Evaluating Test Parameters in an Arc Wind Tunnel
Gennaro Zuppardi – Diego Paterna
Department of Space Science and Engineering “Luigi G. Napolitano”
University of Naples, Piazzale V. Tecchio 80, 80125 Naples, Italy
Abstract. Experimental tests in the arc wind tunnel in Naples (named SPES), aimed at stating the rarefaction level of the jet,
were previously processed by a steady, one-dimensional and non-diffusive code. In this paper the same experimental data
have been processed once again by more reliable codes like the Navier-Stokes (FLUENT) and the Direct Simulation Monte
Carlo (DS2V) solvers, in order to consider the influence of the viscosity, the thermal conductivity and the diffusion of
chemical species. The evaluation of the rarefaction level relies on the computation of a number of parameters and on the
comparison of the measured drag coefficients with the ones reported in the published literature and the ones obtained
reducing the drag computed by DS2V. Even though the revised rarefaction level and the Mach number of the jet are lower
than the ones computed formerly, however the present analysis confirmed the capability of SPES to be a facility for
continuum low-density, high Mach number testing. A proper procedure for the computation of the thermo-fluid-dynamic
parameters of the jet was also established.
INTRODUCTION
It is well known that the tests in a wind tunnel require the evaluation of aerodynamic parameters, like the free
stream density, velocity (or dynamic pressure) and temperature, as well as of the characteristic numbers: Mach
number, Reynolds number and so on. On the other hand, the measurement of physical quantities is very difficult in
an arc wind tunnel. The hostile environment of the jet prohibits from using any probe. Moreover the unknown gas
composition, down-stream the electrical arc, and the strong non-equilibrium state of the gas could make unreliable
any measurement.
The blowdown arc wind tunnel (50 kW) at the Department of Space Science and Engineering in Naples, named
SPES (Small Planetary Entry Simulator), is supplied with a code computing aerodynamic parameters from
quantities that can be measured with a good precision during the test like: the electrical power provided to the arc,
the flow rate of the test gas and the flow rates of the water cooling the torch and the nozzle [1]. The code, labelled
AWT (Arc Wind Tunnel), simulates the working of the arc-heater, the chamber mixing hot gas coming from the archeater, as per nitrogen, and cold gas (if any), as per oxygen, and the conical nozzle. AWT solves a one-dimensional,
steady and non-diffusive flow field. The test gas is simulated in chemical and vibrational non-equilibrium.
Zuppardi [2] already found that in SPES, at some test conditions (flow rate of the gas and electrical power), the
flow field can not be longer considered in continuum, even inside the diverging part of the nozzle. Zuppardi
performed this analysis both numerically, by the computation of a number of rarefaction parameters, and
experimentally, by the comparison of the measured drag coefficients of a sphere, with data reported in the published
literature and with the results of the well validated DS2G code [3], based on the Direct Simulation Monte Carlo
(DSMC) method [4]. A one-dimensional strain-gauge balance (with a full scale capacity of 2 N and an uncertainty
of about ± 0.02 N) measured the drag force of the sphere model. The rarefaction parameters and the free stream
dynamic pressure, reducing the drag force to the related coefficient, were computed by AWT.
Due to the limiting hypotheses on which AWT relies, it is necessary to revise the computation of the aerodynamic
parameters; thus the present work has to be considered as the logical continuation of [2]. The experimental tests
reported in [2] are processed once again; the flow field rarefaction level in the diverging part of the nozzle of SPES
is re-analyzed and the test parameters are re-computed, in order to consider the influence of viscosity, thermal
conductivity and diffusion of chemical species. These phenomena, in fact, are very important at the temperature
levels met in the jet of an arc wind tunnel. This task has been fulfilled both by a Navier-Stokes code (FLUENT [5])
and a DSMC code (DS2V [6]). This revision is necessary also for establishing a proper computing procedure to be
used in a near forthcoming campaign of measurements of the aerodynamic characteristics of a capsule during the reentry.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
Besides the above aim, by means of the present paper, a numerical investigation about the influence of the flow
rarefaction on the computation of aerodynamic parameters by a Navier-Stokes code, already started by Zuppardi [7]
for external flows, is fulfilled also for internal flows.
