Z. Angew. Math. Phys. 63 (2012), 503–520 c 2011 Springer Basel AG 0044-2275/12/030503-18 published online November 30, 2011 DOI 10.1007/s00033-011-0178-4 Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Numerical modelling of rarefied gas flow through a slit at arbitrary pressure ratio based on the kinetic equation Irina A. Graur, Alexey Ph. Polikarpov and Felix Sharipov Abstract. A rarefied gas flow through a thin slit at an arbitrary gas pressure ratio is calculated on the basis of the kinetic model equations (BGK and S-model) applying the discrete velocity method. The calculations are carried out for the whole range of the gas rarefaction from the free-molecular regime to the hydrodynamic one. Numerical data on the flow rate and distributions of density, bulk velocity and temperature are reported. Comparisons of the present results with those based on the direct simulation Monte Carlo method and on the linearized BGK kinetic equation are performed. The conditions of applicability of the linearized theory are discussed. Mathematics Subject Classification (2010). 82B40 kinetic theory of gases. Keywords. Rarefied gas · Kinetic equation · Discrete velocity method · Slit flow · Flow rate · Flow field. 1. Introduction Rarefied gas flows through slits are met in many practical fields such as high altitude flights, molecular beam technology, vacuum sciences, etc. According to Refs. [1–3], flows of rarefied gas through thin slits and orifices are weakly sensitive to the gas–surface interaction that reduces a number of factors affecting the mass flow rate. Since the statement of the problem is simple and the number of parameters deter- mining its solution is small, such a problem is a good example to test new theoretical and experimental methods in rarefied gas dynamics. The slit flow of rarefied gases attracts the attention of many researchers because of its practical and scientific importance, see e.g., Refs. [1, 4–15]. Liepmann [4] was the first to analyze both theoretically and experimentally the flow rate of rarefied gas through a slit into vacuum. His experimental error is rather large and theoretical estimation is rough. Willis [5] used the iterative method to calculate the flow rate through a slit near the free-molecular regime. In this work, the Bhatnagar–Gross–Krook (BGK) model equation [16] is converted into an integral form, and the iterations are performed starting from the free molecular solution. Stewart [6] applied the same iterative procedure to a situation of gas flow driven by a finite pressure drop. Hasegawa and Sone [8] performed calculations of the slit flow driven by a small pressure difference using the linearized BGK kinetic model [16]. Sharipov et al. [1, 7, 10, 12] reported results on the slit flow based on the divers linearized kinetic equations, namely BGK model [16], S-model [17] and McCormack model [18]. The authors of Ref. [9] calculated the mass flow rate trough a slit for several values of the pressure drop based on the BGK and S-model kinetic equations and applying the direct simulation Monte Carlo (DSMC) method [19]. However, their results are restricted by near free-molecular and transitional regimes. As was pointed in Ref. [13], the results of the work [9] are not reliable in the transitional regime. The paper [11] also contains some results on the slit flow obtained by the DSMC method in a restricted range of the gas rarefaction. In Refs. [13, 15], the mass flow rates through a slit and the flow fields are calculated by the DSMC method over the whole range of the gas rarefaction parameter from the free-molecular regime to the hydrodynamic limit and for various values of the pressure ratio. In our previous paper [14], the gas flow through a slit into vacuum was calculated
Zeitschrift fur angewandteMathematik und Physik ZAMP
Numerical modelling of rarefied gas flow through a slit at arbitrarypressure ratio based on the kinetic equation
Irina A. Graur, Alexey Ph. Polikarpov and Felix Sharipov
Abstract. A rarefied gas flow through a thin slit at an arbitrary gas pressure ratio is calculated on the basis of the kineticmodel equations (BGK and S-model) applying the discrete velocity method. The calculations are carried out for the wholerange of the gas rarefaction from the free-molecular regime to the hydrodynamic one. Numerical data on the flow rate anddistributions of density, bulk velocity and temperature are reported. Comparisons of the present results with those basedon the direct simulation Monte Carlo method and on the linearized BGK kinetic equation are performed. The conditionsof applicability of the linearized theory are discussed.
Mathematics Subject Classification (2010). 82B40 kinetic theory of gases.
Rarefied gas flows through slits are met in many practical fields such as high altitude flights, molecularbeam technology, vacuum sciences, etc. According to Refs. [1–3], flows of rarefied gas through thin slitsand orifices are weakly sensitive to the gas–surface interaction that reduces a number of factors affectingthe mass flow rate. Since the statement of the problem is simple and the number of parameters deter-mining its solution is small, such a problem is a good example to test new theoretical and experimentalmethods in rarefied gas dynamics.
