Download Comparison between the Navier-Stokes and the

Computational Methods and Experimental Measurements XII
61
Comparison between the Navier-Stokes and
the Boltzmann equations for the simulation of
an axially symmetric compressible flow with
shock wave using the Monte-Carlo method
S. S. Nourazar1, S. M. Hosseini1, A. Ramazani1,
& H. R. Dehghanpour2
1
Mechanical Engineering Department,
Amirkabir University of Technology, Tehran, Iran
2
Faculty of Physics and Nuclear Sciences,
Amirkabir University of Technology, Tehran, Iran
Abstract
An axially symmetric compressible flow past a flat-nosed cylinder at high
velocity and low pressure with shock wave is simulated using the Navier-Stokes
equation and the Boltzmann equation. The results of the two simulations
(Navier-Stokes and Boltzmann) are compared in terms of different Knudsen
numbers. In the region close to the shock wave the discrepancies between the
results of simulation of the Navier-Stokes equations and the Boltzmann equation
for the temperature, the axial velocity and the radial velocity along the boundary
is not pronounced. However the discrepancies in the region close to the shock
wave for the density is pronounced.
Keywords: Boltzmann equation, DSMC.
1
Introduction
In the simulation of gas dynamics problems where the velocity distribution is a
strict Maxwellian the Euler equation is to be solved. However in the simulation
of gas dynamics problems where the velocity distribution is a moderately
distorted Maxwellian the Navier-Stokes equation is to be solved.
In the gas flow problems where the length scale of the system is comparable
to the mean free path for molecules in the gas flow the concept of the continuum
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62 Computational Methods and Experimental Measurements XII
is no more valid, Knudsen number greater than 0.1, (Bird [1]). In this case the
simulation is done using the Direct Simulation Monte-Carlo (DSMC) or the
Collisional Boltzmann Equation (CBE) methods. In most cases the direct
solution of the CBE is impracticable due to the huge number of molecules,
however most of the time; the implementation of the DSMC is more practicable.
The rarefied hypersonic flow is solved using the DSMC method by Bird [1]. The
comparison between the Navier-Stokes and the DSMC methods for the
simulation of the cicumnuclear coma is done by Crifo et al. [2]. The simulation
of the rarefied gas flow through circular tube of finite length in the transitional
regime at both low Knudsen number and high Knudsen number is done using the
DSMC method by Shinagawa et al. [3]. In this study we would like to investigate
the sensitivity of the flow characteristics such as, velocity, pressure, density and
temperature to the Knudsen number for an axially symmetric compressible flow
past a flat-nosed cylinder at high velocity and low pressure with shock wave.
The results of the simulation solving the Navier-Stokes equation are compared
with the results of the simulation using the DSMC method for different Knudsen
numbers.
2
Mathematical formulations
The Boltzmann equation is written as:
+∞ 4π
∂
∂
∂
nf
V.
nf
F.
nf
(1)
+
+
=
( )
( )
( ) ∫ ∫ n 2 ( f *f1* − ff1 ) Vrσ dΩdV1
∂t
∂r
∂v
−∞ 0
where n is the number density of molecules, f is the velocity probability
distribution function of molecule of class having the velocity of V, f1 is the
velocity distribution function of the molecule of class having the velocity V1, Vr
is the velocity of a test molecule in the class of molecules having the velocity of
V1, f* is the post-collision velocity probability distribution function and f*1 is the
post-collision velocity probability distribution function of the molecule of class
having the velocity V1, F is the external force per unit mass and Ω is the angle
in the spherical coordinates. The conservative form of the continuity, the NavierStokes and the energy equations are solved using the method used by Roe,
1981.The Boltzmann equation (Eq. 1) is solved using the DSMC method.
3
The numerical procedures
In the simulation of the problem using the DSMC we chose two grid systems.
