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Benchmark problemsNumerical modeling of gas flow through a slit:

kitetic approach

Irina A. Graur1, Alexey Ph. Polykarpov1, Felix Sharipov2

1Universite de Provence Aix-Marseille I, IUSTI, UMR 6596 CNRS,Ecole Polytechnique Universitaire de Marseille, France

2 Departamento de Fısica, Universidade Federal do Parana,Caixa Postal 19044, Curitiba, 81531-990, Brazil

64th IUVSTA Workshop, 16-19 May 2011

Graur IA, Polykarpov A, Sharipov F (UP) Benchmark problem IUVSTA 1 / 26

Statement of the problem. Problem 1

y

L1

L2

L1

L2

x

p1,T1

p2,T2Tw

H/2

0

Slit flow is a limit case of the channelflow, when L/H → 0left reservoir: p1, T1,right reservoir: p2, T2,isothermal condition, T1 = T2,Tw = T1vacuum condition p2 = 0 (particularcase)The slit height is equal to H in ydirectionand the slit is infinite in z direction2D in physical space


Imput equation

Since we are going to consider the whole range of the rarefactionparameter, the problem must be solved on the level of the velocitydistribution function f (r′, v), which obeys the stationary Boltzmannequation

v ·∂ f ′

∂r′= Q′, (1)

where r′ is the position vector, v is the molecular velocity, and Q′ is thecollision integral.


Macroscopic parameters

When the distribution function is known, the macroscopic parameters maybe calculated from the following expressions:number density

n′(r′) =

∫f ′(r′, v) dv, (2)

bulk velocity

u′(r′) =1n′

∫v f ′(r′, v) dv, (3)

temperature

T ′(r′) =m

3n′k

∫V′2 f (r′, v) dv, (4)

heat flow vector

q′(r′) =m2

∫V′V ′2 f ′(r′, v) dv, (5)

where V = v − u′ is the peculiar velocity.Graur IA, Polykarpov A, Sharipov F (UP) Benchmark problem IUVSTA 4 / 26

Kinetic models

In the present work, two model equations will be applied, BGK andS-model. For the BGK model the collision integral has the form

Q′BGK =pµ

(f ′M − f ′

), (6)

where

f ′M(n′,T ′,u′) = n′(r′)[

m2πkT ′(r′)

]3/2

exp[−

m(v − u′(r′))2

2kT ′(r′)

](7)

is the local Maxwellian. For the S-model the collision integral reads

Q′S =pµ

{f ′M

[1 +

2mV′ · q′

15n′(kT ′)2

(mV ′2

2kT ′−

52

)]− f ′

}. (8)


Dimensionless quantities

To solve equation (1) numerically the following dimensionless quantitiesare introduced:

r =r′

H, c =

vv1, C =

Vv1, (9)

T =T ′

T1, n =

n′

n1, u =

u′

v1, q =

q′

p1v1, (10)

p =p′

p1, µ =

µ′

µ1. f = f ′

v31

n1, Q =

Hv21

n1Q′. (11)

The reference density n1 is related to p1 and T1 by the state equationn1 = p1/kT1. If we assume the hard-sphere molecular model then thedimensionless viscosity coefficient will be given as µ =

√T .


Dimensionless form of equation

Since we consider the two-dimensional flow, we suppose that the flowcharacteristics do not vary in z direction and for the bulk velocity vector uand the heat flux vector q the component along the z axis is equal to zero.Using the dimensionless variables, the kinetic equation for atwo-dimensional case reads

cx∂ f∂x

+ cy∂ f∂y

= Q, (12)

where

QBGK = δn√

T(

f M − f)

(13)

QS = δn√

T{

f M[1 +

4C · q15T 2

(C2

T−

52

)]− f

}. (14)


Boundary condition

The diffuse boundary conditions at the wall is used here, i.e. the reflectedparticles have the following distribution function

f (0, y, c) =nw1

π3/2 e−c2, cx < 0 (15)

for the left container and

f (0, y, c) =nw2

π3/2 e−c2, cx > 0 (16)

for the right one. The unknown quantities nw1 and nw2 are found from thenon-permeability condition.