The slit flow of rarefied gases attracts the attention of many researchers because of its practical andscientific importance, see e.g., Refs. [1,4–15]. Liepmann [4] was the first to analyze both theoreticallyand experimentally the flow rate of rarefied gas through a slit into vacuum. His experimental error israther large and theoretical estimation is rough. Willis [5] used the iterative method to calculate theflow rate through a slit near the free-molecular regime. In this work, the Bhatnagar–Gross–Krook (BGK)model equation [16] is converted into an integral form, and the iterations are performed starting fromthe free molecular solution. Stewart [6] applied the same iterative procedure to a situation of gas flowdriven by a finite pressure drop. Hasegawa and Sone [8] performed calculations of the slit flow drivenby a small pressure difference using the linearized BGK kinetic model [16]. Sharipov et al. [1,7,10,12]reported results on the slit flow based on the divers linearized kinetic equations, namely BGK model [16],S-model [17] and McCormack model [18]. The authors of Ref. [9] calculated the mass flow rate trough aslit for several values of the pressure drop based on the BGK and S-model kinetic equations and applyingthe direct simulation Monte Carlo (DSMC) method [19]. However, their results are restricted by nearfree-molecular and transitional regimes. As was pointed in Ref. [13], the results of the work [9] are notreliable in the transitional regime. The paper [11] also contains some results on the slit flow obtainedby the DSMC method in a restricted range of the gas rarefaction. In Refs. [13,15], the mass flow ratesthrough a slit and the flow fields are calculated by the DSMC method over the whole range of the gasrarefaction parameter from the free-molecular regime to the hydrodynamic limit and for various valuesof the pressure ratio. In our previous paper [14], the gas flow through a slit into vacuum was calculated
504 I. A. Graur et al. ZAMP
on the basis of the BGK and S-model equations. These results are in a good agreement with the DSMCdata [13], but the numerical solution of the kinetic equations requires less computational effort than theDSMC method, namely it requires a smaller computational memory, shorter CPU time and it is easilyparallelized. Moreover, the DSMC method is affected by the statistical scattering, which can be signif-icant for flows with a small Mach number, e.g., when the pressure difference is small. Thus, a furtherdevelopment of other methods to calculate internal flows of rarefied gas is necessary.
As is known, a numerical solution of the exact Boltzmann equation given in some papers, e.g., Refs. [20–24], is still hard task even nowadays when power computers are available. At the same time, the modelequations like BGK [16] and S-model [17] are effective tools and widely used for practical calculations,see e.g., Refs. [12,25–27]. A detailed comparison of data for internal flows of rarefied gases and for theslip coefficients based on the model equations with those obtained from the linearized Boltzmann equa-tion was performed in Refs. [28,29] and showed that the models provide reliable results with a modestcomputational effort. The same validation is necessary for the nonlinearized model equations.
The aim of this paper is to calculate the flow rate through a slit at different gas pressure ratios overthe whole range of the gas rarefaction embracing both free-molecular and hydrodynamic regimes apply-ing the nonlinear BGK and S-model kinetic equations. The present results include some cases of smallpressure drops, which cannot be calculated by the DSMC method because of the statistical scattering.The obtained results are compared with those obtained by the DSMC method [15] for a large pressuredifference. A comparison of these results with those obtained by the linearized BGK model equation inthe case of the small pressure drop between the reservoirs will be given, and the limits of the applicabilityof the linearized approach will be indicated.
2. Statement of the problem
Consider a flow of monatomic gas through a thin two-dimensional slit fixed in the plane x = 0. Thepressure in the upstream reservoir far from the slit is maintained equal to p0, while the pressure in thedownstream reservoir is equal to p1. The temperature of the gas in both reservoirs and that of the par-tition is maintained equal to T0. The slit height is equal to H in the y direction and it is infinite in thez direction. We are going to calculate the mass flow rate through the slit and flow field in both reservoirs.
Besides the pressure ratio p1/p0, the gas flow through the slit between two reservoirs is determinedby the rarefaction parameter δ defined as follows
δ =p0H
μ0v0, v0 =
√2kT0
m, (1)
where μ0 is the shear viscosity of the gas at the temperature T0, v0 is the most probable molecular speedat the same temperature, k is the Boltzmann constant, and m is the molecular mass. Since the quantity� = μ0v0/p0 represents the equivalent free path of gaseous particles, then the rarefaction parameter δ isinversely proportional to the Knudsen number. Thus, the limit δ � 1 corresponds to the free-molecularregime, while in the opposite limit δ � 1 represents the continuum medium regime.