COMPUTING PROCEDURES
The geometry of the axial-symmetric channel of SPES, downstream the torch, is shown in Fig.1. It is made of: a
constant section chamber (length 22 mm, diameter 8 mm) mixing hot gas (in the present tests nitrogen) from the
torch and cold gas (in the present tests no cold gas was added, thus the gas is a mixture of molecular and atomic
nitrogen), a converging part (length 10 mm) extending to the nozzle throat (5.5 mm) and a diverging part. SPES can
work optionally with two diverging parts: 1) length 155.3 mm, diameter of the exit section 50 mm, 2) length 61 mm,
diameter of the exit section 22 mm. Only the tests made with the first diverging part are considered here.
The numerical simulations of the flow field in the nozzle have been restricted just to the diverging part. In fact,
preliminary computations by FLUENT [8] showed that the flow field properties at the exit of the mixing chamber do
not substantially differ from the properties in the throat section. This can be verified from Fig.1 where, as a typical
example, the distribution of the Mach number at test conditions close to the present ones is shown. Therefore the
computations by FLUENT and by DS2V can start just from the throat, using as boundary conditions the flow field
properties computed by AWT at the exit of the mixing chamber.
The volume of the diverging part is discretized, for both codes, in a system of 140 cells along the axis (x) and 70
cells along the local radius (y). For the computation of the drag of the sphere model (diameter: ds=15.8 mm) by
DS2V, an axial-symmetric volume (length 35 mm, radius 25 mm), including the model, is discretized in a system of
200 cells along x and 100 cells along y.
No temperature of the surfaces of the nozzle nor of the model, was measured in the present tests. The surfaces of
the nozzle and of the model were considered at a reasonable, constant and uniform temperature of 500 K and 700 K,
respectively. This value of temperature of the model surface was formerly measured in SPES on a hemispherical
calorimeter at test conditions similar to the present ones. The runs of FLUENT considered the surface of the nozzle
non catalytic. The runs of DS2V considered the interaction molecules-surface fully accommodate at the surface
temperature (nozzle and sphere model).
Like the runs made by AWT [2], the thermo-fluid-dynamic parameters and gas composition at the exit of the
nozzle, computed both by FLUENT and by DS2V, are supposed to be the free stream ones. The free stream
rarefaction parameters and the dynamic pressure are computed by data at the mid point of the core of the jet (as
shown later). For each test case DS2V ran twice. The first run solved the flow field in the nozzle. The second run
computed the drag of the model, taking as input the thermo-fluid-dynamic parameters computed at the nozzle exit.
All runs of DS2V were made with a number of simulated molecules of about 5×105 m-3 and a simulated time of
about 10-3 s.
Mach Number
2.5
2.3
2.1
1.9
1.7
1.5
1.3
1.1
0.9
0.7
0.6
0.4
0.2
Figure 1. Mach number distribution inside the nozzle computed by FLUENT
RAREFACTION PARAMETERS
A number of parameters has been taken into consideration for fixing the flow rarefaction level: 1) in the diverging
part of the nozzle, by means of local parameters along the axis. Following Boyd [9], these parameters have been
computed by data from the continuum computation (or from FLUENT), 2) in the free jet by means of global
parameters; these parameters have been computed both by data from FLUENT and by data from DS2V at the exit of
the nozzle.
The parameters along the axis of the diverging part of the nozzle are:
1)
local Knudsen number (Kn)LLD, defined as the ratio of the Local free molecular path (λ) and the Local
Diameter (dL) of the diverging part of the nozzle: (Kn )LLD = λ d L . The free molecular path is computed [4] by:
λ=
1
(1)
2πd 2M n
where: n [1/m3] is the molecular number density, dM [m] is the average molecular diameter of the mixture, a bar is
for the average weighted with respect to the gas composition (mass fractions of the species: αN and αN2).
According to the VHS (Variable Hard Sphere) model [4], d M is computed as a function of the relative velocity and
therefore of the temperature. In the present paper the criteria formulated for the overall Knudsen number (i.e. the
ratio of the free stream molecular path and a characteristic dimension of the flow field, as per the characteristic
geometrical dimension of the body), have been supposed to be valid also for the local Knudsen number. According
to Bird [4], the requirement for the failure of the Navier Stokes equations is that the overall Knudsen number should
be larger than 0.1. According to Boyd, the failure of the continuum approach occurs for overall Knudsen number
larger than 0.01.
