Statistical Simulation of Near-Continuum Flows with
Separation
G.N. Markelov , M.S. Ivanov† , S.F. Gimelshein and D.A. Levin
†
AOES, Haagse Schouwweg 6G, 2332 KG Leiden, The Netherlands
Institute
of Theoretical and Applied Mechanics, Novosibirsk 630090, Russia
Pennsylvania State University, University Park, PA 16802, USA
Abstract. Statistical simulation of flow around a hollow cylinder flare was performed for LENS Run 9 and Run 11 cases. A
comparison with experimental and available numerical results shows that the kinetic approach predicts flow properties rather
accurately if all the requirements inherent in the DSMC method (time step, cell size, and the total number of molecules)
are satisfied. A recent study of the flow in the LENS wind tunnel shows that the flow is vibrationally non-equilibrium. A
computation with corrected flow conditions for Run 11 case predicts lower heat fluxes over the forebody and the aft part of
the flare which are in good agreement with experimental data.
INTRODUCTION
One of the most challenging problems of hypersonic aerodynamics is the shock wave/boundary layer interaction. The
interaction is often observed in flows about high-speed vehicles, and one example here is the interaction of bow or
wing shock waves with the boundary layer near the aerodynamic control surfaces. Such an interaction may cause the
formation of the separation zone and, as a consequence, the reduction of the control surface efficiency and an increase
in the heat flux on the wall in the vicinity of the reattachment point.
Numerical solution of this problem for hypersonic laminar flows is traditionally performed using the Navier-Stokes
(NS) equations, sometimes with the initial effects of rarefaction taken into account through the velocity slip and
temperature jump. However, the use of the continuum approach to compute the flow near the leading edge of a slender
body is not justified, even if the velocity slip and temperature jump are taken into account. The rarefaction effects
near the leading edge are significant due to the merge of the shock wave and the viscous layer; the flow physics is
considerably complicated in the case of flow separation. Application of the continuum approach to modeling such a
flow is questionable due to rarefaction effects. To reliably predict hypersonic laminar flows, one should know the area
of applicability of the continuum approach, which depends not only on the free-stream parameters but also on the body
geometry.
The need for reliable predictions of the total and distributed aerodynamic characteristics and heat fluxes in hypersonic flows stimulated research in this field. The main effort was directed toward experimental and numerical studies
of hypersonic flows around simple bodies such as a flat plate. The comparison of results obtained by the continuum
approach (solution of the NS equations) with those of the kinetic approach (direct simulation Monte Carlo method,
DSMC) and experimental data shows that the continuum technique allows an accurate prediction of the distributed
aerodynamic parameters and heat
fluxes over the surface except for the vicinity of the leading edge of the plate where
the Knudsen number Kn λ X is greater than 0.03 [1]. Here, λ is the mean free path and X is the distance from the
leading edge of the plate.
The study of the area of applicability of the continuum approach for more complex phenomena in hypersonic laminar separated flows has been just started. To enhance the understanding of such flows, experimental and computational
studies have been actively promoted by the NATO Research Technology Organization (RTO, formerly AGARD) for
basic axisymmetric configurations. The simplest example of the separated flow is the flow around a hollow cylinder flare. This axisymmetric configuration allows one to avoid three-dimensional effects inherent in the flow near a
compression ramp.
The first example of a detailed experimental investigation of this configuration was the study undertaken in the
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
457
TABLE 1. Separation and reattachment
points on different grids in NS calculations
with no-slip boundary conditions.
