DSMC Calculation of Supersonic Expansion at a Very Large
Pressure Ratio
Koji Teshima1 and Masaru Usami2
1
Department of Industrial Arts Education, Kyoto University of Education, Kyoto JAPAN
2
Department of Mechanical Engineering, Mie University, Tsu, Mie JAPAN
Abstract. Supersonic expansion of room temperature argon from a sonic orifice at a very large pressure ratio up to 16000
for different stagnation Knudsen numbers, 2x10~3 and 4X10"4 is simulated by the DSMC method. In order to calculate a
large flowfield different sized cells and a different time-step scheme were adopted. It was shown that the effects of
rarefaction and background gas to the jet size can be evaluated using a rarefaction parameter or a local Knudsen number.
The calculation was also made for the expansion to a vacuum for a wide range of the stagnation Knudsen number, 4x10"
4
- 0.1. The terminal parallel temperature dependence to the stagnation Knudsen number agrees well with the sudden
freezing model.
INTRODUCTION
The DSMC calculation of a supersonic expansion at a finite pressure ratio of the stagnation to the expansion
chamber pressures, p(/pl9 has shown a good agreement with the experimental observation in the previous paper[l].
However, in the calculation the pressure ratio was limited up to about 100, due mainly to the limitation of the
computer memory and the CPU time. In many applications such as manufacturing facilities of electronic devices,
molecular beam facilities, and molecular spectroscopy apparatus, the working gas is expanded into a vessel not at
vacuum, but still at a finite pressure; at a very large pressure ratio. In such expansion, in addition to the rarefaction
effects such as broadening of the shock waves and the merging of the jet boundary with shock wave the background
molecules which are accommodated with the vessel wall penetrate into the jet core and they collide with expanding
cold molecules, warm up and decelerate them. As a result no distinct jet core cannot be seen in the density field,
whereas the jet core, in which the flow is kept supersonic, exists. This supersonic region must become very much
narrower than the expected one from the well known relation by Ashkenas and Sherman [2] for the continuum
expansion. The jet structure including the length of the supersonic jet core and the distribution of the flow properties
are not known well for these large pressure-ratio jets, although estimation of them might be important for the
applications.
The background gas effect to the jet were studied in relation to molecular beam intensity [3], but these were
indirect measurements. Muntz et al.[4] introduced a rarefaction parameter (=d(pcp1)1/2/T0[dyne/cmK], with orifice
diameter d and the stagnation temperature T0) which correlated certain of the rarefaction phenomena from shock
broadening to the penetration of background gas into the core of the jet. In the previous paper [1] we have shown
that the density distribution on axis can be classified by three flow regimes; continuum, transition, and scattering,
using the rarefaction parameter for a wide range of the flow condition. A theoretical work of the background gas
penetration based on the asymptotic gas kinetic theory for a source flow expansion was made by Brook et al.[5]
However, they were limited only on the jet center line. In some applications the radial distribution of the flow
properties will be as important as the axial ones. Therefore, the whole flowfield of such rarefied jets needs to be
analyzed over a wide range of the pressure ratio and for different stagnation Knudsen numbers.
In this study the calculation method has been improved in order to enable the calculation of a very large pressureratio jet; up to 16000 for the stagnation Knudsen numbers Kn=l/50Q and l/2500.The present calculation enables to
evaluate the jet structure with not only different stagnation Knudsen numbers but also with a wide range of the
pressure ratio.
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
737
It is well known that in a supersonic expansion non-equilibrium in the translational energy modes; between the
translational temperatures in the parallel and perpendicular directions to the expansion, occurs. This phenomena is
one of the rarefaction effect of the supersonic expansion. The former (the parallel temperature T//) does not lower
enough due to not enough collisions in the parallel direction i.e. freezing of the parallel temperature, while the latter
(the perpendicular temperature T 0) continues to decrease with distance. The degree of the parallel temperature
freezing has been measured by the molecular beam time-of-flight method [3] and its dependency on the stagnation
Knudsen number agreed well with an empirical formula based on a sudden freezing model [6]. In order to confirm
the validity of the present calculation method and to demonstrate the effect of the background molecules to the
expansion, expansions into a vacuum for different stagnation Knudsen numbers were calculated. The calculated
freezing parallel temperature was compared with the existing empirical formula[6].