Projection procedure

Introducing new distribution functions we can eliminate the dependence ofthe distribution function on the molecular velocity cz.

cx∂Φ

∂x+ cy

∂Φ

∂y= δn

√T (ΦM − Φ), (17)

cx∂Ψ

∂x+ cy

∂Ψ

∂y= δn

√T (ΨM − Ψ), (18)

where two reduced distribution functions are defined as following

Φ(x, y, cx, cy) =

∫f (x, y, c) dcz, Ψ(x, y, cx, cy) =

∫f (x, y, c)c2

z dcz. (19)

The reduced Maxwellian distribution functions (19) reads

ΦM =nπT

exp− (cx − ux)2 + (cy − uy)2

T

, ΨM =T2

ΦM. (20)


Computational parameters in physical space

Two rectangles of the length L1 = 100H and L2 = 100H. Thedimension of each reservoir is chosen sufficiently large so that itsfurther increase do not change the flow rate within 1%.

The uniform grid is used in the square domain −H/2 ≤ x ≤ H/2,−H/2 ≤ y ≤ H/2. In the other parts of the computational domainsnon-uniform grid is used, where the dimension of cells increasesaccording to the geometric series with increment 1.0125.

The calculations were carried out on the computational grid with412 × 412 cells in each container.

40 cells were used across the slit semi-height.


Computational parameters in velocity space

The polar coordinates (cp, ϕ) related to the Cartesian ones were used

cx = cp cosϕ, cy = cp sinϕ (21)

For the variable cp the Gaussian abscissas, corresponding to thefollowing weight function, were used

w(cp) = e−c2pcp (22)

The nodes of the angle ϕ were uniformly distributed over the intervalfrom 0 to 2π.

The computations were carried out using Nc = 20 points with respectto cp and Nϕ = 100 points with respect to ϕ.

Test calculations carried our for Nc = 25 and Nϕ = 200 showed thatthe uncertainty related to the discretization of cp and ϕ does notexceed 1%.


Numerical scheme

The explicit finite difference scheme (”Fixed point method”) was used forthe approximation of the transport terms in Eq.(17). For the BGK model ithas the form:

− cp∂Φ(k+1)

∂s+ δn(k)

√T (k)Φ(k+1) = δn(k)

√T (k)ΦM(k), (23)

where s is a characteristic directed against the two-dimensional velocity(cx, cy) , k is the iteration number.


Parameters

Solution is determined by

rarefaction parameter

δ =p1Hµvm

, vm =

√2

km

T1

from δ = 0 to δ = 100 hydrodynamic regime

pressure ratio p1/p2

p1

p2= 0, 0.1, 0.5, 0.7, 0.9, 0.99

The mass flow rate M per a length unity in the z direction in thenormalized form

W =MM1

, M1 =Hp1√πv1

(24)


Gas flow into vacuum

Table: Reduced flow rate W vs rarefaction parameter δ

δ BGK1 S-model1 DSMC2

0.01 1.004 1.004 1.0010.1 1.031 1.032 1.0171.0 1.163 1.169 1.13910.0 1.474 1.477 1.479100.0 1.564 1.565 1.566

1 I. Graur, A. Polykarpov, F. Sharipov, Computer and Fluids, 20112 F. Sharipov, D. Kozak, J. Vac Sci. Technol. A 27 (3) (2009) 479–484.


Gas flow into vacuum

0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 4 6 8 10

x

ux

n

T

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-4 -2 0 2 4 6 8 10

x

ux

n

T

Figure: Bulk velocity, temperature and number density distributions along the axisat δ = 0.1 and 100: solid line - DSMC results, circles - BGK, crosses - S model


BGK results, various pressure ratios

Table: Dimensionless flow rate W vs rarefaction parameter δ and pressure ratio,BGK model

Wδ p1/p0 = 0 0.1 0.5 0.7 0.9 0.99

0.00 1.0 0.9 0.5 0.3 0.1 0.010.01 1.004 0.9044 0.5034 0.3023 0.1008 0.010090.1 1.031 0.9322 0.5251 0.3169 0.1062 0.010651 1.163 1.088 0.6651 0.4134 0.1419 0.0143710 1.474 1.463 1.253 0.9626 0.4058 0.04298100 1.564 1.560 1.401 1.177 0.7293 0.1891