Grid
Xsep /L
Xreat /L
160 67
240 100
360 150
480 200
600 300
DSMC
Experiment
0.802
0.748
0.735
0.703
0.703
0.77
0.76 0.01
1.266
1.314
1.327
1.348
1.348
1.32
1.34 0.015
ONERA R5Ch hypersonic wind tunnel at the Mach number M 9 91 and the Reynolds number Re L 18 916. The
pressure and heat fluxes on the model surface were measured, the separation region length was determined using oilflow visualization, and the density profiles in several cross sections were obtained by electron beam fluorescence. These
data were used for verification of numerical codes at the First Europe-U.S. High Speed Flow Field Database Workshop,
1997 and the First Eastern-Western High Speed Flow Field Conference and Workshop, 1998. Several different NS and
DSMC codes were employed to simulate the flow numerically. The review paper [2] gives a synthesis of the results
obtained. It was certain that well-resolved NS computations lead to a larger size of the separation bubble than it was
observed in experiments and DSMC simulations. In order to analyze the reasons for this disagreement, a comparative
study was performed [3] using both continuum and kinetic approaches. The influence of the grid and the number of
particles on the DSMC results was examined, and a grid refinement study for the NS solver was done. As an example,
the grid convergence of the separation and reattachment points is given in Table 1.
Trying to bridge the gap between the results of continuum and kinetic simulations, NS computations with allowance
for the velocity slip and temperature jump on the surface were carried out. The influence of wall boundary conditions
was examined in detail, and it was shown that the agreement between NS and DSMC results could be improved
by using the slip boundary conditions, which account for initial rarefaction effects. In particular, the position of the
leading edge shock wave was predicted in very close agreement
with the DSMC results. Nevertheless, the position of
the separation point was not changed much; it became Xsep L 0 723 instead of 0.703; thus, the difference with the
DSMC results (see Table 1) was still significant.
The existing uncertainty stimulated new experimental studies. They were performed in LENS, an expansion shock
tube situated at CALSPAN (Buffalo, US). The first set of experimental and numerical (both continuum and kinetic)
studies on the flow over a hollow cylinder flare configuration was reported at the 39th AIAA Aerospace Sciences
Meeting in Reno, 2001 (see, e.g., [4] and references therein). Two flare sizes were considered with the Reynolds
numbers based on the cylinder part ranging between 15,000 and 30,000. It was found that the numerical methods
overpredict the experimental heat transfer on the forebody for most cases, and the DSMC solutions underpredict the
measured size of the separation zone. For the lowest Reynolds number case, it was shown [5] that matching DSMC
requirements on the cell size, the number of particles in a cell, and the time step permits one to get good agreement
between calculated and measured values, for example, the size of the separation region and the pressure coefficient.
However, the calculated heat fluxes are higher than the measured ones on the forebody. In [6], it is also shown that
the measured pressure on a sharp cone for the LENS Run 35 case is lower than the value predicted by inviscid gas
dynamics. This raised a question about the reliability of free-stream parameters reported.
The activity was continued in 2001, and the results were presented at the 40th AIAA Aerospace Sciences Meeting in
Reno, 2002 [7, 8, 9]. The last work meeting in Reno (Jan. 2002) showed that the goals of the work of Work Group 10
have not been reached yet. The problem of validation and verification of numerical approaches for hypersonic laminar
flows with separation turned out to be very complex. This is primarily connected with the complexity of obtaining
reliable test cases in the LENS wind tunnel.
To obtain experimental results at low Reynolds numbers (ReL 20 000 100 000), it is necessary to use low
pressures in the settling chamber. This is not regular for this wind tunnel, and several unexpected phenomena have
been observed. The main phenomenon is fast freezing of vibrational temperature of the test gas N 2 . As a result,
hollow cylinder flare and double cone configurations were modeled in a strongly non-equilibrium flow. The impact of
vibrational non-equilibrium in numerical simulation is difficult to evaluate at the moment.
To conduct the final stage of the validation process, it is necessary to perform a new set of DSMC computations that
458
TABLE 2. Free-stream conditions and
wall temperature
p∞ Pa
T∞ K
Tv ∞ K
ReL
M∞
Tw K
Run 9
Run 11
Run 11c
30.4
121.1
121.1
26325
11.44
296.7
21.84
133.32
133.32
15850
11.07
297.2
16.41
97.8
2640
20850
12.23
297.2
L=0.1017 m
0.1183
o
30
0.0325 m
SYMMETRY AXIS
FIGURE 1. Hollow cylinder flare geometry
include a modification of free-stream conditions, which is consistent with the LENS wind tunnel. A large difference
between the free-stream translational temperature (about 100 K) and vibrational temperature (about 2500 K) is inherent
to these conditions. Such a high degree of non-equilibrium complicates the application of NS solvers because a new
surface boundary condition is necessary for vibrational energy. This situation with a strong temperature separation is
not a problem for the DSMC method.