METHOD OF CALCULATION
The position of the Mach disk of a continuum free jet XM can be predicted from the experimental formula
presented by Ashkenas and Sherman [2] as xA/d=Q.61(pQ/p1)l/2. That is, since its distance from an orifice is
proportional to the square root of a pressure ratio, if a pressure ratio is increased by about 250 times from 60 to
16000, the calculation domain whose length is about 16 times larger and whose volume is about 4000 times larger is
required. Of course, since background pressure will become small if stagnation pressure is fixed and a pressure ratio
is enlarged, the number of molecules that exist in the calculation domain becomes only about 16 times larger.
However, since the calculation time which is required for achievement of a steady state is also proportional to the
length of the calculation domain, after all, when a pressure ratio changes 250 times larger, total calculation time is
increased by 100 times or more. This is a reason why a jet calculation with a large pressure ratio is difficult.
Since a free jet reduces its density quickly at a longer distance from an orifice, for a cell network that subdivides
a flowfield, the cell length is made to increase in geometric series not to produce a big difference in the number of
molecules of each cell. For example, on x axis, if the length of the cell just behind an orifice is set to a, and a
common ratio is set to b, the length of the z-th cell will become abl~l. If the position coordinate of a molecule is set to
jc, a cell number n to which the molecule belongs is calculated easily by n=log{x(b-l)/a+l}/log(b)+l (omit under
decimal point). Now, when a molecule passes a large cell, it is not efficient calculation that the distance to which a
molecule moves per time step dt is too small compared with the cell length. Experientially, the moving distance per
step is satisfactory if it is about 1/5 times the cell length. However, since it is difficult to change a time step
according to the position coordinates of a cell (it is difficult to synchronize different time steps), a fixed time step is
adopted usually over all domain in the conventional program. Since this time step is determined on the basis of the
place where density is large, i.e., a small cell, conversely in the cell with large dimension and small density, the time
step becomes short beyond necessity. As a result, the amount of the whole calculation increases and the calculation
time becomes long.
In the present calculation, as shown in Figure 1 (for the case of pQ/pi=16000 and l/Kn=25QQ), the downstream of
an orifice is divided into 12 blocks and they are classified into seven classes (Class 1-Class?), where the far block
from an orifice has a longer time step. If it takes into consideration that a molecule needs to pass through the
boundary of each domain, the value of twice is suitable for the ratio of time step between adjoining blocks.
Therefore, a time step increases by a geometric series of the common ratio 2 with the distance of cell from an orifice.
That is, after molecules that are in the block near an orifice (Class 1) finish a repetition of 64 movements and
intermolecular collisions, molecules in the farthest block (Class?) will perform one long time movement and
intermolecular collision. So, calculation time can be shortened as the farthest block is made large and molecules that
exist there increase. In regard to molecules that move between blocks, the procedure such as stopping molecules on
the boundary of blocks, changing a time step there, and moving them again may be considered. However, this may
become a complicated procedure and has a possibility of preventing shortening of calculation time. If a molecule
goes through a boundary simply using the time step of the block in which the molecule exists before moving,
unbalance of time between "molecular motion" and "intermolecular collision" will arise near the boundary. For
example, the half of the molecules that move from a block with a large time step to a block with a small time step
has a lack of intermolecular collisions.
As will be shown in Figure 3 (left), distortion in the density contour is produced near the boundary of blocks by
the above-mentioned simple processing. The straight line which is visible near y/d=3 is the most remarkable
distortion. Usually, in the DSMC method, the time step dt for molecules that flow in from outside of a calculation
domain is to be multiplied by a random number, and time for molecular motion in the domain is shortened. If this is
738
neglected, the move time of molecules flowing in from a boundary will become larger than collision time, and will
make distortion in the flowfield near a boundary. In figure 3 (left), the phenomenon that is similar with this arises
near a block boundary. In the present calculation, in order to prevent this, the following procedure is considered (See
Fig.2). Numbers given under the arrow express the length of collision time.