In the free-molecular regime (δ = 0), the mass flow rate into vacuum (p1/p0 = 0) per unit length inthe z direction may be calculated analytically [28]
M0 =Hp0√πv0
. (2)
The numerical results for the mass flow rate M at arbitrary pressure ratio and rarefaction parameter willbe given later in the following normalized form
W =M
M0
. (3)
Vol. 63 (2012) Rarefied gas flow through a slit 505
Thus, in the present work, we are going to calculate the dimensionless flow rate W as a function of thepressure ratio p1/p0 and rarefaction parameter δ.
We adopt the diffuse scattering of gaseous particles on the slit walls. As was shown in Refs. [1–3], theslit flow is weakly affected by the gas–surface interaction, that is, the uncertainty of this assumption isless than 1%.
3. Input equation
Since we are going to consider the whole range of the rarefaction parameter, the problem must be solvedon the level of the velocity distribution function f(r′,v), which obeys the stationary Boltzmann equation
v · ∂f ′
∂r′ = Q′, (4)
where r′ is the position vector, v is the molecular velocity, and Q′ is the collision integral.When the distribution function is known, the density n′, bulk velocity u′, temperature T ′ and heat
flux vector q′ are calculated as
n′(r′) =∫
f ′(r′,v) dv, (5)
u′(r′) =1n′
∫vf ′(r′,v) dv, (6)
T ′(r′) =m
3n′k
∫V 2f(r′,v) dv, (7)
q′(r′) =m
2
∫V V 2f ′(r′,v) dv, (8)
respectively. Here, V = v − u′ is the peculiar velocity.In the present work, two model equations will be applied, viz. BGK [16] and S-model [17,28], which
read
Q′BGK =
p′
μ′(f ′M − f ′) , (9)
Q′S =
p′
μ′
{f ′M
[1 +
2mV ′ · q′
15n′(kT ′)2
(mV ′2
2kT ′ − 52
)]− f ′
}, (10)
respectively. Here,
f ′M (n′, T ′,u′) = n′(r′)[
m
2πkT ′(r′)
]3/2
exp[−m(v − u′(r′))2
2kT ′(r′)
](11)
is the local Maxwellian.As was noticed in our previous paper [14], each equation has its own advantages and disadvantages.
That is why both models are used here with the purpose to compare their results.
4. Dimensionless equations
To solve equation (4) numerically, the following dimensionless quantities are introduced:
r =r′
H, c =
v
v0, C =
V
v0, (12)
506 I. A. Graur et al. ZAMP
T =T ′
T0, n =
n′
n0, u =
u′
v0, q =
q′
p0v0, (13)
p =p′
p0, μ =
μ′
μ0. f = f ′ v
30
n0, Q =
Hv20
n0Q′. (14)
The reference number density n0 is related to p0 and T0 by the state equation n0 = p0/kT0. If we assumethe hard-sphere molecular model, then the dimensionless viscosity coefficient will be given as μ =
√T .
We suppose that the flow characteristics do not vary in z direction, then the bulk velocity u and heatflux q are two-dimensional vectors. Using the dimensionless variables (12)–(14), the kinetic equation (9)reads
cx∂f
∂x+ cy
∂f
∂y= Q, (15)
where
QBGK = δn√
T(fM − f
)(16)
QS = δn√
T
{fM
[1 +
4C · q
15T 2
(C2
T− 5
2
)]− f
}, (17)
fM =n
(πT )3/2exp
[− (c − u)2
T
]. (18)
As has been mentioned previously, the diffuse boundary conditions at the wall are used here, that is,the reflected particles have the following distribution function
f(0, y, c) =nw0
π3/2e−c2
, cx < 0 (19)
for the left container and
f(0, y, c) =nw1
π3/2e−c2
, cx > 0 (20)
for the right one. The unknown quantities nw0 and nw1 are found from the nonpermeability condition.
4.1. Numerical scheme
The kinetic equation (15) with the collision integral in the forms (17) and (18) was solved by the discretevelocity method [30]. The numerical scheme used in this work is similar to that applied in Ref.[14] thatis why the details are omitted here, but only the modifications will be given.
