DSMC Simulation with Gas-Surface Interaction Models
in Hypersonic Rarefied Flow
Nobuyuki Tsuboi* and Yoichiro Matsumoto**
* Research Division for Space Transportation, Institute of Space and Astronautical Science, Yoshinodai, Sagamihara,
Kanagawa, 229-8510, Japan
** Department of Mechanical Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Abstract. A DSMC (direct simulation Monte Carlo) simulation using Dynamical Molecular Collision
(DMC) model based on Molecular Dynamics (MD) calculation with various gas-surface interaction models
is applied for solving the two-dimensional nonequilibrium hypersonic rarefied flow over a flat plate with
leading edge angle. The gas surface interaction models adopt the diffuse reflection model, the CercignaniLampis-Lord(CLL) model and the Multi-Stage (MS) model. Numerical results over the flat plate show that
the nonequilibrium between translational and rotational temperature is obtained behind the leading edge
over the plate and that the effect of the gas-surface interaction model is significant.
INTRODUCTION
Many researchers have been studying hypersonic rarefied gas flows around space vehicles with respect to
viscous interaction. Viscous interaction is defined as the mutual interaction between external flow field and
boundary layer growth around a body of given shape. Especially, hypersonic viscous interaction near a leading
edge in rarefied regime shows significant interaction between shock wave and boundary layer growth, and
includes strong nonequilibrium between molecular internal degrees of freedom. Schematic figure of viscous
interaction near a leading edge is shown in Fig. 1. For the continuum flow regime, strong and weak interaction
K
Kinetic »l«Traiis-.»»U—— Merged .
Layer
Flow
I ition I
Interaction
FIGURE 1. Schematic figure of viscous interaction near a sharp leading edge.
is governed near the leading edge. However, a merged layer which is difficult to distinguish between the shock
wave and the boundary layer is appeared near the leading edge for rarefaction parameter V > 0.15 [1]. In the
merged layer, the nonequilibrium between the molecular internal degrees of freedom is occured such as in a free
jet expansion flow. However, the nonequilibrium in the merged layer has not been discussed in detail because
there are less experiments for the nonequilibrium in the merged layer.
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
331
In numerical simulations of the hypersonic flow in rarefied regime, DSMC method [2,3] is valid as the
continuum approach may break down in such as a low density flow. Gas-gas molecular collision and gas-surface
interaction models have significant roles on the accuracy of the simulation. Larsen and Borgnakke constructed
for gas-gas collision model [4] based on the equilibrium property of the molecular energy distributions. Lord
presented for gas-surface interaction model [5] with a lobular scattering distribution. However, the models have
a disadvantage that they are not constituted by the molecular energy exchange process between translational
and rotational energy in micro-scale phenomena.
The multi-scale models are recently developed by molecular dynamics simulation in order to contain the
information about the micro-scale phenomena. The schema of the multi-scale model is shown in Fig. 2. A bow
shock wave around a re-entry vehicle occurs in non-equilibrium rarefied flow. In the shock wave, large amount
of molecular collision relaxations between diatomic molecules exist to exchange between the translational and
rotational energy. With respect to two molecules, interaction between atomic nucleus and electron appears.
Because it needs complicated procedure and computational cost to solve the Shrodinger equations to obtain
the interaction between them, it is useful to assume the potential such as Lennard-Jones potential. Next, the
molecular dynamics simulations with various translational and rotational energies are conducted to obtain the
probability density function of translational and rotational energy exchange before and after the molecular
collisions and to construct gas-gas collision model and gas-surface collision models to estimate aerodynamic
characteristics around the re-entry vehicle. For the gas-gas collision model, Tokumasu and Matsumoto have
constructed a Diatomic Molecular Collision (DMC) model [6] for nitrogen molecule/graphite surface which is
able to capture the nonequilibrium characteristics in rarefied gas flow below 2,000 K. The DMC model is based
on the cross sections and energy distributions after the collisions obtained by molecular dynamics simulations
for diatomic molecules. On the other hand, Yamanishi and Matsumoto have presented a Multi-Stage (MS)
model for gas-surface interaction which is based on molecular dynamics simulations [7].
Molecular
Dynamics
Quantum
Dynamics
Electron
Rarefied Gas
Dynamics
i
Shock wave
x;-.