(a) When moving to the block-B with a large time step from the block-A with a small time step.
(a-1) The molecules that pass through the boundary by the first step dt of the block-A continue their movement at
the second step of the block-A after entering into the block-B. However, intermolecular collisions are not calculated
between both steps. After movement of the second step, the calculation of intermolecular collisions in the block-B
follows it.
(a-2) For the molecules that pass through the boundary at the second step of the block-A, the calculation of
intermolecular collisions in the block-B follows the molecular motion as usual.
(b) When moving to the block-A with a small time step from the block-B with a large time step.
(b-1) The molecules that pass through the boundary earlier than the half of the time step 2dt of the block-B stop
their movement at one half of the time step. From this state, the calculation of intermolecular collisions in the blockA continues.
(b-2) The molecules that could not cross a boundary within the half of the time step of the block-B finish their
movement in the time step 2dt of the block-B. After that, the intermolecular collisions and the molecular motions in
the first step of the block-A are paused once, respectively. And the calculation starts again from the subsequent
intermolecular collisions.
In above (a-1), the collision time (the average of the collision time between pre-movement and post-movement)
is l.5dt on contrary to the move time of molecules through the boundary being dt+dt. However, in (a-2), since the
collision time is l.5dt on contrary to the move time being dt, if both of (a) are averaged, the unbalance between the
move time and the collision time is cancelled. Moreover, in (b-1), the collision time is l.5dt on contrary to the move
time of molecules through the boundary being dt. However, in (b-2), since the collision time is l.5dt on contrary to
the move time being 2dt, if both of (b) are averaged again, the unbalance between the move time and the collision
time is also cancelled. The fundamental idea on the above procedure is such that molecules that flow into a new
block act as if they exist in the block for a long time.
/'/,
''
/
./ClassS
Class4
Downstream boundary
Class6
Block A
Block B
Time step = dt
Time step = 2dt
dt+dt
T o ) ^ 2
Po/Pl=16000
Class?
dt
<a-2).
(2dt)
Classl \xxO-ciass4
Class2 ClassS
Collision time
Figurel. Downstream computational domain divided into 12 blocks.
20
25
Figure!. Molecular motion between different blocks.
Axial distance (x/d)
Axial distance (x/d)
Figure 3. Comparison of density contours with (right) and without (left) considering the time-step difference between different
blocks.
739
Figure 3 (right) is the result obtained by the above-mentioned procedure. Although calculation conditions are the
same as that of the left, the distortion near the boundary is canceled completely. And it agrees well with the result of
conventional calculation obtained using constant time step in all domains. Thus, using the procedure mentioned
above, it is possible to simulate the jet flowfield efficiently with variable time steps without complicated processing.
Figure 3 (right) is the result obtained by the above-mentioned procedure. Although calculation conditions are the
The calculation speed about 8 times faster than that by the conventional program has been obtained (in the case of
same as that of the left, the distortion near the boundary is canceled completely. And it agrees well with the result of
pQ/pl=l60QQ
and l/Kn=25QQ). As large-sized data (a big array), the array which memorizes a name of block in which
conventional calculation obtained using constant time step in all domains. Thus, using the procedure mentioned
each above,
molecule
to arrays
usedwith
by general
calculation
program. processing.
Its size is the
it is exists
possibleistorequired
simulatein
theaddition
jet flowfield
efficiently
variable DSMC
time steps
without complicated
sameThe
as the
number
of
molecules.
However,
since
it
is
memorizable
with
one
byte
integer,
it
does
not
become
calculation speed about 8 times faster than that by the conventional program has been obtained (in the case so
of big
a burden.
p0/p1=16000 and 1/Kn=2500). As large-sized data (a big array), the array which memorizes a name of block in which
Now,
although exists
the above
procedure
has tobeen
devised
the calculation
time balance
between
each molecule
is required
in addition
arrays
used bymaintaining
general DSMC
program.