The first grid system is used to calculate the averaging of flow properties. This
grid system is chosen to be fine enough in order to increase our computational
accuracy. We refined our grid system up to where the variations of the flow
properties were not substantial. The second grid system is chosen to be very fine
(the mesh size is equal to 0.2 times the mean free path of the molecules),
therefore the collision of the molecules are controlled within each mesh
accurately. The second grid system consists of 60 times 80 meshes and the total
number of model molecules is 40000. Each model molecule consists of 10^12
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Computational Methods and Experimental Measurements XII
63
real molecules. The number of real molecules is obtained based on the density of
gas and the Avogadro’s number. For the axially symmetric flows, the
distribution function of molecules is linearly proportional to the radial distance.
Therefore the number of the modeled molecules close to the axis of symmetry is
much less than that of far from the axis of symmetry. This is led to the
introduction of a radial weighting factors such that the number of real molecules
consisting in a model molecule far from the axis is much greater than the number
of real molecules consisting in a model molecule close to the axis (Bird [1]). The
radial weighting factor for our problem is chosen as the one found by Bird [1].
The size of the mesh is in order of the mean free path and the time step in our
simulation is chosen to be 0.2 times the collision time. The molecules are
distributed in the mesh system according to the Normal distribution. The initial
velocity of the molecules is chosen based on the kinetic theory of gases plus the
macroscopic velocity of the flow field. The direction of velocity of the molecules
is chosen randomly based on the uniform random distribution. We then start to
advance with time and the new position of the molecules is designated. The
collisions of the model molecules are done based on the Variable Hard Sphere
(VHS) model proposed by Bird [1]. We then continue advancing in time until the
statistical fluctuations of the flow properties to be minimum. In the boundary
where the axis of symmetry exists the specular reflection rule is used for model
molecules (Fig. 1).
4
Discussions of results
Fig. 1 shows the configuration of flow past a flat-nosed cylinder and the two
boundary sections (ABCD and CE) along which the flow characteristics are
calculated. Figs.2a, 2b, 2c, 2d and 2e show the comparison between the results of
the two simulations (the Boltzmann and the Navier-Stokes equations) for the
variations of temperature, density, axial velocity, radial velocity and pressure
versus the axial coordinates across the boundary section ABCD at different
Knudsen numbers. Table 1 shows the maximum difference between the results
of simulation of the Boltzmann and the results of simulation of the NavierStokes equation. The maximum differences for the temperature, the axial
velocity and the radial velocity are not pronounced. The comparison of the
results of the two simulations for the density and pressure (Figs. 2b and 2e)
shows that in the region close to the shock wave the discrepancies between the
two simulations are pronounced. Table 1 shows that the differences between the
maximum values of the density and the pressure are 28.57% and 25%
respectively. Table 1 shows that the differences between the shock thickness and
the location of shock anticipation are 26.86% and 18% respectively. Figs. 3a, 3b,
3c, 3d and 3e show the comparison between the results of Boltzmann and the
results of Navier-Stokes equations for the variations of temperature, density,
axial velocity, radial velocity and pressure respectively along the axial
coordinate across the section CE at different Knudsen numbers. Figs. 4a, 4b and
4c and Figs. 4d, 4e and 4f show the results of Bird [1], the results of simulation
using the Boltzmann equation and the Navier-Stokes equations for the constant
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64 Computational Methods and Experimental Measurements XII
density contours and for the constant temperature contours respectively. Figs. 5a
and 5b and Figs. 5c and 5d show the comparison between the results of
Boltzmann equation and the results of Navier-Stokes equations for the variations
of the axial velocity and for the variations of the radial velocity respectively
along the axial coordinate.
Table 1:
The comparison between the results of simulations of the
Boltzmann and the Navier-Stokes equation.
Maximum value of T(K)
Maximum value of radial
velocity (m/s)
Maximum value of axial
velocity (m/s)
Maximum value of ρ (Kg / m 3 )
Maximum value of pressure
(Pa)
Location of shock anticipation
(m)
Shock thickness (mm)
Figure 1:
Results of
simulation of
Boltzmann
equation
1063
Results of
simulation of
Navier-Stokes
equations
1059.8
Difference
0.3%
326.65
351.64
7.6%
1000.05
999.94
0.01%
0.00028
0.0002
28.57%
57.651
43.219
25.03%
-0.0134
-0.0098
18%
13.4
9.8
26.86%
Shows the configuration of flow past a flat-nosed cylinder and two
sections, ABCD and CE, along which the flow characteristics are
obtained.