Atomic nucleus
I
Relaxation of
molecular collisions
Interaction betwee
atomic nucleus and electron
Modeling
Modeling |
Energy
j
V
Nonequilibrium flow
I__________I
Probability
Distance
Translational/
Rotational energy
Collision model
Potential
FIGURE 2. Schematic diagram of the present multi-scale analysis model.
The purpose of the present paper is to show the simulation of the hypersonic rarefied gas flow over a flat
plate using the DSMC method with the DMC model for nitrogen molecular collision and the MS model for
the nitrogen/graphite gas-surface interaction.
332
NUMERICAL METHOD
The present DSMC method adopts the DMC model for gas-gas collision model and the diffuse, MS and CLL
[5] models for gas-surface interaction model.
The DMC model of diatomic molecules is based on the collision dynamics and featured by the cross sections
and energy distributions after molecular collisions which are obtained by the MD simulations. The DMC model
is constructed as follows. Two-body collisions of diatomic molecules with various translational/rotational energies are conducted using the MD simulations. Then, probability density functions that represent energy
distributions after collisions and collision cross sections are determined, and the database of those are established. Furthermore, in the DSMC simulations, energy distributions after collisions in each cell are calculated
from those before collisions using the database. The model is valid below 2,000 [K] in which there is small
vibrational excitation effect.
The MS model for diatomic molecules is also based on the database obtained from MD simulations. The
MS model is developed as follows. The MD simulations are used for the numerical analysis on the scattering
of a nitrogen gas molecule from a clean graphite surface. The scattering direction, translational and rotational
energies of gas molecules after each collision obtained by the MD simulations are saved to construct the
database. The basic idea of the model is to separate the collision process into three stages. At each stage, the
energy transfer, the scattering direction and the trapping probability of the gas molecules are determined from
the correlation in the database.
For collision-frequency calculation, the null-collision technique [8] is adopted in the DSMC simulation.
The parallel DSMC method is based on the domain decomposition with load balancing in which a portion
of the physical space along with its associated cells and molecules are allocated to each processor [9]. A
complete DSMC code is loaded on each processor, and the simulation in each sub-domain proceeds essentially
independently of the other domains. Molecules that cross the sub-domain boundaries are sent to the appropriate
processor using MPI message passing scheme based on single instruction multiple data. In each processor, data
of particles (velocity, position and rotational energy) that have to send to other processors are stored in temporal
array, then they send to the corresponding processors. From global point of view, the list of particle's data
builds a matrix to be sent to other processors. The data are stored in the temporal array, then they are sent to
the corresponding processors. Thus the simulations are synchronized between the movement and the collision
routines at each time step.
Inflow conditions are shown in Table 1. The conditions are as the same for Lengrand's experiments. Simu-
Table 1. Simulation conditions.
No.
Leading Edge angle [deg.]
Mach number M^
Working Gas
Stagnation pressure po[Pa]
Stagnation temperatureT0[K]
Free stream temperature ToJK]
Freestream pressure poo [Pa]
Free stream velocity Voo[m/s]
Reynolds number Reoo(based on
L=0.05[m],2L=plate length)
Wall temperature TW[K]
Knudsen number Krioo
(based on L)
Wall boundary condition
Accomm. coeff. (Xn/Ott/OtT
Note
1
2
3
4
5
—
6
20
7
20.2
Nitrogen
3.5xl05
1,100
1332
0.06831
1,503
566
290
0.047
Diffuse
-
MS
with
without
thermal
thermal
equilibrium equilibrium
for MS
for MS
333
CLL
Diffuse
—
1/0.986/1
MS
CLL
—
1/0.986/1
without
thermal
equilibrium
for MS
lations are done for the flat plate without leading edge and with 20 degrees leading edge. For each shape, the
effects of the gas surface interaction model are estimated. Black graphite paint is used for the model surface in
the experiment [10] so that MS model for graphite surface is applied in the present simulations. Two cases of
MS model are used for the DSMC simulations. In one case (no.2), the thermal equilibrium is assumed in the
energy transfer stage on the plate, and the thermal equilibrium is not assumed in another case (no.3 and 6).
One set of normal and tangential accommodation coefficients, an and a$, and rotational accommodation
coefficients, ar in the CLL model is given. For case of no. 4 and 7, an = ar = 1.0 and at = 0.986 are
adopted assuming that the boundary is an engineering surface [11], [12]. The coefficients are decided in their
experiments with molecular beam scattering.
RESULTS AND DISCUSSIONS
The numerical results for no leading edge is shown in Figs. 4, 5. Figure 4 shows density contours over the
flat plate. A thick shock wave is generated near the leading edge and the strength increases as X/L increases.
The influence of the gas-surface model is small in the density contours.
Figure 5(a) shows density profiles over the plate at Jf/,L=1.5. It is shown that the maximum density with
the diffuse reflection model and thermal equilibrium case for the MS model are higher than other cases. The
density distributions near the plate in no. 3 show the least values of others. However, all simulation results
disagree with the experimental results. The pressure distributions (Fig. 5 (b)) and the heat transfer coefficient
distributions on the plate (Fig. 5(c)) also indicate that the results with the diffuse model and MS model(no.2)
are higher profiles than the other two results. Especially, the results of MS (no. 3) in Figs. 5(b) and 5(c) are
close to the experimental results near the trailing edge where the shape of the leading edge has small effects.
The results indicate that there is no domain on the plate where the diffuse reflection and thermal equilibrium
dominate.
1.0
l.O-i
Y/L 0.5 -
0.0
Y/L 0.5 -
I
0.0
I
I
0.5
I
I
I
1.0
X/L
I
1.5
0.0
I
I
0.0
2.0
I
(a) Diffuse model(no.l)
I
0.5
Y/L 0.5 -
I
0.0
I
I
1.0
X/L
I
(b) MS model(no.3)
1.0
0.0^
I
I
0.5
I
I
1.0
X/L
I
I
1.5
I
(c) CLL model(no.4)
Figure 4. Density contours over the flat plate.
334
I
2.0
I
1.5
I
I
2.0
The temperature distributions on the wall in Fig. 5(d) shows that the translational temperature profiles
completely differs depending on the accommodation of the gas molecule, but that the rotational temperature
with the diffuse model is only different results rather than other models.
The above simulations were conducted for the plate without leading edge, however, the leading edge effects
would be considered to be significant. Then the simulations for the plate with the leading edge angle of 20
degrees are carried out and estimated the effects of leading edge angle and gas-surface interaction.
i
20.0
Experiment
[Lengrand et al(1992)"
•9——— Diff.,No.l
-H——— MS,No.2
MS,No.3
CLL,No.4
0.8
0.6
Experiment
[Lengrand et al.(1992)]
0.4
-e—— Diff.,No.l
-B—— MS,No.2
0.2
-e—— MS,No3
-A—— CLL,No.4
I
I
I
I
I
0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
p/p
0.0
0
_L
_L
_L
0.5
1
1.5
X/L
00
(a) Density profiles at X/L=1.5
(b)Pressure distributions on the plate
0.035
0.030
0.025
Ttr,Diff.,No.l
Ttr,MS,No3
Ttr,CLL,No.4
0.020
B 0.015
Experiment
[Lengrand et al(1992)]
Diff.,No.l
• MS,No.2
• MS,No3
CLL,Ncx4
0.010
0.005
0.000
0
0.5
I
I
1
1.5
- - - O- - - - Trot,Diff.,No.l
. _ _ Trot,MS,No.3
X/L
(d)Gas temperature distributions on the plate
(c) Heat transfer rate distributions on the plate
Figure 5. Effects of the gas-surf ace interaction in the case of no leading edge angle.
Figure 6(a) shows density profiles over the plate at X/L=1.5. It is shown that the maximum density with
the diffuse reflection model is higher than other cases. The density distributions near the plate for no. 7 show
the least values of others. Compared the results with Fig. 5(a), there are effects of the leading edge angle
and the results with diffuse model is close to the experimental results in regard to the maximum density. The
pressure distributions (Fig. 6 (b)) and the heat transfer coefficient distributions on the plate (Fig. 6(c)) also
335
indicate that the results with the diffuse model are higher profiles than the other two results. Especially, the
results of MS(no.6) in Figs. 6(b) and 6(c) agree well with the experimental results at X/L > 0.75.