The main objectives of the present paper are as follows.
1. Using the DSMC method, obtain accurate numerical solutions for a hypersonic flow about a hollow cylinder flare
configuration for different Reynolds numbers up to 30,000.
2. Establish the possibilities of the DSMC method for modeling of near-continuum flows, and, what is more
important, restrictions of the continuum approach through the comparison of obtained DSMC results with existing
NS solutions.
3. Study the effect of vibrational non-equilibrium in the free stream on the flow properties.
FLOW CONDITIONS AND NUMERICAL METHOD
Free-stream parameters of the LENS wind tunnel and the wall temperature for Run 9 and Run 11 cases borrowed
from [10] are listed in Table 2. A recent study of the flow in the LENS wind tunnel [7] shows that the flow is vibrational
non-equilibrium and free-stream parameters have to be corrected. New parameters for Run 11 case is called here as
Run 11c. The configuration of the hollow cylinder flare used in these Runs has a rather long flare (Fig.1) to prevent
the influence of flow expansion at the end of the flare on shock interaction and flow separation.
Statistical simulation of the flow is performed with a multi-task DSMC-based computational tool, SMILE [11]. The
following models are used in computations.
•
•
•
Variable Hard Sphere model [12] with parameters corresponding to nitrogen;
Larsen-Borgnakke model [13] with temperature-dependent rotational and vibrational numbers;
diffuse reflection with complete energy accommodation except one case without accommodation of vibrational
energy.
RESULTS
The flow structure over the hollow cylinder flare is shown in Fig. 2 where selected streamlines indicate the separation
region and the pressure gradient flowfield depicts clearly the shock waves. The zoom of the shock-wave interaction
region reveals that the leading edge, L, and separation, S, shock waves coalesce; then a joint shock meets the
reattachment, R, shock wave. The region P is a Prandtl-Meyer expansion. It is well known [14] that an intersection
459
FIGURE 2. Pressure gradient flowfield and selected streamlines (Run 9)
of shocks of the same family produces not only a joint shock but also a Prandtl-Mayer expansion or a weak shock to
match the pressure increase through two shocks and across the joint shock.
Run 9 case
In our previous computations [9] for Run 9 case, up to 80 millions of model particles were used. The size of the
separation region is larger than in all previous DSMC computations performed in [15] but smaller than that obtained
in experiments and in the NS solution (Fig. 3). From our point of view, the attempts to apply the DSMC method to this
case were unsuccessful for numerical rather than physical reasons. Even 80 millions of model particles did not allow
us to satisfy DSMC requirements on the cell size in the shock interaction region. For example, the size was about two
local mean free paths, which is twice larger than the recommended value, and particles experienced two collisions
in average during the time step used whereas “the time step should be much smaller than the local mean collisional
time” [12]. The computations were performed on workstations where there was no possibility of using the number
of model particles and cells required to provide a solution, which is cell-independent and converged in terms of the
number of particles.
The present computation is performed for a reduced size of the flare, and the total length of the configuration is 0.19
m. This reduction together with the use of the “subdomains with different time steps” technique (see details in [16])
allows us to increase the number of model particles in the shock interaction region by a factor of 2 and, hence, to
decrease the cell size by a factor of 2 with a total number of model particles of 65 millions. The use of a smaller cell
size and time step in the present computations significantly decreases the maximum value of heat flux (by 20%), and
the present value is in fair agreement with experimental data (Fig.3). The size of the separation region increases, and
the difference between DSMC and NS [17] predictions of the separation point position is reduced twice compared to
our previous results.
