Rarefaction effects on separation of hypersonic laminar
flows
Gennady N. Markelov, Alexei N. Kudryavtsev, and Mikhail S. Ivanov
Computational Aerodynamics Laboratory
Institute of Theoretical and Applied Mechanics
Siberian Division of Russian Academy of Sciences, Novosibirsk, Russia
Abstract. A hypersonic monoatomic gas (argon) flow around a hollow cylinder flare was numerically
simulated for low-to-moderate Reynolds numbers. A shock wave induced by the flare causes the formation of
a separation region, and flow rarefaction near the leading edge affects the separation extent. The continuum
(Navier-Stokes equations) and kinetic (the DSMC method) approaches were used to study the influence of
the Mach number and wall-to-freestream temperature ratio on the flow structure and separation. To take
into account rarefaction effects, slip boundary conditions are used for the Navier-Stokes solver.
INTRODUCTION
The problem of the shock wave/laminar boundary layer interaction has been actively studied lately both
by numerical and experimental methods. This interaction can cause the formation of a separation region and
lead, for example, to a decrease in the control surface efficiency and to an increase in the heat flux on the wall
in the vicinity of the reattachment point.
The simplest example of the flow near an aerodynamic control surface is the flow around a hollow cylinder
flare. This axisymmetric configuration allows one to study the shock wave/ boundary layer interaction and to
avoid 3D effects inherent in a flow near a compression ramp. Numerical analysis of such flows is traditionally
performed using Navier-Stokes equations, and the initial effects of rarefaction are taken into account through
slip velocity and temperature jump. The use of slip conditions is caused by the fact that rarefaction effects
can be observed for slender bodies even for rather high Reynolds numbers (Re = 20,000 — 30,000). Moreover,
the use of the continuum approach for an analysis of flow in the vicinity of the leading edge of slender bodies
is not justified, even if the slip velocity and the temperature jump are taken into account. The flow physics
is considerably complicated in case when flow separation is formed, for example, by a flare, and simulation of
such flows by the continuum approach is questionable due to the strong rarefaction effects in the vicinity of
the leading edge.
The state-of-the-art of the algorithms of the DSMC method (adaptive grids, variable time step, etc.) [1,2]
allows one to simulate flows at such a rather high Reynolds number Re ~ 20,000. This makes realistic the
study of the area of applicability of the continuum approach for numerical analysis of laminar separated flows.
Such studies were started by the authors of [3] and were continued in [4] where the detailed numerical modeling
of axisymmetric shock wave/ laminar boundary layer interaction for the ONERA R5Ch wind tunnel conditions
by using the continuum (Navier-Stokes equations) and kinetic (the DSMC method) approaches was performed.
The DSMC results revealed the presence of a high slip velocity and temperature jump near the wall. Therefore,
it is necessary to use slip conditions for the Navier-Stokes solver. The application of these conditions has a
substantial effect on the entire flow field and allows one to obtain reasonable agreement with the DSMC results
for flow fields and distributions of the pressure and heat-transfer coefficients. However, the NS solver with slip
conditions predicts a greater length of the separation region than the DSMC method. Probably, this difference
is caused by different descriptions of translational/rotational energy exchange for both methods.
To eliminate the effect of rotational-translational exchange, a continuum and kinetic simulation of a laminar
separated flow of a monoatomic gas (argon) was performed in the paper. The effect of rarefaction, Mach
number, and wall temperature on laminar separation was also studied.
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
707
FLOW CONDITIONS AND NUMERICAL APPROACHES
The hollow-cylinder flare configuration was taken from [5]. The hollow cylinder has a sharp leading edge and
is aligned parallel to the oncoming flow. The compression flare is inclined at 30° to the cylinder and is followed
by a hollow cylindrical section. The total length of the model is 0.17 m. The reference length L = 0.1017 m is
the distance from the leading edge to the flare. The computations were carried out only for the upper part of
the model, and the flow inside the model was not considered.
Two test cases were considered with Mach numbers M^ = 10 and 4, respectively. In these cases, the
same Knudsen number as for ONERA wind tunnel conditions Kn^ = X^/L = 9.35 • 10~3 was used (Aoo
is the mean free path in the freestream flow). The Reynolds number Re^ = 19000 and 7600 was based on
the freestream parameters and reference length L. The computations were conducted for a constant surface
temperature and an adiabatic wall. For ONERA wind tunnel conditions, the wall-to-freestream temperature
ratio was TW/T^ = 5.74. The same value of the temperature ratio was used for case 1. Since the Mach
number is much smaller for case 2 (M^ = 4), the simulation results obtained for TW/T00 = 5.74 are close to
the results calculated for an adiabatic wall. Therefore, additional calculations were conducted for this case
with T^/Too = 1.0 to reveal the effect of the wall-to-freestream temperature ratio.
Continuum approach
The Navier-Stokes code [4] used solves the full laminar compressible Navier-Stokes equations with both
no-slip and slip boundary conditions. The code is based on the finite- volume TVD scheme for the convective
terms and the second-order central difference approximation of the diffusive terms. The inviscid fluxes are
computed by the HLLEM Riemann solver [6]. The computations were conducted using the perfect gas model.
The power-law dependence of the viscosity on the temperature p, oc T" was taken for more straightforward
comparison with the results of DSMC simulations. The relations for the slip velocity and temperature jump
deduced by Kogan [7] were used to model slip boundary conditions. Taking into account slip conditions
somewhat complicates the computations for an adiabatic wall since the heat flux is determined with account
of the non- vanishing gas velocity on the wall
where U denotes the velocity component tangential to the solid wall and k is the heat conductivity. The first
term in this expression is always positive. Therefore, the requirement of zero heat flux can be satisfied only by
a negative temperature jump. In this case, after calculating slip velocity, the temperature jump is determined
by the relation
Kinetic approach
The SMILE computational tool [2] was used. It based on the DSMC method [8] and the exact majorant
frequency scheme [9] for modeling collision processes. The variable hard sphere model [8] was assumed for
intermolecular collisions. Gas/surface interaction was simulated using diffuse reflection with complete accommodation energy and diffuse-elastic reflection. The latter means that the magnitude of the molecular velocity
remains constant during reflection, but an angular distribution of the velocity vector follows the cosine-law of
diffuse reflection. As a result, the heat flux to the body surface equals zero. The diffuse-elastic reflection was
used to model an adiabatic wall with the DSMC method.
