3D DSMC Simulation of Rarefied Hypersonic Flow
over a Sharp Flat Plate
>H
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skHc
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H. Yamaguchi , N. Tsuboi , Y. Matsumoto
* Department of Mechanical Engineering, the University of Tokyo,
7-3-1, Hongo, Bunkyo, Tokyo 113-8656, Japan
** Research Division for Space Transportation, Institute of Space and Astronautical Science,
3-1-1, Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan
Abstract. The flow over a sharp flat plate with a finite leading edge angle located in a rarefied hypersonic flow has been
investigated numerically with the direct simulation Monte Carlo (DSMC) method using the Dynamic Molecular Collision
(DMC) Model based on Molecular Dynamics (MD) simulation of nitrogen molecules. The results of the two-dimensional
numerical simulations of the zero leading edge angle plate agree fairly well with the experimental results [Lengrand, J.C.
et al. (1992)] in the middle cross section of the span direction. Thus to investigate much more precisely about the detailed
physical phenomena in the rarefied hypersonic flow over the sharp flat plate, the three-dimensional numerical simulations
have been made. At first, to compare with two-dimensional simulation, the effect of the existence of the finite leading
edge angle - "the leading edge effect", and the effect of the three-dimensional flow structure due to the side edge of the
plate - "the finite span effect" are investigated. From these numerical simulations, the flow field around the plate is
shown three-dimensionally, and the effects are evaluated to determine the three-dimensional flow structure over the plate.
INTRODUCTION
Many researchers have investigated the rarefied hypersonic flows around space vehicles with respect to the
nonequilibrium characteristics. The external flow field and the boundary layer growth around a body of given shape
have the mutual interaction called viscous interaction. This viscous interaction affects the flight characteristics of the
vehicles. Thus it is important to analyse this phenomenon. In rarefied regime, the shock wave and the boundary layer
"merge" near a leading edge, and in that region, it becomes difficult to distinguish the shock wave from the
boundary layer. This region is called the merged layer. A strong nonequilibrium between translational and internal
degrees of freedom of molecules could be observed in this region. However the nonequilibrium in the merged layer
has not been discussed in detail.
In numerical simulations, the direct simulation Monte Carlo (DSMC) method [1] is widely used to simulate the
rarefied gas flows. The gas-gas interaction model has very important role in this method. To investigate
nonequilibrium between the translational and the internal energy of diatomic molecules, the relaxation of the internal
degrees of freedom should be simulated in detail. Thus the molecular collisions, which is micro-scale phenomena,
and the macro-scale phenomena of the flow field around a body of given shape, should be treated at one time,
because micro-scale phenomena affect macro-scale phenomena and vice versa. The analysis should be done over
multi scales, micro to macro. Thus the Dynamic Molecular Collision (DMC) model [2] is chosen as the collision
model. The DMC model is derived from a lot of Molecular Dynamics (MD) simulations of diatomic molecule (N2)
two-body collisions. It is the statistical model based on micro-scale phenomena for the macro-scale analysing
method. The schematic diagram of their relations is shown in Fig.l.
To make the analysis easier, a sharp flat plate is chosen as an object. The another reason of deciding an object is
that there exists the experimental result [3]. To investigate the merged layer in detail, the numerical simulation is
made and compared with the experimental result. There are two elements that could affect the three-dimensional
flow structure around the object - the finite leading edge angle and the existence of the side edge of the plate. Thus
three-dimensional and two-dimensional numerical simulations have made and compared to investigate these effects
- the leading edge effect and the finite span effect.
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
764
Rarefied Gas
Shock wave
Nonequilibrium flow
Relaxation of
molecular collisions
Interaction between
atomic nucleus and electron
V
Modelling
Probability
Translation^/
Rotational energy
Collision model
FIGURE 1. Schematic diagram of the present multi-scale analysis model.
FIGURE 1. Schematic diagram of the present multi-scale analysis model.
