location in the bottom half of the plume.
There is signicant back
ow observed as the result of the plume-surface interaction.
A strong back
ow will result in a contribution of back
ow molecules interacting with
the plenum surface to the total force. This contribution increases the total force in
x-direction.
The
ow does not change qualitatively when a plate with a triangular surface
roughness is used (see Fig. 3.12). Quantitatively, however, the pressure maximum
at the plate shifts about 1 mm downstream compared to the smooth surface case,
and the maximum value increases by about 10%. The pressure is generally higher for
the rough plate, since most of the plume molecules that collide with the surface are
re
ected backwards in that case. This is especially noticeable in the back
ow region
where the pressure for the rough surface case is about two times higher. Note that
the mean free path of the gas near the plate is on the order of 1 cm, and is an order
of magnitude larger that the roughness size.
In addition to the triangular groove roughness, a rectangular grove shape has
also been examined. The pressure for the latter case is somewhat lower than for the
triangular one, but is still higher than for the smooth surface, as shown in Fig. 3.13.
51
The increase in the angle of the plate measured from the plume direction from 0
to 10 deg signicantly weakens the plume-surface interaction, as shown in Fig. 3.14.
The pressure maximum is more than two times smaller for 10 deg than it was for
0, and the plate no longer has a noticeable eect on the
oweld in the immediate
vicinity of the nozzle exit. The back
ow pressure is also reduced and is only slightly
higher than the corresponding pressure at the bottom half of the plume back
ow.
The eect of the plate surface roughness on the pressure eld for 10 deg is similar to
that for 0 deg, and is not shown here.
Consider now the eect of the plume roughness on surface forces. The distribution
of the forces in X direction (shear force) and Y direction (pressure force) over a smooth
plate is shown in Figs. 3.15 and 3.16. Here, X direction coincides with the direction
of the plume, and Y direction is perpendicular to the plate surface. The maximum in
F
x
force is about 0.26 N and is located close to the plate center, in the region where
both molecular density and axial velocities are suciently large. The maximum in
F
y
force is shifted a few millimeters to the nozzle exit plane, where the local gas
pressure maximum is observed. The maximumF
y
is about two times larger than the
corresponding maximum F
x
, primarily because the force from re
ected molecules is
nite forF
y
and zero forF
x
. Also, the axial velocity component of plume molecules in
that region is somewhat larger than the radial one, with the incidence angle typically
larger than 45 deg.
The next two gures, Figs. 3.17 and 3.18, present the corresponding force distri-
butions for a rough surface of the plate (triangular roughness). They clearly show
the discontinuous structure of the force distributions. TheF
x
values are large on the
sides of the grooves directed toward the nozzle (windside), with the maximum value
almost three times larger than the corresponding maximum on a smooth plate. The
lee sides of the grooves, however, are characterized by forces that act in the direction
52
Figure 3.15: Force in X direction per unit area (N/m
2
) on a smooth plate for the
plate angle of 0 and P
0
= 405 Pa.
opposite to the plume direction, therefore reducing the large force from the wind
sides. The maximum value ofF
x
on a rough plate is close to that ofF
y
. The wind-lee
side structure of the surface is also clearly seen in F
y
, although the direction of the
forces is the same for this case.
The forces on the plate F
x
and F
y
and on the plenum surfaceF
b
are presented in
Table 3.3 for two roughness types and two angles of the plate. For the plate angle of 0
53
Table 3.1: Impact of surface roughness on nozzle properties.
Po;N=m
2
Surface type Mass
ow;kg=s Thrust;N Isp ;s
1.82110
3
Smooth 0.280410
5
0.178010
2
0.647510
2
1.82110
3
Rough 0.280010
5
0.177010
2
0.644910
2
1.82110
2
Smooth 0.230110
6
0.117610
3
0.521310
2
1.82110
2
Rough 0.222610
6
0.112210
3
0.514310
2
1.82110
1
Smooth 0.154810
7
0.742410
5
0.489210
2
1.82110
1
Rough 0.137510
7
0.661410
5
0.490810
2
Table 3.2: The axial force on the cylinder for surface conditions and diameters
d;mm Surface type Temperature;K Length;mm Fx;N
0 Smooth 300 25.5 0.109710
2
0 Rough 300 25.5 0.108010
2
4 Smooth 300 25.5 0.777010
3
4 Rough 300 25.5 0.790610
3
8 Smooth 300 25.5 0.718710
3
8 Rough 300 25.5 0.740810
3
16 Smooth 300 25.5 0.588910
3
16 Rough 300 25.5 0.623610
3
8 Smooth 300 5.0 0.276210
3
8 Rough 300 5.0 0.310110
3
8 Smooth 600 25.5 0.837110
3
8 Rough 600 25.5 0.878010
3
Table 3.3: Surface forces for P
0
= 405 Pa.
Srf Angle F
th
, N F
x;s
, N F
y;s
, N F
b
, N F
tot
, N
Sm: 0 0.30910
3
-0.67610
4
0.10910
3
0.12010
4
2.53410
4
Rect: 0 0.30910
3
-0.73410
4
0.10810
3
0.15210
4
2.50810
4
Tria: 0 0.30910
3
-0.82610
4
0.11210
3
0.18110
4
2.44510
4
Sm: 10 0.30910
3
-0.30810
4
0.72510
4
0.56410
5
2.83910
4
Tria 10 0.30910
3
-0.40610
4
0.74810
4
0.90410
5
2.77510
4
54
Figure 3.16: Force in Y direction per unit area (N/m
2
) on a smooth plate for the
plate angle of 0 and P
0
= 405 Pa.
the magnitude of the forces on the plate is comparable to the plume thrust forceF
th
,
withF
x
andF
y
being about 25% and 30% of the thrust, respectively. Comparison of
rough and smooth surfaces shows that the magnitude ofF
x
is smallest for the smooth
plate and largest for a plate with the triangular roughness shape. The dierence is
about 20% for these cases. The force in Y direction practically does not depend on
surface roughness.
55
Figure 3.17: Force in X direction per unit area (N/m
2
) on a rough plate for the plate
angle of 0 and P
0
= 405 Pa.
Another important contributor to the total X-forceF
tot
is the force on the plenum,
primarily caused by molecules re
ected on the plate. This force is signicantly larger
for rough plates, with the value for the triangular roughness type about 50% higher
than that for the smooth plate. Since the force on the plenum is in the direction
opposite to that on the plate, this 50% dierence considerably reduces the eect of
surface roughness onF
tot
. The dierence betweenF
tot
for a smooth and a rough plate
56
Figure 3.18: Force in Y direction per unit area (N/m
2
) on a rough plate for the plate
angle of 0 and P
0
= 405 Pa.
with triangular grooves amounts to only about 3% of the total force. For the angle
of 10 deg, this dierence is only about 2.5%.
The comparison of contributions to the total force for a plume
ow atP
0
= 155 Pa,
interacting with smooth and rough surfaces, is given in Table 3.4. As compared to
P
0
= 405 Pa, all forces scale approximately with the stagnation pressure, and the
conclusions made for the higher pressure case are applicable for P
0
= 155 Pa.
57
Table 3.4: Surface forces for P
0
= 155 Pa.
Surface F
th
, N F
x;s
, N F
y;s
, N F
b
, N F
tot
, N
Smooth 0.94910
4
-0.236810
4
0.379910
4
0.461810
5
0.758410
4
Rectangular 0.94910
4
-0.259510
4
0.377010
4
0.603610
5
0.749910
4
Triangular 0.94910
4
-0.296010
4
0.382210
4
0.731510
5
0.726210
4
3.7 Comparing numerical results with data
Numerical results will now be compared to available experimental data. These exper-
iments were conducted by Dr Ketsdever and his group at the University of Southern
California.
Comparison of computed and measured total forces versus mass
ow is presented
in Fig.3.19. Here, the lines that show numerical solution were created using the values
ofF
tot
listed in Tables 3.3 and 3.4, that correspond to the chamber pressures of 155 and
405 Pa. The agreement between the experimental and computed force values is good,
and the dierence in all cases does not go beyond a few percent. The experimental
and numerical forces practically coincide both for a smooth polished plate and a rough
plate with triangular roughness. The values for a plate with rectangular grooves are
closer to those for a smooth plate in DSMC, and to triangular grooved plate in the
experiment, although the dierence is rather small and may be attributed to one or
several causes of experimental and numerical inaccuracies.
There are several possible sources of experimental uncertainties in this work. First,
there is always a nite background gas pressure in the chamber that increases with
mass
ow. The background gas may impact the mass
ow measurements only for
plenum pressures larger than 10 torr, although the force (momentum
ux) mea-
surements are aected to some extent at all plenum pressures. A previous study
(Ketsdever (2002)) indicated that the force can be aected by less than 0.5% at the
58
Mass flow, kg/s
Force, N
0 2E-07 4E-07 6E-07
0
0.0001
0.0002
0.0003
0.0004
Experiment, no plate
Experiment, smooth plate
Experiment, triangular grooves
Experiment, rectangular grooves
DSMC, no plate
DSMC, smooth plate
DSMC, triangular grooves
DSMC, rectangular grooves
Figure 3.19: Total force versus mass
ow for free expansion and smooth and rough
surfaces: comparison of numerical and experimental modeling.
experimental conditions of this work. A stand calibration of de
ection angle vs ap-
plied force has been approximated to be within 3%. For a given applied force to
the stand, the standard deviation of the stand's de
ection was less than 1%; how-
ever, the accuracy of the calibration system must also be taken into account. Finally,
there was some error associated with the manufacturing of the nozzle. The nozzle
throat diameter is known only with an accuracy of 1%, and the nozzle surfaces are
signicantly rough. As was mentioned above, the eect of these last issues are minor
59
Mass flow, kg/s
Force, N
2E- 06 4E- 06
0.001
0.002
No plate
Smooth plate
Triangular grooves
Rectangular grooves
Figure 3.20: Measurements of total force versus mass
ow for free expansion and
smooth and rough surfaces.
for the presented results. In addition to the experimental uncertainties, there are a
number of numerical uncertainties. Grid resolution, the maximum number of simu-
lated molecules, eects of the subsonic boundary conditions, and the gas-gas collision
models all account for a numerical uncertainty estimated to be on the order of 1 to
2%.
The experimental results for a wider range of mass
ows that correspond to plenum
pressures up to about 17 torr are shown in Fig. 3.20. The addition of an engineering
60
plate signicantly reduces total force, up to 15%. The surface roughness eect is
much smaller, and the eect of the roughness type is negligible. The small dierence
between rough and smooth surfaces is explained by the eect of the plume molecules
colliding with the plenum, as discussed in the previous section.
3.8 Conclusions
Numerical modeling of a cold gas nozzle plume interacting with engineering surfaces
is performed for nitrogen propellant in the range of nozzle throat based Reynolds
numbers from about 2 to 600. Smooth and rough plates were examined, with surface
roughness introduced through a set of equally spaced 0.5 mm wide grooves perpen-
dicular to the
ow direction. The DSMC method was used in the numerical study,
with the setup corresponding to that in the companion experimental study conducted
in CHAFF-IV. The calculated force vs mass
ow was found to be in a good agree-
ment with the corresponding experimental data. The experiments and computations
showed that there is signicant thrust degradation due to the plume surface interac-
tion, with the total decrease being up to 15%. The force on the plate increases in
magnitude by about 20% for the rough surface as compared to the smooth one. How-
ever, the impact of the surface roughness on total force is small, which is attributed
primarily to the eect of plume molecules re
ected from the plate backwards to the
plenum surface. The number of such molecules is signicantly larger for rough sur-
faces.
The impact of the surface roughness inside the nozzle has been studied numerically.
It was shown that the surface roughness decreases both mass
ow and thrust by over
10% for Reynolds numbers on the order of one. The eect decreases with the increase
of the Reynolds number, and is negligible at Re > 100. The specic impulse is not
61
aected by the surface roughness even at small Reynolds numbers.
62
Chapter 4
Radiometric Forces in Rareed
Gas Flow
The main objective of this part of work work is to examine radiometric forces cre-
ated by rareed gas
ows on heated plates of dierent shapes, and to analyze the
change in the total force as a function of gas pressure. The numerical modeling was
condacted along with companion experimental study conducted by N. Selden. While
experimental measurement of the radiometric force has benets of correctly account-
ing for dierent factors, such as gas-surface accommodation, internal structure of
molecules, and complex three-dimensional geometries, numerical modeling provide
detailed information of gas
ow properties and surface parameters.
4.1 Introduction
The repulsion and attraction of bodies induced by radiation recieved a great deal
of attention from a number of prominent scientists in the 19th and 20th centuries
(Woodru (1966)). The rst published experiment was conducted by Abraham Ben-
63
net (Bennet (1792)) who reported in 1792 the negative result of light shined on a paper
vane suspended by a ber thread in vacuum. At that time, he was unable to see any
motion distinguishable from the eect of heat. The rst successful experiment was
conducted by Fresnel (Fresnel (1825)) who observed in 1825 a repulsion between two
suspended foil vanes when sunlight was focused on them in a low-pressure container.
In the 1870s, William Crookes proposed dierent types of apparatus to investigate
the radiometer eect (Crookes (1872), (1875)); one of them became known as the
Crookes radiometer. It consists of an airtight glass bulb containing a partial vacuum
with a set of vanes mounted inside the bulb on a spindle; the vanes rotate when
exposed to light or another heat source. Crookes incorrectly suggested that the force
causing the vanes to move was due to photon pressure. This theory was originally
supported by J. Maxwell who had predicted this force. O. Reynolds had initially
proposed a reasoning based on surface outgassing, and then presented a more rigorous
explanation based on kinetic theory (Reynolds (1876)). According to the latter theory,
the gas in the partially evacuated bulb is the main driving force responsibe for the
rotation of the vanes.
Reynolds also took part in the experiments conducted by Schuster (Schuster
(1876)) that turned up the rst experimental evidence of gas forces being the dom-
inant cause of the radiometric eect. In this experiment, the radiometer case was
suspended by parallel bers and light was directed onto the vanes. The radiometer
case was pushed in the direction opposite the vanes, proving that the radiometric
phenomenon is caused by the interaction between the heated side of the vane and
the gas. The kinetic theory explanation given by Reynolds is in fact a free molecule
approximation of the radiometric eect: the molecules leaving the hot side leave with
an increased velocity relative to those leaving the cold side. This leads to a larger
momentum change on the hot side, and results in the motion of the vanes with the
64
hot side trailing.
The situation is, however, dierent in transitional or near-continuum
ow. The
molecules with higher velocities leave the hot side of the vane and collide with in-
coming molecules. These collisions cut the surface
ux more eciently than those
re
ected on the cold surface. Essentially, this means that these eects compensate
each other, and pressures in the center of the vane are equal. This theory was rst
proposed by Reynolds (Reynolds (1879)). At about the same time, Maxwell (Maxwell
(1879)) also showed that an unbalance force exists near the edge of the heated side
of the vane, where the heat
ow in the gas in non-uniform. Almost fty years later,
Einstein presented a simple theory (Einstein (1924)) that related the force on the
vanes to their perimeter. This edge dependence of the vane force has found partial
conrmation in experimental work (Marsh (1926)), where the force was found to de-
pend on perimeter, although not to the extend Einstein has predicted. Since about
that time, the edge theory has become widely accepted.
The inversely proportional dependence of the radiometric force of a vane placed
in a temperature gradient, deriven in Einstein (1924), is similar to the high-pressure
part of the general dependence proposed by Br uche and Littwin (1928) that combines
both high and low pressure regimes as
F =
1
(a=p) + (p=b)
: (4.1)
This expression re
ects the fact the the radiometric force has a maximum at some
pressure that depends on gas and geometric properties, quantitatively shown as early
as 1919 (Westphal (1920)). At low pressures, a free-molecular area force is the dom-
inant one, with the force increasing with pressure. At high pressures, the collisional
edge force becomes dominant, and the force decreases as pressure increases.
65
The strong interest in the radiometer problem since 1873 declined steadily after
1928, mostly because the issue of force production was considered closed, and no direct
application for radiometric forces had been identied at that time. The phenomenon
that drives Crookes' radiometer has been summarized by Draper (1976) who in eect
described our present understanding of it. A temperature gradient exists on the
surface if tangential stresses are to arise. These stresses are the result of thermal
transpiration, with the gas moving over the surface from the cold to the hot side.
Following this explanation (Draper (1976)), the principal force that contributes to the
rotation of the vanes in the pressure regime where the radiometer is most eective, is
the force created near the edges (a zone with the dimensions of a mean free path ,
according to Einstein (1924).
Lack of applications may explain little interest to the problem in the second half
of the last century, when only a few research papers on the subject have been pub-
lished (Mason (1966) and Crawford (1985)), with other publications being historic
analyses and overviews. The situation started to change over the last decade, when
the radiometric phenomena was found to be useful in a number of dierent micro- and
large scale devices. The radiometric force has been shown to be applicable to modern
microactuators in Wadsworth and Muntz (1996), where the direct simulation Monte
Carlo (DSMC) method was used to model forces on vanes mounted on an armature.
This method, along with experimental measurements, have been employed in Ota
et al (2001) to study a concept of an opto-microengine that uses radiometric forces.
Subsequently, a series of papers by Passian with co-workers have been published (see,
for example, Passian et al (2002), (2003)), where radiometric phenomena was studied
experimentally and analytically, mostly with application to microcantilevers. The use
of radiometric forces as an approach to study gas-surface translational energy accom-
modation has also been mentioned by Passian et al in Passian and Ferrell (2003).
66
A new concept of a high-altitude aircraft supported by microwave energy that uses
radiometric eects has also been put forward in Benford and Benford (2005).
The new studies of the old radiometric phenomena have been supported on the
one hand by modern technologies that allow more accurate measurements, and on
the other hand by state-of-the-art numerical methods that rely heavily on parallel
computing. These two factors, along with the revived interest in the application of
radiometric phenomena, has prompted the authors to revisit the contribution of the
\collisionless" (area) versus the \collisional" (edge) forces to the total radiometric
force.
4.2 Numerical approach
The direct simulation Monte Carlo computational tool, SMILE, was used in all DSMC
computations presented in this work. The variable hard sphere model with param-
eters from (Bird (1994)) was used for the intermolecular potential, and the Larsen-
Borgnakke model with variable rotational relaxation number used for translational-
internal energy transfer. The gas-surface model was assumed to be diuse with full
energy and momentum accommodations. The two-dimensional module of SMILE was
used in this work.
The computations have been conducted for a 0.9x3.9cm heated plate with a xed
temperature gradient immersed in an initially uniform stagnant gas
ow in a cham-
ber. The third (Z axis) dimension of 12.7cm was assumed when calculating forces
consistent with the actual size of the plate in the present experimental study. The
main surfaces of the plate were heated to 410K (cold) and 450K (hot), and the side
walls were 430K. The chamber wall temperature was assumed to be constant at 300K.
The computations were performed for chamber pressures ranging 0.006Pa to 6Pa and
67
three gases, helium, nitrogen, and argon.
4.3 Translational temperature and streamline
ow-
elds
The
ow eld structure typical for a transitional
ow around a plate is shown in
Fig. 4.1, where the translational temperature and streamlines are shown for helium
at a chamber pressure of 2Pa. The hot site is on the left hand side of the plate. There
are four vortices created by the temperature gradients, two at each side of the plate.
The vortices at the hot side of the plate are noticeably stronger than at the cold side.
For the case under consideration, the maximum
ow speed in the hot surface vortices
is about 5m/s, and in the cold surface vortices it is about 2m/s.
Figure 4.1: Translational temperature and streamlines shown for helium at a chamber
pressure of 2Pa
68
4.4 Radiometric forces as a function of pressure
The forces from gas on hot and cold surfaces were also computed, and the dierence
between the two (the net force) was analyzed for three carrier gases, helium, argon,
and nitrogen. The net force for these gases as a function of pressure is given in Fig.
4.2. It is seen that for every xed pressure under consideration the force is maximum
for the gas with maximum mean free path for a given pressure, helium; it is minimum
for nitrogen, that has the smallest mean free path. This is related to the fact that the
force per molecule is maximum in the free molecular regime, and it decreases as soon
as molecules start colliding with each other, moreover, it tends to zero as the Knudsen
number decreases. This is illustrated in Fig. 4.3 where the net force normalized by
the force on the hot plate is shown as function of the Knudsen number calculated
near the hot surface. It is also important to note that the normalized force for three
gases is within the error bars of the computations, that are estimated to be about
5statistical scatter of the surface properties.
Pressure, Pa
Force, N
1 2 3 4 5 6
0.0E+00
4.0E- 05
8.0E- 05
1.2E- 04
Nitrogen
Helium
Argon
Figure 4.2: Net radiometric force as a function of chamber gas pressure
69
Kn
d F/F
hot
10
- 2
10
- 1
10
0
0.005
0.01
0.015
0.02
0.025
Nitrogen
Helium
Argon
Figure 4.3: Net radiometric force normalized by the force on the hot plate as function
of the Knudsen number calculated near the hot surface as a function of chamber gas
pressure
4.5 Eects of viscosity index, molecular mass and
diameter
The impact of viscosity index, mass and diameter of the gas molecules on radiometric
forces was investigated for a helium and argon gases at 3.04Pa. In these calculations,
the viscosity index, mass and diameter of helium and argon molecules were inter-
changed. The results of the study are shown in Table 4.1. From the results it is seen
that varying the mass of a molecule or the viscosity index does not have much impact
on net radiometric forces. When the magnitude of mass of the helium molecule is
changed to that of an argon molecule as shown in the second row of Table 4.1 the net
radiometric forces are not very dierent from those of a normal helium mass shown
in the rst row.
However when the value of the diameter of the helium molecule is changed to that
of an argon molecule as shown in the third row of the table, the resulting radiometric
70
Table 4.1: Eects of viscosity index mass and diameter of the molecules on net
radiometric forces
Gas Mass Diameter Index,! F
hot
, N F
cold
, N F
net
, N
He He mass He dia. He ! 1.55110
2
-1.53910
2
1.26010
4
He Ar mass He dia. He ! 1.58610
2
-1.57410
2
1.26110
4
He He mass Ar dia. He ! 1.49810
2
-1.49510
2
3.58110
5
He He mass Ar dia. Ar ! 1.50210
2
-1.49710
2
4.42010
5
Ar Ar mass Ar dia. Ar ! 1.57110
2
-1.56710
2
4.35610
5
forces are much closer to that of a normal argon gas in the fth row of the table. This
is a clear indication that molecular diameter plays a much bigger role than mass in
radiometric forces.
The impact of varying the viscosity index is shown by comparing the net forces
in the third and fourth rows of the table.
X, m
Temperature, K
- 1 - 0.5 0 0.5 1
280
300
320
340
360
380
400
420
0.45m domain
0.9m domain
1.35m domain
1.8m domain
Figure 4.4: Temperature proles are shown for dierent chamber sizes for Helium gas
71
4.6 The impact of chamber size on radiometric
forces
The impact of the chamber walls has been stdied for a helium
ow at 2Pa with the
chamber size varied from below 0.2m to about 2m with the same temperature gradient
of 40K between the plate sides and the chamber walls of 300K. The analysis of the