RESULTS AND DISCUSSIONS
The general structure of the flow is shown in Fig. 1 (Numerical Schlieren picture) with clearly visible shock
waves formed in the flow around a hollow-cylinder flare. The separation shock wave intersects and coalesces
with the reattachment shock wave. The new shock, in turn, intersecs and coalesces with the leading-edge shock
708
0.6
Y/L
0.4
0.2
o
30
0
¡”É\¸+¯§¨‚¬®­z°iºCËm­w£+º²¬®§¨‚§«Ž¦¬®­±‚¸O¨‚§`°i«¹Ð©x§cº®§c­±‚§z¹“©†±x¨‚§z°i¯ º®¬ «§c©R´qÉ`Ë&©xªBº®Ä[§¨zÅ
0
0.5
1
’µ
1.5
X/L
X/L
®¿–À~Á]Â
·“¸º¹§»§¼3½¥¾
FIGURE 1. Numerical Schlieren picture and selected stream lines (NS solver, M^ = 4)
Y/L
Y/L
0.25
Y/L
0.10
0.15
0.20
X/L=0.6
X/L=0.3
X/L=0.9
0.15
0.10
0.05
0.10
0.05
DSMC
NS, np-slip
NS, slip
DSMC
NS, no-slip
NS, slip
0.05
DSMC
NS, no-slip
NS, slip
”¡ ͔§c«+©x¬Ñ±M¥›¦+¨‚ª (º®§c©¾´ [µ
ë\ãÒï—ðHì[Þí+ÞOé¼÷4á+ܔïÒðCÝHé¼Þ‹ï„ódã—Þiï ðHèÞë`èwïÒÞVçwßCÝìmçwàcÝñì’Þçwï—ðHãÒÞBø›ìOçwódÝHï—ðHÞø â7†ÞiáOóHòHøOðñގóñìmè‰àÝHôÐádécã„ì[êôOë”Þià\ì[íOìOÞÐðñóïÒðçÝHçދÝHÞûñá’í(ë;ï—éVïÒâ7ðHÞVïÒç‚ÝHô/ï—ðñáOóæ>çwçÝHÝHÞÞèwãÒÝHÞBádìOódécêÐï—ðHë”øì[ÞiíOóHÞBøOèTÞOï„ùgè`ë`ìmã„ÝHè‰ï—áÐécÝé¼ã—Þiï—ðñìOódàwã—ï„ôŽéVìmíçwï„ÞièwèïÒâHìã—ÞOè‰÷KçwàܔáOÝHðHï„ø è
íè‰Þiï„èßñé¼ìmï„àcó ì’Mçï—ð(ïÒá+í(ðï—èé¼àÞVï„óüøOï—áOï—ððçwÞVé¼àcã—ìOÞiìOé¼çwàwï—ã—áOôðí(ïÒï—ðèwï—âHçwãÒÝñÞï—èï—ð ûCPá’ë ï—øñ÷ àwÞi+øOùHï—ë`áOðYÝH÷Žï„écÝܔÝHìOÞã—èwáèwÝHè‰ádÝñécá’êüëRë\è\ì[èwÞVí+ã—Þ[Þiî’é¼ã—çwìOÞBäÐóï—è‰ðñçwìOàà¾ÞiìOâ7äŽáOòHãÒï—ðñðHóñÞièiìm÷ àôã—ì[ô+ÞVàïÒðçÞVàcìOézçïÒá+ð/ï—ðñódòñéVÞièì
Û
Constant
surface temperature
éëa¼áOÞ&ðCܔéVè‚ÝHçcáOÞ/ìmðñðèwàwç”ï—ÞBóHèwè‰ÞVòHòHà`àwãÒçæoçèìOÝHéVޛßHÞRàéVçwÞiì+Þièwè‰äŽÞVÞðßCë`çwÞiÞBïÒàó#çwìmÝ çwïÒòHð àÞ çÝHï„èLèwÞiHé¼÷çwï—9áOð ï—øOòHë\àÞV“Þ àÞüìmŽðCéVèwó ÝHìmã„á’é¼ëRòñèRã—ìmìçwÞiódó#ÞVçòCìmè‰ï—ï—ã ðHÞi&øõóæ}á+é¼çá+ÝHà äŽÞßñéVìmáOàðï—çwèwï—áOð(ðLòHÐòHámìmäæTðñódó Þiìmðñðñè‰ó ïÒç‚ôê(ï—ßñðHàwÞ¼CáOçùï—ýñé/àãÒÞBÞiìmè`èwßHß7ï—ßHÞiðüézàá+çè‰ïÒÞiì+í+íOécÞVÞVÝHã—àcôOÞiìm÷èãY9æ}éVáOïÒàwàcàáè‰èwçiì è ù
è‰ódÞBÞVézðñçwèwï—áOïÒç‚ðCô&è ïÒðCé¼àÞiìOèwÞiè9ï—ðñþ èwï—ódޔHçùÝHìmޔðñâCó á+òHðñHóH÷ìOܔàwô›ÝHÞÐã—ì[ã—ô+ì+ÞVè‚à4çë`é¼ÝHàá+ÞVèðŽè¾ìmèwßñÞißHé¼çwàwï—ááOìOð/écÝHßCï—ìOðHèø‹è‰ÞBçwè`ÝHÞ`çwÝñâ7àwádá+ódòHô›øOÝèwòHçàwÝHæoìOÞé¼è‰Þ+Þi÷9ßñܔìmàcÝñì’ï—è4çïÒï„á+è9ðç‚ô(àwßHÞiøOï—éiï—áOìmã(ðYæ}÷ á+àT¾ì‹ámçwÝ(ޛô(çwß7ÝñÞVìmàcçè‰á+çwðHÝHï—Þé
ûñìá’è‰ëÓã—ï øOìOÝàwçwá+ã—òHôLðñÝñóÐïÒø+ìÝHé¼ÞVá+à‹ã—óÐßCáâ7è‰á(ïÒçwóHï—ôOáO÷ðYáOï—æ9êOÞ`çÝHï—Þ›ðŽã—çwÞiÝHì+ÞRódéiï—ðHìOøèwÞ`†Þië`ódø+ï²çÞ&ݎè‰ìÝñá(óHécï—ìmêçwáOë”äŽì[íOï„é”Þø+çwÝñì+èìOðMçwùOÝñçwޓÝHÞÿ Ž]è‰á+ãÒäŽí+ÞVÞ¼à4çÝHë`ádïÒóWçwÝ÷¾ðHܔá ÝH†Þ&èwãÒï—òCßè‰Þ“é¼á+ámðñæTódè‰ïÒã—çwï ï—ß/áOðñé¼è4á+ðñßHódàÞiïÒçwóHï—ï—áOé¼ðñçèè
æ}àwáOÞBà\è‰òHçwãÒÝHçÞè¾æ}áOLàódèwáOÞiã—ðñíOèwÞVï²àaç‚ôàwÞBßHódàòñáméVýCÞiãÒèaÞBè‹çwÝHìmÞç èwãÒá+ßCÞámæPçwÝHþÞìOãÒÞBðñìOó ódï—ðHøñM÷ ÞBód¾øOá’Þëaè‰ÞiÝHíOádÞiécàiêŽùìë”ì[èwíOïÒø+ދðHìOïÒýCðñéVóìOð+ô(çï—ÞVódã„óHï 7Þè\iøOàwá(Þiádðñóé¼ÞŽìmø+éVìOàwÞið/ÞVäŽâ7ޓÞVððHç\áOë`çwÞBï²óüçݐï—ðüçÝHçwÞÝñÿގéVàwáèwè
ï²è‰ódçcÞBï èÞézçwVäŽï—àáOÞVìOð ðøOçRðHïÒã—çwÞVòCðHódøOޓçwÝñï„è”èáOódæPÞ¼çë`çÞVÝHïÒàÞ&çwäŽÝèwïÒÞVðHç‚ßñëaÞBìOó/áàìmâ(ódçwôÞVï—áOðñçðèwÝHïÒç‚àÞôÞVø+è‰ßCÞiïÒá+ßñÞBðYìmìmêdùdàcì’èVìmç÷&ðñïÒá+óÜ”ðüÝñçwÝñގàwÞiï—è`èwøOÞiï—ß7áOéVÞiáOðüìOðñêã—óüÞVðñï„ßCè`ømÞBçwìmìmÝgã—äŽêü÷“á+é¼Ü”è‰á+ç`ÝHàwàÞ þ Þièw%ß7áOðñÝñè‰óHïÒá+ø+èRãÒÝHí+çwÞVÞVáLàà”æ}çwìmá+ÝHðñà ÞóüèwçwÞVLÝHßñÞàwìOÞBàÿ è‰ìmòñçwã²ï—çcáOèVðü'÷ èwÝHäÐádÞVécçwêÝHádë\óì[í+ßHÞOàùÞiódìmðñï„ézó ç
0.00
0.0
0.5
1.0
1.5
2.0
2.5
0.00
0.0
0.5
1.0
1.5
2.0
2.5
0.00
0.5
1.0
1.5
ρ/ρ
2.0
2.5
8
ρ/ρ
Ä
¶¿–À
ÅCÆ
· ¸º¹§»§¼3½2.à Density profiles
“
FIGURE
(Moo
= Á\
10)
8
8
ρ/ρ
wave. The line separating the boundary layer and the flow behind the shock waves is also clearly visible. This
6
line coincides with the leading-edge
shock wave in the vicinity of the leading edge, which indicates a strong
®
viscid-inviscid interaction in this flow region. The shock wave/laminar boundary layer interaction induces a
‚
†1,
- which also shows selected streamlines.