NUMERICAL METHOD AND CONDITIONS
NUMERICAL METHOD AND CONDITIONS
To make numerical simulations, the DSMC method is applied. In the DSMC method, relaxation of diatomic
To make numerical simulations, the DSMC method is applied. In the DSMC method, relaxation of diatomic
molecules in nonequilibrium flow is the state of art. From the view of the multi scale analysis, the DMC model,
molecules in nonequilibrium flow is the state of art. From the view of the multi scale analysis, the DMC model,
which is derived from a lot of MD simulations of diatomic molecules (N2) two-body collisions, is chosen as a
which is derived from a lot of MD simulations of diatomic molecules (N ) two-body collisions, is chosen as a
collision model of diatomic molecules. The collision frequency is calculated2with Null-Collision method [4]. For the
collision model of diatomic molecules. The collision frequency is calculated with Null-Collision method [4]. For the
gas-surface interaction, the diffuse model is applied as the conclusion of Ref.5.
gas-surface interaction, the diffuse model is applied as the conclusion of Ref.5.
The
flow conditions and the plate sizes are same as those of experiment in Ref.3, and listed in table 1. The
The flow conditions and the plate sizes are same as those of experiment in Ref.3, and listed in table1. The
coordinate
all over
over the
the plate
plate belongs
belongstotothe
themerged
mergedlayer
layerregime
regimeininthis
thiscondition.
condition.
coordinateisistaken
taken as
as Fig.l.
Fig.1. Almost
Almost all
The
numerical
simulation
is
made
with
four
ways.
The numerical simulation is made with four ways.
a)a) 2-dimensional
edge angle
angle (infinitely
(infinitely thin
thinplate
plateapproximation)
approximation)
2-dimensional with
with zero
zero leading
leading edge
b)b) 2-dimensional
with
finite
leading
edge
angle
2-dimensional with finite leading edge angle
c)c) 3-dimensional
edge angle
angle (infinitely
(infinitely thin
thinplate
plateapproximation)
approximation)
3-dimensional with
with zero
zero leading
leading edge
d)d) 3-dimensional
leading edge
edge angle
angle
3-dimensional with
with finite
finite leading
Three-dimensional
much memory
memory and
and time.
time.Thus
Thusthree-dimensional
three-dimensionalcase
caseisissimulated
simulatedwith
with
Three-dimensional calculation
calculation needs
needs too
too much
parallel
computing.
parallel computing.
Y
X
Z
FIGURE
for the
the numerical
numerical simulation.
simulation.
FIGURE2.
2. Coordinate
Coordinate system for
765
TABLE 1. Flow Conditions and Plate Sizes.
TABLE 1. Flow Conditions and Plate Sizes.
Properties
Properties
M∞
TTolK]
0 [K]
Pol[Pa]
Pa]
P
0
TT»[K]
[K]
∞
P*[Pa]
P
∞ [Pa]
TT*[K]
w [K]
Re
(based
on 50mm)
50mm)
Re (based on
Kn
(based
on
50mm)
Kn (based on 50mm)
Length
of
the
plate
[mm]
Length of the plate [mm]
Width
of
the
plate
[mm]
Width of the plate [mm]
Thickness of
of the
the plate
plate [mm]
[mm]
Thickness
Leading edge
edge angle[deg.]
angle[deg.]
Leading
Moo
Ref.[3],[6]
Ref.[3],[6]
20.2
20.2
1100
1100
3.5-1055
3.5·10
13.32
13.32
0.06839
0.06839
290
290
566
566
0.047
0.047
100
100
100
100
55
20
20
PARALLEL COMPUTING
COMPUTING
PARALLEL
ItIt takes
cost when
when three-dimensional
three-dimensional simulation
simulationisis done.
done.Thus
Thusparallel
parallelcomputing
computingisis
takes much
much time
time and
and computing
computing cost
applied
DSMC code
code isis modified
modified with
withMPI
MPI(Message
(MessagePassing
PassingInterface)
Interface)library.
library.
applied to
to that
that case.
case. The
The three-dimensional
three-dimensional DSMC
When
there
is
a
need
to
transfer
data
from
one
process
to
another
process,
the
data
is
restructured
as
a
matrix
and
When there is a need to transfer data from one process to another process, the data is restructured as a matrix and
total
transfer
data
length
is
shorten.
total transfer data length is shorten.