ow eld shows that even when the chamber is about a hundred times larger than the
heated plate, there is still pronounced eect of the location of the chamber walls on
the
ow properties not only near the chamber walls, but also near the plate. This is
illustrated in Fig. 4.4 where the temperature proles are shown for dierent chamber
sizes in the cross section perpendicular to the plate and coming through its center.
The plate center is at X=0. Generally, the increase in the chamber size results in
decrease of temperature and pressure gradients near the plate. As a result, the net
force exerted on the plate decreases for larger chamber sizes, as shown in Fig. 4.5.
The trend of the force decrease clearly shows that the force has an asymptotic value
of about 65mN.
4.7 Impact of surface area and edge eects
To examine the contribution of the edge eects to the net force, and analyze the
relative importance of the forces on the edges of plate versus the area forces, the
computations were performed in helium, for two geometric congurations, (i) a solid
innitely thin 3.9cm plate (ii) a 3.9cm plate that has ten 1mm holes. Consider rst
the surface pressure distribution over the cold and hot sides of the solid plate, shown
in Fig. 4.6 for a chamber pressure of 2Pa. The pressure has a minimum near the
center of the plate both for the hot and cold sides. The values near the edges are
72
Chamber size, m
Force, N
0 0.5 1 1.5
6.0E- 05
8.0E- 05
1.0E- 04
1.2E- 04
1.4E- 04
1.6E- 04
1.8E- 04
2.0E- 04
Figure 4.5: Net radiometric force as a function of chamber size for Helium gas
Distance, m
Pressure, Pa
0 0.01 0.02 0.03 0.04
2.13
2.14
2.15
2.16
2.17
2.18
Hot Surface
Cold Surface
Figure 4.6: surface pressure distribution over the cold and hot sides of the solid plate.
visibly larger than that near the center. The net force, however, is produced by the
dierence of pressure forces on the hot and cold sides of the plate, and this dierence
is nearly constant for dierent distance stations along the plate. This is an indication
that the area, and not the edge, forces are the ones that control the net force. An
additional conrmation to this statement is given in Table 4.2. The force on a hollow
73
Table 4.2: Radiometric forces on solid and hollow plates
Plate Pressure, Pa Force
hot
, N Force
cold
, N Net Force, N
Solid 0.006 2.884297010
5
-2.812034010
5
7.2263000010
7
Hollow 0.006 2.158603610
5
-2.103664710
5
5.495788310
7
Solid 2.000 1.018273310
2
-1.100774510
2
1.052830010
4
Hollow 2.000 6.859194510
3
-6.770459910
3
8.857232010
5
Solid 6.092 3.003296010
2
-2.993009010
2
1.028700010
4
Hollow 6.092 2.045564710
2
-2.034866610
2
1.061452010
4
plate is generally lower than that on a solid plate, except for the 6Pa case where it is
within the error bar of the computations.
Pressure, Pa
Force/dT, N/K
1 2 3 4 5 6
0
2E- 07
4E- 07
6E- 07
8E- 07
1E- 06
DSMC
Experiment
Figure 4.7: Comparison of the DSMC and experimental predictions of the net force
on the plate in nitrogen
4.8 DSMC and experimental measurements
Companion experiments were conducted by A. Ketsdever and N. Selden. Comparison
of the DSMC and experimental predictions of the net force on the plate in nitrogen
74
is shown in Fig. 4.7 for dierent pressures. The force is normalized by the tem-
perature dierence between the cold and hot surfaces, since the actual values of the
temperature somewhat varied in the experiment. The comparison shows that both
measurements and DSMC predict pressure maximum at about 1.3Pa. The numeri-
cal results noticeably over predict the measurements, with the dierence attributed
primarily to the chamber walls eect and an incomplete energy accommodation that
was not accounted for in DSMC.
4.9 Conclusions
A DSMC study has been conducted to study net radiometric forces on a thin plate
with with a temperature gradient along its thickness. Three gases - helium, nitrogen
and argon - were investigated for dierent pressures. It has been shown that for every
xed pressure under consideration the net radiometric force is maximum for the gas
with maximum mean free path for a given pressure.
The role of viscosity index, molecular mass and diameter was also investigated. It
has been shown that viscosity index and molecular diameter play a much greater in
controlling radiometric forces than molecular mass
The controversy about whether surface area or edge eects control radiometric
eects has been addressed. It has been shown that for a chamber comparable to the
radiometer size in the rareed regime, radiometric forces were proportional to the
plate surface area. Thus, in this regime, surface area eect and not the edge eect,
is the main contributor to the radiometric force. For all gases considered, maximum
radiometric forces were also observed in the rareed regime.
The size of chamber was found to play a considerable role in controlling the forces.
As the chamber size increases, the net forces decrease until they asymptote to some
75
value.
76
Chapter 5
Acceleration of Polarizable
Molecules in Low density gases
5.1 Introduction
An optical lattice is created by two intersecting counter propagating laser elds,
and is characterized by an interaction between polarizable particles and the eld of
the optical interference pattern. With increasing laser beam intensities, the optical
lattice potential depth increases and at relatively low gas densities a large number
of gas particles can be trapped and accelerated. Particle velocities can be increased
from zero up to the 10 - 100 km/s range using nanosecond duration laser pulses
over distances of 100's of microns (Barker and Shneider (2001)). Such hyperthermal
molecular beams are of great importance for material processing such as etching,
deposition, as well as for studying the relaxation processes in gases and gas-surface
interactions.
Generally, synchronous acceleration of charged particles to energies in excess of
100 GeV can be achieved using electrostatic and Lorentz forces, and accelerated
77
neutral atomic beams can be created from ion beams by charge capture, (Livingston
and Blewett (1962)). There are gas dynamic methods that accelerate molecules to
velocities greater than 10 km/s (14.5 eV for N
2
), (Neely and Morgan (1994)), but
it is extremely challenging to accelerate neutral molecules above this energy range
without a large fraction of the gas being thermally ionized and dissociated. Linear
acceleration within the time varying electric eld of an accelerated optical traveling
wave has been proposed as a means to accelerate atoms to high velocities,(Kazantaev
(1974), (1978)) and more recently molecular acceleration has been proposed, (Corkum
et al (1999)). Linear acceleration using optical elds is attractive because extremely
large dipole forces can be produced by high electric eld gradients that can be created
within an optical traveling wave. The electrodeless electric eld gradient produced by
a focused laser beam can be orders of magnitude greater than electrostatic gradients,
allowing acceleration of not only polar, but also polarizable molecules and atoms.
This concept has already been used to accelerate ultracold atoms up to the velocities
in the m/s range using very weak optical periodic potentials, which are called optical
lattices, (Madison et al (1997) and Peik et al (1997))
In contrast to acceleration of ultracold atoms in weak lattices (Madison et al
(1997) and Pek et al (1997)), Barker (2001) and Shneider (2005) have studied the
acceleration of polarizable gas particles, both molecules and atoms, at much higher
temperatures (5-300 K) and to velocities in the 10 to 100 km/s range by application of
large lattice potentials created by pulsed lasers. The latter work follows on the original
work of Kazantsev,(Kazantsev (1974), 1978)) and investigates the motion of trapped
and untrapped particles in the velocity phase space of the accelerated optical dipole
potential. The dynamics of the accelerating ensemble of polarizable particles under
the in
uence of large dipole or stark forces is examined, and the velocity distribution
function of both trapped and untrapped particles is predicted, (Barker and Shneider
78
(2001)). More recently, deceleration and acceleration of molecular and atomic species
produced in a supersonic beam (I 10
12
W/cm
2
) has been demonstrated using deep,
100 K, optical lattices (Fulton et al (2006)).
The ability of the optical lattice forces to eectively manipulate molecular gases
has stimulated the study of transport of gas species in the collisionless and collisional

