Supersonic Flow Through a Permeable Obstacle
M. Yu. Plotnikov, A. K. Rebrov
Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
E-mail:[email protected]
Abstract. Study of one-dimensional supersonic monatomic gas flow through an infinite permeable surface - obstacle of
zero thickness was performed by the direct simulation Monte Carlo method. The distinguishing features of the shock
disturbance formation by the heat and impulse interaction of a flat-parallel flow with a permeable obstacle and the
influence of this disturbance on flow parameters behind the obstacle were ascertained. The shock disturbance has been
studied at different laws of gas-surface interaction.
1. INTRODUCTION
The behaviour of a gas under the transition through a one-dimensional disturbance on molecular scale, which is
provoked by action of a light stream, stream of particles or heat-mechanical permeable shield, has fundamental
importance as far as physical phenomena of a flow transformation in this case are not shadowed by any influence of
geometry. Gas flows through permeable obstacles are specified by variety of regimes and practical applications. At
the simplified classification one can define followings:
1) flows of gases and gas mixtures through porous partitions, for example, in the apparatus with a porous
cooling;
2) flows of gas mixtures in porous and capillary membranes of separation units;
3) a supersonic inleakage of gas mixture on an obstacle, related to aerodynamic problems of high altitude flights
and gas mixture separation on a permeable obstacle with a thickness comparable with a molecular mean free path;
4) subsonic flows of gas mixtures through obstacles with a high permeability at a localised strong translational
nonequilibrium, related to the problems of new vacuum technologies.
Some intermediate place between flows, determined in the points 3 and 4, is occupied by supersonic flows
around permeable obstacles of limited sizes, when the shock wave forms close to the obstacle. Such flows have been
studied in [1-4] with an application to the aerodynamic computation of lattice aerial. Results of the experimental
study of separating gas species of disparate mass by the interaction of a hypersonic gas mixture jet with a permeable
obstacle are given in [5]. These works concern the situation when a part of the gas is flowing around the grid, and
even in the simplest formulation the problem occurs to be either two-dimensional [5] or three-dimensional [3,4]. In
this case the subsonic flow, disturbed by the detached shock wave, enters to the obstacle and penetrates through it. In
[6] the one-dimensional supersonic water vapour flow through a hypothetical grid was considered, using the direct
simulation Monte Carlo (DSMC) method [7]. These calculations were performed for the case when the flow
disturbance at the obstacle is distorted by constraint of calculation region, and shock wave is artificially retained by
the obstacle. The subsonic flow through the capillary sieve [8] is of special interest from the point of view of
accommodation coefficient measurements .
In the presented paper the one-dimensional supersonic flow through an infinite permeable surface, as an obstacle
of the zero thickness, was investigated by the DSMC method. It has allowed: a) the detail presentation of a shock
disturbance structure in the vicinity of the obstacle; b) the ascertainment of the influence of the scattering indicatrix
on dissipative processes; c) the ascertainment of conditions with retention of a supersonic flow behind the obstacle;
d) the conceptual evaluation of possibilities to determine accommodation coefficients when studying the gas –
obstacle interaction.
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
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2. THE FORMULATION OF THE PROBLEM AND DSMC SOLUTION
It is supposed that the permeable obstacle of the zero thickness is a flat surface of a given permeability, located
perpendicular to a flow. The Cartesian coordinates are used: gas flows in the direction of X-axis and Y- and Z-axes
are perpendicular to a flow. Let the undisturbed upstream supersonic flow of a monatomic gas with the translational
temperature T1, Mach number M1, density n1 is defined as directed to the permeable obstacle. It was supposed for
the DSMC calculation that the surface of the flux source, surface of the permeable obstacle, and surface of the full
flow absorption are located in sections x = x1, x = 0 and x = xe, correspondingly. If a particle reaches the section x=
0, it collides with the surface with the probability (1- P) or flies through it with the probability P. The value P in this
case plays the role of geometrical permeability. If the particle turns back to the source surface, it is absorbed.