separation region clearly visible in9Fig.
)Š‹=njEK‹Œ\Ç!Ž=’ÈC7”}°_ŒE°sÉ‘&°©’G¢ŒŽ=’°
The results presented in this section were calculated using the continuum and kinetic approaches for a
constant surface temperatureT^/T^
1.0
M^
10-
and M^
4,: respectively. First,
„}…7w1„ i =~5.74
Af/c8: andTw/T^
„†…7w1„ i =T
-f for ?
g i = Ÿ
g i = @
7‚
we consider the case withRM^
g i =‰10.
-˜ Figure
‚
32 shows a detailed comparison of density profiles in several cross
sections=Ê
X/L
= 0.3,0.6,
w8d‰
f f  and)0.9.
f x The last cross section passes through the separation region. Note
& that the
density increases inside the boundary layer when approaching the body surface. This is typical for a hypersonic
flow around a cold body.KLike
in the case with a diatomic gas ´
[4],
NS
š
+ :0 the =
solver with no-slip
conditions predicts
a slightly higher position of the leading-edge
shock wave than the DSMC
6
E")( method. The use of slip conditions
for the%
NS
shock wave and yields good agreement with the DSMC
solver reduces the slope of the leading-edge
E")(
results for density profiles at%X/L
However,
a significant difference
can be noted in the cross
Êw8d=Ë0.3
f andR0.6.
Gf  J
.
section
X/L
= 0.9
)Ê
w8dË
f x with two density peaks. The second peak corresponds to the separation shock wave, and
its magnitude is determined by the separation region length. The Ì
NS
% solver and the DSMC
E")( method predict
different
lengths
of
the
separation
region,
and
this
peak
is
almost
30%
higher
for
NS
results.
ƒ
!Í
?%
709
0
2
4
6
T/T <«
8
1 0
1 2
FIGURE 3. Temperature profiles (Moo = 10)
TABLE 1. Separation region length.
Moo = 10
Moo =4
method
NS, no-slip
NS, slip
DSMC
NS, no-slip
NS, slip
DSMC
constant surface temperature
Xreat/L
Xsep/L
0.740
1.30
0.755
1.30
0.79
1.285
0.640
1.28
0.654
1.28
0.69
1.28
adiabatic wall
0.896
0.930
0.85
0.692
0.745
0.71
1.15
1.15
1.22
1.265
1.25
1.265
The results for density and velocity (not shown) profiles are rather close inside the boundary layer. At the
same time, a comparison of the temperature profiles in Fig. 3 shows that the difference between the continuum
and kinetic results reaches 40%. The use of slip conditions allows one to obtain identical values of temperature
near the body surface and decrease the difference inside the boundary layer. However, the temperatures inside
the boundary layer for DSMC and NS with slip conditions still differ by 20%. Obviously, the differences observed
in the boundary-layer structure affect its stability to the action of the pressure gradient. The coordinates of the
points of boundary-layer separation Xsep and reattachment Xreat are listed in Table 1, and it is seen that the
NS solver predicts a greater length of the separation region than the DSMC method. The use of slip conditions
leads to some displacement of the coordinate Xsep downstream and does not affect Xreat. Nevertheless, the
separation region length is still greater than the DSMC results by 10%.
Figure 4 shows the distributed aerothermodynamic characteristics obtained using the continuum and kinetic
approaches. The differences in the pressure Cp and skin friction Cf coefficients, and Stanton number are mainly
determined by two factors. The first one is the different length of the separation region. An earlier separation
of the flow in NS results leads to a lower pressure peak and to a smaller heat flux on the flare surface. The
second factor is the strong nonequilibrium of the flow near the leading edge. Its effect is noticeable up to
X/L = 0.1 and can be related to the fact that the continuum approach is inapplicable even with account of slip
conditions in the vicinity of the leading edge. The measure of this nonequilibrium is the ratio of the streamwise
temperature to the total translational temperature. A 30% deviation of TX/T is observed up to X/L = 0.1
(Fig. 5). The magnitude of TX/T decreases downstream, but is still noticeable at a distance X/L = 0.4, which
corresponds to 400AQQ.
Strong flow nonequilibrium is the reason that the NS solver with no-slip and slip conditions predicts a
significantly greater heat flux (up to a factor of four) than the DSMC method in the vicinity of the leading
edge (Fig. 4). This also leads to a significant decrease in flow temperature downstream and an earlier separation
of the boundary layer.
As the freestream Mach number decreases, the level of flow nonequilibrium decreases too (see Fig. 5). This
can be expected to lead to a better agreement between the NS and DSMC results. The flow around a hollowcylinder flare for M^ = 4 was calculated, and fairly good agreement was actually obtained in terms of density,
velocity and even temperature profiles (Fig. 6). A small difference is observed only in the value of the heat
flux to the surface for 0.6 < X/L > 0.9 (similar to Fig. 4) and in the separation region length (~ 6%). The
NS solver with no-slip and slip conditions predicts a greater length of the separation region than the DSMC
method (see Table 1).