ow regimes. It has been shown (Shneider and Baker (2004), (2005)) that when an
optical periodic potential created by a light eld in
uences neutral molecules with
a nite polarizability in the collisionless regime (collision time is large compared to
the pulse duration), these particles undergo a process similar to a Landau damping
(Landau (1946)). This process was called optical Landau damping (Shneider and
Baker (2005)); it has been shown to be similar to the collisionless Landau damping
observed in plasmas. In the optical Landau damping, the dissipation of the optical
wave is transformed into particle motion via the dipole force. The momentum and
energy can be transferred from an optical lattice to gas also in the case when the gas
mean free path is smaller than the period of the optical lattice potential, (Shneider
and Baker (2005)), (Shneider, Baker and Gimelshein (2006)).
The energy and momentum transfer from lattice to gas in the collisional regime
was used as a driving force of a new type of microthrusters proposed in Shneider,
Gimelshein and Baker (2006)). The main objective of this work is to study the
interaction of a pulsed optical lattice with gas molecules in the
ow regimes from
free-molecular to near-continuum and, in particular, examine the impact of lattice
and gas properties. Two applications will be considered, energy and momentum
deposition from optical lattice to molecular gases, and the propagation of an acoustic
signal created by an optical lattice.
To study the bulk gas
ow induced by the lattice requires calculation of the
molecular velocity distribution functions created by the lattice potential. Accurate
79
numerical modeling of the interaction process has therefore to be conducted at the
kinetic level. In this work, the direct simulation Monte Carlo (DSMC) method is
utilized as it is currently the most powerful and widely used kinetic approach. The
principal computational tools used in this work is SMILE, an advanced code based
on the DSMC method.
5.2 Theoretical background
It has been shown (Barker and Shneider (2001)) that the equation of motion of a
particle moving in a pulsed accelerated or decelerated optical lattice is given by
d
2
x
dt
2
=
1
2
[qE
1
(t)E
2
(t)=m] sin(qxt
2
); (5.1)
where is the molecular polarizability, q is the wave number, E
1
and E
2
are the
amplitudes of the electric elds of the two laser beams,m is the particle mass, and
is the frequency chirp. In a reference frame that accelerates with the optical lattice,
this equation may be conveniently re-written in non-dimensional units,
d
2

dT
2
=
aq

sin() 2; (5.2)
where =XT
2
is the phase of the particle with respect to the accelerated frame,
and T =
p
t and X = qx are the nondimensional temporal and spatial variables,
respectively. The phase space analysis (Barker and Shneider (2001)) shows that par-
ticles may be trapped when the requirement

2
aq

< 1 is satised; in this case, particle
velocities increase with the lattice velocity. With increasing laser beam intensities,
the optical lattice potential depth increases and at relatively low gas densities a large
number of gas particles can be trapped and collisionless accelerated. Particle veloc-
80
ities can be increased from the room temperature level to 10 - 100 km/s range in a
short laser pulse over distances of 100's microns.
When the gas density is high enough so that the mean collision time is much
smaller than the pulse duration, it is not possible to eectively trap particles and
accelerate them to high velocities with the lattice; it is still possible however to
increase the particle thermal velocity as well as transfer momentum from the lattice
to the gas. In this case, the chirping of the lattice is not needed, and a constant lattice
velocity should be used. Let us now consider the eect of lattice on gas and associated
energy deposition in more detail. In what follows, it is assumed thatl
c
= 1, where
is period of the optical lattice potential and l
c
is the gas mean free path.
In a single interaction of a molecule with the lattice potential the change in mo-
mentum and energy of the molecule are given by p = 2m(v); $ = 2m(v):
Here, andv are the lattice phase velocity and molecule velocity, respectively. In the
collisionless case, the corresponding trajectory length for the momentum or energy
exchange with the moving potential well is about . In high density gas, a particle
travels only a small part of its trajectory, approximately l
c
=, over a collision time
given by
col
= l
c
=jvj. Therefore, the eective rates of momentum and energy
exchange are

p (p=
col
)l
c
= = 2m(v)jvj=
and

$ ($=
col
)l
c
= = 2m(v)jvj=
Integrating these expressions with the Maxwellian distribution function over velocity
space from to + for all particles inside the potential well, and expanding the
distribution function into a Taylor series up to the rst order,f(v)f
0
()+
df
0
dv
(v);
we obtain the total rate of energy and momentum exchange between the optical lattice
81
and gas given by

W =P
d

m
2

2
k
B
T
f
0
()
4
; (5.3)

1

P
d
=
m
2

k
B
T
f
0
()
4
; (5.4)
where

W is the rate of the optical lattice electromagnetic energy exchange in inter-
actions with gas molecules (W =
"
0
E
2
a
2
=
I
c
), P
d
is the power transfered to the gas
due to the optical wave dissipation (Schneider (2005)) (P
d
=
dW
dt
),

is the rate of
the momentum transfer, =
p
2
m
=m and
m
is the potential well depth. For two
counter-propagating laser beams with the combined intensity I =I
1
+I
2
, the maxi-
mum potential well depth is
m
=IZ
0
=n; whereZ
0
=
p

0
="
0
= 376:73
and n is
the index of refraction. From equations (5.3) and (5.4) follows (Shneider et al (2005))
that the phase velocity, corresponding to the maximum rate of momentum transfer
is
m
max
=
p
k
B
T=m, and maximum rate of energy transfer is at
E
max
=
p
2k
B
T=m.
It may also be shown that P
d
/

/NI
2
;where N is the gas density.
Note also that the longitudinal force acting on the gas isF
x
=

. If gas is moving
along the optical lattice axis with the velocity v
x
, the corresponding F
x
and P
d
are
P
d

m
2
(v
x
)
2
k
B
T
f
0
(v
x
)
4
F
x

m
2
(v
x
)
k
B
T
f
0
(v
x
)
4
and the optimum phase velocities for the energy and momentum transfer from the
traveling optical lattice to gas molecules are
E
max
=
p
2k
B
T=m +v
x
and
m
max
=
p
k
B
T=m +v
x
: For a stagnant gas of methane at 300 K this corresponds to 558 m/s
and 394 m/s, respectively.
82
5.3 Numerical approach
The DSMC-based software system SMILE was used in all DSMC computations.
The majorant frequency scheme (Ivanov and Rogasinsky (1988)) was used to cal-
culate intermolecular interactions. The intermolecular potential was assumed to be
a variable hard sphere (Bird (1994)). Energy redistribution between the internal and
translational modes was performed in accordance with the Larsen-Borgnakke model.
Temperature-dependent rotational and vibrational relaxation number were used. The
axisymmetric capability of SMILE has been used as the principal kinetic approach.
The important features of SMILE that are relevant to this work are parallel capability,
dierent collision and macroparameter grids with manual and automatic adaptations,
and spatial weighting for axisymmetric
ows.
The parallel capability was used in all computations presented below. The sep-
aration of collision and macroparameter sampling grid was especially important for
acoustic signal simulations, when good statistic is needed with a relatively coarse sam-
pling grid and very ne collision grid to properly resolve collision relaxation in high
density gases. In one dimensional computations, periodic boundary conditions were
used to simulate the core
ow of the optical lattice. In two-dimensional simulations, a
large computational domain was used, so that the disturbances arising downstream
do not signicantly in
uence the
ow near the subsonic boundaries. For the
ow in a
conned gas cell, the re
ection of molecules on the surface was assumed to be diuse
with complete energy accommodation.
83
5.4 Particle trapping and acceleration in weakly
collisional regime
A deep optical lattice with a well depth in the 100 K range can induce signicant
perturbations to the molecular velocities. To study the eect of collisions on particle
velocities in optical lattice, computations have been performed for a pure methane
carrier gas perturbed by an optical lattice eld. An 800 nm laser pulse intensity with
a Gaussian temporal prole, a peak intensity of 1.310
13
W/cm
2
and a FWHM pulse
duration of 10 ns was used in the simulation. An initial lattice velocity of -6.5 km/s,
and a chirp of 10
19
rad/s
2
was used. The
ow was computed in one dimension,
assuming no potential in radial direction, and periodic boundary conditions at the
in
ow boundaries. Gas was initially stagnant at a temperature of 300 K. Two gas
pressures were considered, 10
5
torr (free-molecular
ow) and 3 torr. For the latter
case, the gas is in a weakly collisional regime. For these pressures, this laser pulse is
well below the breakdown. The initial simulated particle population was the same in
these two cases, which allowed us to examine the trajectories of the same particles
with and without collisions.
The periodicity of the optical lattice is 400 nm, and it is necessary consider the
eect of the lattice potential on particles at all phases and all initial velocities, since
particles can not be injected in the lattice at the necessary phase, unlike macro-
scopic acceleration in electrostatic traveling potentials. To understand motion of
both trapped and untrapped molecules within the lattice, it is useful to examine the
trajectories of particles in the velocity phase space [;] derived from Eq. 5.2. The
corresponding system of equations is given by
d
dT
=
aq