Its size"molecular
is the
movement"
"intermolecular
collisions",
paying
attention to with
a motion
of integer,
each molecule,
same as and
the number
of molecules.
However,ifsince
it is memorizable
one byte
it does notunnaturalness
become so bigwill
a burden.
remain
in the calculation processing in the case of moving of molecules from a block with large time step to a block
Now,
the Since
above aprocedure
devised maintaining
time through
balance abetween
"molecular
with small
timealthough
step in (b).
molecule has
stopsbeen
its movement
for dt whenthe
passing
boundary,
small time
andhere.
"intermolecular
collisions",
if payingofattention
a motion
molecule,On
unnaturalness
will for
delaymovement"
is generated
If a calculation
is an analysis
a steadytoflow,
thereofis each
no problem.
the other hand,
remain inofthe
processing in the
case of moving
a blockHowever,
with largesince
time step
to a block
the analysis
ancalculation
unsteady phenomenon,
a simulation
will of
bemolecules
distorted from
delicately.
the time
scale of
with small
time step in (b).
Sincethan
a molecule
stops
movement
for dt when
passing
a boundary, small time
the unsteady
phenomenon
is larger
time step
dt its
generally,
it hardly
becomes
thethrough
problem.
delay
is
generated
here.
If
a
calculation
is
an
analysis
of
a
steady
flow,
there
is
no
problem.
On the other
hand,
for
The Knudsen number is defined by a mean free path at stagnation divided by an orifice diameter
d. As
boundary
the analysis of an unsteady phenomenon, a simulation will be distorted delicately. However, since the time scale of
conditions applied to molecules that flow in from an upstream boundary, the Maxwell distribution with some flow
the unsteady phenomenon is larger than time step dt generally, it hardly becomes the problem.
velocityThe
perpendicular
to theis boundary
assumed.
On at
thestagnation
other hand,
on by
a downstream
boundary,
the Maxwell
Knudsen number
defined byisa mean
free path
divided
an orifice diameter
d. As boundary
distribution
at background
pressurethat
without
velocity
is assumed.
Thethe
diffuse
reflection
is assumed
on aflow
wall of
conditions
applied to molecules
flow flow
in from
an upstream
boundary,
Maxwell
distribution
with some
an orifice.
The
collision between
molecules
performed
by the
null-collision
method using
VHS molecular
model
velocity
perpendicular
to the boundary
is is
assumed.
On the
other
hand, on a downstream
boundary,
the Maxwell
for argon.
For
pjpi=16000,
the
size
of
a
flowfield
(axially
symmetric
field)
is
0.75d
x
0.75J
for
an
upstream
of an
distribution at background pressure without flow velocity is assumed. The diffuse reflection is assumed on a wall of
orifice,
and 247d
54d forbetween
a downstream
orifice. by
The
of molecules
is about
millionsmodel
and the
an orifice.
The xcollision
moleculesofisan
performed
thenumber
null-collision
method using
VHS16
molecular
number
of cellsFor
is about
140,000.
for argon.
the size of a flowfield (axially symmetric field) is 0.75d x 0.75d for an upstream of an
p0/p1=16000,
orifice, and 247d x 54d for a downstream of an orifice. The number of molecules is about 16 millions and the
number of cells is about 140,000.
RESULTS AND DISCUSSION
RESULTS AND DISCUSSION
The Mach number profile of a jet at pressure ratio of 16000 for l/Kn=25QQ and 500 are shown in Figure 4. The
Mach number
reaches about 20 for l/Kn=25QO and begins to decrease gradually at a distance much shorter than Xy/d
The Mach number profile of a jet at pressure ratio of 16000 for 1/Kn=2500 and 500 are shown in Figure 4. The
and eventually
thereaches
flow becomes
No distinct
shock
or jet core
exists. The
supersonic
expansion
region
Mach number
about 20subsonic.