For description of intermolecular collision, the model of hard spheres was used. The particle-surface interaction
was considered for two models of reflection: specular and diffuse with the surface temperature. For the definition of
a particle-surface energy exchange the accommodation coefficients α were used: α = 0 for the specular reflection
and α = 1 for the diffuse one.
The given boundary conditions at the source surface are followings: density n1, temperature T1, Mach number
M1. The obstacle temperature is denoted as Tp. The molecular mean free path L is determined by the density n1. The
value n1, T1, L and the most probable molecular thermal velocity at the temperature T1 were used as reference values
to reduce the problem to non-dimensional one. After that, the problem is determined only by parameters M1, Tp/T1,
P, α.
It is obvious that at some set of these parameters the flow stalls. It corresponds to the case, when the flow
through the obstacle (up- and downstream) becomes subsonic and a shock wave moves upstream with a constant
velocity. It was important to show that beginning from a definite magnitude of x1 the structure of the shock wave
corresponds to the classic one and its movement upstream does not depend on the distance between the source and
the obstacle chosen for modelling. Just that distances were used in calculations. A stationary solution of the problem
was of the prime interest. The cases and conditions of the supersonic flow stall were considered separately.
During computations the macroscopic parameters such as density, velocity, Mach number, temperatures on
directions: Tx (along the flow) and Ty (perpendicular to the flow), and total temperature T=(Tx+2Ty)/3 were
calculated.
While modelling, from 20 to 100 thousands particles were used on every time step. The stationary solution was
as realisation of a large amount of repetitions. The computation accuracy was controlled by calculation with
different mesh widths and time steps of the DSMC algorithm.
3. RESULTS OF COMPUTATIONAL EXPERIMENTS
The computational experiments were performed for the following set of parameters: M1 = 3; 5; 10; P∈[0; 1], α∈[0;
1], Tp/T1∈[1; 2.4×(1+(M1)2/3 ]. Selection of the temperature Tр needs a special explanation. The ratio Tp/T1=1
2
corresponds to a “cold” obstacle. The value of T p = T1 × 1.2 × (1 + M 1 / 3) is close to the recovery temperature for
monatomic gas, and
2
T p = 2 × T1 ×1.2 × (1 + M 1 / 3) corresponds to a “hot” obstacle.
The Fig.1 presents the structure of the usual shock wave (a) and shock disturbance (b, c) in the flow with M1 = 5.
The changing of density, Mach number, total temperature, temperature along and perpendicular to the flow in the
shock region is shown. The obstacle has the permeability P=0.95, Tp/ T1 = 1 for α = 1 (b) и α = 0 (c).
The location of the obstacle in Fig.1, b, c corresponds to x = 0. The location of an “obstacle” for the usual shock
wave is attributed by the point, where the tangent to a density curve in the point of inflexion intersects the horizontal
line on the level of density behind the shock wave. As it is seen from the figures, shock disturbance structure at the
obstacle is different from that one for the usual shock wave by more strong non-equilibrium. The last one is
pronounced in more non-isotropic temperature distribution and larger steepness in changing of parameters in the
relaxation zone immediately behind the obstacle.
158
12
а
1
2
3
4
5
9
6
3
0
-15
X/L
0
12
b
10
1
2
3
4
5
8
6
4
2
0
-15
X/L
0
10
c
1
2
3
4
5
8
6
4
2
0
-15
0
X/L
FIGURE 1. Structure of the shock wave (a) and disturbance around the obstacle with α = 1 (b) and with α = 0 (c) for M1 = 5,
P=0.95, Tp/ T1 = 1. 1 - density; 2 - Mach number; 3 - Tx; 4 – Ty; 5 - T.
159
In the considered case, the total energy of the flow behind the obstacle at diffuse reflection constitutes 0.956E,
where E is the specific total energy of the undisturbed flow. The flow losses its energy. The loss of momentum at the
obstacle constitutes 0.06I and 0.09I for the diffuse and specular reflection, correspondingly. Here I is the
non-disturbed flow momentum. It is worth to note a higher intensity of the dissipative processes in the case of a
specular reflection due to molecular reflections with higher energy.