710
Cp
0.5
Cf
0.10
DSMC
NS, no-slip
NS, slip
0.4
St
DSMC
NS, no-slip
NS, slip
0.100
0.08
0.3
DSMC
NS, no-slip
NS, slip
0.05
0.2
0.03
0.1
0.010
\
0.00
0.0
0.0
0.3
¨‚§c©x©x¸+¨‚§R­cª’§ ­c¬®§c«’±
0.6
í
0.9
X/L
1.2
1.5
î
¡\͔¬®©†±x¨‚¬²Á+¸+±‚§z¹“mË ÃO°i¬®«›§¨‚ªV½—¨‚±‚¬®£­§±‚¬®¨‚ªB¯«ªm­c¹mª’¥O§ «(°i­c¯¬®§c¬®­”«’± ­‰£(°¼¨‰°i­±‚§w¨‚¬²©†±‚¬®­c©¾´ mË ±‰°V«’[±‚µªB«Ð«m¸+¯Ád§¨
0.0
0.3
0.6
0.9
0.9
X/L
X/L
1.2
0.0
1.5
0.3
0.6
î
0.9
0.9
X/L
X/L
1.2
1.5
Pressure coefficient
Skin friction coefficient
Stanton number
¶¿–À~ÁµÅºÆ
·“¸C¹§»,D
¼ ½\ï
FIGURE 4. Distributed aerothermodynamic characteristics (Moo = 10)
0. 900
0. 925
1. 025
1. 050
1. ISO
1. 175
·“¸º¹3»,¼D½ð
¡`È4£+§¨‚¯&°iºñ«ªi«§z̒¸+¬²º®¬ Á+¨‚¬®¸+¯´q­cªB«+©†±‰°i«’±T©x¸+¨x½„°i­c§`±‚§c¯¦d§¨‰°¼±‚¸+¨‚§BÅOº®§½Ò±>¿
mŨ‚¬®³B£’±T¿
K¿–ÀÁµÅºÆ
[µ
#¿–ÀÁ2Â
FIGURE 5. Thermal nonequilibrium (constant surface temperature, left - MOO = 10, right - MOO = 4)
Y/L
Y/L
0.35
Y/L
0.25
0.15
Y
0.15
f
X/L=0.3
X/L=0.6
0.20
DSMC
NS, no-slip
NS, slip
0.30
0.25
0.10
X/L=0.9
0.15
0.20
0.15
0.10
0.05
0.10
0.05
DSMC
NS, no-slip
NS, slip
0.00
0.8
1.0
1.2
1.8
2.0
2.2
0.00
0.8
1.0
¡”È7§c¯¦d§¨‰°¼±‚¸+¨‚§”¦O¨‚ª (º®§c©R´
1.2
1.5
T/T
1.8
2.0
Ä
2.2
’µ
0.00
0.8
¶¿–ÀÁ]Â
1.2
1.6
2.0
T/T
2.4
2.8
8
·o¸C¹§»,¼D½~ñ
0.05
8
8
1.5
T/T
DSMC
NS, no-slip
NS, slip
FIGURE 6. Temperature profiles (Moo = 4)
U@ò=±Kó%¢Œ”\ôF7õõ
Adiabatic wall
‹
P
è
„
ï
9
è
‰
è
H
Ý
’
á
`
ë
“
ð
m
ì
7
â
’
á
O
í
O
Þ
i
ù
w
ç
ñ
Ý
\
Þ
w
è
Ò
ã
—
ï
“
ß
V
é
O
á
ñ
ð
H
ó
²
ï

ç
Ò
ï
+
á
ñ
ð
P
è
O
ì
Ò
ã
—
’
á
ë
óçwdÝHÞVÞÐðñèwâCïÒç‚á+ô›òHìmðñðñóHó“ìOàwíOôÞiãÒã„ádì[éVôOï²Þiç‚àiô÷ßHܔàámÝHýCÞÐãÒÞBÝHègÞiæ}ìmá+ç¾à ûñòdöüðñÞiìmà‹ñçwùOÝñìmÞÐðCó&ã—áOÞiçì+ðHÝHódÞaÞ`ï—çwðHódá‹øï áOÞÞBâdVódçcàøOÞVìmޓðCï—ðé¼ßHޔæoàwìmÞBï—ï—è4ódà4á+ï„ìmézâñøOçwèwÞBàÞVÞVóüàÞiíOäŽâ(Þiôó›ÞVðçwáOçYÝHðHÞæ}ã—áOôàéVáOï—ð“ð›çwçï—ÝHð(ìmޔòHðñòHí’ó“ìmä ÿ ã—òHìmޔßHámßñæñàwçáÞVàìOäŽÞiécèwÝß7òHÞVã²ï„çcàcèègì’ï—èwçwð&ÞVòñí+àwçÞVޔÞVàcàï—ìmäðñãÒè‰æ}ègáOï„ódáOã„óÞ æ
ÝHódïÒï Þø+ÝHVàÞVÞVàaðCçwé¼ÝCÞiìmèiðùæ}çáOÝñà\ì’Þ¼ç\öHßHìmàwäŽÞBódßHï„ã—ézÞOçwùÞBçóÝHÞâ(ôÐódï çwÞÝHVދàÞVê(ðïÒðHç”ÞVãÒçwÞiï„ðHé‹ømìOçßHݐßHáOàæYá+çwì+ÝHécÞÝY÷ èwÞV†ßñç`ìOéVàìOìmðçwï—âCáOÞðìOàèÞVè‰ø+òñïÒá+äÐðYÞB÷ óŽ‹çwâ(ÝCí(ì’ï—çaáOÞVòñöHèwìOãÒô+ézù+çï—ãÒðôÐççÝHÝHï—Þè\éVï„ìOèTèwçÞ¾ÝHáOÞ¾æPàé¼Þiá+ì+ðè‰á+çwï—ðŽðòñæ}áOòHà\ä ámçìmÝHðñÞVó à
êܔï—ÝHðHÞVÞVàçwÞ¼ï„é&æ}á+è‰àwï—Þ+ä›ù(òñìã—ìméVçwáOï—äŽáOðßHã—ámÞ¼æ4çÞçÝHéVÞ&áOï—ûñðñá’é¼ë'ï„ódÞiìOðñàwé¼á+ÞòHðñámæKóé¼ìá+ðâ7çwá(ï—ðóHòñôLòHë`ä ïÒçwìmÝ/ðCìOóLðüê(ìOï—ðHódÞ¼ï„çìmï—âCéì’àçwÞiï„éèwòHë”ãÒçìmè`ã—FéiùCìmçwðÝHޛâCÞÝHÞiÞVìmö(çRß7Þiûñé¼òdçwöÞBóWï„èR÷ ï„ódÞVðçwï„éVìOã oðñìmäŽÞVã—ôOù VÞiàwá z÷
>
K%
")(
og i ˆallow
-
ƒ
As is shown above, the slip conditions
one to obtain
fair agreement for NS and DSMC results in terms of
density and velocity profiles for MQQ = 10, and the difference is observed only in the value of temperature inside
m
the boundary layer. The heat flux near the leadingsedge
predicted by the continuum approach is severalfold
ƒ
ƒ kinetic approach. It can be assumed
7; that exactly this is the reason for other
higher
than that predicted by the
é¨ and
®
differences, for example, the different length of the separation region. Obviously, in the case ofcontinuum
kinetic simulation of the flow around a body with an adiabatic wall, the heat flux is identical (namely, zero).
Therefore, a complete coincidence of continuum and kinetic results can be expected.