The
are Hitachi
Hitachi SR2201,
SR2201, IBM
IBM RS/6000
RS/6000 SP-2,
SP-2, Visual
Visual Technology
TechnologyAlpha-21164A
Alpha-21164A600MHz.
600MHz.
The parallel
parallel computers
computers are
The
last
computer
is
not
parallel
machine,
but
workstation
cluster,
which
are
connected
with
100Base-TX
Ethernet
The last computer is not parallel machine, but workstation cluster, which are connected with 100Base-TX Ethernet
network.
to calculate
calculate speed
speed up
up is
is only
only CPU
CPU time.
time. ItIt does
doesno
no include
includeI/O
I/Oprocess.
process.Almost
Almostallall
network. The
The computing
computing time
time to
cases,
achieved. From
From this
this result,
result, this
this DSMC
DSMC parallel
parallel computing
computingcode
codeshows
showsvery
verygood
good
cases, the
the linear
linear speed-ups
speed-ups are
are achieved.
performance
on the
the system
system nor
nor architectures
architectures of
of the
the machine.
machine.
performance and
and does
does not
not depend
depend on
This
used for
for the
the three-dimensional
three-dimensional simulation
simulationwritten
writtenbelow.
below.
This parallel
parallel computation
computation is
is used
1
I I I I I 111
SR2201 " :
SP-2
VT-Alpha
ideal
a.
•o
CD
10
8.
03
1
1
2
4 6 8
46
10
number of processors
FIGURE 3.
3. Parallel speed-up of
FIGURE
of the
the 3D
3D DSMC
DSMC code
codeto
tothe
thenumber
numberofofprocessors
processorsused
usedtotocalculate
calculate(log
(logscale).
scale).
766
RESULTS
RESULTS AND
AND DISCUSSIONS
DISCUSSIONS
ToTocompare
that the
the condition
condition of
of the
the numerical
numerical
comparethe
theexperimental
experimental results
results and
and the
the numerical
numerical results,
results, itit is
is better
better that
simulation
is
simple.
Thus
ordinary
simulation
is
made
with
some
approximations.
In
this
case,
the
hypersonic
simulation is simple. Thus ordinary simulation is made with some approximations. In this case, the hypersonic
rarefied
infinitely thin
thin plate
platecondition.
condition.Now,
Now,
rarefiedgas
gasflows
flowsaround
aroundthe
theflat
flatplate
platecould
could be
be simulated
simulated with
with two-dimensional
two-dimensional infinitely
from
simulation is
is checked.
checked.
fromthe
theview
viewofofthe
themulti
multiscale
scaleanalysis,
analysis,the
the validity
validity of
of numerical
numerical simulation
Four
Fourcases
casesare
aresimulated
simulatedon
onthe
thesame
samecondition
condition as
asthe
the experiment
experiment [3] -–
a)a) 2-dimensional
approximation)
2-dimensionalwith
withzero
zeroleading
leadingedge
edge angle
angle (infinitely
(infinitely thin plate approximation)
b)b) 2-dimensional
with
finite
leading
edge
angle
2-dimensional with finite leading edge angle
c)c) 3-dimensional
approximation)
3-dimensionalwith
withzero
zeroleading
leadingedge
edge angle
angle (infinitely
(infinitely thin plate approximation)
d)d) 3-dimensional
3-dimensionalwith
withfinite
finiteleading
leadingedge
edge angle.
angle.