The character of the molecular reflection from the obstacle strongly influences on the shock disturbance structure
and gas parameters in the equilibrium flow behind the obstacle. The disturbance appears to be more intense (with
more steep gradients) at α = 1, possibly, because of the flow cooling. Also, the Mach number in this case is
essentially higher then for specular reflection.
12
M,T
1
2
3
4
9
6
3
0
3
5
7
9
M1
FIGURE 2. Mach number and temperature behind the obstacle in dependence on M1 for P = 0.95 and Tp/ T1 =1. 1 - Max number
for α=0, 2 - Max number for α=1, 3 - T for α=0, 4 - T for α=1.
The influence of the reflection laws on the parameters behind the obstacle, always behind the relaxation zone, is
clear from Fig.2, 3. The dependencies of Mach number and temperature on M1 for P = 0.95 and Tp/ T1 =1 are shown
in Fig.2. The behaviour of Mach number is rather conservative as regards the reflection laws by variation of M1 from
3 to 10. The temperature change substantially increases with the growth of M1. It is more pronounced for α = 0,
when the dissipation of momentum in the formed disturbance is more strong.
The dependence of temperature and Mach number on the obstacle permeability is given in Fig.3 for α = 0 and 1,
M1 = 3 and 10. One can see substantially greater influence of accommodation coefficients on the temperature at the
high Mach number. The presented data are obtained for Tp = T1. At the free molecular flow through the obstacle, the
flow is cooled comparing to the recovery temperature. The series of calculations for Tp = T1 × 1.2× (1 + M12/3), i. e.
temperature, close to recovery one, has shown attenuation of the influence of the scattering law. It is in agreement
with the known insight that the recovery temperature does not depend on accommodation coefficient.
The computational modelling was performed for the hot obstacle as well, particularly, with the temperature
roughly twice as large as the recovery temperature. The corresponding dependencies of temperature and Mach
number on the obstacle permeability at Tp=T1×2.4×(1+M12/3) are shown in the Fig.4. Qualitatively the results
presented in Figs 3 and 4 are similar.
The general conclusion can be state that the temperature behind the obstacle is a sensible parameter to determine
accommodation coefficients on a result of gas flow - obstacle interaction.
160
T
а
16
8
0
M
1
2
3
4
8
b
4
1
0.92
P
0.96
FIGURE 3. Dependencies of temperature (a) and Mach number (b) behind the obstacle on P for Tp/ T1 =1. 1 - α=0 for M1=3, 2 α=1 for M1=3, 3 - α=0 for M1=10, 4 - α=1 for M1=10.
T
а
20
10
0
M
9
b
1
2
3
4
3
1
0.92
0.96
P
FIGURE 4. Dependencies of temperature (a) and Mach number (b) behind the obstacle on P at Tp = T1 ×2.4× (1 + M12/3). 1 α=0 for M1=3, 2 - α=1 for M1=3, 3 - α=0 for M1=10, 4 - α=1 for M1=10.
161
The presented data show that behind a hot obstacle at α = 1 parameters change more than at α = 0. The situation
is reverse one in the case of a cold obstacle. With the growth of the obstacle temperature at α > 0 the ultimate low
value of a permeability P* for maintaining the supersonic flow behind the obstacle is increased. One can see it from
the comparison of Fig.3 and 4. Rather high value of the ultimate value P* at M1 = 3 (Figs. 4) engages ones attention.
4. COMMENTS TO THE TRANSITION OF THE FLOW BEHIND THE OBSTACLE TO
SUBSONIC ONE
The transition from a supersonic flow to a subsonic one behind the obstacle may take place in cases of Mach
number or permeability decreasing, the obstacle temperature increasing, and such changing accommodation
coefficient when the reflected molecule energy increases. The transition can be result of mutual effect of above
mentioned factors. It is obvious, that in general, the loss of momentum and heating of the gas are responsible for the
approach of this critical situation. The knowledge of these processes is important by elaboration grid antenna and
high altitude parachutes.