Run 11 case
Run 11 case has the lowest Reynolds number over all LENS test cases. Therefore, the grid and number of particles
requirements are less severe in this case and are satisfied in the modeling. Figure 4 shows a comparison of available
experimental data and numerical results obtained with the DSMC method and NS equations. The DSMC method
predicts the pressure distribution very accurately.
The calculated heat fluxes are also in good agreement with the
measured values except the forebody (X L 0 8) and the aft part of the flare (X L 1 5) where the difference is
about 15 20%. This difference is caused mostly by incorrect free-stream conditions and it will be discussed in the
next section.
The NS solution [5] with no-slip boundary conditions predicts a larger size of the separation region (1.6 of DSMC
value), which shifts the peaks of Cp and St downstream (see Fig. 4). This situation was studied in detail for the ONERA
R5Ch case, where the Reynolds and Mach number are close to the correspondent values for Run 11. It was shown
that inapplicability of the NS equations at the leading edge affects flow properties far downstream, for example, the
separation region size [3]. Taking into account the initial rarefaction effects (slip velocity and temperature jump) for
460
0.06
1.2
Experiment
DSMC [15]
NS [17]
SMILE [9]
SMILE (Present)
1
0.05
0.04
St
Cp
0.8
Experiment
DSMC [15]
NS [17]
SMILE [9]
SMILE (Present)
0.6
0.4
0.03
0.02
0.2
0.01
0
0
0.5
1
1.5
0
2
0
0.5
X/L
FIGURE 3.
1
1
1.5
2
X/L
Pressure coefficient and Stanton number (Run 9)
Experiment
SMILE
NS [5]
Experiment
SMILE
NS [5]
0.05
0.8
0.6
St
Cp
0.04
0.4
0.02
0.2
0
0.03
0.01
0.5
1
1.5
0
2
0.5
X/L
FIGURE 4.
1
1.5
2
X/L
Pressure coefficient and Stanton number (Run 11)
the NS equations reduces the separation-region size and decreases the difference between NS results and experimental
data. However, a slight difference remains for conditions considered.
461
1
0.11
Experiment
SMILE (Run 11)
SMILE (Run 11c)
0.8
0.1
0.09
Cp
Y, m
0.08
0.07
0.6
50
0.4
0.06
100
0.05
50
50
3
0.2
20
0.04
10
3
3
5
8
0.03
0
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.5
1
FIGURE 5.
1.5
2
X/L
X, m
Dimensionless pressure contours (dashed line is Run 11, solid line is Run 11c) and pressure coefficient
Run 11c case
All numerical solutions presented above and in [4] overpredict heat fluxes (by 15 30%) on the forebody for most
LENS cases. An additional investigation [7] of flow properties in the LENS wind tunnel revealed that the flow is
vibrationally non-equilibrium under the conditions considered. As a result, the translational temperature and the
velocity in the LENS test section are lower than the values reported previously. Corrected free-stream conditions
for Run 11 case (Run 11c in Table 2) have higher Reynolds and Mach numbers by 30 and 10 %, respectively.
A computation for Run 11c conditions predicts flow properties close to Run 11 values (see dimensionless pressure
contours in Fig. 5). The size of the separation region does not change because increases in Reynolds and Mach numbers
have opposite effects on the separation region. Higher Reynolds and Mach numbers slightly increase the pressure peak
on the flare, but this insignificantly affects the agreement between numerical results and experimental data (see Fig. 5).
Figure 6 shows the translational temperature flowfield for Run 11c case. The value of temperature is about 1,000
K for most part of the disturbed flow with the maximum value slightly above 1,800 K in a narrow region behind
the reattachment shock wave. For this temperature range, the vibrational relaxation number is 196 0 2 3 10 6
(see e.g. [12]), and there is practically no energy exchange between the vibrational mode and the rotational and
translational modes of particles under conditions considered. For example, the vibrational temperature is constant
over the computational domain, and it is equal to the free-stream value for diffuse reflection without accommodation
of vibrational energy. Therefore, only the gas/surface interaction can change the vibrational energy of particles and
affect the heat fluxes.