2)
Local Knudsen number (Kn)LLG, defined as the ratio of the Local free molecular path and the Gradient
Length (LG) of a flow quantity G, as per density (ρ) or temperature (T): (Kn )LLG = λ L G
LG =
G
dG dl
(2)
where l is the distance between two points in the flow field.
According to Bird [4], requirement for the failure of the Navier-Stokes equations is fixed by (Kn)LLG > 0.1, the
beginning of the discrete molecule regime is fixed by (Kn)LLG > 0.2. According to Boyd, the failure of the NavierStokes equations is fixed by (Kn)LLG > 0.05.
3)
The “continuum breakdown” parameter P of Bird for expanding flows [4]:
π1 2 λ dρ
(3)
S
2 ρ ds
where: s is the distance along the stream line, S is the “speed ratio” ( S = V 2RT , V is the velocity, R is the gas
constant).
As suggested by Boyd, a continuum breakdown parameter P based on the gradient of temperature, has been also
considered; this parameter will be labelled PT. According to Bird [4], the beginning of “continuum breakdown” in
expansions starts from P=0.02 (in the present paper the same limit has been assumed also for PT). Boyd suggests a
value of 0.05 for both breakdown parameters (P and PT) to identify the failure of the Navier-Stokes equations. In the
present application s and l coincide with the distance along the nozzle axis (s= l =x); thus P and PT include the
scale lengths of density and of temperature, therefore include also (Kn)LLG for density and temperature, respectively.
In the present computations, derivative d / dx has been approximated numerically by a forward, first order, finite
difference.
Besides the above parameters a criterion proposed by Boyd, identifying the failure of the continuum approach
when the local difference of density and/or of temperature between FLUENT and DS2V is at least 5%, has been also
taken into account. Generalizing the Boyd’s criterion, the departure of other fluid-dynamic parameters computed by
FLUENT from the ones computed by DS2V, can be considered indicative either of the failure of the Navier-Stokes
equations or even of the continuum breakdown. The Mach number is particularly meaningful for this purpose,
because it includes velocity, temperature and gas composition.
Two global rarefaction parameters have been considered: the free stream (∞) Knudsen number Kn∞ and the
Reynolds number behind a normal shock wave Re2:
Kn ∞ = λ ∞ d s , where λ∞ is computed by Eq.1 at the free stream (or the jet) conditions,
Re2 = ρ∞ V∞ d s µ 2 , where µ 2 is the viscosity behind a normal shock wave.
The viscosity of each chemical specie is computed by means of the Chapman-Enskog theory [10]. The viscosity of
the mixture is then evaluated by the Wilke rule [10]. The transitional regime for a sphere is defined in terms of Kn∞
[11] as 10-2<Kn∞ <1, in terms of Re2 [12] as 10-1<Re2<103. A value of Re2>10 identifies a continuum low-density
regime and a value of Re2<10 identifies a near free molecule regime [13].
P=
ANALYSYS OF RESULTS
& (g/s), the electrical powers (kW), provided to
The tests, re-processed here, are related to five mass flow rates m
the heater, are reported in brackets: 0.5 (14.1), 0.75 (21.6), 1 (22.8), 1.5 (28.2) and 2 (44.6). Figures 2 and 3 show
the profiles of P and PT along the axis for each mass flow rate, respectively. The limit of “continuum breakdown” is
& =0.5 g/s.
barely achieved at the end of the nozzle (x=155 mm) only for the mass flow rate m
The parameter PT is very far from providing the same indication like the parameter P does; the maximum value of
PT at the end of the nozzle with the minimum mass flow rate is 6.8×10-3. The same indications are provided also by
(Kn)LLG based on density and temperature (to save space, no plot of these parameters is reported here). Maximum
& =0.5 g/s and at the end of the nozzle are 6.7×10-3 and
values of (Kn)LLG, for density and temperature, even with m
2.3×10-3, respectively, namely much less than the limit value (10-1) of the failure of the Navier-Stokes equations.
Profiles of (Kn)LLD (Fig.4) indicate that the failure of the continuum approach is achieved at about x=46, 104 mm
& =0.5, 0.75 and 1 g/s, respectively.
and at the end of the nozzle for m
0.030
.
.
m=0.75 g/s
.
m=0.5 g/s
P
0.025
m=1 g/s
.
m=1.5 g/s
x=155 mm
.
m=2 g/s
0.020
"continuum breakdown" limit
0.015
0.010
0.005
0.000
0
40
80
120
x [mm] 160
Figure 2. Profiles of the parameter P along the axis of the diverging part of the nozzle
0.010
.
m=0.5 g/s
PT
.