711
·“¸C¹§»,¼D½~ö
6+000
7.000
¡”È7§c¯¦d§¨‰°¼±‚¸+¨‚§\¶ª¼· §cº²¹L´qÉ`Ë&©xªBº®Ä[§¨zÅ
jÄ
E¿
Oź®§½—±T¿P­cªB«+©†±‰°i«’±T©x¸+¨x½„°i­c§`±‚§c¯¦d§¨‰°¼±‚¸+¨‚§BÅm¨‚¬®³B£’±T¿P°B¹O¬Ò°VÁ(°¼±‚¬²­`·4°iº®Òµ
À
ÁµÅºÆ
FIGURE 7. Temperature flow field (NS solver, MOO = 10, left - constant surface temperature, right - adiabatic wall)
Y/L
0.30
Y/L
0.20
Ojfk
Y/L
0.40
0.35
0.25
X/L=0.3
0.30
0.20
0.10
0.25
0.20
0.15
0.15
0.10
0.05
0.10
DSMC
NS, no-slip
NS, slip
0.00
X/L=0.9
X/L=0.6
0.15
0
5
DSMC
NS, no-slip
NS, slip
0.05
10
20
25
30
0.00
0
5
10
15
T/T
20
25
30
Bµ
0
5
Ä
¶¿ À µ
Á źÆ
·“¸C¹§»,¼D½÷
FIGURE 8. Temperature profiles (Moo = 10)
Y/L
0.30
Y/L
0.20
0.00
DSMC
NS, no-slip
NS, slip
DSMC
NS, no-slip
NS, slip
0.25
X/L=0.3
0.20
10
15
T/T
20
25
30
8
¡”È7§c¯¦d§¨‰°¼±‚¸+¨‚§”¦O¨‚ª (º®§c©¾´
8
8
15
T/T
0.05
Y/L
0.40
DSMC
NS, no-slip
NS, slip
0.35
0.15
DSMC
NS, no-slip
NS, slip
0.30
0.25
X/L=.06
0.10
X/L=0.9
0.20
0.15
0.15
0.10
0.05
0.10
0.05
0.05
¡ g§cº®ª’­c¬Ñ±M¥›¦O¨‚ª (º®§c©R´ Bµ
ë\ìLìOéVÜPãÒáOWá`ðñæ}íOá+è‰çÞià ìOàwð+ïÒæ}çôRè‰çòñÝHà‰ï—èPæoì+ìO鼎èޓè‰òñã—çwÞiäÐÞVìOäŽóñßHè`ß7çwï—ÞVçwáOàcáLðYì’ù[çìòHé¼á+è‰àwï—Þ+äÐøOù7ðñßñçwï²òdÝHýCçÞÐéiìmìmçwçwðï—ÞiáOçRäŽðñï—ßCèðñÞié¼æ}á+ààìmàPÞiì+çwì+òHè‰ódÞàï—Þ“ìOï—ðâñï—ðüì’ûñççá’ï—é9ÝHë?ގë”çwìmâCÞiã—á+äÐ+òHéVß7ðñáOÞVðñóHàcódìOì’àwïÒççwôòHï—áOàã„ðCÞ&ì[ôOèWðHÞiëaÞià‹ìOÞií’àwà`ÞTìmçwàéVÝñï—áOÞiޛðñóâ7óHæ}á(òñàóHézáOçôä ÞióWèwòH÷9àwܔæoìOÝHé¼ÞKÞ+òñ÷ðHèwÞBHÞTìmáOà‹ámà¾ædçwÞ¼ìmÝHöHð&ÞÐìmì+äŽæ}áOódßHàï—ìOÞVã—ÞOâ7âñùCádì’æ}çódáOï—ô éà
çwïÒðüá ìŽðñìOàwàá’ðHë;ÞiìOàcàìmðHçwÝHø+ގÞæ}û7àwìmá+àä ގèwòHàwæoìO»éVÞOçùYá ë`ÝHÞiàwLÞBìOùCèë`æ}ÝHá+à&ï—écÝìmðï„è¾ìOèwódãÒï„ï—øOìmâñÝçìmãÒçwôLï„éÐãÒá’ë\ë\ìOÞVãÒà”MùYççÝñÝHìmގðçwçÞiÝHäÐÞß7çwÞVáOàcçì’ìOçãWòHçwàÞiÞÐäÐï—ðß7ÞVçwàcÝHì’ÞçòHâ7àÞáOòñámðñæ4óHçwìmÝHàÞ&ôüûñã„á’ì[ë?ôOÞiàþ í’ìOþ àwï—Þiè
áçwmÞV9æäŽïÒïÒøCðß7÷ çwÞVÞiàcàwz÷ì’ÞB/çè‚òHçŽÜ”àwޓçÝñáõÞðHÞiðHçwìOámÞiàRäŽçÞçßCÝHçwÞiޛÝñàìmìmë`çwçÝHòHçwá+àÝHÞãÒÞÐÞ ßHè‰àòñámà‰ýñæoì+ã—èwÞié¼áOè&Þ&ã—íOæ}áOáOÞiæKà›àŽçwçwë`ÝHÝHÞÐï²ÞçÝâCé¼ádáOðHódðá ô+çx÷ïÒè‰ð(㗾òHï ßÓòHð/ä éVï—áOðñðñìOé¼àódðñÞiïÒóçwì+ï—è‰áOê(Þ&ðCïÒðHï—èðÞVçwìmçï„ðñéÞVó äŽìOßHßCçwÝñßHÞiÞüààìmá+çwÿìOòHécàÝñޛÞiè&ï—ðìOàwçäŽÞÝHÞVޓéVçwáOÝHðñäŽádÞiìmóÓßñà ìm†ßHë\ààÞiìOÞióãÒódYï„ï—ézàð çÞVø+9ìOïÒá+ï—ð øñð÷ ï„ãÒódñÞBÞV÷ìOðóHç†èRï—ç“éiçwìmï„á è ã
è‰é¼ï—ã—øOÞiðHìmàïÒýCã—ôéiìmè‰ðÞiç\ÞVðécÝñï—ðLìOðHçøOÝHÞBÞèTçwï—ÞiðäŽçÝHßCÞ¾ÞiàûCìmá’çwòHë;àÞ&è‚çìmàwòñðñé¼óçwòHí+àÞVÞOã—á(ù(éVï—ðï²ç‚ôßñìOßHà‰àwçáOï—ýñéVòHã—Þiã—èìOàiù+9çwï—áŽøñ÷ ìmðzùïÒñðñìOéVðñàwóLÞBìOçwèwáÞRï—ìÐðøOçwàÝñÞiދìmçwâ7ÞiáOà”òHèwðCã—áOóHß7ìmÞàô ámMæ9ã„ì[çwôOÝHÞiÞàKã—çwÞiÝHì+ï„ódécê(ïÒðñðHø Þi†èÞièiódùø+ë`ÞÝñèwï—ÝHécÝádécï„ê è
ë\ì[ñí+áOÞOà>÷ ìOðŽì+ódï„ìmâñìmçwï„éaë”ìmã—MùmçwÝñÞRódï ÞVàÞVðCé¼Þ`ï—ðŽçwÝHÞRß7á+èwïÒçwï—áOðŽámæ7çwÝHÞ¾ãÒÞBìOódï—ðHø †ÞióHøOޔè‰ÝHádécê“ë\ì[í+Þaá+âdçìOïÒðHÞBóÐâô›çwÝHÞ ŽèwáOã—íOÞVà
ë`è‰òHï²àwçæoÝìOéVðHÞ&á çxèwÞVãÒäŽï—ßßCÞié¼àá+ìmðñçwódòHïÒàçwÞï—áOðñoézè“æ‚÷›ìmðñçwÝHóõގçwódÝHÞiÞðñÿè‰ïÒç‚ôßH%àámäŽýñã—Þ¼ÞiçwèÝñï—á(ð ó9ï—ï—è“øñäŽ÷ áOàìOސðñóè‰ï—øOðñï²¼÷ýC&éiìmܔðÝHç&ގçÝñòñìmè‰ÞÐðõámæ}æ\á+à›è‰ã—ïÒçßÝHÞLé¼á+ìOðñâCódá’ïÒí+çwސï—áOðñéVì+èè‰è‰Þçwàë`áOï²ðHçø+ÝãÒôì/ódéVÞBáOé¼ðñàè‰ÞiçìOìOèwð+Þiçè
çwäÐÝHÞVÞçwÝHèwãÒádá+óWßC÷ ÞRáOáOæaçwçÞÝHçÞÝñã—ì’ÞiçRìOóHçÝHïÒðHޛø ódMÞBÞiódðñøOèwÞï²ç‚ôLèwÝHðHádÞiécìOêüà”ë”çwÝHì[Þ&íOގâ7ádè‰áódôçÝñè‰ì’òHç&àwæoìOïÒç&éVÞï—è&æ}áOãÒádà¾éVìOìmðçwÞBì+óódÞiï„ìmíOâñÞiìmðçwï„ã—éá’ëaë\ÞiìOàãÒYçwÝñï„è`ìOï„ðódÞVçwðÝñçìmï—ç&éiìmßHãWàwæ}ÞBá+ódà`ï„ézçwçwÝHÞBޓóé¼âáOô/ðççwïÒð(ÝHòHÞòHÿ ä ìmðñó
êï—ðHÞVçwï„éìmßHßHàá+ì+écÝHÞièi÷
0.00
0.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
U/U
0.6
Ä
0.8
1.0
0.00
¶¿–À~ÁµÅºÆ
0.0
0.2
0.4
U/U
0.6
0.8
1.0
8
·“¸C¹§»,¼D½~ø ù
8
8
U/U
0.00
0.0
FIGURE 9. Velocity profiles (M^ = 10)
To verify this assumption, computations for adiabatic wall conditions were conducted. The use of an adiabatic
g i Ë-
‚
wall for
MOO — 10 leads to a significant increase in flow temperature near the body surface.´For
example, for
ú-˜6„ i
a constant surface temperature, the temperature in the boundary layer varied from lOToo near the forebody