sin 2;
84
d
dT
=
The critical points of this equation may be found equating these expressions to zero,
which leads to sin =2=aq, and = 0: The linear stability analysis shows that
the points [;] = [2n arcsin(2=aq); 0] are stable, and [;] = [(2n 1) +
arcsin(2=aq); 0] are not (saddle points). The velocity phase diagrams obtained in
DSMC computations presented in Fig. 5.1 for two selected molecules and two pres-
sures under consideration. The trajectory of a trapped molecules is given in Fig. 5.1a
(red line). It is clearly seen that the molecule becomes trapped, and while being
trapped its phase velocity oscillates around a stable point. After the laser intensity
decreases, the potential well becomes shallow, and the molecule exits the well and
becomes untrapped. The trajectory of the same molecule in the collisional case (green
line) shows that it collided at some early time moment before trapping, and as the
result its phase velocity changed and the molecules was not trapped. Particle colli-
sions, changing molecular velocities, may also lead to the trapping of particles that
would not have been trapped without collisions. This is illustrated in Fig. 5.1b. In
the free molecular regime, the considered molecule is not trapped. For a pressure
of 3 torr, this molecule experienced a collision that changed its velocity so that is
became trapped by the lattice.
Consider now the evolution of particle velocities for these two cases. The evo-
lution of particle velocities shown below corresponds to the rst 20 ns of the
ow
development. Note that the phase at the initial time moment is calculated from the
nearest potential well, therefore falling in the interval from 0 to 2. Typical molecular
velocities of methane are in the range of a few hundred meters per second, and the
lattice starts with a velocity of6; 500 m/s that quickly rises with time and surpasses
particle velocity att
s
10 ns. As a result, the phase of untrapped particles relative to
85
q, rad (adjusted scale)
dq/dt (relative units)
Free Molecular
3 Torr
q, rad (adjusted scale)
dq/dt (relative units)
10
- 1
10
0
10
1
10
2
3 Torr
Free Molecular
(a) (b)
Figure 5.1: Phase diagrams for two particles in free molecular and collisional regimes.
the lattice increases until t
s
, reaching tens and hundreds of lattice periods, and then
decreases after t
s
, when the lattice velocity surpasses that of a particle. During this
time, the velocity of untrapped particle oscillates due to the oscillations of the lattice
potential, and the amplitude of the oscillations is proportional to the eld intensity.
It peaks at 10 ns, when the intensity is maximum.
The molecular velocities in the direction of the lattice acceleration as a function
of the phase coordinate (relative to the accelerating lattice) are presented in Fig. 5.2
for two molecules that are not trapped by the lattice, and two gas pressures. The
molecular velocity of the rst molecule, Fig. 5.2a, red line, oscillates due to the impact
of the lattice, and increases from the initial value of -10 m/s to almost 600 m/s. In
the collisional case, Fig. 5.2a green line, this molecule experiences a collision at the
upper branch that results in the decrease of its velocity to about 100 m/s. Another
untrapped molecule trajectory is presented in Fig. 5.2b. In this case, the velocity
in the free molecular regime decreases from the initial value of about 10 m/s to
about -400 m/s due to the lattice impact. It is therefore clear that the optical lattice
signicantly changes velocities even those particles that were not trapped.
In the collisional case, the molecule experiences two collisions, one is during the
86
q, rad (adjusted scale)
U, m/s
0
200
400
600
800
3 Torr
Free Molecular
- 100 0 200 300
q, rad (adjusted scale)
U, m/s
- 500
0
500
1000
1500
2000
2500
3000
Free Molecular
3 Torr
0 200 290
(a) (b)
Figure 5.2: Velocities of two untrapped molecules in the free molecular and transi-
tional regimes.
transition of the molecule to the lower branch, and second is at second branch. Note
that the second collision increases the molecule velocity to over 3000 m/s, which is
much higher than the average thermal speed of 558 m/s. Such a high value means
that the molecule collided with a trapped particle that was accelerated by the lattice
to a very high velocity, such as one of those shown in Fig. 5.3, where the evolution
of velocities of trapped particles is presented. The rst molecule, Fig. 5.3a, was
not disturbed by a collision during its trapping and acceleration. This resulted in
an increase of the particle velocity to almost 12 km/s over the pulse period. The
collision occurred for the 3 torr case close to the end of the pulse, which resulted in
the instantaneous velocity decrease to about 1 km/s. The trajectory of a particle that
experiences a collision during its trapping and acceleration is illustrated in Fig. 5.3b;
the velocity drops from 9 km/s to about 5.5 km/s due to the collision. Note that since
particles of the same mass collide, all collisions are relatively strong, and particles that
collide while trapped usually become untrapped.
The situation is dierent if a trapped heavy molecule collides with surrounding
light molecules, since in this case the velocity does not change signicantly, and it
may still be kept by the lattice. This is demonstrated in the computations performed
87
q, rad (adjusted scale)
U, m/s
0
2000
4000
6000
8000
10000
12000
Free Molecular
3 Torr
0 150 240
q, rad (adjusted scale)
U, m/s
2000
4000
6000
8000
10000
Free Molecular
3 Torr
0 200 290
(a) (b)
Figure 5.3: Velocities of two trapped molecules in the free molecular and transitional
regimes.
for a gas mixture of 99% He and 1% Xe. The gas and lattice conditions were the
same as in the previous case of methane. The molecular velocity in the longitudinal
direction (direction of the lattice motion) is shown in Fig. 5.4 as a function of the
phase for three dierent Xe atoms.
A typical velocity of an untrapped atom is shown in Fig. 5.4a (red line), and
is similar to those given above for methane. The phase relative to the lattice rst
increases to about 255 rad, and then moves in the opposite direction, decreasing to
a negative value of -2100 rad due to lattice acceleration. In the collisional regime
(green line), the atom experiences a collision with a helium atom at 100, and a
small velocity change due to this collision is sucient for this Xe atom to be trapped
by the lattice. The trapping results in strong increase in particle velocity along with
the lattice, that begins when the lattice velocity becomes close to that of the atom.
This Xe atom has two more collisions when its velocity reached 11 km/s at 255.
These collisions result in the vertical decrease of particle velocity by a few hundred
meters per second, and subsequent escape of the atom from the potential well of the
lattice. The escape is also stimulated by the decrease in the lattice intensity. The
atom has also experienced two more collisions afterwards, as the corresponding drops
88
q, rad (adjusted scale)
U, m/s
0
2000
4000
6000
8000
10000
Free molecular
3 torr
- 2000 - 1000 - 500 0 100 200 250
q, rad (adjusted scale)
U, m/s
0
2000
4000
6000
8000
10000
12000
Free molecular
3 torr
- 500 0 100 200 250 260 265
(a)
(b)
q, rad (adjusted scale)
U, m/s
0
2000
4000
6000
8000
10000
12000
Free molecular
3 torr
- 100 0 100 200 250 255
(c)
Figure 5.4: Velocities of three dierent atoms in the free molecular and transitional
regimes.
in velocity show.
Particle collisions may easily result in an abrupt untrapping of colliding particles,
as illustrated in Fig. 5.4b. First, the atom under consideration (green line) collided
with a helium atom at 120. This collision did not change the Xe velocity sig-
nicantly due to large mass dierence of colliding particles, and it has been trapped
similarly to its uncollided counterpart (red line). However, when another collision
occurred during rapid particle acceleration (atom velocity was about 6 km/s at that
time), the change in atom velocity was relatively large. As a result, the atom was lost
by the lattice. The nal velocity after 20 ns in the free molecular regime was close to
12 km/s, and in the collisional regime was only about 6 km/s.
89
Collisions may not only remove atoms from the potential well, but also move
molecules closer to the well depth. This eect is illustrated in Fig. 5.4c. In the
free molecular regime, the atom becomes trapped when the lattice velocity gets close
enough to the particle velocity. Particle is then quickly accelerated to 11 km/s.
When the lattice intensity decreases, the potential well becomes more shallow, which
in turn leads to the atom's exit from the well. In the collisional regime, the atom
collides twice when being trapped. The collisions are clearly seen at about 8 km/s and
9 km/s. The collisions shift the atom closer to the potential well, which allows it to
stay longer in the well, and reach 13 km/s (note that two more collisions occur when
the atom velocity reaches 13 km/s). These three examples show that the collisional
regime diers signicantly from the free molecular in terms of particle trapping. Some
particles escape the lattice due to collisions, some stay longer in the potential well,
and some become trapped due to velocity change in collisions.
To study the eect of a pulsed lattice on the velocity distribution functions at
dierent pressures, one-dimensional computations have been performed . The test
gas of methane was used which was initially stagnant at 300 K. The laser parameters
were the same as in the above computations of the Xe-He mixture. The molecular
distribution function of the gas at the end of the pulse is shown in Fig. 5.5 for two
pressures, 0.01 torr, which is an essentially collisionless regime over the pulse duration,
and 3 torr case where the mean collision time of about 17 ns is comparable to the
pulse duration.
In the absence of molecular collisions, the molecules that are being trapped in the
potential well of the lattice can escape the well only after the laser intensity decreases
enough. All of them acquire the lattice speed and are signicantly accelerated. As
a result, a bimodal distribution function is observed by the end of the pulse formed
by untrapped molecules peaked at zero velocity and trapped molecules peaked at
90
U, m/s
F
0 5000 10000 15000
10
- 7
10
- 6
10
- 5
10
- 4
10
- 3
Free molecular
3 torr
Figure 5.5: The velocity distribution function for methane gas at two pressures for
an accelerating lattice.
14 km/s. The distribution function between the peaks is 0. The distribution func-
tion of both trapped and untrapped molecules oscillates corresponding to dierent
positions of molecules relative to the lattice. The maximum value of the distribution
function of trapped molecules and accelerated molecules is an order of magnitude
lower than that of untrapped molecules.
When molecular collision time becomes close to the pulse duration, the collisional
relaxation causes a three-fold degradation of the trapped molecules peak. It is however
remarkable that even at a pressure of 3 torr there is a signicant number of molecules
(about 3%) with velocities over 10 km/s. The collisions of trapped and untrapped
molecules produce a distribution of molecules with velocities between 0 and 14 km/s.
Note also that due to these high-energy collisions there are more molecules with large
negative velocities at a pressure of 3 torr. At even higher gas pressures, corresponding
to 1 atm and higher, molecular collisions result in a rapid gas thermalization during
the pulse, and ecient dissipation of laser energy.
91
5.5 Conclusions
The eect of high intensity optical potentials on atomic and molecular gases is stud-
ied with the DSMC method. The trajectories of dierent atoms and molecules in
the phase velocity space and molecular velocity space are examined both in the free
molecular and collisional regimes. Trapping of particles by optical lattices in a col-
lisional regime obtained in a numerical simulation is shown for the rst time. It is
also shown that in single species gases the intermolecular collisions mostly deplete
the population of trapped particles; they also may lead to trapping of particles that
would not have been trapped in a free molecular regime. In a gas mixture, collisions
of heavy particles with lighter ones may result in heavy particles leaving the lattice
potential well as well as staying in it longer than without collisions.
92
Chapter 6
Interaction of Molecular Gases
with Pulsed Optical Lattices in
Collisional Regime
6.1 Introduction
In the previous chapter it was shown that that when an optical periodic potential
created by a light eld in
uences neutral molecules with a nite polarizability in
the collisionless or weakly collisional regime (collision time is large compared to the
pulse duration), these particles neutral polarizable molecules can be trapped and
accelerated by the optical lattice to very high velocities. In relatively higher density
gases, where gas mean free path is smaller than the period of the lattice, it not
possible to trap molecules. However, momentum and energy can be transferred from
an optical lattice to the gas.
The main objective of this work is to study the interaction of a pulsed optical
lattice with gas molecules in the continuum
ow regime, and examine (1) the energy
93
deposition from lattice to gas, 2) propagation of acoustic signal created by the lattice,
and (3) gas mixing caused by the lattice.
A two-step kinetic/continuum approach is applied. The kinetic step uses the
DSMC method to model the gas-lattice interaction over the pulse period. The con-
tinuum step based on the solution of Navier-Stokes equations utilizes the kinetic
solution as initial condition, and predicts the gas evolution at spatial and temporal
scales larger than those associated with the optical lattice. The direct simulation
Monte Carlo method tool SMILE will be used to simulate the interaction of an op-
tical lattice with the target gas, and a Navier-Stokes solver CFD++ will be used to
model the following evolution of the pressure and velocity perturbation in an open
volume.
6.2 Numerical approach
Accurate numerical prediction of an interaction between the gas molecules and an
optical lattice generally requires a kinetic approach to be used even when gas is in
continuum regime relative to the lattice width. There are two reasons for this. First,
the lattice period is usually comparable to gas mean free path. For example, for
800 nm laser beams propagating in gas at standard atmospheric conditions, the gas
mean free path of60nm is on the order of the lattice period of 400 nm. Second, the
gas collision time is often same order of magnitude or longer than the lattice pulse
period. For example, in a 1 atm gas the mean collision time of200 ps is signicantly
longer than a 50 ps laser pulse. In this case, the change in the distribution function
is important, and needs to be properly accounted for.
A suitable approach for modeling gas/lattice interaction was found to be the direct
simulation Monte Carlo (DSMC) method. However, the application of this approach
94
to compute continuum gas
ows over times much longer than the lattice pulse and
spatial scales much larger than the lattice size is not practical due to computational
limitations of the DSMC method. One alternative would be to use a deterministic
kinetic solver, such as one based on the solution of model kinetic equations. The
other alternative is to use a kinetic tool to model the gas/lattice interaction, and
then a continuum solver to predict further development of the perturbation in gas
macroparameters caused by the lattice. In the latter case, the kinetic solution is used
as initial condition for the subsequent continuum simulation.
In the present work, the above two-step kinetic-continuum approach is applied.
The DSMC-based software system SMILE was used in all DSMC computations. The
majorant frequency scheme was used to calculate intermolecular interactions. The
intermolecular potential was assumed to be a variable hard sphere. Energy redistribu-
tion between the internal and translational modes was performed in accordance with
the Larsen-Borgnakke model. Temperature-dependent rotational and vibrational re-
laxation number were used. The axisymmetric capability of SMILE has been used.
The important features of SMILE that are relevant to this work are parallelization,
dierent collision and macroparameter grids with manual and automatic adaptations,
and spatial radial weighting for axisymmetric
ows.
The parallel capability was used in all computations presented below. The sep-
aration of collision and macroparameter sampling grid was especially important for
acoustic signal and gas mixing simulations, when good statistic is needed with a rel-
atively coarse sampling grid and very ne collision grid to properly resolve collision
relaxation in high density gases. In one dimensional computations, periodic boundary
conditions were used to simulate the core
ow of the optical lattice. In two-dimensional
simulations, a large computational domain was used, so that the disturbances arising
due to the lattice impact do not have time to reach the subsonic boundaries. For the
95