To evaluate conditions of the formation of a departing upstream shock wave we consider the limiting critical case
when the Mach number behind the obstacle is equal to 1 and the shock wave is supposed to depart off the obstacle
with the infinitesimal velocity. When the shock wave and the disturbance at the obstacle are separated, the
parameters of the equilibrium flow behind the obstacle are connected with the parameters of the given supersonic
flow by conservation laws, describing the transformation of the flow through two disturbances: shock wave and an
obstacle. The corresponding conservation laws in non-dimensional form are followings:
nl M l Tl = n j M j T j
(1)
nT
∆I
= (1 + γM 2j ) j j
nl kTl
nlTl
T
∆E
1 + (γ − 1) M l2 / 2 +
= (1 + (γ − 1) M 2j / 2) j
γkTl
Tl
1 + γM l2 +
(2)
(3)
Here ∆I and ∆E are change of momentum and energy at the obstacle, γ is specific heat ratio. The variables with
indexes 1, 2, 3 denote the parameters before the shock wave, between the shock wave and the obstacle, and behind
the obstacle, correspondingly. It is obvious that the set of equations (1) - (3) with ∆I = 0 and ∆E = 0 describes the
shock wave. In this case l = 1 and j = 2. For a flow through the obstacle l = 2 and j = 3.
The solution of above system for the localised disturbance gives the gas parameters of state in the region between
the shock wave and the obstacle. Now one can use three equations for five unknowns for the description of the
obstacle disturbance. The system is closed, if at M3 = 1 and the value of one of disturbances, heat ∆E or momentum
∆I is known or can be determined. So solution of complemented systems allows to define the conditions of
transition to subsonic velocity behind the obstacle.
The case ∆E = 0 in some sense is the simplest one. It is realised for the specular obstacle surface or when the
obstacle temperature is equal to the recovery one. It is possible to determine the value ∆I for any given set of
parameters n1, T1, M1. For aerodynamic applications the knowledge of coefficient
Cx =
2 ∆I
(1 − P ) ρ u
*
2
1 1
≡
2 ∆I
1
n1kT1 (1 − P * )γM 12
is important.
This value can be treated as drag coefficient of a hypothetical obstacle with the critical permeability P*
corresponding to the conditions of the studied transition. Figs. 5 shows the dependencies Cx and C1=Cx (1 – P*) on
Mach number M1 for the specular obstacle ( ∆E = 0 ). The value P* was determined by the DSMC method. It is
interesting to note, that Cx is slightly higher than those values for convex bodies in a free molecular flow. The
behavior of values C1 shows that ∆I increases faster than M12 in regimes of critical permeability.
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5. CONCLUSION
The front of the shock permeable obstacle disturbance at diffuse as well as at specular reflection is distinguished
from the usual shock wave viscous front, particularly, by more strong translational non-equilibrium
The transition from a supersonic flow to a subsonic one behind the obstacle may take place in cases of Mach
number or permeability decreasing, the obstacle temperature increasing, and such changing accommodation
coefficient, when the reflected molecule energy increases. All these effects correspond to the increasing of the
momentum loss and heating of the gas.
The structure of the disturbance, inevitably depending on scattering indicatrix of molecules, is sensible to integral
characteristic of reflection in the form of accommodation coefficients. Last ones can be determined in experiments
with nets or grids in a supersonic flow, basing on measurements of downstream temperatures.
The providing of a supersonic flow behind an obstacle needs very high permeability. Even weak permeability
attenuation (about 5-10%) can block the flow and reduce it to subsonic one.
This work was supported by INTAS (№ 99-00749), by Presidium RAS (№ 57) and by integration grant of SB
RAS 2000 (№ 43).
C1,Cx
12
10
5
3
3
5
7
9
M1
FIGURE 5. Dependencies coefficients Cx (solid curve) and C1 (dashed curve) on M1 for the specular obstacle ( ∆E = 0).
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