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Computational Methods and Experimental Measurements XII
65
c
a
b
d
e
Figure 2:
a, b, c, d and e show the comparison between the results of
Boltzmann equation and the results of Navier-Stokes equations for
the variations of temperature, density, axial velocity, radial velocity
and pressure respectively along the axial coordinate across the
section ABCD at different Knudsen numbers.
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66 Computational Methods and Experimental Measurements XII
a
b
c
d
e
Figure 3:
a, b, c, d and e show the comparison between the results of NavierStokes equations and the results of Boltzmann equation for the
variations of temperature, density, axial velocity, radial velocity
and pressure respectively along the axial coordinate across the
section CE at different Knudsen numbers.
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Computational Methods and Experimental Measurements XII
a
d
b
e
c
f
Figure 4:
67
a, b and c show the results of Bird [1], the results of simulation
using the Boltzmann equation and the Navier-Stokes equations for
the constant density contours respectively. d, e and f show the
results of Bird [1], the results of simulation using the Boltzmann
equation and the Navier-Stokes equations for the constant
temperature contours respectively.
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68 Computational Methods and Experimental Measurements XII
a
c
b
Figure 5:
5
d
a and b show the comparison between the results of Boltzmann
equation and the results of Navier-Stokes equations for the
variations of the axial velocity respectively along the axial
coordinate. c and d. show the comparison between the results of
Boltzmann equation and the results of Navier-Stokes equations for
the variations of the radial velocity respectively along the axial
coordinate.
Conclusions
In the flow region where the Knudsen numbers are greater than 0.1 there are
some discrepancies between the results of simulation of the Boltzmann equation
(DSMC) and the results of simulation of the Navier-Stokes equations. The
comparison between the results of simulation for the Boltzmann equation
(DSMC) and the results of simulation of Bird [1] for the temperature and the
density shows a reasonable agreement, however the results of simulation of the
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Computational Methods and Experimental Measurements XII
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Boltzmann equation shows better agreement than the results of simulation of the
Navier-Stokes equation in comparison with the results of Bird [1]. Across the
boundary (section ABCD) the discrepancies between the results of the two
simulations (Boltzmann and Navier-Stokes) for the temperature, the axial
velocity and the radial velocity are not pronounced, however the discrepancies
between the results of the two simulations for the density and pressure are
pronounced. The maximum value of discrepancies occurs in the region where the
shock is occurred. The results of simulation of the Boltzmann equation for the
pressure in the shock region show higher values than the results of simulation of
the Navier-Stokes equations for the pressure by 25% in the shock region. The
thickness of the shock region in the results of simulation of the Boltzmann
equation shows greater value than the thickness of the shock region in the results
of the simulation of Navier-Stokes equations by 26.86%. The location of shock
anticipation in the results of simulation of the Boltzmann equation occurs earlier
than the location of shock anticipation in the results of simulation of the NavierStokes equations by 18%.
References
[1] Bird, G. A. 1994, Molecular Gas Dynamics and the Direct Simulation of
Gas Flows, Oxford Univ. Press, London.
[2] Crifo, J. E., Lukianov, G. A., Rodionov, A. V., Khanlarov, G. O. and
Zakharov, V.V. 2002, Comparison between Navier-Stokes and Direct MoteCarlo Simulation of the Circumnuclear Coma, Icarus 156, 249-268.
[3] Shinagawa, H., Setyawan, H., Asai, T., Sugiyama, Y. and Okuyama, K.
2002, An Experimental and Theoretical Investigation of Rarified Gas Flow
Through Circular Tube of Finite Length, Pergamon, Chemical Engineering
Science, 56, 4027-4036.
[4] Roe, p. L., 1981, The Use of the Reimann Problem in Finite Difference
Scheme, Lecture Notes in Physics, 141, pp. 354-359.
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