ow in a conned gas cell, the re
ection of molecules on the surface was assumed to
be diuse with complete energy accommodation.
The macroparameters obtained with the DSMC have been used as initial condi-
tions for the continuum solver CFD++. CFD++ is a
exible computational
uid
dynamics software suite for the solution of steady and unsteady, compressible and
incompressible Navier-Stokes equations, including multi-species capability for perfect
and reacting gases. In this work, a perfect gas compressible Navier-Stokes solver
was used with second order spatial discretization and explicit time integration. The
results presented below were performed for a rectangular grid with a total of 4,000
(gas mixing) to 10,000 (acoustic signal propagation) nodes. For acoustic signal sim-
ulations, the boundary conditions were stagnation pressure and temperature at the
outer boundary and symmetry planes at the perpendicular boundaries. For gas mix-
ing, back pressure imposition was assumed at the three free stream boundaries. The

ow is modeled as axisymmetric, and a symmetry condition was used at the lattice
axis.
6.3 Energy and momentum deposition in collisional
regime
The eciency of energy and momentum deposition is analyzed theoretically in the
previous chapter. In order to verify these theoretical predictions and associated as-
sumptions, and provide more detail on energy and momentum deposition, 1D DSMC
computations have been performed for the following laser parameters. The laser pulse
duration was 1 ns, the maximum single laser beam intensity was 0:25 10
17
W/m
2
and 0:5 10
17
W/m
2
, the optical lattice wave length was 400 nm, and a Gaussian
temporal prole of intensity was used with a maximum at 1 ns. The lattice velocity
96
was varied from 0 to 1,500 m/s. The gas was methane, stagnant at T
0
= 300 K, and
four pressures from 3 torr to 10 atm were considered.
The energy and momentum deposition per molecule is presented in Fig. 6.1 as
a function of the lattice velocity for I
max
= 0:5 10
17
W/m
2
. The momentum is
normalized by the molecular mass. Several conclusions can be drawn from these
gures. First, the values of
max
are close to those predicted analytically. Second,
the energy, as well as momentum, deposition per molecule slightly increases with
pressure, while the theory predicts a linear dependence on pressure (P
d
/ NI
2
).
This small dierence is related to the fact that the theory does not include all the
details of the interactions between the molecules inside and outside the potential well,
when some molecules may leave the well while other may become trapped in the well
due to molecular collisions. The third conclusion is that the energy and momentum
deposition are generally less ecient for 10 atm than for lower considered pressures
at high lattice velocities. This is related to the fact that the quick maxwellization at
10 atm, when the collision time is over an order of magnitude shorter than the pulse
duration, acts to prevent the deposition as compared to the velocity distribution that
has a plateau formed due to the optical Landau damping at lower pressures. This
is conrmed by the fact that deposition at high lattice velocities becomes relatively
more ecient for shorter pulses.
The above computations show that it is possible to increase temperature of stag-
nant gas by a few hundred K if a nanosecond pulse is applied. The temperature
increase may be much more signicant if shorter (picosecond) and more intensive
pulses are used to create an optical lattice. This is clearly shown in Fig. 6.2a where
an increase in gas translational temperature due to an optical lattice formed by two
50 ps laser pulses at I = 10
18
W/m
2
and zero phase velocity is given. Two gases are
considered here, methane and nitrogen; initially they are at 1 atm and 300 K and
97
Lattice velocity, m/s
Normalized energy per molecule, K
0 500 1000 1500
0
100
200
300
400
500
600
10 atm
1 atm
30 torr
3 torr
Lattice velocity, m/s
Normalized momentum per molecule, m/s
0 500 1000 1500
0
50
100
150
200
250
300
10 atm
1 atm
30 torr
3 torr
Figure 6.1: Energy and momentum deposition in an initially stagnant gas for dierent
pressures and a long pulse with I
max
= 0:5 10
17
W/m
2
.
(a) (b)
have no
ow velocity. According to the estimates of high-intensity limits of break-
down for neutral molecules, the above laser intensity is at the breakdown threshold
for methane, and still lower than the breakdown for nitrogen. The after-pulse tem-
perature increases by 2000 K for methane and 1000 K for nitrogen; the higher tem-
perature for methane is related to an almost three times larger polarizability-to-mass
ratio. The ability of an optical lattice to increase gas temperature to thousands of K
is comparable to that employed in the infrared laser powered pyrolysis that is being
widely used to study initiation and development of chemical reactions and various re-
laxation processes. Note also that gas temperature may be increased to signicantly
higher values if polarized molecules, such as water, are used. The increase in the laser
beam intensity results in signicantly stronger momentum deposition to the gas (see
Fig. 6.2b). In this case, the gas velocity reaches 1000 m/s for methane at the phase
velocity of V
0
= 2000 m/s. The value of V
0
= 2000 m/s is much higher than that
predicted in Sec. II for small laser intensities; this is related to the fact that velocity
distribution function signicantly widens during the pulse, therefore increasing the
associated thermal velocity.
98
V
0
, m/s
Methane Temperature, K
Nitrogen Temperature, K
- 10000 - 5000 0 5000 10000
0
500
1000
1500
2000
2500
200
400
600
800
1000
Methane
Nitrogen
V
0
, m/s
Methane Momentum, m/s
Nitrogen Momentum, m/s
- 10000 - 5000 0 5000 10000
- 1000
- 500
0
500
1000
- 150
- 100
- 50
0
50
100
150
Methane
Nitrogen
(a) (b)
Figure 6.2: Translational temperature and momentum increase in an initially stagnant
gas after a short pulse with I
max
= 10
18
W/m
2
.
Consider now the impact of gas pressure on energy deposition from lattice to gas.
The change in translational temperature is given in Fig. 6.3a as a function of ini-
tial gas pressure. As gas pressure increases, the translational temperature recorded
immediately after the pulse decreases. This decrease is due to the energy transfer
from translational modes of molecules quickly excited by the lattice to the rotational
modes. The drop in translational temperature for higher pressures is therefore com-
pensated by an increase in the rotational temperature; the overall temperature does
not depend on pressure (the momentum change is not aected by gas pressure either).
The temporal relaxation of translational and rotational temperatures is presented in
Fig. 6.3b for nitrogen gas at 1 atm impacted by a short-pulsed optical lattice (50 ps
pulse duration andI
max
= 10
18
W/m
2
at a lattice wavelength of 400 nm). The Gaus-
sian pulse starts at t = 0, and has maximum intensity at 50 ps. The translational
temperature increases due to lattice-gas interaction, with maximum gradient observed
for maximum pulse intensity. By the end of the pulse, t 90 ps, the translational
energy loss due to the transfer to rotational modes exceeds the gain due to the lattice
energy deposition.
99
Pressure, atm
Temperature, K
0.5 1 1.5 2 2.5 3
0
200
400
600
800
1000
1200
Translational T
Rotational T
Time, s
Temperature, K
0 5E- 11 1E- 10 1.5E- 10
0
200
400
600
800
1000
1200
1400
1600
Translational T
Rotational T
(a) (b)
Figure 6.3: (a) Translational and rotational temperature increases in nitrogen, (b)
Transient evolution of nitrogen temperatures at 1 atm. A 50 ps pulse with I
max
=
10
18
W/m
2
.
It is also important to note that for high pressures the energy deposition as a
function of laser intensity is in good agreement with the theory that predicts thatI
2
dependence. It is also clearly seen in Fig. 6.4 where the energy deposition is shown for
gas pressure of 1 atm and lattice velocity of 500 m/s. The line in this gure represents
E(I
0;max
)
Imax
I
0;max
, where E is the normalized energy and I
0;max
= 0:25 10
17
W/m
2
.
Intensity at the axis, W/m
2
Normalized energy per molecule, K
2E+16 4E+16
100
200
300
DSMC results
Theory
Figure 6.4: Energy deposition for dierent laser intensities.
100
6.4 Acoustic signals induced by optical lattices
The results of the previous section demonstrate the ability of an optical lattice to
eectively deposit momentum and energy into gas molecules. It would be highly de-
sirable to compare these results with experimental measurements, and thus validate
the theoretical approach and numerical models used. Experimental data on transient