for 1/Kn=2500
and begins
to decrease
gradually
at a distance
much
shorter than
xM/d has
been and
narrowed
due
to
the
rarefaction
effects
by
the
broaden
and
merged
shock
waves
with
the
jet
boundary
and to
eventually the flow becomes subsonic. No distinct shock or jet core exists. The supersonic expansion region has
the background
gasdue
effect
byrarefaction
penetration
of the
molecules
into the
jet core
and/or
collisionsand
of towith
been narrowed
to the
effects
by background
the broaden and
merged shock
waves
with the
jet boundary
expanding
molecules.
l/Kn=5QO
the attainable
number
becomesinto
lower
andcore
the supersonic
regionofbecomes
the background
gasFor
effect
by penetration
of the Mach
background
molecules
the jet
and/or collisions
with
narrower.
expanding molecules. For 1/Kn=500 the attainable Mach number becomes lower and the supersonic region becomes
narrower.
1/Kn=250G, pressure ratio=16000
1/Kn=500, pressure ratio=16000
20
15
10
5
0
Figure
4. Mach
number
distribution
a jetat/?^
at p07/p
for 1/^=2500
1/Kn=2500 (left)
1=16000 for
Figure
4. Mach
number
distribution
ofofa jet
=16000
(left)and
and500
500(right).
(right).
740
In Figure 5 the density distributions on axis at different pressure ratios for l/Kn=25QQ are shown. The expected
In Figure
5 the
density
onby
axis
at different
for with
1/Kn=2500
are shown.
The expected
locations
XM of the
Mach
diskdistributions
are indicated
lateral
lines. pressure
It can be ratios
seen that
increasing
the pressure
ratio the
locations
of
the
Mach
disk
are
indicated
by
lateral
lines.
It
can
be
seen
that
with
increasing
the
pressure
ratiowith
the
x
density begins
M to increase upstream of XM and the supersonic region has been relatively narrowed in comparison
begins
to increase
of xtheory.
and
the
supersonic
region
has
been
relatively
narrowed
in
comparison
with
thedensity
expected
region
from theupstream
continuum
This
is
more
clearly
seen
from
the
temperature
distribution
shown
in
M
the expected
from the
continuumbetween
theory. the
Thisparallel
is moreand
clearly
seen from the
distribution
in
Figure
6, whereregion
an average
temperature
perpendicular
onestemperature
is shown, and
from theshown
velocity
Figure 6, where
an average
between
the parallel
and perpendicular
ones isinto
shown,
and from
the velocity
distribution
on axis,
shown intemperature
Fig.7. In these
figures
flow properties
for expansion
vacuum
are also
shown.
distribution
shown into
in Fig.7.
In these
figures
flowway
properties
forbackground
expansion into
vacuum
are The
also average
shown.
They
follows on
the axis,
expansion
vacuum
in almost
same
until the
effects
appear.
They follows
thetoexpansion
into vacuum
in almost
samethe
way
until begins
the background
temperature
begins
increase upstream
the location
where
density
to increase.effects appear. The average
temperature begins to increase upstream the location where the density begins to increase.
EH
0.1
EH
I 0.01
£
>1
3rd 0.1
M
CD
0.001
a
-P
i 0.0001
0.01
Q
1e-05
10
100
10
Axial distance (x/d)
Axial distance (x/d)
Figure
5. 5.
Density
profile
ononaxis.
Figure
Density
profile
axis.l/Kn=250Q.
1/Kn=2500.
Figure
l/Kn=25QO.
Figure6.6.Average
Averagetemperature
temperature profile
profile on
on axis.
axis. 1/Kn=2500.
=
In In
thethecase
16000, when
caseofofl/Kn=25QQ
1/Kn=2500and
andP(/PI
whenthe
theorifice
orificediameter
diameter d=l
mm the
the pressure
pressure in
in the
the expansion
expansion
p0/p1=16000,
d=1 mm
chamber
will
be
1
Pa.
At
this
background
pressure
the
effect
to
the
supersonic
jet
is
still
severe;
the
attainable
Mach
chamber will be 1 Pa. At this background pressure the effect to the supersonic jet is still severe; the attainable Mach
number
will
be
less
than
20.