AtAtfirst,
numerical simulation
simulation with
with finite
finite
first,the
thedensity
densitydistributions
distributionsofofone
onecase
case of
of the
the results
results of
of three-dimensional
three-dimensional numerical
leading
clear and
and the
the size
size of
of the
the
leadingedge
edgeangle
angleare
areshown
shownininFig.4.
Fig.4.From
Fromthem,
them, the
the special
special density
density distributions become clear
leading
leadingedge
edgeeffect
effectand
andthe
thefinite
finitespan
spaneffect
effectmight
might be
be appears.
appears.
To
and the
the finite
finite span
span effect,
effect, the
the
Todetermine
determinethe
thesize
sizeof
of three-dimensional
three-dimensional effects
effects -– e.g.
e.g. the leading edge effect and
density
left side
side four
four contours,
contours,not
not aa
densitycontours
contoursofofthe
theflow
flowfield
field are
are compared
compared between
between four
four cases (Fig.5). From the left
large
shows different
different
largedifference
difference could
could be
be seen.
seen. But
But the
the right
right side
side contours
contours of
of the cross-sections at XX/LL ==1\.5
.5 shows
distribution
of the
the contour
contour map
map should
shouldbe
be
distributionaround
aroundZ/L
wherethe
the side
sideedge
edge of
of the
the plate
plate exists. The difference of
Z L==1.0,
1.0 ,where
the
comparison isis made
made atatthe
themiddle
middle
theeffect
effectofofboundary
boundarycondition
condition that
that isis applied
applied to
to the
the numerical
numerical simulation. The comparison
thusititisisnot
not affect
affect the
the conclusion.
conclusion.
crosssection
sectionofofthe
thespan
span Z/L
Z L==0.0,
0.0 ,thus
cross
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.2
0.0
0,5
0.00
1.0
0.25
0,50
0.75
1.00
1.25
Z<t
X/L
ML
Dens ity distributions
FIGURE4.4. Density
Density contours
contours over
over the
the flat
flat plate
plate (upper
(upper left:
L ==00,, upper
FIGURE
left: cross
cross section
section atatZZ/L
upper right:
right: cross
cross
section atat X/L
lower left:
left: cross
X L==1.5
1.5 ,, lower
L ==00 ,, lower
section
cross section
section at
at YY/L
lower right:
right:special
specialdistribution
distributionover
over
the
half
width
of
the
plate).
the half width of the plate).
767
•'$>r
1.0
0.8
,1 *0
O>
!
0.6
^IIE^* """~~'"T~Z^^$;i.4:t5.
0.4
^^lt-.4- -..-....- -^g^.™ ^^5 ^
Jf
'
^^^^^^.feSf
0.2
^?/
>^i
0.0
_.....^!!....|t
0.0
1.0
0.5
1.5
2.0
1.5
2.0
X/L
a) 2D
2D infinitely
infinitely thin
thin plate
plate
a)
1.0
0.8
0.6
0.4
0.2
o.o
0.0
1.0
0.5
X/L
b) 2D
2D with
with leading
leadingedge
edgeangle
angle 20deg.
20deg.
b)
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
9
:o^°;ro.7^
.....i......gr:...............l.
o.o
o.o
1.0
0.5
1.5
2.0
_rr-rr1.4
0.2
1 2
-:i......-3:^T
- q
..':!; 0.9 ..::0
1
::::t.^Sr::: ^^^
0.0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
1.50
1.75
Z/L
X/L
c) 3D infinitely thin plate
c) 3D infinitely thin plate
c) 3D infinitely thin plate
c) 3D infinitely thin plate
1.0
1.0
0.8
1.1-
0.8
0.6
0.6
0.4
0.4
0.2
0.2
o.o
o.o
1.0
0.5
1.5
2.0
0.0
0.00
0.25
0.50
0.75
1.00
1.25
Z/L
X/L
d) 3D with leading edge angle 20deg.
d) 3D with leading edge angle 20deg.
d) 3D with leading edge angle 20deg.
d) 3D with leading edge angle 20deg.