.
m=0.75 g/s
m=1 g/s
0.008
.
.
m=1.5 g/s
m=2 g/s
0.006
0.004
0.002
0.000
0
40
80
120
x [mm] 160
Figure 3. Profiles of the parameter PT along the axis of the diverging part of the nozzle
0.018
(Kn)LLD
0.016
.
.
m=0.5 g/s
m=0.75 g/s
.
m=1 g/s
.
0.014
m=1.5 g/s
.
m=2 g/s
0.012
"continuum
0.010
x=155 mm
breakdown" limit
x=104 mm
x=46 mm
0.008
0.006
0.004
0.002
0.000
0
40
80
120
x [mm] 160
Figure 4. Profiles of the local Knudsen number along the axis of the diverging part of the nozzle
The criterion based on the 5% difference of the ratio of temperature between FLUENT and DS2V showed that the
& =0.5 and 0.75 g/s, at x=50 mm for m
& =1 and
failure of the continuum approach is achieved at about x=34 mm for m
& =2 g/s. Figure 5 reports the ratios of temperature and density (to evaluate
1.5 g/s and finally at x=110 mm for m
these ratios, data from DS2V have been interpolated by 6 degrees best fit polynomials). For the sake of clearness the
& =0.5 and 2 g/s. Profiles of the ratio of density are quite entangled, therefore not
plots are only for the tests with m
meaningful. A numerical value of the criterion based on the departure of the values of the Mach number is not
established; it can be considered just qualitatively through a graphical evaluation. It looks from Fig.6 that the
departure from the continuum approach could be localized at x=23 mm, for each mass flow rates. For this reason
Figure 6 reports the Mach number profiles only for the extreme mass flow rates.
Three rarefaction parameters (P, (K n ) LLD , TFLUENT T
) as well as the indication from the comparison of the
DS2V
Mach number profiles agree on fixing the failure of the continuum approach inside the nozzle only for the test with
& =0.5 g/s, even though at different locations along the axis. No other local parameter seems to provide reliable
m
indication of a continuum breakdown for tests with larger mass flow rates. Even though the local rarefaction
parameters and the criteria provided indications pretty confused and in disagreement, concerning the continuum
breakdown, however they could be considered indicative, at least, of the failure of the Navier-Stokes equations.
1.8
1.8
.
ρ FLUENT
ρ DS2V
TFLUENT/TDS2V (m=0.5 g/s)
TFLUENT
T DS2V
.
TFLUENT/TDS2V (m=2 g/s)
.
ρFLUENT/ ρDS2V (m=0.5
g/s)
ρFLUENT ρDS2V .
/
(m=2 g/s)
1.4
1.4
x=110 mm
x=34 mm
1.05 line
1.0
1.0
0.95 line
0.6
0.6
0.2
0.2
0
40
80
120
x [mm] 160
Figure 5. Ratios of temperature and density along the axis of the diverging part of the nozzle
4.5
M
4.0
3.5
3.0
2.5
2.0
.
FLUENT m=2 g/s
.
x=23 mm
FLUENT m=0.5 g/s
1.5
.
DS2V
m=2 g/s
DS2V
m=0.5 g/s
.
1.0
0
40
80
120
x [mm]
160
Figure 6. Profiles of the Mach number along the axis of the diverging part of the nozzle
25
y
[mm]
20
.
FLUENT m=2 g/s
.
DS2V m=2 g/s
.
FLUENT m=0.5 g/s
.
15
DS2V
m=0.5 g/s
core of the jet
10
reference line
5
0
0
1000
3000 V
2000
[m/s] 4000
Figure 7. Profiles of velocity along the exit section of the nozzle
Figure 7 reports the velocity profiles, computed by both codes at the exit of the nozzle, for tests with minimum
and maximum mass flow rates. These profiles show that it is reasonable to consider the radius of the core of the jet
of about 10 mm. The free stream aerodynamic parameters are the ones at a reference line, located at the middle of
the core (y=5 mm). Tables 1 and 2 report the test parameters computed by FLUENT (labelled a,…,e) and by DS2V
(labelled A,…,E), related to the mass flow rates 0.5,…,2 g/s, respectively. h0∞ and M∞ are the total enthalpy and the
Mach number of the jet, respectively. The global rarefaction parameters (Kn∞ and Re2) and the dynamic pressure,
reducing the measured drag to the related coefficient are computed by these data.