Basically, the difference between the present scheme from that used in Ref. [14] is the computationaldomain shown in Fig. 1. It represents two squares of the same side length L = 100H. The dimensionL is chosen sufficiently large so that its further increase does not change the flow rate within 1%. Theuniform grid is used in the square domain −H/2 ≤ x ≤ H/2, −H/2 ≤ y ≤ H/2, with forty cells acrossthe slit semi-height. In the other parts of the computational domains, nonuniform grid is used, where thedimension of cells increases according to the geometric series with an increment 1.0125. The calculationswere carried out on the computational grid with 412 × 412 cells in each container.
5. Results and discussion
5.1. Flow rate
The numerical calculations were carried out for several values of rarefaction parameter in the range fromδ = 0.01 (free-molecular regime) to δ = 100 (hydrodynamic regime). Five values of the pressure ratiop1/p0 were considered: 0.1, 0.5, 0.7, 0.9 and 0.99. The data of the dimensionless flow rate W obtained
Vol. 63 (2012) Rarefied gas flow through a slit 507
H
H/2
L
L L
y
x0-L L
Fig. 1. Computational domain
Table 1. Dimensionless flow rate W versus rarefaction parameter δ and pressure ratio p1/p0, BGK model andDSMC [15]
Fig. 2. Reduced flow rate W ′ versus rarefaction parameter δ: curve results of Ref. [1], symbols present results
δ=1
-5 0 5 10
x
0
5
y
δ=10
-5 0 5 10
x
0
5
y
δ=100
-5 0 5 10
x
0
5
y
Fig. 3. Streamlines for p1/p0 = 0.1
from the BGK model are presented in Table 1. In the free-molecular regime (δ → 0), the numerical valueof W tends to its theoretical value (1 − p1/p0). In the hydrodynamic regime (δ → ∞), the flow rateW also tends to a constant value, which depends on the pressure ratio p1/p0. A significant variation ofthe flow rate W occurs for the rarefaction parameter δ from 0.5 to 50 for all pressure ratios in the range
Vol. 63 (2012) Rarefied gas flow through a slit 509
δ=1
-5 0 5 10
x
0
5
yδ=10
-5 0 5 10
x
0
5y
δ=100
-5 0 5 10
x
0
5
y
Fig. 4. Streamlines for p1/p0 = 0.9
0.1 ≤ p1/p0 ≤ 0.7. For the pressure ratios p1/p0 = 0.9 and 0.99, the flow rate W does not reach itsmaximum value in the interval of the rarefaction parameter considered here. The relative variation ofthe flow rate W increases by increasing the pressure ratio. In the case of the small ratio, i.e., p1/p0 = 0.1,the value of the flow rate W in the hydrodynamic regime (δ � 1) is 1.73 times larger than its value in thefree molecular regime (δ � 1), while for the pressure ratio p1/p0=0.7 the flow rates increase 3.92 timesin the transition from δ = 0 to δ = 100.
The values of the dimensionless flow rate W obtained from S-model are presented in Table 2. Com-paring these data to those obtained from the BGK model, it can be seen that the S-models provide thesame flow rate W within the numerical accuracy 1%.
The comparison between the present BGK results and the results obtained by the DSMC [15] aregiven in Table 1. Analyzing these data, we conclude that for the small values of the rarefaction parameterδ = 0.1, the discrepancy between the BGK and DSMC results is less than 1% for all pressure ratios givenin Table 1. For the large values of rarefaction parameter δ = 100, the maximal discrepancy is around 2%for the pressure ratio 0.7. In the transitional flow regime (δ ∼ 1), difference in the flow rate given by eachmethod reaches 4% for the pressure ratio 0.7, which exceeds the numerical accuracy of the methods.