ow properties inside an optical lattice are currently unavailable, and obtaining them
is a obviously very dicult task; it is much easier to examine the propagation of acous-
tic perturbations caused by a lattice, at some distance from its center. Measurements
of an acoustic signal at dierent locations from the lattice would provide excellent
ground for studying the lattice-gas interaction and numerical prediction validations.
In what follows, the initial stage of the evolution of a
ow aected by an opti-
cal lattice and the development of acoustic perturbations is evaluated numerically.
Axisymmetric DSMC computations have been performed where macroparameters in
cells were recorded versus time. Over 160 million simulated molecules were used in
order to maintain suciently low level of statistical
uctuations (the error associ-
ated with the statistical scatter is below 0.5%). The after-pulse transient pressure
variation in the center of the lattice and at 1 mm in the radial direction is shown
in Fig. 6.5a (red line). Gas is nitrogen with initial pressure of 0.1 atm, temperature
300 K, and zero
ow velocity. The laser pulse parameters are I
max
= 10
18
W/m
2
,
V
0
= 0, the wave length is 800 nm, the beam diameter is 50 micron, and the Gaussian
pulse duration is 50 ps. The coordinate ofY = 0 corresponds to the lattice centerline.
The pressure at the center of the lattice quickly decreases in the rst 300 ns after the
pulse due to the expansion and diusion of high-velocity molecules created by the
lattice, which results both in temperature and density decrease at that time. The
over-expansion of these molecules results in pressure decrease to values lower that the
initial value of 0.1 atm. This is also clearly seen in Fig. 6.5b where the corresponding
101
density proles are shown. The pressure at the radial station ofY = 0:975 mm oscil-
lates near its equilibrium value during the rst 2 s after the pulse; after that, there
is a clear perturbation that correspond to an acoustic signal caused by the lattice.
Note that the mentioned small
uctuations at Y = 0:975 mm are due to statistical
scatter. For Y = 0, though, the
uctuations at t> 3s are clearly stronger than the
statistical noise, and are attributed to residual after-pulse
ow oscillations near the
lattice centerline. The density oscillations at t > 3s are also noticeable (Fig. 6.5b,
red line).
Time, s
Pressure, Pa
0 5E- 06 1E- 05 1.5E- 05
10000
15000
20000
25000
Y= 0.025mm
Y= 0.975mm
acoustic signal
Time, s
Number Density, molecule/m
3
0 5E- 06 1E- 05 1.5E- 05
1.4E+24
1.6E+24
1.8E+24
2E+24
2.2E+24
2.4E+24
2.6E+24
Y= 0.025mm
Y= 0.975mm
(a) (b)
Figure 6.5: Pressure (a) and number density (b) as a function of time at two radial
locations.
The propagation of an acoustic signal created by the lattice is clearly seen in
Fig. 6.6a where the pressure is shown as a function of time at several radial stations.
The signal reaches Y = 0:525 mm at a time of about 0.8 s, which corresponds to
the average signal speed of about 650 m/s, and eective gas temperature of about
1000 K. The temperature decrease downstream from the lattice center results in the
decrease of the acoustic signal speed; the average speed is about 500 m/s when it
reaches 1 mm (see Fig 6.5a).
The results results presented above were obtained with the equilibrium free stream
102
boundary conditions to approximate the acoustic signal development in a free space.
If a gas cell is studied where the target gas is bound by solid walls, the acoustic signal
will experience multiple re
ections on these walls. The rst of these re
ections is
illustrated in Fig. 6.6b where the pressure versus time is shown atY = 0:025 mm for
two boundary conditions, solid wall and equilibrium free stream (without wall). The
acoustic signal re
ected on the wall returns back to the centerline at about 5s, which
corresponds to an average signal velocity of 400 m/s, and an eective temperature of
375 K.
Time, s
Pressure, Pa
0 1E- 06 2E- 06 3E- 06 4E- 06 5E- 06
10000
15000
20000
Y= 0.025mm
Y= 0.125mm
Y= 0.225mm
Y= 0.325mm
Y= 0.425mm
Y= 0.525mm
Time, s
Pressure, Pa
0 2E- 06 4E- 06 6E- 06 8E- 06 1E- 05
10000
15000
20000
25000
with wall
without wall
(a) (b)
Figure 6.6: (a) Pressure at dierent radial locations. (b) Pressure in the center of the
lattice for bounded and unbounded gas cell.
6.5 Two-step kinetic/continuum approach
Let us now compare the acoustic signal propagation results obtained with the DSMC-
only and the two-step kinetic/continuum approaches. Note rst that when the length
of the lattice is much larger than its width, the initial development of the acoustic
signal may be well approximated by a one-dimensional
ow in radial direction. The
calculations have therefore been conducted for a pseudo-1D
ow (axisymmetric
ow
103
with contained between two vertical symmetry planes, with no gradients in the axial
direction). Two 50 ps laser pulses at I = 5 10
17
W/m
2
and zero phase velocity
were simulated in stagnant nitrogen kept initially at 0.1 atm and 300 K. These laser
intensities are noticeably lower than the breakdown threshold (Dewhurst (1978)).
The pressure evolution obtained with the two approaches is presented in Fig. 6.7a
for two time moments, 0.5 and 1 s. A limited number of sampling cells was used
in DSMC in order to reduce the statistical scatter to the level of about 0.1-0.2%.
Accounting for the spatial averaging eect of these cells, it can be seen that there
is a good agreement between the two approaches, which essentially validates further
application of the kinetic/continuum approach.
Y, m
Pressure, Pa
0 0.0002 0.0004
6000
8000
10000
12000
14000
NS, 0.5 ms
NS, 1.0 ms
DSMC, 0.5 ms
DSMC, 1.0 ms
(a) (b)
Figure 6.7: Comparison of DSMC and NS pressure proles for two time moments at
P=0.1 atm (a), and pressure proles for four time moments 0, 0.05, 0.5, and 1s at
P=1 atm (b).
The acoustic signal development obtained with the two-step approach for the same
laser parameters in nitrogen at 1 atm is given in Fig. 6.7b. The gure also shows
the computational domain used; the problem is essentially one-dimensional with no
gradients in the axial direction due to the presence of specularly re
ecting walls at
X=0 and X=1 mm. The acoustic signal reaches Y=0.2 mm in the rst 0.5s, which
corresponds to the propagation speed of about 400 m/s. The decrease in temperature
104
due to viscous dissipation decreases the propagation speed to about 300 m/s over the
next 0.5 s. Note also that the pressure perturbation caused by the optical lattice,
recorded at a distance of 0.5 mm from the lattice axis, is rather strong, over 5% of the
free stream value. This means that the acoustic signal should be easily measurable
even 10 mm away from the lattice.
(a) (b)
Figure 6.8: The impact of pressure (a) and beam intensity (b) on acoustic signal
propagation. (a) From left to right, pressures of 0.1, 0.333, and 1 atm. (b) From left
to right, intensities of 1:7 10
17
, 2:5 10
17
, and 5 10
17
W/m
2
.
Consider now the impact of carrier gas pressure on the acoustic signal propagation.
The pressure elds obtained at 0.5 s after the pulse and normalized by the free
stream values are shown in Fig. 6.8a for three initial gas pressures. It is seen that
while the decrease in pressure results in a wider perturbation front, it does not have
a visible eect on the location of the pressure maximum and minimum. The absolute
magnitudes of the maximum and minimum pressure also does not depend on gas
pressure. While the variation in gas pressure has relatively small eect on the acoustic
signal propagation, change in the laser pulse intensity result in a noticeable change of
the magnitude of the signal. The pressure elds for initially stagnant 1 atm nitrogen
are shown in Fig. 6.8b for a time moment of 1 s and three laser intensities. The
maximum pressure strongly increase with the increase in intensity (nearly asI2). The
signal speed is also visibly larger for stronger beams.
105
6.6 Gas mixing
Ecient mixing of
uids in the low Reynolds number
ows typical of micro total
analysis systems, chemical microreactors, or microscale combustion devices, requires
novel approaches. Among the approaches emerged recently, we mention here magne-
tohydrodynamic mixing (Gleeson and West (2002)) and gas mixing induced by bubble
bursting. Generally, it is critical to have robust mixing technologies at micro or mil-
liscale that would allow orders of magnitude improvement over usual processes of
diusion and convection. The non-resonant interaction of gas molecules with optical
lattices may potentially provide such a capability.
To examine the mixing ability of optical lattices, an Ar-N
2
gas mixture is consid-
ered. The gas is initially stagnant at 1 atm and 300 K. In order to better illustrate
the gas mixing process, it is assumed that at the initial time moment all molecular
nitrogen is conned inside a cylindrical volume of 100m in diameter and 200m
long. This corresponds to the dimensions of the optical lattice, that is formed by two
counter-propagating laser beams with an intensity of 510
17
W/m
2
, phase velocity
of 1000 m/s, and a duration of 50 ps. The nitrogen mole fraction inside the above
volume is unity. Outside the volume, gas is argon. The two-step kinetic/continuum
approach is used, and the
ow is modeled as axisymmetric.
The evolution of pressure elds is shown in Fig. 6.9a for dierent time moments
after the pulse. The lattice is centered at X=0 and aligned along X axis. As men-
tioned above, the result of energy and momentum transfer to the target gas is the
increase in gas temperature and bulk velocity. The interaction of high-temperature
gas moving along X axis, with stagnant gas that was not in
uenced by the lattice,
results in a pressure increase in front of the moving gas, and decrease in the region
behind it, which is clearly seen at 0.25s. At larger times, gas collisions smear the the
dierence between the upstream and downstream regions, and the pressure perturba-
106
(a) (b)
Figure 6.9: Gas mixing: pressure elds at dierent time moments for a single lattice
(a) and two counter lattices (b). From top to bottom: 0.25s, 0.5s, 1.0s, 1.5s.
(a) (b)
Figure 6.10: Gas mixing:
ow velocity elds at dierent time moments for a single
lattice (a) and two counter lattices (b). From top to bottom: 0.5s, 2s, 5s, 10s.
tion propagates circularly with the speed of sound (about 350 m/s). In the vicinity
of X=0, where the lattice was centered, the pressure is close the non-perturbed value
for t> 1:5 s.
In addition to the calculations with a single lattice, computations have also been
performed for two lattices. The counter-propagating lattices are aligned along X axis
and centered at X=0 and X=0.3 mm. The lattice intensities and dimensions are the
same as before, and the phase velocities are 1000 m/s and -1000 m/s, respectively.
Nitrogen is contained in two cylindrical volumes located and sized accordingly to the
107
two lattices. The evolution of gas pressure for this case is presented in Fig. 6.9b. The
interaction region of the counter propagating gas regions is observed near X=0.15 mm.
Due to larger pressure rise in the interaction region relative to the single lattice
case, the pressure perturbation moves outward at a larger velocity of about 500 m/s.
Pressure in the interaction region returns to the free stream value at t 1:5 s,
similar to the single lattice case.
The velocity damping is a much slower process than the pressure equilibration.
This is illustrated in Fig. 6.10a where the axial velocity elds are shown at dier-
ent time moments for the single lattice case. Starting from about 80 m/s in the
center of the lattice immediately after the lattice pulse, the maximum
ow velocity
gradually decreases to about 40 m/s at 10 s. The location of the maximum shifts
downstream by about 0.3 mm over that time. As expected, the velocity damping is
somewhat faster when two counter propagating lattices are considered, as illustrated
in Fig. 6.10b. In this case, the magnitude of velocity maximum is only about 20 m/s
at 10 s. Note also that the interaction of two counter
ows results in the forma-
tion of two vortex-like structures, with
ow velocities at Y=0.1 m having opposite
direction to those at Y=0.
Finally, let us consider the evolution of nitrogen mole fractions, that serve as
an indicator of
ow mixing. In order to provide the reference point, in addition to
the single and dual lattice cases shown above, a no-lattice computation has been con-
ducted to show the mixing due to the diusion-only process. Note that the convection
at these small scales is orders of magnitude slower than the diusion. The results
for these three cases are presented in Fig. 6.11. The main conclusion here is that
the optical lattice allows for signicant increase in mixing eciency compared to the
diusion process. An about 50% mixing is achieved at 10 s, which is better than
that for the diusion-only case at 30s. Also note that the application of the optical
108
(a) (b)
(c)
Figure 6.11: Gas mixing: nitrogen mole fraction at 10s (a), 30s (b), and 100s (c).
Top, no lattice; middle, single lattice; bottom, two counter lattices.
lattice results in larger mixing region, mostly in horizontal direction for the single
lattice and vertical direction for the dual lattice case. Even more ecient mixing is
expected when several non-aligned lattices are created.
6.7 Conclusions
The in
uence of high intensity optical gradient forces on atomic and molecular gases
in continuum
ow regime is studied with a two-step kinetic/continuum approach. The
formation and development of an acoustic signal created by the lattice is analyzed.
For lower pressures, a single kinetic approach (the DSMC method) was also used.
The solutions obtained with the DSMC method agree with those computed with
the two-step approach. The latter one was shown to be able to capture both the
propagation of an acoustic signal in high-density gases and after-pulse
ow oscillations
observed near the center of the lattice. An experimental investigation of the acoustic
109
signal created by an optical lattice is highly desirable, and may provide additional
insight into gas-lattice interaction and numerical analysis validation.
The transfer of energy and momentum from an optical lattice to a stagnant gas
is examined numerically. It is shown that an optical lattice created by two 800 nm,
50 ps laser beams with an intensity at the axis of 10
18
W/m
2
increases translational
temperature to 2300 K in methane and 1300 K in molecular nitrogen. The use of
shorter and stronger pulses, shorter laser wavelengths, or polarized molecules such as
H
2
O, are all expected to produce even higher temperatures. Due to a small loss of
laser energy to gas after a single pass, the use of a multi-pass optical system, with two
laser beams being recycled many times, is also expected to produce signicantly higher
temperatures. The interaction of an optical lattice moving with a nite velocity, with
an initially stagnant gas, leads to the momentum deposition, and an increase of gas
velocity to hundreds and thousands of meters per second. This gas motion was shown
to eectively mix gases in an open space. For a single 50 ps pulse, a 50% mixing of
initially unmixed gas species was achieved several times faster than the conventional
diusion process. A stronger mixing eect will be achieved when several non-aligned
optical lattices are used, and a factor of ten or better is expected for the optical lattice
gas mixing compared to the diusion or convection mixing.
110
Chapter 7
Optical Lattice Operated
Micropropulsion Devices
The direct simulation Monte Carlo method is used to study the feasibility of new
propulsion concepts based on the interaction of an optical lattice with gas molecules.
Two regimes are considered, high density and low density. In the rst one, a de Laval
nozzle is examined with the carrier gas driven by energy and momentum deposition
from the lattice to the region near the nozzle throat. Analytical expressions are
developed and compared with the numerical predictions, that describe the energy and
momentum energy transfer between the lattice and the gas. In the second regime,
a multiple orice
ow is considered with molecules accelerated to high velocities by
a chirped lattice potential. Specic impulse of about 500 is obtained with the total
thrust of over 10 N per single 100 m orice.
111
7.1 Introduction
The main goal of this chapter is to numerically study new propulsion concepts based
on the interaction of an optical lattice with gas molecules in two
ow regimes, nearly
free-molecule and continuum. Two new microthruster schemes are considered for
these regimes that use (1) acceleration of molecules to very high velocities with a
linearly chirped lattice, and (2) energy and momentum deposition from an optical
lattice traveling at a constant speed into the throat of a microscale nozzle. The
estimates of thruster performance characteristics are presented and possible ways to
improve them are outlined. The direct simulation Monte Carlo method (DSMC) is
used in this study.
7.2 Laser beam propulsion
Propulsion concepts based on laser energy deposition, primarily to the solid matter,
were proposed as early as forty ve years ago. One of the rst studies in the eld
is presented in Askar'yan and Moroz (1962) where the thrust produced by molecules
vaporized from a target surface due to laser radiation was examined. The process
of surface ablation and thrust generation by pulsed laser beams has been extensively
studied experimentally and theoretically starting in early 70s (Prokholov et al (1973),
Pirri et al (1974)) for dierent chemical composition of the surface. The use of laser
beam propulsion with a terrestrial source of laser energy was proposed in Pirri et al
(1974). More recent research (Smith et al (2001)) includes the use of high-energy
lasers to transmit solar power through free space in the form of coherent light.
The laser ablation of the surface is not the only known laser-based mechanism
of spacecraft thrust generation. The laser beam energy may also be used for liquid
propulsion thrusters, where the laser energy plays role of the oxidizer, and for air-
112
driven thrusters (Bunin and Prokhorov). In the latter concept, the thruster is powered
by atmospheric air eectively heated by a laser beam. At present, although lasers
did not take on a thrust-producing role of ion engines, the idea of beaming energy
from both space- and terrestrial-based lasers to power spacecraft continues to draw
interest. A radiatively driven hypersonic wind tunnel has being proposed in Princeton
ten years ago (Miles et al (1995)) to provide high enthalpy, high Mach number true
air
ows for durations of seconds with high power laser or electron beams as suitable
energy sources. The concept has recently been demonstrated at 1 MW of electron
beam energy power (Girgis et al (2002)).
The principal dierence of the present work from previous studies is that the
non-resonant impulse and energy deposition is used here. This means that only
translational and not internal energies of molecules are directly changed by laser
radiation, and it is therefore possible to avoid or minimize a number of undesirable
eects such as molecular dissociation and ionization, and associated charging and
degradation of working surfaces.
7.3 Optical lattices in free molecular and weakly
collisional regimes
With increasing laser beam intensities, the optical lattice potential depth increases
and at relatively low gas densities a large number of gas particles can be trapped and
collisionless accelerated. Particle velocities can be increased from the room level to
10 - 100 km/s range in a short laser pulse over distances of 100's microns (Barker
and Shneider (2001)). The evolution of particles that are distributed over a range of
velocities, and over all phases, is described by the Boltzmann equation. For short ac-
celeration periods, acceleration without collisions between particles becomes possible
113
at pressures in the 100 torr range, and the collision integral in this equation can be
set to zero. The results of calculations of particle acceleration using the solution of
the collisionless Boltzmann equation are given in Barker and Shneider (2001).
To model the evolution of particles at higher pressures, when the collision time
in gas becomes on the order of the pulse duration, and molecular collisions are im-
portant, the full integro-dierential Boltzmann equation has to be solved. One of
the most convenient and widely used approaches to the solution of the Boltzmann
equation is the DSMC method. This method has been used in all presented computa-
tions. The principal computational tool used in this work is SMILE, an advanced code
based on the DSMC method. The variable hard sphere model is used for intermolec-
ular interactions. The discrete Larsen-Borgnakke model with temperature-dependent
rotational relaxation number is utilized for rotation-translation energy transfer.
Modeling of particle acceleration due to the impact of an optical lattice in a weakly
collisional regime is performed in one dimension, with periodic boundary conditions
applied at the in
ow/out
ow boundaries. Gas is methane, initially stagnant at tem-
perature of 300 K. Two gas pressure are considered, 3 torr and 10 torr. The maximum
laser beam intensity is assumed to be 6 10
15
W/m
2
, the optical lattice wave length
is 400 nm, and a Gaussian temporal prole of intensity is used with a maximum
at 10 ns. The computation is performed with about 3 million molecules to provide
an acceptable accuracy of the distribution functions, and the computational domain
spans over ten wave lengths. The unsteady
ow development is modeled with a time
step of 10
12
s. The in
uence of molecular collisions on acceleration is illustrated
in Fig. 7.1, where the distribution function of molecular velocities in the laser beam
direction is shown.
Over the rst few nanoseconds, the distribution function is close to Maxwellian
since the beam intensity is relatively weak at this stage. Then, by forth nanosecond,
114
a bimodal distribution is starting to appear. The separation of trapped molecules
from untrapped ones becomes clear at times larger than 6 ns. It is also clearly seen
that there is a considerable amount of trapped molecules accelerated up to velocities
of 15 km/s over the time period of 16 ns for a pressure of 3 torr. The larger number of
molecular collisions at 10 torr generally reduces the number of trapped particles, since
almost all trapped particles that collide with other particles leave the potential well
as the result of collisions. Still, there is a signicant amount of molecules accelerated
to velocities over 10 km/s, which means that such an acceleration is possible even for
collisional gas with pressures on the order of 10 torr.
Figure 7.1: Distribution function of molecular velocities at dierent time moments
for 3 torr (left) and 10 torr (right).
7.4 Low density microthruster
The acceleration of molecules to very high velocities, possible when the gas collision
time is on the order of or smaller than the pulse duration, may be used in propulsion
devices with the primary goal of obtaining high specic impulses. One of the possible
designs of this device (Shneider, Gimelshein and Barker (2005)) is shown in Fig. 7.2.
115
In this gure, the transparent plates denote windows, and grey plates denote mirrors.
Carrier gas is entrapped between two transparent windows, one of which has multiple
openings to allow for gas expansion. The gas density between the mirrors and the
windows is negligibly small. The principle of operation is based on acceleration of
molecules by multiple optical lattices created with two laser beams that have multiple
intersections in the gas section. Gas molecules accelerated to very high velocities
leave through the openings, thus creating net thrust force. The left window separates
the carrier gas from the left branch of optical lattices, that would have otherwise
accelerated molecules in the direction opposite to the openings. The optical lattices
are created through multiple re
ections of the two counter-propagating beams on the
opposite windows. Due to the recent progress in laser related optics technologies that
resulted in incredible quality improvements of low absorbing mirrors, re
ectivity on
the order of 99.999% is no longer surprising. This, in conjunction with the fact that
only a small fraction of the laser energy is absorbed by the gas, makes possible the
use of hundreds and thousands of thrust-producing openings in a single propulsion
device fed by two laser beams.
gas flow
gas flow
gas flow
optical lattice
beam 2
beam 1
Figure 7.2: Schematic of a low-density micropropulsion device powered by the optical
lattice/gas interaction.
The performance characteristics of the above low-density propulsion device has
116
been analyzed with the DSMC method for dierent stagnation pressures. The simu-
lations have been performed for an axisymmetric domain that represents a part of the