In
case
of
l/Kn=5QQ
the
background
pressure
will
be
0.2
Pa
with
the
same
orifice,
number will be less than 20. In case of 1/Kn=500 the background pressure will be 0.2 Pa with the same orifice, the
the
attainable
Mach
number
becomes
This
rarefaction
and
background
gas
effect
can
be
evaluated using
using
attainable
Mach
number
becomesabout
abouthalf
half(11).
(11).
This
rarefaction
and
background
gas
effect
can
be
evaluated
05
a local
Knudsen
number
rarefaction
a local
Knudsen
numberdefined
definedbybyKn^Knfa^pj)
theinverse
inverseof
ofwhich
which isis essentially
essentially the
the same
same as
as the
the rarefaction
KnL=Kn(p0/p' 1,)0.5,the
parameter
.
Here,
we
assume
the
supersonic
expansion
distance
as
a
distance
x
,
where
the
velocity
10%
09
falls 10%
parameter ξ. Here, we assume the supersonic expansion distance as a distance x0.9, where the velocity falls
from
the
value
obtained
for
the
expansion
into
a
vacuum.
The
relative
supersonic
expansion
distance
on
axis
to
the
from the value obtained for the expansion into a vacuum. The relative supersonic expansion distance on axis to the
expected
location
of
the
Mach
disk
X
is
plotted
as
a
function
of
l/Kn
and
in
Figure
8.
It
can
be
seen
that
the
L L and ξ in Figure
seen that the
expected location of the Mach disk MxM is plotted as a function of 1/Kn
rarefaction
and
background
gas
effects
on
the
jet
core
length
can
be
evaluated
by
Kn
or
for
wide
ranges
of
the
rarefaction and background gas effects on the jet core length can be evaluated by KnLL or ξ for wide ranges of the
stagnation
Knudsen
number
stagnation
Knudsen
numberand
andthe
thepressure
pressureratio.
ratio.
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Axial distance (x/d)
Figure8.8.Rarefaction
Rarefaction effect
effect to
tojet
jet core
core length.
length.
Figure
Pressureratio
ratioranges
ranges from
from 15
15 to
to 16000.
Pressure
16000.
Figure
Axial
velocity
profile.l/Kn=2500.
1/Kn=2500.
Figure
7. 7.
Axial
velocity
profile.
741
The density field of the supersonic jet is often approximated by the following equation (Boyntons's formula) [7],
(x,0) = cos2
cos7(
12 J,
where =tan1(y/x), / = 2/( -1) , and m= {[( +!)/( -l)]1/2-l}/2 for the gas with the specific heat ratio .In
Figure 9 radial density distributions at different axial distances for the jet of p(/p1=l6QOQ and l/Kn=25QO are
compared with above equation. It can be seen that the Boyntons's formula is a good approximation for the off-axis
density distribution in a certain region, which depends on the flow condition.
20
40
radial distance (y/d)
Figure 9. Radial density distribution at different axial locations.
and l/Kn=25QQ.
In order to compare the calculation with the experimental observation using the laser-induced fluorescence (LIF)
method [8] the calculated results were used to produce a pseudo-LIF intensity distribution. In Figure 10 the pseudoLIF intensity is compared with the LIF visualization. Although the comparison is only relative, because the
sensitivity of the film in not linear, the pseudo-LIF intensity distribution gives a similar image as observed by the
LIF method.
PQ/P=8000
20
60
x/d
80
100
120
J__|
10 20 30 40 50 60 70 80 90 100 110
x/d
Figure 10. The pseudo-LIF intensity compared with the corresponding LIF photograph.