TABLE 1. Test Parameters from FLUENT
Test
a
b
c
d
e
h0∞
[MJ/kg]
7.7
7.6
8.5
10.9
9.8
V∞
[m/s]
3355
3366
3542
3976
3828
ρ∞
[kg/m3]
1.2×10-4
1.7×10-4
2.1×10-4
2.6×10-4
3.6×10-4
λ∞
[m]
8.2×10-4
6.0×10-4
4.9×10-4
4.2×10-4
3.0×10-4
T∞
[K]
1625
1523
1701
2242
1901
M∞
3.8
3.9
4.0
4.0
4.1
n
[m-3]
2.90×1021
3.92×1021
4.89×1021
5.94×1021
8.24×1021
αN
αN2
0.172
0.180
0.148
0.104
0.146
0.828
0.820
0.852
0.896
0.854
TABLE 2. Test Parameters from DS2V
Test
A
B
C
D
E
h0∞
[MJ/kg]
5.4
5.8
6.8
9.0
8.6
V∞
[m/s]
2808
2952
3203
3662
3606
ρ∞
[kg/m3]
1.4×10-4
2.1×10-4
2.2×10-4
2.8×10-4
3.8×10-4
λ∞
[m]
6.7×10-4
4.4×10-4
4.4×10-4
3.8×10-4
2.7×10-4
T∞
[K]
1196
1128
1336
1795
1600
M∞
3.0
3.2
3.3
3.5
3.5
n
[m-3]
3.34×1021
5.02×1021
5.14×1021
6.26×1021
8.91×1021
αN
αN2
0.197
0.206
0.156
0.110
0.172
0.803
0.794
0.844
0.890
0.828
The profiles of Re2 and Kn∞, as a function of the mass flow rate, are shown in Fig.8. Re2 and Kn∞, computed by
data from DS2V (table 2), are in the ranges 57-150 and 4×10-2-2×10-2, the ones from FLUENT (table 1) are in the
ranges 41-119 and 5×10-2-2×10-2. Both parameters indicate that the flow rarefaction is in continuum low-density
regime but, contrarily to what predicted in [2], the flow rarefaction is pretty far from the discrete molecules
regime. Re2 and Kn∞ from AWT (not shown in figure) are in the ranges 26-98 and 8×10-2-2×10-2.
This remark is confirmed, experimentally, by the comparison of the present drag coefficients, obtained by
reducing the measured drag by data from DS2V, from FLUENT and, for comparison, also from AWT [2] with the
ones from the literature (Fig.9). The drag coefficients from AWT [2] are labelled 1,…,5. The drag coefficients
from DS2V range between 1.25 and 1.02. The ones from FLUENT range between 1.06 and 0.96. The ones from
AWT range between 1.56 and 1.31. The drag computed by DS2V and reduced to the related coefficient by data in
table 2 range between 1.32 and 1.16. As already pointed out in [2], the drag measurement from test 4 (or d, D in
table 1, 2) is inaccurate.
The trend of the drag coefficients obtained by data from table 2 agrees fairly well both with the ones of
experimental data and the one computed by DS2V. On the opposite the trend of the drag coefficients from
FLUENT seems to be inaccurate. For this reason the error bars are reported only on the coefficient from DS2V.
The incorrect working of FLUENT agrees with the analysis by Zuppardi [7]. This analysis fixed the limit of a
correct working of this code to flows with Kn∞<7×10-5 and/or Re2>3×104.
CONCLUSIONS AND FURTHER DEVELOPMENTS
Processing once again test parameters in the arc wind tunnel in Naples (SPES), by means of two advanced codes,
the Navier-Stokes (FLUENT) and the Direct Simulation Monte Carlo (DS2V) solvers, showed that even though the
flow rarefaction level and the Mach number in SPES are lower than the ones formerly computed by a onedimensional and non-diffusive code (AWT), however in SPES it is possible to obtain continuum low-density and
high Mach number flow.