ow adjacent to an opening. The computational domain is shown in Fig. 7.3. The
equilibrium stagnation conditions were set at the upstream boundary (left) bound-
ary, the lower boundary is the axis of symmetry, specular conditions are specied
at the upper boundary to simulate the eect of a large array of openings, and the
vacuum condition was used at the downstream boundary. The gas was methane,
the stagnation temperature was 300 K, and the three stagnation pressures used are
0.01 torr, 0.1 torr, and 1 torr. The optical lattice was created with two 800 nm wave-
lenth 50 m radius beams with a chirp of 10
19
rad/s
2
, the lattice center was located
0.5 mm upstream from the opening, the maximum laser intensity at the axis was
6.410
16
W/m
2
, and the pulse duration was 10 ns. The initial lattice velocity, V
0
,
varied from -5 km/s to -7.5 km/s in order to select the optimum value.
Figure 7.3: Axial velocity elds (m/s) in a low-density micropropulsion device at two
dierent pressures and two time moments.
117
The evolution of the axial
ow velocity eld after a single pulse is presented in
Fig. 7.3 for pressures 0.1 and 1 torr and V
0
=6:5 km/s. Two time moments are
shown here, t = 0 which corresponds to the time immediately after the pulse, and
t = 0:04s. During the pulse, the trapped molecules start moving toward the opening,
and the location of the maximum
ow velocity shifts downstream by about 0.1 mm
by the end of the pulse. The molecules with high velocities start passing through
the opening, and create a region of elevated velocities next to its exit plane. After
0.04 s, most of the molecules have passed through the opening, and their relatively
small radial velocities lead to the formation of high-velocity region closer to the axis
downstream from the orice. For the lower pressure of 0:1 torr, the velocities in that
region exceed 10 km/s and reach a maximum of 13.5 km/s. The twenty-fold increase
of average
ow velocities in the core
ow after the opening exit plane as a result of
the lattice/gas interaction is related both to the high number
ux of the trapped
molecules and their low divergence from the centerline. For this pressure, the mean
free path is larger than the distance from the center of the lattice to the exit plane,
and molecular collisions do not reduce signicantly the number of particles trapped
by the optical lattice. For the higher pressure of 1 torr the molecular collisions visibly
reduce the number of trapped particles, therefore decreasing the average velocity in
the high-velocity region at and after the exit plane.
The temporal change of the thrust force computed using instantaneous values of

ow velocity and density are given in Fig. 7.4 for pressures 0.1 and 1 torr. It is clearly
seen that the thrust force has a maximum at about 0.04s for 0.1 torr and 0.02s
for 1 torr. The value of 0.04s corresponds to the time when the trapped molecules
initially located at the center of the lattice reach the orice exit plane. Forp = 1 torr,
the mean free path is too small for these molecules to travel to the exit plane without
collisions, and the largest contribution to the thrust force give trapped molecules
118
Time, m s
Thrust, m N
0 0.02 0.04 0.06
0
2
4
6
8
10
12
14
0.1 torr, X
c
=-0.5mm
1 torr, X
c
=-0.5mm
1 torr, X
c
=-0.1mm
Figure 7.4: Temporal change in thrust force for dierent pressures and center of the
lattice locations X
c
.
located closer to the exit plane. The center of the lattice generally corresponds to
the maximum laser intensity, and therefore the largest portion of trapped molecules
compared to the regions o center. It is therefore benecial for the higher pressure
to move the center of the lattice closer to the exit plane. Even though the number
density is somewhat smaller in that region compared to the stagnation conditions, it
results in a signicantly larger number of trapped molecules that reach the exit plane
compared to the original location. This is illustrated in Fig. 7.4 for the optical lattice
center located at X
c
=0:5 mm and X
c
=0:1 mm. Note, the maximum thrust in
the latter case is about ten time higher than that for 0.1 torr.
For pressures lower than 0.1 torr the thrust as a function of time increases almost
linearly with pressure, as shown in Table 7.1. It is also possible to preserve this
linear behavior even for higher pressures, but moving the center of the lattice closer
to the exit plane. The increase of thrust due to the optical lattice is very signicant,
about 20 times for the range of pressures considered. The maximum specic impulse
observed is about 450 s, which is a seven-fold improvement compared to the
ow
119
Table 7.1: Maximum observed thrust and specic impulse for various pressures and
X
c
.
Pressure, X
c
, Mass Flow Thrust I
sp
torr mm kg/m
2
N s
0.01 -0.5 3:899 10
11
1:794 10
7
469
0.1 -0.5 3:400 10
10
1:472 10
6
441
1 -0.5 1:670 10
9
5:092 10
6
311
1 -0.3 2:443 10
9
9:174 10
6
383
1 -0.1 3:360 10
9
1:474 10
5
448
not aected by the optical lattice. Note also that the results shown above are for
the initial lattice velocity of -6.5 km/s, which was found to give maximum particle
trapping. This is because at time 10 ns, that corresponds to the maximum of the
laser intensity, the lattice velocity is about 0 and therefore the maximum number
of molecules become trapped. For lattice velocities of -5 km/s and -7.5 km/s, the
maximum thrust decreases by over 40% compared to the -6.5 km/s case.
7.5 High density microthruster
When the gas density is high enough so that the mean collision time is much smaller
than the pulse duration, it is not possible to eectively trap particles and accelerate
them to high velocities with the lattice; it is still possible however to increase the
particle thermal velocity as well as transfer momentum from the lattice to the gas.
In this case, the chirping of the lattice is not needed, and a constant lattice velocity
should be used. The magnitudes of the change in momentum and energy of the
molecule interacting with a lattice potential are given in Section 5.2
The non-resonant deposition of laser radiation energy into high density gas with
the consequent increase of both gas momentum and energy, examined in the previous
section, may be used to construct a thruster driven by the non-resonant optical lattice
120
/ gas interaction. Although there is a number of possible congurations for such a
thruster, the most straightforward seems to be a converging-diverging de Laval nozzle
with the interaction region located near the nozzle throat. A schematic of such a
conguration is shown in Fig. 7.5. The rst laser beam passes through the plenum
and the diverging section of the nozzle, and then re
ects back on the mirror mounted
on the upper part of the nozzle. The second beam enters the diverging section through
the after re
ecting on the mirror located in the lower part of the nozzle, and intersects
with the rst beam near the nozzle throat, thus creating an optical lattice in that
region. Since the energy consumed by gas molecules over a single pulse represents a
small fraction of the total pulse energy, the beams should be re-utilized using a mirror
system, with an optical lattice repeatedly formed in the throat region.
beam 2: k ,
2 2
w
1 1
beam 1: k ,w
mirror
optical
lattice
gas flow gas flow
phase velocity
mirror
window window
Figure 7.5: Schematic of microthruster integrated with an optical lattice, operating
in the continuum regime.
The eciency of the proposed scheme has been studied for a cold gas microthruster
in
uenced by intersecting 690 nm laser beams with a pulse duration on the order of
510
11
s, maximum intensity at the axis of 510
17
W/m
2
, the beam radius of 40m,
and the initial lattice velocity of 1,300 m/s. The ten 20 ns pulses were repeated every
30 ns, and such a high repetition rate may be achieved with the above mentioned
multiple use of the laser pulse power. A conical de Laval nozzle was used with the
throat radius of 50 m, the length of the diverging part of 1 mm, and the diverging
121
half-angle of 15 deg. The stagnation pressure was assumed to be 1 atm, temperature
was 300 K, and the carrier gas was nitrogen. For this pressure, the considered laser
pulse is well below the breakdown.
The results of the unsteady
ow development after ten successive laser pulses are
shown in Fig. 7.6 for the axial velocity and translational temperature elds. Imme-
diately after the pulses, t = 0, there is a high velocity region (Fig. 7.6, left) observed
at the location of the optical lattice near the nozzle throat. The maximum velocity
reaches 2,500 m/s in this region, compared to about 400 m/s if there where no optical
lattice. The velocity perturbation propagates downstream, and the elevated velocity
front spreads from the nozzle centerline out to the nozzle surface. The velocity maxi-
mum in this front decreases with time, and leaves the nozzle after 1s from the pulses.
The gas translational temperature (Fig. 7.6, right) also increases due to the lattice
/ gas interaction. Initially (t = 0), the temperature in the throat is approximately
700 K, which is about 450 K higher than in the
ow without the lattice. During the
subsequent propagation of the elevated temperature front through the diverging part
of the nozzle, the maximum temperature is observed near the surface and not at the
nozzle axis.
Figure 7.6: Temporal evolution of the axial velocity elds (m/s) inside a micronozzle.
Left, axial velocity (m/s); right, translational temperature (K).
122
The temporal change of the axial velocity prole extracted along the nozzle axis
is presented in Fig. 7.7. In this gure, X=0 and X=1 mm correspond to the nozzle
throat and exit planes, respectively. Although most of the velocity increase due to
the lattice-gas interaction occurs downstream from the throat, there is a signicant
part of molecules being accelerated by lattice in the converging section of the nozzle,
where the
ow is subsonic. This perturbation therefore expected to aect both nozzle
mass
ow and thrust. Positioning the lattice fully in the supersonic region further
downstream would minimize the impact of the lattice on the mass
ow, but on the
other hand, reduce the momentum and energy deposition into the gas. Generally, at
high pressures the energy deposition increases nearly as the square of the gas density,
and the interaction of the lattice with a higher-density gas near the throat is thus
benecial.
The axial velocity maximum at the axis decreases from over 2,500 m/s att = 0 to
about 1,500 m/s at 0.3s and further down to 1,300 m/s at 0.6s. The half-velocity
front reaches the nozzle exit at t 0:65 s. After 1 s, the front has mostly left
the nozzle, but the velocity at the exit plane is still over 300 m/s higher than for an
undisturbed
ow. The large values of axial velocity at the exit result in an almost
three-fold increase in thrust due to the lattice-gas interaction. The thrust of the nozzle
increases from 1.19 mN for a
ow without a lattice to 2.99 mN for the
ow. The lattice
eect on the specic impulse is smaller since the mass
ow is also increases in the
present conguration, but still signicant, 92 s versus 72 s. The better performance
of a cold gas thruster integrated with an optical lattice demonstrates the feasibility
of such an integration. It is also important that larger gas density, laser energy, and
the number of pulses will all result in subsequent increase in device performance both
in terms of thrust and specic impulse.
123
X, m
U, m/s
0 0.0002 0.0004 0.0006 0.0008 0.001
1000
1500
2000
2500
t=0
t=0.3ms
t=0.6ms
t=1ms
Figure 7.7: Axial velocity proles along the nozzle axis at dierent time moments.
7.6 Conclusions
The non-resonant interaction of gas molecules with the optical lattice potential is
studied with application to rocket propulsion. Two propulsion concepts are examined,
high density and low density, based on the gas
ow regime. The direct simulation
Monte Carlo method is used in all presented computations.
The principal driving force for the non-resonant laser propulsion at high densities
is the energy and momentum deposition from optical lattice to gas. Analytical model
has been developed to qualitatively characterize this deposition; the results of the
model are in good agreement with the DSMC predictions. Note that the analytical
model may be used to specify initial conditions for optical lattice/gas interactions in
the continuum regime using continuum approaches, such as the solution of Navier-
Stokes equations.
The high-density propulsion device considered in this work represents a de Laval
nozzle with power deposition in the region of the nozzle throat. The stagnation
pressure of 1 atm was considered, and the DSMC computations of a 1 mm long nozzle
124
have shown an improvement of about 300% in thrust and 20% in specic impulse for
a 50 mJ pulse. It is possible to use higher pressures and larger nozzle dimensions
to further increase the thrust and specic impulse. It is also benecial to recycle the
laser beams using mirrors and an optical cavity, since the laser energy absorption by
gas is very small, on the order of 10
6
for a pressure of 1 atm.
The non-resonant acceleration of molecules to velocities of 10 km/s and higher
in a chirped frequency optical lattice is used in the proposed low-density propulsion
device. The stagnation pressures from 0.01 to 1 torr are investigated, and a seven-
fold increase in the specic impulse is obtained. The maximum thrust and specic
impulse for 1 torr are 16N and 448, respectively. There is a number of possible ways
to increase these numbers, such as to (i) further increase stagnation pressure, (ii) use
larger lasers and bigger openings, (iii) use the multiple openings setup to recycle laser
power, (iv) optimize the device in terms of lattice location, chirp, and velocity, (v)
utilize polar molecules, such as H
2
O. Only the latter option alone may give several
times larger specic impulse due to the larger forces on molecules.
The shown ability of the optical lattice to accelerate molecular beams to extremely
high velocities in weakly collisional regime without ionization and dissociation of
molecules may be used not only in propulsion, but also in various material processing
devices. The non-resonant energy deposition at high pressures may also be used
to create high-temperature gas pockets in arbitrary points of space. Note that a
variable lattice velocity needs to be used for this purpose in order to account for gas
temperature increase and provide optimum energy deposition.
125
Chapter 8
Concluding Remarks and Summary
8.1 Summary
The objective of this thesis was to perform a detailed study of rarefaction eects in
gas
ows with particle approaches. Understanding rarefation eects in gases is very
important because most aerospace systems operate in the rareed gas regime. The
main tool for the analysis has been SMILE computational tool based on the DSMC
method. The main conclusions are summarized below.
8.1.1 The CHAFF-IV ground testing facility
The performance of CHAFF-IV as a cryogenic pump was studied numerically. Flat
panel and radial n array congurations were considered. It was found was that
radial n arrays performed better when the sticking coecient between the particles
and the pump surface was low, as in the case of high energetic ions. Particles, such
as neutrals, with high sticking coecients will be pumped better with the
at wall
conguration.
126
DSMC investigation also veried that there are no pressure gradients in the cham-
ber and hence no induced velocities. This condition is very important because it
describes a space environment. Any ground based testing facility must meet this
condition in order to faithfully reproduce space environment.
8.1.2 Eects of surface roughness in plume impingement on
spacecraft
Numerical modeling of a cold gas nozzle plume interacting with engineering surfaces
was performed for nitrogen propellant in the range of nozzle throat based Reynolds
numbers from about 2 to 600. A companion experiment was performed by A. Ketsde-
ver and T. Lilly. In the experiment, nano-Newton resolution force balance was used to
measure thrust force of a plume expanding from a conical nozzle, and then the total
force resulted from the interaction of the plume with aluminum plates attached to the
same force balance. Smooth and rough plates were examined, with surface roughness
introduced through a set of equally spaced 0.5 mm wide grooves perpendicular to the

