Comparison between the Navier-Stokes and the Boltzmann ...€¦ · an axially symmetric compressible flow with ... 1 Introduction ... velocity of the molecules is chosen based on

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 1Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran 2Faculty 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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Computational Methods and Experimental Measurements XII 61

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:

( ) ( ) ( ) ( )4

2 * *1 1 r 1

0

nf V. nf F. nf n f f f V d dVt r

fv

π

σ+∞

−∞

∂ ∂ ∂+ + = − Ω

∂ ∂ ∂ ∫ ∫ (1)

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 Navier-Stokes 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


62 Computational Methods and Experimental Measurements XII

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 Navier-Stokes 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



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.

The comparison between the results of simulations of the Boltzmann and the Navier-Stokes equation.

Results of simulation of Boltzmann equation

Results of simulation of Navier-Stokes equations

Difference

Maximum value of T(K) 1063 1059.8 0.3% Maximum value of radial

velocity (m/s) 326.65 351.64 7.6%

Maximum value of axial velocity (m/s) 1000.05 999.94 0.01%

Maximum value of )m/Kg( 3ρ 0.00028 0.0002 28.57% Maximum value of pressure

(Pa) 57.651 43.219 25.03%

Location of shock anticipation (m) -0.0134 -0.0098 18%

Shock thickness (mm) 13.4 9.8 26.86%

Figure 1: Shows the configuration of flow past a flat-nosed cylinder and two sections, ABCD and CE, along which the flow characteristics are obtained.

Table 1:



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.

a c

b d

e



Figure 3: a, b, c, d and e show the comparison between the results of Navier-

Stokes 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.

a c

b d

e



Figure 4: 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.

a d

b e

c f



Figure 5: 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.

5 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

a c

b d



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 Navier-Stokes 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 Mote-Carlo 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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Comparison between the Navier-Stokes and the Boltzmann ...€¦ · an axially symmetric compressible flow with ... 1 Introduction ... velocity of the molecules is chosen based on