In a rarefied jet it is well known that the parallel translational temperature does not lowered from a certain value,
which depends on the stagnation condition p^d or the stagnation Knudsen number Kn, due to not enough collisions
during the expansion. The terminal speed ratio S// ^ defined by this terminal parallel temperature (S// =u /(2kT// ^
742
/m)°'5, with u ^is the isentropic terminal velocity, k the Boltzmann constant and m the atomic mass) is given as S// L
=l.QlKnQA [6] using a sudden freezing model. In order to confirm the validity of the present calculation method and
to demonstrate the effect of the background molecules to the expansion, calculation was made for the expansion into
vacuum; without interaction with background molecules for a wide range of the stagnation Knudesn number, 4x10"40.1. The calculated parallel and perpendicular temperatures are shown in Figure 11. Both temperature can be
assumed to be equilibrium at the orifice for Knudsen numbers smaller than 0.01. The parallel temperature tends to
freeze and the attainable parallel temperature (the terminal parallel temperature, T// ) decreases with decreasing Kn,
whereas the perpendicular temperature continues to decrease in an almost similar manner for different Kn smaller
than 0.01. For expansions with the stagnation Knudsen number larger than 0.01 the temperature non-equilibrium
already occurs in the upstream region of the orifice. By using the definition of the speed ratio and by assuming that
the flow velocity has reached to its terminal value, the terminal parallel temperature can be written as T// L
/T0=2.18Kn°-8 from the above equation. In Figure.12 the terminal parallel temperatures are plotted against l/Kn and
are compared with the equation reduced from the sudden freezing model. The calculated values agree well with the
existing relation and then the present DSMC calculation method proves to be a good simulation of the supersonic
jets in a wide flow conditions.
100
10
10000
Axial distance (x/d)
Figure 12. Terminal parallel temperature as a function of
inverse of Knudsen number; comparison with the sudden
freezing model.
Figure 11. Parallel and perpendicular temperature on axis
for expansion into vacuum.
CONCLUSIONS
We have applied the DSMC method to simulation of the supersonic free jet issuing from a thin circular orifice
and have obtained the following conclusions.
1. By adopting the different time-step scheme a large flow field could be treated and a jet of a very large pressure
ratio, up to 16000, could be simulated.
2. The rarefaction and background gas effects to the jet size can be evaluated quantitatively by the local Knudsen
number defined by Kn^Knfa^pjf'5 or the rarefaction parameter introduced by Muntz[4].
3. The empirical equation (Boyntons's formula[7]) for the radial density distribution is a good approximation for the
off-axis density in a limited region, which depends on the flow condition.
4. Computation of the expansion into vacuum for a wide range of the stagnation Knudsen number showed that the
dependence of the frozen parallel temperature to the stagnation Knudsen number agrees well with the sudden
freezing model [6].
5. Above conclusions give the validity of the present DSMC calculation method to simulate the supersonic jets in
wide ranges of the pressure ratio and the stagnation Knudsen number.
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REFERENCES
1. Teshima, K. , and Usami, M., An Experimental Study and DSMC Simulation of Rarefied Supersonic Jets, Rarefied Gas
Dynamics, ed. by C. Shen, Beijin University Press, pp.567-572, 1997.
2. Ashkenas, H. and Sherman, F.S., The Structure and Utilization of Supersonic Free Jets in Low Density Wind Tunnel,
Rarefied Gas Dynamics, Fourth Symposium, Vol.11, Academic Press, New York, 1966, pp.84-95.
3. Anderson J.B. , Molecular Beams from Nozzle Sources, in Molecular Beams and Low Density Gas Dynamics, Wegener, P.P
ed., Dekker, N.Y. 1974.
4. Muntz, E.P., et al., Some Characteristics of Exhaust Plume Rarefaction, AIAA J. 8, 1984, pp.1651-1658.
5. Brook, LJ.W., Hamel, B.B., and Muntz, E.P., Theoretical and Experimental Study of Background Gas Penetration into
Underexpanded Free Jets, Phys. Fluids, Vol.18, 1975, pp.517-528.
6. Anderson, J.B. and Fenn, J.B., Phys. Fluids, 8, 1965, p.780.
7. Boynton, F.P. AIAA J. 5, 1967, p. 1703.
8. Teshima, K. and Nakatsuji, H., Visualization of Rarefied Gas Flows by a Laser Induced Fluorescence Method, Rarefied Gas
Dynamics, ed. by Oguchi, H., University of Tokyo, 1984, pp.447-454.
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