Comparison with linearized BGK model

The particular interest is the comparison of the present results with thoseobtained earlier applying the linearized BGK model where the reducedflow rate is introduced as

W′ =MM′1

, M′1 =H(p1 − p2)√πv1

, (25)

where M′0 is the mass flow rate in the free molecular regime (δ � 1) due toa pressure drop (p1 − p2).The relative difference of the flow rate W′ obtained from the linearizedBGK model and that obtained here from the full BGK model does notexceed the quantity δ(p1 − p2)/p1. Thus, if the rarefaction parameter issmall the linear theory provides reasonable results even if the pressuredrop (p1 − p2)/p1 is large. However, for large values of δ the linear theorycan fail even for a small pressure drop.


Comparison with linearized BGK model

0

5

10

15

20

25

1 10 100δ

W ′

p0 /p1=Linear theory

0.99

0.9

Figure: Reduced flow rate W ′ vs rarefaction parameter δ: curve - results oflinearized BGK, symbols - present results


Number density p2/p1 = 0.1 and 0.9

0

0.2

0.4

0.6

0.8

1

-5 0 5 10 15 20

n

x

δ=1

δ=10

δ=100

δ=1

δ=10

δ=100

0.9

0.92

0.94

0.96

0.98

1

-5 0 5 10 15 20

n

x

δ=1

δ=10

δ=100

δ=1

δ=10

δ=100

Figure: The comparison of results BGK and S-model for p2/p1 = 0.1 and 0.9(lines - BGK, symbols - S-model)


Temperature p2/p1 = 0.1 and 0.9

0

0.2

0.4

0.6

0.8

1

-5 0 5 10 15 20

T

x

δ=1

δ=10

δ=100

δ=1

δ=10

δ=100 0.965

0.972

0.979

0.986

0.993

1

-5 0 5 10 15 20

T

x

δ=1

δ=10

δ=100

δ=1

δ=10

δ=100



Velocity p2/p1 = 0.1 and 0.9

0

0.4

0.8

1.2

1.6

-5 0 5 10 15 20

u x

x

δ=1

δ=10

δ=100

δ=1

δ=10

δ=100

0

0.07

0.14

0.21

0.28

0.35

-5 0 5 10 15 20

u x

x

δ=1

δ=10

δ=100

δ=1

δ=10

δ=100



X/H

Y/H

0 5 10 15 200

2

4

6

8

Velocity flow field δ = 100, p2/p1 = 0.1


Gas flow into vacuum. Computational efforts

Table: Comparison of computational efforts

DV M DS MCδ Iter. CPU(h) Samp. CPU(h)

0.1 135 1.8 13000 1891 193 2.5 11700 16310 1925 22 9400 127100 19193 140 9200 130


Conclusion

The flow of rarefied gas through a thin slit at an arbitrary pressuredrop was calculated applying the BGK and S model equations forwide ranges of the gas rarefaction. The results obtained from the twomodel kinetic equations were compared between them and to theDSMC results.

The present results for small pressure differences were comparedwith whose obtained from the linearized BGK model. It has beenshown that the linear theory works well in a wider range of thepressure drop than that indicated earlier.

Comparison with the experimental results?


Acknowledgements

The authors are grateful toThe research leading to these results has received funding from theEuropean Community’s Seventh Framework Programme (ITN -FP7/2007-2013) under grant agreement N 215504.Programme Action en Region de Cooperation Universitaire etScientifique (ARCUS, France)the IDRIS (Institute for Development and Resources in IntensiveScientific computing, France) for the supercomputing resources underthe project number i2010022168


Numerical schliren at δ=100 and p2/p1 = 0.1

X

Y

0 2 4 60

0.5

1

1.5

2

2.5

BarrelShock

NormalShock

Zone ofSilence

Figure: Numerical schliren at δ=100 and p2/p1 = 0.1


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