Both the rarefaction level and the Mach number can be considered sufficient to perform successful measurements
for studying the dynamics of a space vehicle for various entry conditions. The present computing procedure will be
used to characterize the aerodynamic parameters in a forthcoming campaign of measurements of forces on a model
of sphere-cone capsule. The diameter of the core of the jet has been fixed on about 20 mm; the diameter of the cross
section of the model will be 20 mm, too. For this model Kn∞ and Re2 will range between 3×10-2-1×10-2 and 72-190
for the mass flow rates from 0.5 to 2 g/s. A new and more accurate three components balance has been already made
ready. This is characterized by full scale capacities of 0.7 N and 2.5 N along the x and y axes, with an uncertainty of
about ± 0.004 N and ±0.01 N, respectively.
Furthermore the present analysis showed that DS2V, using as input parameters the ones computed by AWT at the
throat of the nozzle, is proper for solving the flow field in the diverging part of the nozzle, at each test conditions,
and therefore for computing the parameters of the jet, making possible to avoid a complex, hybrid continuummolecular computing procedure. At the same time FLUENT proved to be inadequate to work at the present
rarefaction levels.
REFERENCES
1. Esposito A., Monti R., Zuppardi G., “The Atmospheric Reentry Simulator in Naples”, in 20th Congress of the
Irnational Council of the Aeronautical Sciences (ICAS 96), Sorrento (Italy), 1996, pp.1044-1051.
2. Zuppardi G., Esposito A., Journal of Spacecraft and Rockets 38, 946-948 (2001).
3. Bird, G.A., The DS2G Program User’s Guide, Version 3.2, G.A.B. Consulting Pty Ltd, Killara (Australia),
1999.
4. Bird G.A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford
(England), 1998, pp. 1-23 and pp. 40-41.
5. Fluent 5 User’s Guide, Fluent Inc., Lebanon (U.S.A.), 1998.
6. Bird G.A., The DS2V Program User’s Guide, Version 1.1, G.A.B. Consulting Pty Ltd, Killara (Australia), 2000.
7. Zuppardi G., Paterna D., The Journal of Aerospace Engineering, (submitted for publication).
8. Monti R., Paterna D., Savino R., Esposito A., International Journal of Thermal Sciences 40, 804-815 (2001).
9. Boyd I.D., Chen G., Candler G.V., Physics of Fluids 7, 210-219 (1995).
10. Bird, R.B., Stewartson, W.E., Lightfoot, E.N., Transport Phenomena, 1st ed., Ambrosiana, Milano (Italy), 1960,
pp. 518-523 (in Italian).
11. Whitfield, D.L., AIAA Journal 11, 1666-1670 (1973).
12. Koppenwallner, G., “The Drag of Simple Shaped Bodies in the Rarefied Hypersonic Flow Regime”, in AIAA
20th Thermophysics Conference, Paper 85-0998, Williamsburg (U.S.A.), 1985.
13. Vallerani, E., “A Review of Supersonic Sphere Drag from the Continuum to the Free Molecular Flow Regime”,
AGARD CPP 124, Paper 22, 1973, pp. 1-15.
14. Phillips, W. M., Kuhlthau A.R., AIAA Journal 9, 1434-1436 (1971).
15. Kussoy, M.I., Horstman C.C., AIAA Journal 8, 315-320 (1970).
16. Wegener, P.P., Ashkenas H., Journal of Fluid Mech. 10, 551-562 (1961).
1E+0
1E+3
FMF
contin.
Kn
Re 2
beginning of "discrete
molecules" regime
1E+2
limit of
N-S eq.
1E-1
contin.
low-den.
contin.
1E-2
1E+1
near
free
mol.
Re 2 DS2V
1E+0
Re2 FLUENT
Kn
DS2V
Kn
FLUENT
FMF
1E-3
0.25
0.50
0.75
1.00
1.25
1.50
1.75
.
2.00
1E-1
2.25
m [g/s]
Figure 8. Profiles of the free stream Knudsen number and Reynolds number behind a normal shock as a function of the mass
flow rate
2.00
CD
Continuum low-density
1.80
1.60
Near free molecules
1.40
Phillips (Ref.14)
Kussoy (Ref.15)
1.20
Weger (Ref.16)
AWT (M
DS2V (M
1.00
FLUENT (M
5, Ref.2)
3)
4)
DS2V
0.80
1E-1
FMF
1E+0
1E+1
1E+2
Re 2 1E+3
contin.
Figure 9. Sphere drag coefficients as a function of the Reynolds number behind a normal shock