FIGURE 5.
5. Density
Density contours
contours over
over the
the flat
flat plate
plate (from
(from top:
top: condition
condition a),
a),b),
b),c),
c),d);
d);left:
left: cross
crosssection
section
FIGURE
at
Z/L
=
0,
right:
cross
section
at
X/L
=
1.5
).
at Z L = 0 , right: cross section at X L = 1.5 ).
768
1.0
«lr- 2DleOO~
2Dle20
0.8
SDIeOO3Dle20
Exp.
I
0.6
0.4
0.2
0.0
0.8
1.0
1.2
1.4
P/Pinf
--Jr- 2DleOO
-A- 2Dle20
34x10"
16
SDIeOO
3Dle20
Exp.
32
14
30
12
28
10
8
Jr- 2DleOO
2Dle20.
- SDIeOO
26
3DI020Exp.
22
6
I
0.0
0.5
1.0
1.5
24
20
2.0
0.0
X/L
0.5
1.0
1.5
2.0
X/L
FIGURE 6.
6. Density
Density profiles
profiles over
over the
the plate
plateatat XX/L
(upper),Pressure
Pressureprofiles
profilesononthe
theplate
plateatatZ Z/L=Q
FIGURE
L ==11.5
.5 (upper),
L=0
(lower left),
left), Heat
Heattransfer
transferrate
rateprofiles
profileson
onthe
theplate
plateatatZZ/L
(lowerright).
right).
(lower
L ==0 0(lower
(Exp.:[Lengrand,
[Lengrand,J.C.
J.C.etetal.
al.(1992)
(1992)[3]]).
[3]]).
(Exp.:
Now quantitative
quantitative comparisons
comparisonsbetween
betweenfour
fourcases
casesand
andexperimental
experimentalresults
results[3]
[3]are
aremade
madetotodetermine
determinethethesize
sizeofof
L ==1.1.5
5 , ,pressure
L ==
0 ,0,heat
the effects. Density
Density profiles
profiles over
over the
theplate
plateatat XX/L
pressureprofiles
profilesononthe
theplate
plateatatZ Z/L
heattransfer
transferrate
rate
L ==00 are
profiles on
on the
the plate
plate atat ZZ/L
are compared
compared ininFig.6.
Fig.6.The
Thepressure
pressuremeans
meansthe
thenormal
normalstress
stressFnFndeduced
deducedfrom
from
molecular momentum
momentum exchange,
exchange, not
notthermodynamic
thermodynamicpressure
pressurep.p.ItItisissame
sametreatment
treatmentasasRef.3.
Ref.3.From
Fromthem,
them,several
several
things could be
be apparent.
apparent. The
The density
density profiles
profiles show
show very
very good
good agreement
agreementbetween
betweenexperimental
experimentalresults
resultsand
and
numerical results.
results. The
The properties
propertieson
onthe
theplate
plateagree
agreewell
wellnear
nearthe
theleading
leadingedge
edgebut
butlarge
largediscrepancy
discrepancycould
couldbebeseen
seeninin
most part of
And
it it
of the
the downstream
downstreamregion.
region.The
Theprofile
profileofofthe
theexperimental
experimentalpressure
pressureshows
showsa trend
a trendofofdiscontinuity.
discontinuity.
And
is very rarefied condition
condition on
on and
andnear
nearthe
theplate.
plate.Thus
Thusthe
theresults
resultsofofexperiment
experimentand
andnumerical
numericalsimulation
simulationagree
agreewell
well
in reasonable
reasonable way
way for
for this
this condition.
condition.