2-
²„ i
to ISToo near the flare surface, whereas for an adiabatic wall, the temperature in the boundary layer varies
`!27^
c1„©i
h6291^,
x„†i
:Gf 6„©i
in a narrow range from
to
which is slightly lower than the total temperature of the flow 34.3^
¶‚ (Fig.
7c7). The temperature profiles for the continuum and kinetic approaches are compared ineFig.
‚
¢8.
² rItm is
r
solver with no-slip
6
E"F( method predict an identical
of interest to note that the
NS
conditions and the DSMC
%
>
temperature near the whole surface of the body. An increase in temperature in the near-wall
region leads to
!
significant changes in the flow structure, in particular, to an increase in the boundary-layer thickness, which is
‚
Gx!
6
clearly seen in the temperature and velocity profiles&¶(Fig.
9), and to a greater slope of the leading-edge
shock
wave.
‚ For an adiabatic wall, the difference
ƒ
6
% solver
in the position of the leading-edge
shock wave obtained by the%NS
6
E"F( method is more significant than for the above case with a constant
with no-slip
conditions and the DSMC
– (cf. the density profiles in
)‚ Fig.© 2 and
j-˜10).
The use of slip conditions strongly decreases
surface temperature
E")(
the slope of the leading-edge shock wave so that it is located even lower than that predicted by the DSMC
method. Note that the density near the body surface for an adiabatic wall is identical for the continuum and
kinetic approaches.
712
Y/L
0.30
Y/L
0.20
X/L=0.3
0.25
Y/L
0.40
0.35
X/L=0.6
X/L=.09
0.30
0.15
0.20
0.25
0.15
0.10
0.20
0.15
0.10
0.05
0.10
DSMC
NS, no-slip
NS, slip
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.00
0.0
0.5
DSMC
NS, no-slip
NS, slip
0.05
2.5
0.00
0.0
¡`Í\§c«©x¬Ñ±M¥›¦O¨‚ª (º®§c©R´
1.0
1.5
ρ/ρ
2.0
3.0
Bµ
DSMC
NS, no-slip
NS, slip
0.5
1.0
1.5
ρ/ρ
2.0
Cf
0.20
Cp
0.30
DSMC
NS, no-slip
NS, slip
0.25
2.5
3.0
8
À Áµ=ÅºÆ 10)
Ä
¶¿ (M^
·“FIGURE
¸º¹3»,¼D½¥¾610.
û
Density profiles
8
8
ρ/ρ
0.05
— DSMC
DSMC
— -NS,
no-slip
NS,
no-slip
—• slip
NS, slip
NS,
0.15
0.20
0.10
0.15
0.05
0.10
°i¹+¬²°iÁ°V±‚¬®­`·K°Vº²Òµ¡ÐÈ4£+§¨‚¯&°iºP«ªi«§z̒¸¬®º ¬®Á+¨‚¬®¸¯ ´qº®§½Ò±wµ¾°V«(¹¦+¨‚§©x©x¸+¨‚§´q­c§c«’±‚§¨wµ`°i«¹©xÃm¬²«½Ò¨‚¬®­±‚¬®ªB«´„¨‚¬®³B£’±wµR­ª’§ ­c¬®§c«’±‚©Ð´ mÅ
éïÒ¼ð/ô(ãÒܔçwï—ðñÝHÝHódÞÞàé¼ï„òCéVàè‰ìmá+ÞãCèè¾áOè‰òHæ4èwàwÞièwæoé¼ãÒìOï—çwßéVï—ÞáOé¼ð á+oéVðñàwódáïÒèwçwè>ï—áOè‰ðñÞBézè`çwã—ï—ÞiáOì+ðCóHèìmè”àޛçáï—ð/ìÐàèwÞiï—ì+øOè‰ðHá+þ&ïÒýCðñìOìméVðñìOâHóðã—Þ›ç`ìOçwÞVøO䎼÷àÞVß7ÞV¾äŽÞVá’àcÞiì’ëað+çÞiòHç‹íOàwæ}ÞiÞ áOàià‚ù’òHçâCäŽÝHáOÞ`ßüçwÝûñìOá’ìmðñë#ßHóßñßñè‰àwìOã—áïÒàßìOìOécäŽí+ÝHÞVÞ¼ÞBã—çwèVádÞi÷ é¼àïÒè4xç‚ðüôï—ðñçáOèwÝHï—ðóHï—èÞRçwÝñéVçwÞàwÝñáÞRæ}èwàwèâCá+á+ð+è‰ÞBòHçRézðñßCçóHïÒìmá+ìOàwðYàwçRôÐù7ámçwã„æ9ÝHì[çwôOÞiÝHàwÞVÞÞà
ï—è‰è“ï—äŽðHïÒáüã„ìmûCàá’è‚ëçàwòñè‰é¼ÞiçwßñòHìmààcÞiì’è¾çïÒámá+æKðçÝHæ}áOޛà â7áOòHõðCàóHÞiìmèwàòHôLãÒçã„èiì[ùTôOìmÞiðñà¾óõìmÝñçwÞiÝHìOÞóÿ áOæKçwÝH%ÞÐè‰äŽÞißñÞ¼çìmÝHàcì’ádçóïÒá+ßHðàÞißCódá+ï„ïÒézðçcçiè&ù7ççÝHÝHÞޓÿâCÞiøOï—ðH?ðHï—ðHäŽø/ÞVçwáOÝHæ‹ádè‰óÞißñßHìmààcÞiì’óHçï—ïÒé¼á+çðYè¾÷ìÿ‹øOàÞiÞièwìmßHçwïÒçwÞVÞà
çwãÒôÞiáï—ðHÞVìŽã„ømóçæ}Ýõòñìmà‰ð/ámçæ”ÝHï„ódÞVçwàÝHÞið+ÞódçÞiï—è‰éiéVÞiìmàwßñãKÞBìmìOã—àcÞVèwì’ÞðHçøOïÒïÒá+ðüçwðÝçwÝHàámÞVæTޛø+çèwïÒÝHá+ÞVðÞßCìmèwçwàcÞVÝñì’ßñìOçwìOï—ðáOàðìmçwçwÝHï—àÞáOÞVðüø+ïÒá+àÞVðø+èwã—ïÒáOÞVá+ã—ðñðíOømÞVæ}çwà áOÝgà÷¾oèwçwÞVܔÝñސÞÝ(òñÜ9é¼èVìmáOùCâHðçwã—çÝñÞ ïÒð(ޓòHé¼¼÷òHá+üä äŽÜPßHìOìOòdðñê(çóïÒìmðHçwøêï—áOï—ðHï—ðñðÞVè¾çwçwá/ï„ë`éÐì+ï²çìOéVÝ/ßHéVáOßHìmòHàðüá+ðì+ç›ì+écódèwÝHï„ãÒÞiìmï—ßèiâñù7ìmé¼ìOçwá+ðñï„ðñéóódë\ïÒÞVçwìOíOï—ãÒáOÞi9ðñð/ódè›ï—ã—óüÞiãÒÞBó/ðHìOóHçwámá çè
çwßñÝHì’ÞçÝá+ßHï—ðßCáçwè‰ÝHïÒÞçwÞðHèwÞBïÒìmçwòñàìmë”çwï—ìmáOã—ðYK÷àwÞiܔøOï—ÝñáOï—ðèódï„èiòHùPގìmßñçwáßñìmÝHàï—ÞVøOðÝñçwÞVã—ôOàù9ë”éVìmìOã—òñ4è‰çwÞBÞVó/äŽâ(ß7ô/ÞVàcç‚ì’ëaçòHáàwæoÞìOé¼ódçwï„á+óàèiðH÷ ámç&9écïÒàcÝñè‚ìOçBðHù9øOìOÞÐðûñï—ðñá’ë¤éVàwÞBðHìOáOèwðñÞÐÞ ï—ðïÒã—òHçwÝñâHÞàï—òHãÒádä éVìOãKoé¼äŽæ‚÷ Þiìm9ðïÒøæ}èVà÷ ÞVÞ
âûCCìmðñìmá+àóòHÞðñóHèwòHìOìàwàwæoô OìO†ðñéVã—ì[Þ¾óô+çÞVódÝñàaï„ìmóõçwðÝHðHï„ïÒécðLáOê(ç“çwðHÝñï—Þiðñދèé¼è”ûCàãÒÞiá’ÞBì+ë$óLè‰ÞŽçwìmáàçáOÝHìŽòHÞðCßCìmóáàè‰Þiì“ïÒìçwï—ÝHáOámáOðæ`ã—Òá’ámçÝHëæ9ÞLçwxé¼ÝñìOô(Þ&ßHã—ïÒè‰ßHðCÝHã—ódï ádéiÞVécìmàaê âHûC†ï—ë\ãìmï²ç‚ì[àôÞí+ÞRë`ámïÒæ”ï²ðçççÝÞVÝHàcìŽÞ ìOézé¼çá+ïÒðñá+è‰ðLÞ çìmàòñðÞVç`ø+ìmçwïÒè‰á+ï—òñáOðLà‰ðñæoã—èiì+ád÷ é¼éVދìm(çwçwÞBÞiÞVé¼óLäŽá+ðñä›ß7óWÞVòCùKàcécì’ìmݐçðòHæoàwìOï—Þ+à‰ðñ÷çéVÝHxàwÞVðÞBà”ìOçèwæ}ÝHÞàáOï—è`ïÒä ðéiìOçwçwèwÝHÝHÞOÞÞ ù
çwßHÝHàwÞRÞBèwÞVèwödòHßñàÞRìmðñøOèwàcï—ìOáOóHðïÒÞiì’ð+çTç\çìmÝHðñÞ`óWûCù(ìOÝHàwÞiÞRðñèwé¼ÝHÞá+é¼òHá+ã—óHðñÞVè‰à>Þ è‰òHçwàÞVáOððHçwã—ø+ôOÞVù+àTçwìÝñ7ÞދBèwézÞVçßñèKìOçwàÝHìmÞ¾çwï—èwáOÝHðádàécÞVê›øOë”ï—áOì[ðíOÞ`ã—ÞVï—ðñðømçwçwÞiÝgàì+÷4ézܔçwï—ÝñáOï—ðÐèaàï—è\ÞVøOàï—Þ¼áOûñðÞBézìOçðñÞióóódï—ðÞiéVçwàwÝñÞBÞìOèwódÞiï„è9è‰çwçàÝHïÒÞ¾âHòHìOçwóHï—íOáOÞVðàcèwáOÞ æ
ßHàwŽÞBøOèwë`àèwÞiòHÝHì’àá+çÞèwÞVÞàRìOðñäèwãÒóLá+ìmßCøOæ}àwÞðñï„ézï²áOççòñæPïÒá+ódçðLÝHÞÞá+áOðã—ðÞiìOççwÝHóHÝñÞïÒÞðHæ}øûCáOMìmàÞBàÞVódÞâ7øOádâCÞódÞBèwôOéVÝHìm÷ ádäŽécêÞáOë”æPì[çíOÝHދÞ&æ}áOèwìOà`äŽçÝHÞÞ á+àóHÞVà”èwáOë`ã—íOï²çÞiÝà\ë`ï²çcè`ïÒçwÝí’ìmðñã—òHá Þxè‰ðHã—ïÒÞBßìmà\é¼á+çwðñÝHódޛïÒçwãÒï—ÞBáOìOðñódè¾ï—ðHéVøŽìmòCÞiè‰ódÞBø+è`Þ ìmoðèwÞVïÒÞ ðCé¼9àïÒÞiøCìO÷ èwÞïÒz÷ð
çwâCï—ÞVð†ç‚òñçëaòHÞië\ä ÞVì+ðè¾ìmçwðCèwÝñÝHóÞá’ê(ë`ûñïÒðñðá’Þ¼ë%çwçwÝñï„é‹ßñìmìmçàwàcÞBììmè‰äŽòHóHãÒçÞ¼Þièiçé¼ÞV÷ààcÞiaèVì+÷ áOè‰ÞäŽRï—ßHðá’òdë\ççcÞVÝHì’í+çÞ&ÞVïÒá+àBæ}àðñùPÞVè\ì ÞBë`è‚çï²àwòñçÞBÝìmìmã—äïÒìOçðì’/çìOïÒí+ódìOÞï„écìmÝódâCï ì’ð(7ÞçwòHiï„ä›é‹àwÞiâ7ë\ðñÞVé¼ìOàÞãÒCã—ï—æ}ÞiðáOìOà óñçwè`ÝñÞLçwáLèwãÒè‰á+äßCÐìmÞã—ìmáOÞVã„æ\à‹è‰á“çóHÝHàwï ÞÞÞiíOVã—àÞBÞiÞVìmì+ðñã—ódÞiéVïÒóðñÞiè¾ø èw†äâCÞiÞVódìmç‚ø+ã—ëaÒÞÞiÞià\ÞVè‰ÝHódðïádÞçécÝHVêàޓÞVë\ðñéVì[áOéVí+ðÞiÞè
ooèwÜ9ÞVìmދâHçwã—ÞÝHދaïçwÞi„äÐè”ß7àwÞVÞVçàcìmì’ï—çðHòHÞBàóWދ÷ ßHàámýCãÒÞBè>ï—ð 9ï—øñ÷ ¼ù(ïÒðçwÝHÞódï„è‚çàwï—âHòdçÞióLécÝñìmàcìOé¼çwÞVàï„è‚çï—éièVùìmðñóïÒðçwÝHÞè‰Þißñìmàcì’çïÒá+ðàwÞiøOï—áOðã—ÞVðñømçwÝ
0.00
·“¸C¹§»,¼D½Ë¾®¾
0.05
0.0
0.3
0.6
0.90.9
X/L
X/L
1.2
1.5
0.0
0.3
0.6
0.9 0.9
X/L X/L
î
1.2
¶¿
1.5
À
ÁÅºÆ
FIGURE 11. Thermal nonequilibrium (left) and pressure (center) and skin friction (right) coefficients (Moo = 10,
adiabatic wall)
$
The use of slip3conditions
leads
significant
jump and slip velocity on the front part of the
7Êto
w8dja@
Gf
Pf temperature
#J
cylindrical surface )
(cross
X/L = 0.3 and 0.6). However, the flow parameters
inside the boundary layer
ÊHw6dsections
TGf x
m
in the cross section X/LÌ%
= 0.9 are in reasonable
agreement
for both approaches. In this cross section, there
")
(
is no flow separation for NS results, and the DSMC method predicts thebeginning
of separation. Despite
E")(
similar structures of the boundary layerahead
of h
the
%
separation
9-˜ point, the DSMC method predicts a greater
length of the separation region than the NS solver (see Table 1). Taking into account slip conditions leads
to a further decrease in the separation region length. Thus, the computations with an adiabatic wall did not
H‚ and kinetic approaches, and even led to
yield an identical length of the separation region for the continuum
in the ü
Hmean
‚
sfree
A
the opposite situation. This is, apparently, caused by two factors. First, an increase
local
@-in
-˜ the near wall region due to higher wall temperature did
change
) noneqilubrium (cf. Figs. 5
path
not
flow
!
and 11) !and
did not increase the area of the applicability
of the NS equations. Second, an increase in the
the shock-wave interaction region located much farther
Km
boundary-layer thickness led to a position of
from the
6
flare surface than in the flow around a hollow-cylinder
flare with a constant surface temperature. In this case,
˜ stronger affects the shock wave interaction region and decreases the adverse
the expansion at the flare shoulder
¬pressure
7­
P distribution
D‚
©--1 of
gradient and, hence consequently, the separation region length. This is reflected in the
>Cp whose magnitude on the flare
D
%
6
became of the same order with its value near the leading edge (see Fig. 11).
A greater slope of the leading-edge shock wave for the NS solver with no-slip conditions causes an increase in
pressure and friction on the forebody.
m
B"
ƒ
E
o
(
?
j
g

i
@
:
.
It was shown that a decrease in the freestream Mach number leads to smaller differences between the conJ
9 with an adiabatic
.
6 smaller differences
tinuum and kinetic results. Computations
wall for M^ = 4 also revealed
h‚
é-˜
between the flow parameters. However, a qualitative difference in the slope of the leading-edge shock wave
P-˜
(see the temperature profiles in Fig. 12), in the distributed characteristics, and in the separation region length
(Table 1) is retained.
713
FIGURE 12. Temperature profiles (Af*, = 4)
CONCLUSIONS
The computations for the Mach 10 flow of a monoatomic gas (argon) show that the continuum approach
predicts a greater length of the separation region than the DSMC method. The use of slip boundary conditions
for taking into account initial rarefaction effects allowed us to decrease the difference between the continuum
and DSMC results. A comparison of the density, temperature, and velocity profiles shows that significant
differences are observed only for the temperature inside the boundary layer. This is probably caused by strong
flow rarefaction near the leading edge and, as a consequence, by a significant difference in heat fluxes near
this edge predicted by the NS solver and DSMC method. A decrease in the freestream Mach number leads to
smaller flow nonequilibrium in the vicinity of the leading edge and to better agreement between the continuum
and kinetic results. In particular, the difference in the separation region length was reduced from 10% for
MOO = 10 to 6% for MOO = 4.
The use of adiabatic wall boundary conditions for flow simulation near a hollow cylinder flare induced
a significant increase in the gas temperature near the body surface, which led to substantial changes in the
boundary-layer structure. The NS solver with no-slip conditions and the DSMC method predict almost identical
values of density and temperature near the body surface. In this case, however, the DSMC method predicts a
greater length of the separation region than the NS solver. The use of slip conditions leads to more considerable
changes than in the case of a constant surface temperature. For example, the slope of the leading-edge shock
wave becomes even smaller than the DSMC prediction. The qualitative effect of slip conditions on the separation
region length is retained, the latter decreases in size, and the difference between the DSMC and NS results
increases.
To study in more detail the above-described quilitative difference between the DSMC and NS predictions of
the separation region length for cold and adiabatic walls experimental data are indispensable. In the term of
RTO Working Group 10 an experimental study of hypersonic laminar flow around a hollow-cylinder flare with
a heated wall is planned and it will clarify the situation.
Acknowledgement This work was supported by the INTAS Grant No. 99-00749 and RFBR Grant No.
00-01-00824. This support is gratefully acknowledged.
REFERENCES
1. Moss, J.N., Olejniczak, J., AIAA Paper 98-2668, June 1998.
2. Ivanov, M.S. , Markelov, G.N., Gimelshein, S.F., AIAA Paper 98-2669, June 1998.
3. Markelov, G.N., Kudryavtsev, A.N., Ivanov, M.S., Proc. of 21 Int. Symp. on Rarefied Gas Dynamics, Cepadues
Editions, 1, 647 (1999).
4. Markelov, G.N., Kudryavtsev, A.N., Ivanov, M.S., AIAA Paper 99-3527, June-July, 1999.
5. Chanetz, B., TR RT 42/4362 AN, ONERA (1995).
6. Einfeldt, B., Munz, C.D., Roe, P.L., Sjogren, B.,J. Comput. Phys., 92, 273 (1991).
7. Kogan, M.N., Rarefied gas dynamics, New York: Plenum Press, 1969, ch. 5.
8. Bird, G.A., Molecular gas dynamics and the Direct Simulation of gas flows, Oxford: Clarendon Press, 1994.
9. Ivanov, M.S., Rogasinsky, S.V., Proc. of 18 Int. Symp. on Rarefied Gas Dynamics, VCH, 629 (1991).
714