ow direction.
The setup in the numerical study corresponded to that of the experiment. The
calculated force vs mass
ow was found to be in good agreement with the corre-
sponding experimental data. The experiments and computations showed that there
was signicant thrust degradation due to the plume surface interaction, with the total
decrease being up to 15%. The force on the plate increased in magnitude by about
20% for the rough surface as compared to the smooth one. However, the impact of
the surface roughness on total force was small, which was attributed primarily to the
eect of plume molecules re
ected from the plate backwards to the plenum surface.
The number of such molecules was signicantly larger for rough surfaces.
The impact of the surface roughness inside the nozzle has been studied numerically.
127
It was shown that the surface roughness decreases both mass
ow and thrust by over
10% for Reynolds numbers on the order of one. The eect decreases with the increase
of the Reynolds number, and is negligible at Re > 100. The specic impulse is not
aected by the surface roughness even at small Reynolds numbers.
8.1.3 Radiometric Forces
A numerical investigation of the causes of radiometric forces was performed. Maxi-
mum radiometric forces were observed in the rareed regime. In this regime, radio-
metric force is nearly proportional to the surface area of the radiometric plates.
The contribution of molecular mass, diameter and viscosity index on radiometric
forces was investigated. Molecular diameter has far much in
uence on radiometric
forces than molecular mass. The role of viscosity index has also been found to be
more signicant.
It has also been shown that the size of the vacuum chamber strongly aects
radiometric forces, with the force increasing when increasing the chamber size. The
role of chamber size is so signicant that even when the chamber is hundred times
larger than the heated plate, the relative position of the wall still aects radiometric
forces. Gas rarefation was found to be an important factor aecting the radiometric
forces.
8.1.4 Optical lattices in high and low density gases
The eect of high intensity optical potentials on atomic and molecular gases has been
studied. The trajectories of dierent atoms and molecules in the phase velocity space
and molecular velocity space were examined both in the free molecular and collisional
regimes. Trapping of particles by optical lattices in a collisional regime obtained in
a numerical simulation is shown for the rst time. It was also shown that in single
128
species gases the intermolecular collisions mostly deplete the population of trapped
particles; they also may lead to trapping of particles that would not have been trapped
in a free molecular regime. In a gas mixture, collisions of heavy particles with lighter
ones may result in heavy particles leaving the lattice potential well as well as staying
in it longer than without collisions.
The deposition of energy and momentum in high density gases from an optical
lattice to a stagnant gas has been examined numerically. Optical lattices were shown
to increase translational temperature to values as high as 2300 K in methane and
1300 K in molecular nitrogen. Even larger temperatures may be obtained for shorter
and stronger pulses, shorter laser wavelengths, or, especially, if polarized molecules
such as H
2
O are used. Since only a small fraction of laser energy is deposited to gas,
the use of a multi-pass optical system, with two laser beams being recycled many
times, is also expected to produce signicantly higher temperatures. The interaction
of an optical lattice moving with a nite velocity, with an initially stagnant gas,
leads to the momentum deposition, and an increase of gas velocity to hundreds and
thousands of m/s.
In the study of the formation and development of an acoustic signal created by
the lattice, the DSMC method was shown to be able to capture both the travel of an
acoustic signal in high-density gases and after-pulse
ow oscillations observed near
the center of the lattice.
The ability of optical lattices to increase the mixing rate of gases has also been
demonstrated. In the numerical analysis, optical lattices improved mixing rates by
an order of magnitude.
129
8.1.5 Micropropulsion devices based on pulsed optical lat-
tices
The non-resonant interaction of gas molecules with the optical lattice potential was
studied with application to rocket propulsion. Two propulsion concepts are examined,
high density and low density, based on the gas
ow regime.
The principal driving force for the non-resonant laser propulsion at high densities
is the energy and momentum deposition from optical lattice to gas. The results of
DSMC predictions were found to be in good with analytical models.
The high-density propulsion device considered in this work represents a de Laval
nozzle with power deposition in the region of the nozzle throat. The stagnation
pressure of 1 atm was considered, and the DSMC computations of a 1 mm long nozzle
have shown an improvement of about 300% in thrust and 20% in specic impulse for
a 50 mJ pulse. It is possible to use higher pressures and larger nozzle dimensions
to further increase the thrust and specic impulse. It is also benecial to recycle the
laser beams using mirrors and an optical cavity, since the laser energy absorption by
gas is very small, on the order of 10
6
for a pressure of 1 atm.
The non-resonant acceleration of molecules to velocities of 10 km/s and higher
in a chirped frequency optical lattice is used in the proposed low-density propulsion
device. The stagnation pressures from 0.01 to 1 torr are investigated, and a seven-
fold increase in the specic impulse is obtained. The maximum thrust and specic
impulse for 1 torr are 16N and 448, respectively. There is a number of possible ways
to increase these numbers, such as to (i) further increase stagnation pressure, (ii) use
larger lasers and bigger openings, (iii) use the multiple openings setup to recycle laser
power, (iv) optimize the device in terms of lattice location, chirp, and velocity, (v)
utilize polar molecules, such as H
2
O. Only the latter option alone may give several
130
times larger specic impulse due to the larger forces on molecules.
The shown ability of the optical lattice to accelerate molecular beams to extremely
high velocities in weakly collisional regime without ionization and dissociation of
molecules may be used not only in propulsion, but also in various material processing
devices. The non-resonant energy deposition at high pressures may also be used
to create high-temperature gas pockets in arbitrary points of space. Note that a
variable lattice velocity needs to be used for this purpose in order to account for gas
temperature increase and provide optimum energy deposition.
8.2 Closing Remarks
A wide ranging eects of rarefaction in gas
ow have been succesfully investigated
and analysed using DSMC. Where possible DSMC results were compared with ex-
perimental data and were found to be in reasonably good agreement.
However, there is currently no experimental data available for optical lattices.
Thus it was not possible to compare the simulation results with emperical data. This
is probably an area where more work needs to be done. It will be very interesting to
see how these results compare with empirical data. Such data may soon be available in
the near future because Dr Andrew Ketsdever of the University of Southern California
and Dr Peter Barker of Edinburgh University are, independently, considering to do
experimental work on this subject.
131
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Abstract (if available)

Abstract The objective of this study is the numerical analysis of gas flow rarefaction phenomena with application to a number of aerospace-related problems. The understanding and accurate numerical prediction of rarefied flow regime is important both for aerospace systems that operate in this regime, and for the development of new generation of gas-driven nano- and micro-scale devices, for which the gas mean free path is comparable with the reference flow scale and rarefaction effects are essential. The main tool for the present analysis is the direct simulations Monte Carlo (DSMC) method.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Ngalande, Cedrick Goliati (author)

Core Title Study of rarefaction effects in gas flows with particle approaches

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace Engineering (Astronautics)

Publication Date 04/23/2007

Defense Date 03/19/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Aerospace,CHAFF-IV,DSMC,nozzle,OAI-PMH Harvest,optical lattice,radiometric forces,surface roughness

Language English

Advisor Kunc, Joseph (committee chair), Erwin, Daniel A. (committee member), Gimelshein, Sergey (committee member), Gruntman, Michael A. (committee member), Judge, Darrell L. (committee member)

Creator Email ngalande@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m439

Unique identifier UC1464606

Identifier etd-Ngalande-20070423 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-494791 (legacy record id),usctheses-m439 (legacy record id)

Legacy Identifier etd-Ngalande-20070423.pdf

Dmrecord 494791

Document Type Dissertation

Rights Ngalande, Cedrick Goliati

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu