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Kinetic Model Solution for Micro-Scale Gas Flows

Chan H. Chung*

Daegu University, Gyungsan, Gyungbuk, 712-714, Korea

A kinetic theory analysis is made of low-speed gas flows around micro-scale flat plates. The Boltzmann equation simplified by a collision model is solved by means of a finite-difference approximation. Calculations are made for flows around a flat plate with zero thickness and a 5% flat plate. Results are compared with those from the IP-DSMC method and numerical solutions of the Navier-Stokes equations. The calculated results show good agreement with those from the IP method except for some differences in details near the leading and trailing edges. The pressure on the surface and at a short distance away from the surface showed quite different behavior especially in the region close to the leading edge where rarefaction effects play an important role.

Nomenclature cA = collision frequency

d = characteristic length of the flow field f = distribution function F = Maxwell-Boltzmann distribution

,g h = reduced distribution functions Kn = Knudsen number R = gas constant t = thickness of the plate U∞ = free stream velocity

xV = speed of molecule

xV = velocity of molecule in the x direction

yV = velocity of molecule in the y direction

zV = velocity of molecule in the z direction l = normal distance from the surface λ = mean free path

I. Introduction IMULATION of the flow of rarefied gases around micro-scale structures such as micro-electro-mechanical systems(MEMS) and micro air vehicles(MAV) is gaining importance due to the emerging miniaturization

technologies. While the fast-growing micro-scale fabrication technologies result in a wide variety of practical micro-scale devices, very little numerical work has been performed to analyze the gas flows in and around practical micro-scale structures. In micro-scale structures the Knudsen number of the flow field, the ratio of the mean free path to the characteristic dimension of the flow field, is usually not negligibly small even at atmospheric operating conditions. Hence, conventional computational fluid dynamics (CFD) methods which are based on continuum assumptions may not be appropriate and a method based on kinetic gas theory is required to describe the flows accurately.

Of the various methods available for the analysis of micro-scale flows, the direct simulation Monte-Carlo (DSMC) method1 has been used by many researchers.2-5 The DSMC method has been known to be a robust and accurate method because it is based on kinetic gas theory and does not rely on the continuum assumption that is not valid for high Knudsen number gas flows. Even though the DSMC method has been successfully applied for the * Professor, Department of Chemical Engineering, Member AIAA.

S

37th AIAA Thermophysics Conference28 June - 1 July 2004, Portland, Oregon

AIAA 2004-2590

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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analysis of various kind of rarefied gas flows, simulation of low-speed micro-scale flows using the DSMC method suffers from several difficulties including large statistical scatter that has not been encountered in the high speed gas flow simulations. To get a meaningful result by reducing the large statistical noise, the DSMC method requires huge amount of computational effort due the number of time steps to reach the steady-state flow condition and the large number of sample size.4 These computational demands can render the standard DSMC method impractical given current computing power limitations. An alternate approach is the information preservation (IP) method developed by Fan and Shen.6 The IP method preserves macroscopic information while the flow field is simulated by the DSMC method. The method has been applied to various kind of low-speed flow problems.7-9 Even though the IP method has been reported to reduce the sampling size by orders of magnitude, the method still requires a large amount of computing effort. In the present study a finite-difference method coupled with the discrete-ordinate method10,11 is employed to analyze low-speed gas flows around micro-scale flat plates. In the method, Boltzmann equation simplified by a collision model (BGK equation)12 is solved by means of a finite-difference approximation. The physical space is transformed by a general grid-generation technique. The velocity space is transformed to a polar coordinate and the concept of the discrete ordinate method is employed to discretize the velocity space. The modified Gauss-Hermite quadrature13,14 and Simpson’s rule are used for the integration of the discretized velocity space. To assess the present method, calculations are made for flows around a flat plate with zero thickness and a 5% flat plate. Results are compared with those from the IP-DSMC method and numerical solution of the Navier-Stokes equations with slip boundary conditions.

II. FINITE DIFFERENCE METHOD

Governing Equation

We consider the steady-state Boltzmann equation with the BGK model12 in a two-dimensional Cartesian coordinate system

( )x yx y cf fV V A F f∂ ∂+ = −

∂ ∂ (1)

where ( , , , , )x y zf x y V V V is the distribution function, x and y are Cartesian coordinates of the physical space, xV ,

yV , and zV are the velocity components of the molecules, and cA is the collision frequency. The equilibrium Maxwell-Boltzmann distribution F is given by

3/ 2 2(2 ) exp[ ( ) / 2 ]F n RT V U RTπ −= − − (2)

The moments n , U , and T can be obtained by integrating the distribution function over the velocity space:

n f dV= ∫ (3a)

nU V f dV= ∫ (3b)

23 ( )nRT V U f dV= −∫ (3c)

where R denotes the gas constant, n the particle density, U the macroscopic flow velocity. The following reduced distribution functions are introduced to reduce the number of independent variables:

( , , , ) ( , , , , )x y x y z zg x y V V f x y V V V dV+∞

−∞= ∫ (4a)

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2( , , , ) ( , , , , )x y z x y z zh x y V V V f x y V V V dV+∞

−∞= ∫ (4b)

These kinds of reduced distribution functions were first applied by Chu15 and employed by many investigators. The corresponding equations for the reduced distribution functions are obtained by integrating out the zVdependence with the weighting functions 1 and 2

xV , respectively:

( )x yx y cg gV V A G g∂ ∂+ = −

∂ ∂ (5a)

( )x yx y ch hV V A H h∂ ∂+ = −

∂ ∂ (5b)

where

( , , , )x y zG x y V V F dV+∞

−∞= ∫ (5c)

2( , , , )x y z zH x y V V V F dV+∞

−∞= ∫ (5d)

Using the characteristic length of a flow field d and the most probable speed oV defined as

2o oV RT= (6)

the following dimensionless variables are introduced:

ˆ /x x d= , ˆ /y y d= , ˆ / on n n= , ˆ /i i oV V V= , ˆ /i i oU U V= , ˆ /i i oT T T= , ˆ ˆ /c c oA A d V= ,

2ˆ /o og gV n= , ˆ / oh h n= , 2ˆ /o oG GV n= , ˆ / oH H n= , 2ˆ /(1/ 2 )omn Uτ τ ∞= (7)

where the subscript o refers to a reference condition . By introducing a polar coordinate system, which is defined as

ˆ sinxV V φ= , ˆ cosyV V φ= , 1 ˆ ˆtan ( / )x yV Vφ −= (8)

and applying general transform rules, the governing equations in the new coordinate system ( , )ξ η are written as16

ˆ ˆ ˆ ˆ ˆ( )cg gB C A G gη ξ∂ ∂+ = −∂ ∂

(9a)

ˆ ˆ ˆˆ ˆ( )ch hB C A H hη ξ∂ ∂+ = −∂ ∂

(9b)

where

ˆ ˆ( cos sin ) / tB x y V Jξ ξφ φ= − (9c)

ˆ ˆ( sin cos ) / tC y x V Jη ηφ φ= − (9d)

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Here, tJ denotes the Jacobian of the transformation.

Discrete Ordinate Method

In order to remove the velocity-space dependency from the reduced distribution functions, the discrete ordinate method10,11 is employed. This method, which consist of replacing the integration over velocity space of the distribution functions by appropriate integration formulas, requires the values of the distribution functions only at certain discrete speeds and velocity angles. Employing discrete distribution functions ˆ ( , , , )g Vδσ δ σξ η φ and ˆ ( , , , )h Vδσ δ σξ η φ for the discrete speed Vδ and velocity angle σφ , the macroscopic moments given by integrals over

the molecular velocity space can be substituted by the following quadratures:

ˆ ˆn P P gδ σ δσδ σ

= ∑∑ (10a)

ˆˆ ˆsinxnU P P V gδ σ δ σ δσδ σ

φ= ∑∑ (10b)

ˆˆ ˆcosynU P P V gδ σ δ σ δσδ σ

φ= ∑∑ (10c)

2ˆˆ ˆ3 / 2 ( )nT P P h V gδ σ δσ δ δσδ σ

= +∑∑ 2 2ˆ ˆˆ( )x yn U U− + (10d)

where Pδ and Pσ are weighting factors of the quadratures for the discrete speed Vδ and velocity angle σφ ,respectively. Thus instead of solving the equations for a function of space and molecular velocity, the equations are transformed to partial differential equations, which are continuous in space but are point functions in molecular speed, Vδ , and velocity angle, σφ , as follows

ˆ ˆ ˆ ˆ ˆ( )cg g

B C A G gδσ δσδσ δσ δσ δση ξ

∂ ∂+ = −

∂ ∂ (11a)

ˆ ˆ

ˆˆ ˆ( )ch hB C A H hδσ δσ

δσ δσ δσ δση ξ∂ ∂

+ = −∂ ∂

(11b)

where

ˆ ˆ( cos sin ) / tB x y V Jδσ ξ σ ξ σ δφ φ= − (11c)

ˆ ˆ( sin cos ) / tC y x V Jδσ η σ η σ δφ φ= − (11d)

Collision Frequency

The simplest model for the collision integral is the BGK model12 which has been widely used and generally gives reasonable results with much less computational effort. In the BGK model the collision frequency is given by

cPA ψµ

= (12)

where the quantity ω is the viscosity index, and ψ is a numerical parameter. The coefficient of viscosity, µ , is assumed to have a temperature dependency16

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0o

TT

ωµµ

=

(13)

The equilibrium mean free path for the VHS model17 is employed

165 2

k oo

o o

Fmn RT

µλ

π= (14)

where the quantity cF is given by

(5 2 )(7 2 )24kF ω ω− −= (15)

Combining Eqs. (12) to (15), we obtain

1ˆˆ8ˆ

5k

co

F nTAKn

ωψπ

−

= (16)

where oKn is the Knudsen number at the reference condition based on the characteristic length of the flow field, d .

Numerical Procedure

Equations (11a) and (11b) are solved by means of finite-difference approximations in physical space using simple explicit and implicit schemes depending on the characteristics of physical and velocity space. Details of the method can be found elsewhere.18 Resulting system of nonlinear algebraic equations is solved by means of successive approximations. In the iterative procedure, only the values of ˆ

cA , Gδσ , and Hδσ have to be determined from moments of the previous iteration, and the values of distribution functions do not need to be stored. Convergence is assumed to have occurred when the relative differences in the x-velocties of two successive iteration steps are less than 510− for all spatial grid points. As a proper quadrature formula for the for the discrete speed Vδ , the modified Gauss-Hermite half range quadrature for integrals of the form13,14 is used:

20

exp( ) ( )jV V Q V dV∞

−∫ (17)

III. RESULTS AND DISCUSSION

Flow around a Flat plate with zero thickness

Consider the steady flow of air around a flat plate with zero thickness and a length of 20L mµ= . The free stream velocity and temperature are 69 /m s and 295K, respectively. The free stream Mach number is 0.2. The computational domain is divided into 241( ) 161( )x y×non-uniform rectangular grids that are clustered to the plate. Figure 1 shows the computational domain for the simulation. The VHS molecular parameter ω was chosen to be 0.7 and the numerical parameterψ was chosen to be 1.

For the case of free-molecular flow, there are no collisions between molecules, and molecules are moving without collision until reaching the surface of a body. Thus at any point in the physical space, the velocity space can be divided into two regions, one for molecules coming from the free-stream and the other for molecules emitted from the surface. Since exact values of the distribution

x/L

y/L

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

1

2

3

4

Figure 1. Computational domain and grid.

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function can be obtained for each region, theoretical values of moments for free-molecular flow conditions maybe obtained by integrating the distribution function. The free-molecular solution may serve as a good standard for checking the accuracy of a finite-difference approximation. In case of the flow around a flat plate with zero thickness, macroscopic moments along the plate surface may be calculated by the following equations:

2, , ,exp( ) 1 ( )w y y yn U U erf Uπ∞ ∞ ∞ = − − − (17)

( ),1 1 12 2w yn n erf U∞

= + −

(18)

( ), , ,1 12w x x ynU U erf U∞ ∞

= −

(19)

( ),3 3 1 12 4 2w ynT n erf U∞

= + − ( ) ( )2

, ,1 1 1 24 y xerf U U∞ ∞ + − +

( )2, ,

1 12 y yU erf U∞ ∞

+ − ( ), 2 2

, ,exp2

yy w x

UU nU

π∞

∞+ − − (20)

( ) ( ), 2, , ,2

2 1 exp 12 2

xw y y y

UU U erf U

Uπτ

π∞

∞ ∞ ∞∞

= − − − + − (21)

Figures 2 and 3 show surface pressure and shear stress, respectively, at / 2x L= at the free-molecular condition. In the figures, the results of the present method at Knudsen numbers of 10Kn = and 100Kn = are compared with the theoretical values at the free-molecular flow condition. The Knudsen number is based on the free-stream mean free path and the length of the plate, /Kn Lλ∞= . Good agreement is obtained between the calculated results and theoretical values at the free-molecular flow condition.

Figure 4 shows the comparison of the drag coefficient of the plate for both sides at low Reynolds numbers from the IP-DSMC8 method and the numerical solutions of the full Navier-Stokes equations of incompressible flows19 together with experimental data.20,21 Good agreement is observed among several different methods with

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50

free-molecular theorypresent, Kn=100present, Kn=10

Skin

fric

tion

coef

ficie

nt,C

fAngle of attack (deg.)

Figure 3. Comparison of shear stress distributions at free-molecular flow.

0.75

0.8

0.85

0.9

0.95

1

1.05

0 10 20 30 40 50

free-molecular theorypresent, Kn=100present, Kn=10

Surf

ace

pres

sure

,P/P

o

Angle of attack (deg.) Figure 2. Comparison of surface pressure distributions at free-molecular flow.

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experimental data for the Reynolds numbers greater than 3. However the drag coefficient from the IP method is greater than that from the present method for the Reynolds numbers smaller than 2. The difference between the two methods becomes larger as the Reynolds number decreases. The drag coefficient from the present method approaches the free-molecular value around Re 0.02= ( 13Kn ≅ ) while that from the IP method approaches the free-molecular value around Re 0.2= ( 1.3Kn ≅ ). Figure 5 shows the effect of Reynolds number on the pressure contours. It can be seen that as Reynolds number decreases, i.e., as the flow becomes more rarefied, the disturbance of the flow caused by the solid object becomes larger. This is because as the flow becomes more rarefied the /P P∞ molecules collided with the surface can travel more distance without experiencing collision between molecules.

Flow around a 5% flat plate

To compare results from the present method with those from other methods, a low-speed flow around a 5% flat pate has been chosen. Consider the steady flow of air around a 5% flat plate with an angle of attack of 10 degree. The length of the plate is 20L mµ= . The free stream velocity and temperature are 30 /m s and 295K, respectively. The free stream Mach number is 0.087 and the Reynolds number is 4. The computational domain is divided into 41,000 non-uniform rectangular grids that are clustered to the plate. The calculated pressure contours are compared with those from the IP method8 and the continuum approach.8 It can be seen that the results from the present and the IP methods show good agreement except for the small differences in the magnitude and shape of the contours. For example, the /P P∞ = 1.006 contour in front of the leading edge and the /P P∞ = 0.994 contour behind the trailing edge from the present method are larger than that from the IP method. The contours from the present method have circular shapes while those from the IP method are closer to ellipses. The pressure contours from the continuum method in overall show similar behavior to the other two methods. However there are distinct differences between the results from the continuum and molecular approaches. The /P P∞ = 0.996 contours from the molecular approaches are attached to the upper leading and lower trailing edges while that from the continuum approach is attached to the upper surface at / 0.2x L ≅ and to the lower surface at / 0.85x L ≅ .

Figures 7 and 8 show shear stress and pressure distributions, respectively, on the surface. The calculated pressure contours are compared with those from the IP method8 and the continuum approach.8 In Fig. 7, it can be seen that calculated shear stresses from the present and the IP methods show good agreement except for some differences near the leading and trailing edges. The shear stresses from the continuum approach8 are much smaller than those from the molecular approaches.8

0

2

4

6

8

10

0.001 0.01 0.1 1 10 100

free-molecular theorypresent methodIPcompressible experimentincompressible experimentNumerical N-S

Dra

gco

effic

ient

,C

D

Reynolds number, Re Figure 4. Drag coefficient of flat plate with zero thickness at low Reynolds numbers.

1

1.005

0.991.01

0.995

x/L

y/L

-1 -0.5 0 0.5 1 1.5 20

0.5

1

1.5

2L=20 micronMa = 0.2Re = 1.

P/Po

(a) Re = 1.

1

1.0050.995

1.01 0.99

x/L

y/L

-1 -0.5 0 0.5 1 1.5 20

0.5

1

1.5

2L=20 micronMa = 0.2Re = 10

P/Po

(b) Re = 10.

1

1.0050.995

0.991.01

x /Ly/

L-1 -0.5 0 0.5 1 1.5 2

0

0.5

1

1.5

2L=20 micronMa = 0.2Re = 100

P/Po

(c) Re = 100.

Figure 5. Effect of Reynolds number on the pressure contours.

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(a) present method

(b) IP method(Ref. 8)

(c) continuum method(Ref. 8)

Figure 6. Comparison of pressure contours from various methods.

-2

0

2

4

6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

present(lower side)present(upper side)IP(lower side)IP(upper side)NS(lower side)NS(upper side)

Skin

fric

tion,

Cf

x/L

Re = 4α = 10

Figure 7. Shear stress distributions on the surface at Re = 4.

-6

-4

-2

0

2

4

6

8

10

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

present(upper side)present(lower side)IP(upper side)IP(lower side)NS(upper side)NS(lower side)

Pres

sure

coef

ficie

nt,C

P

x/L

Re = 4α = 10

Figure 8. Pressure distributions on the surface at Re = 4.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

present(on the surface)

present(l/t = 0.08)

present(l/t = 0.17)

IP (Ref. 8)

Pres

sure

coef

ficie

nt,C

P

x/L

Re = 4α = 10Upper side

Figure 9. Comparison of pressure distributions on and away from the upper surface at Re = 4.

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The calculated pressure coefficients from the present and the IP methods also show good agreement except for some differences near the leading and trailing edges. The pressure coefficient from the continuum approach8 shows different behavior from those calculated by the molecular approaches. To investigate the differences in the pressure coefficients from the present and IP methods near the leading and trailing edges, the pressure coefficients on and at a short distance away from the upper surface are compared in Fig. 9. The quantity t is the thickness of the plate and z is the normal distance from the surface. It is interesting to see that the pressure coefficients from the present method along the lines of /l t = 0.08(0.08 mµ away from the surface) and 0.17 decrease near the leading edge, show local minimum, and increase abruptly. Near the trailing edge, the pressure coefficients near the surface decrease abruptly. However, the pressure coefficient on the surface decreases abruptly and shows a minimum at the leading edge while decreasing smoothly near the trailing edge, which is physically more plausible. The pressure coefficient from the IP method show similar behavior to those along the lines of /l t = 0.08 and 0.17 from the present method.

The aerodynamic characteristics of the 5% plate are shown in Fig. 10. It can be seen that both lift and drag from the present method are greater than those from the IP and continuum methods21 and the differences become larger as the angle of attack increases. The reason for the higher lift in the results from the present method can be explained from Figs. 8 and 9. As it can be seen in the figures, the pressure from the present method is slightly higher on the lower surface and slightly lower on the upper surface than that from the IP method. It also can be seen that the lower pressure region near the upper leading edge has a significant effect in the lift. The ratio of lift to drag from three methods is shown in Fig. 11. For comparison, the theoretical ratio of lift to drag for free molecular flow and that from the present method at Re = 0.015 are shown together. The ratio of lift to drag from the present method is greater than those from the IP and continuum methods21 and more closer to those for free molecular flow.

IV. Conclusion A finite-difference method coupled with the discrete-ordinate method is employed to analyze low-speed gas

flows around micro-scale flat plates. Calculations are made for flows around a flat plate with zero thickness and a 5% flat plate. Results are compared with those from the IP-DSMC method and numerical solutions of the Navier-Stokes equations with slip boundary conditions. The calculated results show good agreement with those from the IP method except for some differences in details near the leading and trailing edges. The calculated pressure on and at a short distance away from the surface showed quite different behavior especially in the region close to the leading edge where rarefaction effects play an important role. The behavior of the surface pressure from the IP method was closer to that of the present method at a short distance away from the surface rather than that on the surface. The advantage of the present method is that it does not suffer from statistical noise which is common in particle based methods and requires much less amount of computational effort. All the calculations in the present work are performed in a desktop computer with a Pentium IV 3.0GHz processor.

-4

-2

0

2

4

6

0 10 20 30 40

CD

present

CL

present

CD

IP

CL

IP

CD

N-S

CL

N-S

Coe

ffici

ent,

CD

orC

L

Angle of attack (deg.)

Re = 4Ma = 0.087

Figure 10. Characteristics of 5% flat plate at Re = 4.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40

free molecular theorypresent, Re = 0.015presentIP (Ref. 22)N-S (Ref. 22)

Rat

io,C

L/C

D

Angle of attack (deg.)

Re = 4Ma = 0.087

Figure 11. Comparison of /L DC C at Re = 4

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