Comparisons
Comparisons should
should be
be done
done between
between numerical
numericalresults
resultsininthe
thenext
nextstep.
step.From
Fromthe
thefigures,
figures,it itisisclear
clearthat
thatthethe
discrepancy between
between four
four lines
linesisis not
notsosolarge
largeasasthat
thatbetween
betweenexperimental
experimentalresult
resultand
andnumerical
numericalresult.
result.But
Buttotoseesee
much more
more closely,
closely, itit appears
appears that
thatfour
fourlines
linesare
areseparated
separatedtototwo
twosets,
sets,continuous
continuousand
anddotted
dottedline,
line,not
nottriangle
triangleand
and
diamond-shaped markers.
markers. ItIt means
meansthat
thattwo-dimensional
two-dimensionaland
andthree-dimensional
three-dimensionalresults
resultsdodonot
notshow
showany
anydiscrepancy
discrepancy
and zero
zero degrees
degrees leading
leading edge
edgeangle
angleand
andfinite
finiteleading
leadingedge
edgeangle
anglesimulation
simulationresult
resultdifferent
differentininthe
thecross
crosssection
sectionofof
769
0.75
0.50
0.50
0.25
0.00
0 . 2 5 0.50
0.75 1.00 1.25 1.50 1.75
0.00
0.25 0.50
0.75 1.00 1.25 1.50 1.75
FIGURE7.7. Pressure
Pressurecontour
contour map
map on
on the
the plate (left), Heat transfer
FIGURE
transfer rate
rate contour
contour map
map on
onthe
theplate
plate(right).
(right).
thecentre
centreofofthe
theplate
plateinin span
span direction.
direction. This
This means
means that
that the
the leading
the
leading edge
edge effect
effect could
could be
be affect
affectthe
theflow
flowfield.
field.But
Butinin
thiscondition,
condition,the
the external
external flow
flow isis rarefied
rarefied to
to Kn=0.047,
Kn=0.047, hypersonic
hypersonic to
this
to M-20.2
M=20.2 and
and the
the leading
leadingedge
edgeangle
angleisissmall
small
20deg.
deg.and
andthe
theaspect
aspectratio
ratioisis1.7.All
All of
of them
them make
make these
these effects
effects small.
asas20
small. Thus
Thus itit could
could be
beconcluded
concludedthat
thatthe
theleading
leading
edgeeffect
effect and
and the
the finite
finite span
span effect
effect isis negligible
negligible in
in this
this condition.
condition. The
edge
The leading
leading edge
edge effect
effect isis larger
largerthan
thanthe
thefinite
finite
span
effect.
But
the
leading
edge
effect
could
be
negligible
around
the
angle
20
deg.
as
Ref.7.
In
Ref.7,
the
span effect. But the leading edge effect could be negligible around the angle 20 deg. as Ref.7. In Ref.7, theleading
leading
edgeangle
angleisisset
setasas15,
75,30,
30,45,
45,60
60and
and80
80deg.
deg. ItItisisconcluded
concluded there
there that
that only
only the
the case
case 15
75 deg.,
edge
deg., the
the properties
properties on
on the
the
plate
is
not
affected.
Thus
in
the
middle
cross
section
of
the
span
direction,
the
flow
could
be
treated
with
plate is not affected. Thus in the middle cross section of the span direction, the flow could be treated withthe
the
approximationofofthe
thetwo-dimensional
two-dimensional flow.
flow.
approximation
But
it
is
clear
that
without
the
middle
cross section
section of
of the
the span
span direction,
But it is clear that without the middle cross
direction, the
the flow
flow field
field isis different
differentfrom
fromthe
theresult
result
of
two-dimensional
simulation.