The temperature distributions on the wall in Fig. 6(d) show that the translational temperature profiles
completely differs depending on the accommodation of the gas molecule, but that the rotational temperature
with the diffuse model is only different results rather than other models. The same tendency exists in the case
of leading edge angle of 20 degrees as compared with the case of no leading edge.
i
20.0
Experiment
[Lengrandetal.(199
Diff,no.5
0.8
15.0
MS,no.6
CLL,no.7
0.6
10.0
-
0.4
Experiment
[Lengrand et aL(1992)]
-e—— Diff,no.5
5.0
0.2
0
0.6 0.8
I
1
I
I
I
I
1.2 1.4 1.6 1.8
0.0
2
-e—— MS,no.6
-ft—— CLL,no.7
0
I
I
1
X/L
1.5
(b)Pressure distributions on the plate
(a) Density profiles at X/L=1.5
0.035
60
0.030
50
0.025
40
0.020
Ttr,Diff,no.5
Ttr,MS,no.6
Ttr,CLL,no.7
30
0.015
0.010
0.005
0.000
I
0.5
0
0.5
Experiment
[Lengrand etal.(1992);
Diff,no.5
MS,mx6
CLL,no.7
_|______|______
1
1.5
20
- - - - € > - - - - Trot,Diff,no.5
- - - - « - - - - Trot,MS,no.6
- - - - A - - - - Trot,CLL,no.7
10
0
X/L
J_____I_____I
0
0.5
1
X/L
1.5
(d)Gas temperature distributions on the plate
(c) Heat transfer rate distributions on the plate
Figure 6. Effects of the gas-surf ace interaction in the case of 20 deg. leading edge angle.
Figure 7 shows that the translational and rotational energy contribution for the heat transfer rate on the
plate. It is shown that the translational energy contributes more than the rotational energy on the heat transfer
rate on the plate. The results indicate that heat transfer rate is significantly dependent on the translational
energy for such the flow parallel to the plate. Heat transfer rate contributed from the rotational energy has
minus value near the leading edge because the rotational temperature near the leading edge is lower than the
336
0.035
0.030
0.025
CHtr,Diff,no.5
CHtr,MS,no.6
0.020
CHtr,CLL,no.7
g 0.015
0.010
- - - CHrot,Diff,no.5
... cHrot,MS,no.6
0.005
... CHrot,CLL,no.7
0.000
-0.005
I
0
0.5
1
X/L
1.5
Figure 7. Translational and rotational energy contribution on heat transfer rate on the plate in the case of 20 deg.
leading edge angle.
wall temperature 290[K]. Gas surface interaction models are small influence for the rotational temperature
contribution on the heat transfer rate, however, they are large for the translational temperature contribution.
The tendency is similar to the results for gas temperature distributions on the plate. The reason would be
considered to be the difference of the translational accommodation coefficients.
The above comparisons show that the DSMC results did not coincide with the experimental results. The
factors for the discrepancy in the experiment side are considered: (i) non-uniformity flow at nozzle exit, (ii)
three-dimensional effects, (iii) rotational temperature freezing in the nozzle, (iv) vibrational excitation. For
non-uniformity flow at the nozzle exit, Allegre [13] measured density distributions downstream the nozzle exit.
The results shows that the density gradient exists at the nozzle exit due to the thick boundary layer developed
in the nozzle and the use of conical nozzle. In regards to the three-dimensional effects, Yamaguchi simulated
in the present inflow conditions (but uniform inflow) and revealed that three-dimensional effects were small
[14]. For the rotational temperature freezing in the nozzle, the influence would be considered to be significant,
however, the value was not estimated. Finally, the vibrational excitation for TO = 1,100[,K"] is considered to
be small, but the excitation rate for TO = 1,100[,K"] would be about 10%. Furthermore, degree of vibrational
temperature freezing is larger than that of rotational temperature and is thought to be freezing completely.
Therefore, the above effects in the experiment should be estimated in order that the experimental data are
utilized for the validation of the simulation.
CONCLUSIONS
The DSMC simulations with the DMC model for the gas-gas collision and the MS model for gas-surface
interaction are conducted for the non-equilibrium hypersonic gas flow over the flat plate. The effects of gassurface interaction model are examined and it reveals that the MS model can predict heat transfer coefficient on
the plate accurately rather than the diffuse reflection and the CLL models, and that the effects are significantly
large for the nonequilibrium profiles near the plate.
ACKNOWLEDGEMENTS
One of the auther(N.Tsuboi) is supported by Research Fellowships of the Japan Society for the Promotion
of Science for Young Scientists.
337
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