Study of the Shock Wave Structure about a Body
Entering the Martian Atmosphere
M.S. Ivanov∗ , Ye.A. Bondar∗, G.N. Markelov∗ ,
S.F. Gimelshein† and J.-P. Taran∗∗
∗
Institute of Theoretical and Applied Mechanics, Novosibirsk 630090, Russia
†
Penn State University, University Park, PA 16802
∗∗
ONERA, France
Abstract. The flow structure along the stagnation line for a blunted body descending in the Martian atmosphere
was simulated by the DSMC method for Knudsen numbers ranging from 10 −2 to 10−3 . Along with the conventional continuous approach for translational-internal energy transfer and chemical reactions, the new discrete
internal energy model with corrected reaction rates was used in the computations. The impact of chemical reactions on the structure of the shock-wave front was examined. The nonequilibrium inside the bow shock was
studied both in terms of macroparameters and distribution functions of translational, rotational, and vibrational
energy modes. It is shown that vibrational relaxation and chemical reactions have a significant effect on the
structure of the shock-wave front.
INTRODUCTION
Over thirty missions to Mars have been attempted by space agencies of the U.S., Japan, and the former Soviet Union.
During last years, NASA has been developing a long-term Mars exploration program that charts a course for the
next two decades. Several new Mars missions for 2003, 2005, 2007 and beyond are now taking shape. An important
condition of successful maneuvers in the Martian atmosphere is the detailed knowledge of physics of the flow over
spacecraft at different flight altitudes.
Numerical modeling is important for obtaining credible information on flow features and not only supplements
the experimental research but may also provide unique data in flow regimes where the use of ground-based facilities
is difficult and the flight data are limited. While the principal tools used for a multiparametric study of spacecraft
aerodynamics may still be fast engineering approaches, the utilization of computationally intensive but much more
accurate CFD techniques is necessary.
A sophisticated CFD technique that is well suited for a detailed modeling of chemically reacting carbon dioxide
flows around a spacecraft descending in the Martian atmosphere is the direct simulation Monte Carlo (DSMC)
method [1]. In mid-90s, the method has extensively been used to calculate hypersonic flows of CO 2 over the Pathfinder
and Microprobe Mars entry capsules (see Ref. [2] and references therein). The flow was calculated in a wide range of
altitudes from about 50 km to 140 km, with and without chemical reactions.
An accurate account for real gas effects, namely, the excitation of internal degrees of freedom and chemical
reactions, was shown to be important for aerothermodynamics of various bodies in the Earth atmosphere [3, 4].
Similarly, it may turn out to be important for the Martian atmosphere as well. The DSMC simulation of energy
transfer between the internal and translational molecular modes and chemical reactions is complicated for the Martian
atmosphere. This is because CO 2 is a triatomic molecular system with no existing comprehensive DSMC collision
model.
The Larsen-Borgnakke (LB) model is the most common in the DSMC method for the translational-internal energy
exchange. The key advantage of this model is that it is relatively simple and can be applied to polyatomic molecules [1].
Both continuous and discrete representations of rotational and vibrational energy of molecules in the LB model are
currently used in the DSMC method. As it was shown in [5], the use of discrete models in the DSMC method is
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
481
-0.2
0.0
0.2
0.4
0.6
0.8
-0.2
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
FIGURE 1.
0.2
0.4
0.6
0.2
0.4
0.6
0.8
1.2
2
3
5 1.0
8
12
17
22
29 0.8
36
45
54
64 0.6
75
87
99
113
127 0.4
143
159
176
194
213 0.2
232
253
0.0
-0.2
0.0
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.8
0.0
-0.2
0.0
0.2
0.4
0.6
0.8
Translational temperature Tt /T∞ flowfields. Non-reacting (left) and reacting (right) discrete cases
preferable. The chemical reactions are usually taken into account in the DSMC method by using the Total Collision
Energy (TCE) model derived for continuous internal energies [1]. This model was recently modified for the discrete
case in [5].
The main objective of this work is to use continuous and discrete models for internal energy transfer and chemical
reactions of the DSMC method to analyze their impact on the structure of the shock wave in front of a body entering
the Martian atmosphere.
STATEMENT OF PROBLEM AND FREE STREAM CONDITIONS
The structure of a bow shock in front of a 1 m radius cylinder has been investigated for three Knudsen numbers (0.01,
0.0033,and 0.001) with and without chemical reactions. The computations were performed using the 2D version of the
SMILE software package [4]. The diffuse reflection model with complete accommodation of translational and internal
energy was used at the surface.
Two models of internal (rotational and vibrational) energy modes, continuous and discrete, were employed in the
computations. The LB model with temperature-dependent rotational and vibrational collision numbers was used for
energy transfer between the internal and translational modes. The vibrational modes of CO 2 were considered as a
whole in the continuous case with a characteristic temperature of vibration of 945 K. Three separate vibrational modes
of CO2 with characteristic temperatures of vibration (945, 1903, and 3329 K) were used in the discrete internal energy
model.
The TCE model was applied for simulation of chemical reactions in a CO 2 /N2 mixture. A reduced set consisting of
11 dissociation and exchange reactions [6], which is sufficient for flow conditions considered, is listed below.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
O +CO
N2 +CO
CO +CO
CO2 +CO
O +CO2
N2 +CO2
CO +CO2
CO2 +CO2
CO +CO
O +CO
O +CO2
→
→
→
→
→
→
→
→
↔
↔
↔
482
O +C + O
N2 +C + O
CO +C + O
CO2 +C + O
O +CO + O
N2 +CO + O
CO +CO + O
CO2 +CO + O
CO2 +C
O2 +C
O2 +CO
T / T_inf
T / T_inf
300
T / T_inf
300
300
250
250
200
200
150
150
100
100
100
50
50
50
200
Tt
Tr
Tv
Tt
150
Tr reacting
Tv reacting
250
0
-0.6
non-react
non-react
non-react
reacting
-0.5
-0.4
-0.3
X, m
-0.2
-0.1
0
0.0 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05
X, m
0
0.00 -0.25
-0.20
-0.15
-0.10
X, m
-0.05
0.00
FIGURE 2. Distributions of translational, rotational, and vibrational temperatures along the stagnation line. Continuous case (left
– Kn = 0.01, center – Kn = 0.003, right – Kn = 0.001)
The chemical rate constants were taken from [6], and for the discrete internal energy model, they were modified to
match the correct reaction rates according to the procedure proposed in [5].
Free stream parameters for a body entering the Martian atmosphere were assumed as given in [2]. A free-stream
velocity of 7400 m/s and temperature of 137.4 K were taken. The mole fractions of CO 2 and N2 were assumed
to be 0.9537 and 0.0463, respectively. Under these conditions, the free-stream density values were 4.231 × 10 −6 ,
1.2693 × 10 −5 , and 4.231 × 10 −5 kg/m3 for the three considered Knudsen numbers of 0.01, 0.0033, and 0.001,
respectively. A temperature value of 1000 K was set at the surface for all the cases.
The center of a cylinder with a radius of 1 m was located at the point (x, y) = (1, 0). The stagnation point corresponds
to (x, y) = (0, 0).The average number of model molecules in a typical computation was 7 million (for Kn = 0.001),
and the number of collision cells was about 1 million.
RESULTS
The general flowfield structure for Kn = 0.001 is illustrated in Fig. 1, where the translational temperature of the
mixture is given for non-reacting (left) and reacting (right) flows with the discrete model for internal energy transfer.
A significant effect of chemical reactions both on the temperature behind the shock wave and on the stand-off distance
is clearly visible.
An analysis of sensitivity of the shock wave structure along the stagnation line in front of the cylinder to models of
internal energy transfer was made for the following cases.
•
•
•
•
non-reacting gas with the continuous model of internal energy exchange
non-reacting gas with the discrete model of internal energy exchange
reacting gas with the continuous model of internal energy exchange
reacting gas with the discrete model of internal energy exchange
The profiles of translational, rotational, and vibrational temperatures along the stagnation line are plotted in Fig. 2
for reacting and non-reacting flows. The translational temperature has a prominent maximum inside the shock. Farther
downstream, the rotational temperature, in turn, reaches a maximum. After that, all three temperatures relax to the
same value. It is evident that, for the non-reacting flow, there is a visible plateau in the temperature distributions
behind the shock wave front, which becomes longer as the Knudsen number decreases. For the reacting flow, there
is no such plateau, and the temperature monotonically decreases toward the stagnation point. The flow along the
stagnation line can be divided into three zones: shock-wave front (up to the point of equalization of translational and
rotational temperatures), viscous shock layer (finalization of vibrational relaxation and thermally equilibrium viscous
flow), and boundary layer. For the greatest Knudsen number considered (Kn = 0.01), the zone of the shock layer is
absent, and the shock-wave front transforms directly to the boundary layer.
The impact of the internal energy model for the reacting flow is illustrated in Fig. 3. It is clear that there is no
significant difference between the results obtained using the continuous and discrete models of internal energy transfer
483
T / T_inf
T / T_inf
300
T / T_inf
300
300
250
250
200
200
150
150
100
100
100
50
50
50
200
Tt
Tr
Tv
Tt
150
Tr discrete
Tv discrete
250
0
-0.6
continuous
continuous
continuous
discrete
-0.5
-0.4
-0.3
X, m
-0.2
-0.1
0
0.0 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05
X, m
0
0.00 -0.25
-0.20
-0.15
-0.10
X, m
-0.05
0.00
FIGURE 3. Distributions of translational, rotational, and vibrational temperatures along the stagnation line. Impact of the internal
energy models on the reacting flow (left – Kn = 0.01, center – Kn = 0.003, right – Kn = 0.001)
T / T_inf
T / T_inf
300
300
Tt Kn=0.01
Tr Kn=0.01
Tv Kn=0.01
250
250
200
200
Tt Kn=0.0033
Tr Kn=0.0033
150
150
Tv Kn=0.0033
Tt Kn=0.001
Tr Kn=0.001
Tv Kn=0.001
100
100
50
50
0
-0.35
-0.30
-0.25
-0.20 -0.15
X, m
FIGURE 4.
-0.10
-0.05
0.00
0
0
Tt 1D shock
Tr 1D shock
Tv 1D shock
10
20
30
40
X / lambda_inf
50
60
70
Shock wave structure similarity for the non-reacting flow. Continuous model
for all Knudsen numbers considered. This result confirms the applicability of the TCE discrete model with corrected
reaction rates.
Figure 4 (left part) shows the profiles of translational, rotational, and vibrational temperatures for the continuous
non-reacting case. The effect of rarefaction on the shock-wave front thickness is clearly seen. The temperature
gradients inside the front increase significantly, and the size of the disturbed region (i.e., the position of the fore
front of the smeared shock) decreases with increasing free-stream density. The processes of internal energy relaxation
behind the shock wave finish rather rapidly for low Knudsen numbers (0.0033 and 0.001). Then, the temperature is
constant inside the shock layer and drastically decreases in the boundary layer, approaching the wall temperature. For
the high Knudsen number (Kn = 0.01), relaxation of vibrational energy is completed inside the boundary layer.
If the distance along the stagnation line is non-dimensionalized with the free-stream mean free path for the
corresponding Knudsen number, then the temperature profiles become self-similar. To compare the temperature
distributions for different Knudsen numbers, the profiles were shifted so that the translational temperature maxima
be at the same non-dimensional location. The shifted profiles versus the non-dimensional distance x/λ ∞ are plotted
in Fig. 4 (right part). The same figure also shows the results of solving the 1D shock-wave structure problem in a
relaxing gas for the continuous model . In this case, the boundary conditions behind the shock are determined by
Rankine—Hugoniot relations for the gas with the excited vibrational degrees of freedom,
n1 u1 = n 2 u2
K
K
2
2
n1 u1 ∑ mi ri + kT1 = n2 u2 ∑ mi ri + kT2
i=1
i=1
484
(1)
(2)
T / T_inf
T / T_inf
300
300
Tt Kn=0.01
Tr Kn=0.01
250
250
200
200
Tr Kn=0.0033
Tv Kn=0.0033
150
150
Tt Kn=0.001
Tr Kn=0.001
100
100
50
50
0
-0.35
-0.30
-0.25
-0.20 -0.15
X, m
FIGURE 5.
-0.10
-0.05
0
0
0.00
Tv Kn=0.01
Tt Kn=0.0033
Tv Kn=0.001
10
20
30
40
X / lambda_inf
50
60
70
Shock wave structure similarity for the chemically reacting flow. Discrete model
K
K
K
K
i=1
i=1
i=1
i=1
u21 ∑ mi ri + kT1 ∑ (7 + ξi1vib )ri = u22 ∑ mi ri + kT2 ∑ (7 + ξi2vib )ri ,
(3)
where mi and ri are the molecular mass and the mole fraction of the ith species, respectively, n is the number density,
T is the temperature, u is the velocity, and k is the Boltzmann constant. The subscripts 1 and 2 denote the upstream and
downstream values, respectively. The number of effective vibrational degrees of freedom of the ith species is given by
the relation based on the Simple Harmonic Oscillator approximation,
ξivib = Ji
2θi /T
,
exp[θi /T ] − 1
where Ji is the number of vibrational modes and θ i is the characteristic vibrational temperature.
This 1D solution agrees well with the temperature distributions along the stagnation line outside the boundary layer.
These results confirm that the stand-off of the bow shock is long enough (in mean free paths) for rather low Knudsen
numbers, and complete relaxation of internal energy occurs. For a larger Knudsen number (Kn = 0.01), complete
relaxation is not observed, and all three temperatures become equal inside the boundary layer.
The temperature profiles for the chemically reacting flow are also self-similar in non-dimensional coordinates (see
Fig. 5). Note, the influence of chemical reactions is already observed inside the shock wave front. The maximum
values of all three temperatures inside the shock wave are somewhat lower as compared to the non-reacting flow.
Behind the shock wave, chemical reactions result in a significant decrease in temperature inside the shock layer along
the stagnation line.
Recall that the use of the discrete model with corrected reaction rates does not lead to significant differences in
temperature distribution along the stagnation line from the continuous case for the chemically reacting flow. Moreover,
the study shows that the distributions of mass fractions of all the species of the flow, which are plotted in Fig. 6 (left
part) for Kn = 0.0033, are similar for both cases. A significant effect of chemical reactions on the flow already inside
the shock-wave front should also be noted. For instance, at the point x = −0.175, which corresponds to the maximum
of rotational temperature, the mass fraction of CO 2 is more than 20% lower than in the free stream.
The profiles of translational, rotational, and vibrational temperatures, and also different components of the translational temperature (perpendicular Tty and parallel Ttx to the flow direction) of CO 2 are shown in Fig. 6 (right part).
It is of interest to note that translational-rotational relaxation inside the shock-wave front proceeds at the background
of translational nonequilibrium (Ttx is not equal to Tty ). The profiles of parallel, perpendicular, and rotational temperatures converge at one point (x ∼ −0.175 m). At this point, excitation of vibrational modes is already significant
(Tv ∼ 0.5Tt = 0.5Tr ). Vibrational energy relaxation continues further downstream and ends at the point x = −0.075 m,
where the region of an equilibrium viscous flow in the shock and boundary layer begins.
Flow nonequilibrium inside the shock-wave front can be demonstrated in more detail at the level of distribution
functions. The distribution functions of longitudinal velocities, rotational energy, and vibrational energy (for the first
485
Mass fraction
T / T_inf
1.0
Continuous
O
0.8
0.6
Discrete
O
500
Tt
450
C
O2
N2
C
O2
N2
400
CO
CO2
CO
CO2
300
0.4
Tr
Tv
Ttx
350
Tty
DF POINTS
250
200
150
0.2
100
50
0.0
-0.25
-0.20
-0.15
-0.10
-0.05
0
0.00 -0.35
-0.30
X, m
-0.25
-0.20 -0.15
X, m
-0.10
-0.05
0.00
FIGURE 6. Mass fraction of different species along the stagnation line (left) for Kn = 0.0033. Distributions of CO2 temperatures
(Tt , Ttx , Tty , Tr , and Tv ) for Kn = 0.0033 and points of distribution functions sampling (right)
mode with a characteristic temperature of 945 K) of CO 2 have been obtained in different locations along the stagnation
line (see the right part of Fig. 6, squares). At a large distance from the cylinder, in the free-stream, all the distribution
functions correspond to equilibrium (see the first row of plots in Fig. 7). At the point x = −0.25, where the rotational
and vibrational temperatures start increasing, the distribution functions come into strong nonequilibrium. The velocity
distribution function appears as a delta peak corresponding to the free-stream velocity. There exist a small but finite
number of molecules with large negative velocities. These molecules are generated in the hot portion of the shock
wave and propagate upstream into the cold portion of the flow. At the point x = −0.2, which corresponds to the
peak of the translational temperature, the distribution function of velocity starts to become Maxwellian, but the delta
peak still exists. The rotational and vibrational energy distribution functions also correspond to strong nonequilibrium.
At the point x = −0.175 corresponding to the maximum of the rotational temperature, the distribution functions of
velocity and rotational energy are close to the Maxwellian shape. The vibrational energy distribution function is not
in equilibrium at this point. This nonequilibrium is manifested in the overpopulation of the zeroth vibrational level,
whereas the energy redistribution over the remaining levels has an almost Boltzmann form. Complete equilibrium is
observed at x = −0.075, where all three temperatures become identical.
CONCLUSIONS
•
•
•
•
The DSMC method was used to simulate the shock structure about a cylinder for Knudsen numbers ranging from
10−2 to 10−3 in the Martian atmosphere. Computations were performed with and without chemical reactions for
continuous and discrete internal energy transfer models.
The impact of chemical reactions on the flowfield structure was examined; the stand-off distance of the bow shock
decreases by 30 % for the reacting flow. The results for a chemically reacting gas with discrete and continuous
internal energies are in a good agreement, which may be considered as the verification of TCE model for the
discrete case with corrected reaction rates. The advantage of the discrete model is the fact that it allows one to
correctly reproduce the distribution functions of rotational and vibrational energies and to accurately capture the
Jeans and Landau—Teller relaxation rates.
The macroparameter profiles along the stagnation line plotted versus the distance normalized by the free stream
mean free path were found to be identical for different Knudsen numbers for reacting and non-reacting flows.
The computations showed strong thermal nonequilibrium between translational and internal energy modes inside
the shock wave front both in terms of macroparameters and distribution functions.
Vibrational relaxation begins inside the shock-wave front and partly occurs at the background of translationalrotational nonequilibrium.
486
F_Ux
0.6
FREE STREAM
F_Er
FREE STREAM
FREE STREAM
1e+00
SIMULATED DF
ELLIPSOIDAL DF
0.5
F_Ev
1e-01
SIMULATED DF
SIMULATED DF
EQUILIBRIUM DF
1e-02
MAXWELL DF
EQUILIBRIUM DF
1e-01
0.4
1e-02
0.3
1e-03
1e-03
0.2
1e-04
1e-04
0.1
0.0
1e-05
25 26 27 28 29 30 31 32 33 34 35
0
Sx
F_Ux
X = -0.25 m
50
F_Er
0.5
100
150
LEVEL
200
250
300
1e-05
0
X = -0.25 m
F_Ev
1e-01
0.4
5
10
15
LEVEL
20
25
30
X = -0.25 m
1e+00
1e-01
1e-02
0.3
1e-02
1e-03
0.2
1e-03
1e-04
0.1
0.0
-10
0
F_Ux
10
20
Sx
30
40
50
1e-04
1e-05
0
50
X = -0.2 m
F_Er
0.090
100
150
LEVEL
200
250
300
1e-05
0
10
X = -0.2 m
F_Ev
1e-01
20
30
40
LEVEL
50
60
70
X = -0.2 m
1e+00
0.075
1e-01
1e-02
0.060
1e-02
0.045
1e-03
1e-03
0.030
1e-04
1e-04
0.015
0.000
-40
-20
0
20
40
60
1e-05
0
50
100
Sx
F_Ux
X = -0.175 m
F_Er
150
LEVEL
200
250
300
X = -0.175 m
0.04
10
F_Ev
1e-01
0.05
1e-05
0
20
30
40
LEVEL
50
60
70
50
60
70
50
60
70
X = -0.175 m
1e+00
1e-01
1e-02
0.03
1e-02
1e-03
0.02
1e-03
1e-04
0.01
0.00
-40 -30 -20 -10
F_Ux
0
10
Sx
20 30
40
50
1e-04
1e-05
0
X = -0.075 m
50
F_Er
0.06
100
150
LEVEL
200
250
300
1e-05
0
X = -0.075 m
10
F_Ev
1e-01
20
30
40
LEVEL
X = -0.075 m
1e+00
0.05
1e-01
1e-02
0.04
1e-02
0.03
1e-03
1e-03
0.02
1e-04
1e-04
0.01
0.00
-30
-20
-10
0
Sx
10
20
30
1e-05
0
50
100
150
LEVEL
200
250
300
1e-05
0
10
20
30
40
LEVEL
FIGURE 7. Distribution functions F_Ux , F_Er , and F_Ev of axial velocity, rotational energy, and vibrational energy of CO2
√
respectively. Sx is given by Sx = Ux / 2RT∞
487
ACKNOWLEDGEMENT
Reaserch performed at the Institute of Theoretical and Applied Mechanics (Russia) and ONERA (France) was carried
out under INTAS Grant 99-0701.
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1.
2.
3.
4.
5.
6.
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12, No. 4, 489–495 (1998).
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Gimelshein, S. F., Gimelshein, N. F., Levin, D. A., Ivanov, M. S., and Markelov, G. N., AIAA Paper 2002-2759 (2002).
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488