To
see
these
three-dimensional
effects,
the
contour
maps
of
the
pressure
and
heat
of two-dimensional simulation. To see these three-dimensional effects, the contour maps of the pressure and heat
transfer rate on the plate are shown in fig.7. It is interesting that the region of high value in maps is narrow toward
transfer rate on the plate are shown in fig.7. It is interesting that the region of high value in maps is narrow toward
the leeward of the plate. It is because the external flow passed the side edge affect the flow over the plate. The Heat
the leeward of the plate. It is because the external flow passed the side edge affect the flow over the plate. The Heat
transfer rate contour map shows that the heat transfer is much larger around the side edge. In Fig.5 - right side of
transfer rate contour map shows that the heat transfer is much larger around the side edge. In Fig.5 – right side of
case d) 3D with leading edge angle 20deg. of cross section at X/L = 1.5 , the density around the side edge is higher.
case d) 3D with leading edge angle 20deg. of cross section at X L = 1.5 , the density around the side edge is higher.
It shows another proof that the flow from the side edge affects the flow field over the plate.
It shows
another proof thatflow
the flow
frombecomes
the side edge
affects
field over of
thethe
plate.
The three-dimensional
structure
apparent
withthe
theflow
investigation
three-dimensional numerical
The
three-dimensional
flow
structure
becomes
apparent
with
the
investigation
of
the
three-dimensional numerical
simulation.
simulation.
CONCLUDING REMARKS
CONCLUDING REMARKS
The rarefied hypersonic flows over a sharp flat plate have been investigated with the 3D DSMC numerical
The rarefied
hypersonic
over around
a sharptheflat
plate
with the From
3D DSMC
numerical
simulations.
From
them, the flows
flow field
plate
hashave
been been
showninvestigated
three-dimensionally.
the comparison
simulations.
From them,
the flow fieldresult,
aroundthethevalidity
plate has
beennumerical
shown three-dimensionally.
From though
the comparison
between numerical
and experimental
of the
simulation is confirmed,
there is
between
numerical
and
experimental
result,
the validity
numerical
simulationtheis leading
confirmed,
there
discrepancy
between
them.
To compare
the four
cases of of
thethe
numerical
simulations,
edgethough
effect and
theis
discrepancy
between
To compare
four cases
numerical
leading
edgeofeffect
and the
finite span effect
arethem.
investigated.
Thesetheeffects
couldofbethenegligible
in simulations,
the conditionthe
here,
because
rarefaction,
finite
span effect
investigated.
These
effects
couldaspect
be negligible
in leading
the condition
here, isbecause
of rarefaction,
hypersonic
flow, are
small
leading edge
angle,
and large
ratio. The
edge effect
much lager
than the
hypersonic
edge angle,
large aspect ratio.
leading
edge effect
much lagerFrom
than the
the
finite span flow,
effect.small
The leading
approximation
of theandtwo-dimensional
flowThe
could
be applied
to thisis condition.
profiles
on effect.
the plate
from
the three-dimensional
numerical simulation,
the three-dimensional
flow
structure From
over the
finite
span
The
approximation
of the two-dimensional
flow could
be applied to this
condition.
the
plate is on
affected.
Thefrom
three-dimensional
flow structure
over simulation,
the plate is analysed.
profiles
the plate
the three-dimensional
numerical
the three-dimensional flow structure over the
plate is affected. The three-dimensional flow structure over the plate is analysed.
770
ACKNOWLEDGMENTS
The authors would thank Professor J.C. Lengrand (C.N.R.S - France) for his kindly offer of information about his
experiments.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Bird, G.A., Molecular Gas Dynamics, Clarendon Press, Oxford, 1976.
Tokumasu, T. and Matsumoto, Y., Phys. Fluids 11 (7), 1907-1920 (1999).
Lengrand, J.C., Allegre, J., Chpoun, A., and Raffm, M., 18th Int. Symp. on Rarefied Gas Dynamics, 1992, pp.276-284.
Koura, K., Phys. Fluids 29 (11), 3509-3511 (1986).
Tsuboi, N. and Matsumoto, Y., 22^ Int. Symp. on Rarefied Gas Dynamics, to appear.
Lengrand, J.C., private communications.
Heffner, K.S., Gottesdiener, L., Chpoun, A. and Lengrand, J.C., AIAA 91-1749, (1991).
771