The heat fluxes for diffuse reflection without accommodation of vibrational energy
are presented in Fig. 6. The use
of corrected
free-stream
conditions
decreases
heat
fluxes
both
on
the
forebody
(X
L
0 8) and on the aft part of the
flare (X L 1 5) leading to better agreement with experimental data. Now the calculated values on the forebody are
higher than the measured ones by 5%, which is comparable with the measurement errors. The NS solution [8], which
takes into account vibrational non-equilibrium of the flow and velocity slip and temperature jump on the surface, is
also shown in Fig. 6. This solution predicts the largest separation-region size, which increases with free-stream nonequilibrium and slip conditions. This is confusing because vibrational energy is completely frozen, and it is possible
to neglect it under conditions considered.
The next DSMC computation is performed for complete accommodation of vibrational energy during gas/surface
interaction. The vibrational temperature (Fig. 7) varies from the free-stream value to the wall temperature, and this
variation is due to diffusion of reflected particles. Complete accommodation increases heat fluxes (about 10%), and
the ratio of heat fluxes obtained with and without accommodation is constant over the hollow cylinder flare surface
462
Experiment
SMILE (Run 11)
SMILE (Run 11c)
NS [8] (Run 11c)
0.05
St
0.04
0.03
0.02
0.01
0
0.5
1
1.5
2
X/L
FIGURE 6. Translational temperature flowfield (Run 11c) and comparison of heat fluxes
0.06
2
no accommodation
full accommodation
ratio of heat fluxes
St
0.04
1.5
0.03
1
0.02
ratio of heat fluxes
0.05
0.5
0.01
0.05
0.1
0.15
X, m
FIGURE 7. Vibrational temperature for complete accommodation of vibrational energy (left) and influence of vibrational energy
accommodation on heat fluxes (right)
(Fig. 7). Computations with different values of accommodation of vibrational energy are conducted on a coarse grid,
and they show that the heat flux monotonically grows from the value obtained without accommodation up to that
corresponding to complete accommodation, and there is no double peak observed in heat flux measurements. Possibly,
the double peak appears due to non-uniformity of free-stream conditions. For example, Ref. [8] shows that even a small
level of non-uniformity in the free-stream conditions is important for the flow over the double cone configuration under
the LENS Run 35 conditions.
463
CONCLUSIONS
Statistical simulation of near-continuum flow around a hollow cylinder flare has been performed using the DSMCbased software. Three different test cases were considered, Runs 9, 11, and 11 with vibrational non-equilibrium. The
Reynolds numbers based on the hollow-cylinder length vary within the range from 15,000 to 26,000, which leads to
a large number of model particles (tens of millions) to match DSMC requirements on the cell size and the number of
particles in a cell. DSMC requirements on the time step lead to large CPU time, and it is rather difficult to obtain a
time-converged solution using today’s workstations and computational clusters.
A comparison with experimental and available numerical results shows that the DSMC method is capable of
predicting separated flow properties in the near-continuum regime rather accurately only if all the requirements
inherent in the DSMC method (time step, cell size, and the total number of molecules) are satisfied. Very good
agreement between DSMC results and experimental data for the pressure and heat transfer distributions was observed
for the least dense case, Run 11.
The vibrational relaxation number is extremely high (from 2 to 200 millions) for Run 11 case, and the vibrational
energy of molecules affects only the heat flux through accommodation of vibrational energy during gas/surface
interaction. The ratio of heat fluxes obtained with complete accommodation and without accommodation is constant
over the hollow cylinder flare surface, and it is about 1.1. The heat fluxes calculated without accommodation of
vibrational energy are in very good agreement with the measured values. Therefore, one can assume that there is no
accommodation of vibrational energy under Run 11 case conditions.
ACKNOWLEDGMENTS
The authors thank Alexei Kudryavtsev (ITAM, Novosibirsk) for his comments and fruitful discussions and Pavel
Vashchenkov (ITAM, Novosibirsk) for his help in Run 9 case computation.
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