The particular interest is the comparison of the present results with those obtained earlier applyingthe linearized BGK model in the paper [1] where the reduced flow rate is introduced as
W ′ =M
M ′0
, M ′0 =
H(p0 − p1)√πv0
, (21)
where M ′0 is the mass flow rate in the free-molecular regime (δ � 1) due to a pressure drop (p0 − p1), so
that W ′ = W (1−p1/p0). The quantity W ′ corresponding to the linearized BGK model is plotted in Fig. 2
510 I. A. Graur et al. ZAMP
δ=1
-5 0 5 10
x
0
5
y
δ=10
-5 0 5 10
x
0
5
y
δ=100
-5 0 5 10
x
0
5
y
Fig. 5. Streamlines for p1/p0 = 0.99
by the solid line. The same quantity obtained from the nonlinearized BGK model is presented by thesquare and circle symbols for p1/p0 = 0.99 and 0.9, respectively. It can be seen that for the small valuesof the rarefaction (δ < 10), the results based on the linearized equation are in a good agreement with thesolution of the nonlinear BGK equation even for the pressure ratio p1/p0 = 0.9 that corresponds to 10%of the relative pressure drop. But when the rarefaction parameter increases (δ > 10), the results basedon the linearized BGK equation diverge from those based on the nonlinear BGK even for the pressureratio close to unity p1/p0 = 0.99 that corresponds to 1% of the relative pressure difference. The conditionof the applicability of the linearized kinetic equation to the slit flow was estimated in Ref. [1] departingfrom the assumption that both Mach and Reynolds numbers must be small. Then, it was deduced thatthe linear theory is valid if (p0 − p1)/p0 � 1 for δ < 1 and if δ2(p0 − p1)/p0 � 1 for δ � 1. An analysisof the numerical values of the flow rate obtained here shows that these conditions are very strong. Inreality, the range of the applicability of the linear theory is wider. For the rarefaction interval consideredhere, i.e., 0.01 ≤ δ ≤ 100, it is possible to state that the relative difference of the flow rate W ′ obtainedfrom the linearized BGK model [1] and that obtained here from the full BGK model does not exceed thequantity δ(p0 − p1)/p0. Thus, if the rarefaction parameter is small, the linear theory provides reasonableresults even if the pressure drop (p0 − p1)/p0 is large. However, for large values of δ, the linear theorycan fail even for a small pressure drop.
5.2. Flow field
The Figs. 3, 4 and 5 present the streamlines for p1/p0 = 0.1, 0.9 and 0.99, respectively. One can see thatfor each value of the pressure ratio, the behavior of the streamlines is qualitatively different when δ = 10and 100. For the small value, i.e., at p1/p0 = 0.1 and δ = 100, the gas forms a jet past the slit with a
Vol. 63 (2012) Rarefied gas flow through a slit 511
0
0.2
0.4
0.6
0.8
1
-5 0 5 10 15
n
x
0.2
0.4
0.6
0.8
1
-5 0 5 10 15
T
x
0
0.3
0.6
0.9
1.2
1.5
-5 0 5 10 15
u x
x
Fig. 6. (Color online) Distributions of dimensionless density, temperature and bulk velocity [Eqs. (13), (14)] along thex-axis for p1/p0 = 0.1. Lines DSMC [15] (dotted line δ = 0.1, dashed line δ = 10, solid line δ = 100), circles BGK, crossesS-model; red δ = 0.1, green δ = 10, blue δ = 100
512 I. A. Graur et al. ZAMP
0.5
0.6
0.7
0.8
0.9
1
-5 0 5 10 15
n
x
0.75
0.8
0.85
0.9
0.95
1
-5 0 5 10 15
T
x
0
0.15
0.3
0.45
0.6
0.75
-5 0 5 10 15
u x
x
Fig. 7. (Color online) Distributions of dimensionless density, temperature and bulk velocity [Eqs. (13), (14)] along thex-axis for p1/p0 = 0.5. Lines DSMC [15] (dotted line δ = 0.1, dashed line δ = 10, solid line δ = 100), circles BGK, crossesS-model; red δ = 0.1, green δ = 10, blue δ = 100
deviation from the straight line near the point x ≈ 5. For the value p1/p0 = 0.9 and δ = 100, the jet pastthe slit is straight. At p1/p0 = 0.99, no jet is observed past the slit. In the transitional regime (δ = 1),the streamlines are similar for all values of the pressure drop.
Vol. 63 (2012) Rarefied gas flow through a slit 513
0.7
0.76
0.82
0.88
0.94
1
-5 0 5 10 15
n
x
0.88
0.91
0.94
0.97
1
-5 0 5 10 15
T
x
0
0.15
0.3
0.45
0.6
-5 0 5 10 15
ux
x
Fig. 8. (Color online) Distributions of dimensionless density, temperature and bulk velocity [Eqs. (13), (14)] along thex-axis for p1/p0 = 0.7. Lines DSMC [15] (dotted line - δ = 0.1, dashed line - δ = 10, solid line δ = 100), circles BGK,crosses S-model; red δ = 0.1, green δ = 10, blue δ = 100
The axial distributions of the dimensionless number density n, bulk velocity ux and temperature Tdefined by Eq. (14) and obtained from both BGK and S-models are shown in Figs. 6, 7, 8, 9, and 10.One can see that for the small values of the rarefaction (δ ≤ 10), the behaviors of the distributions are
514 I. A. Graur et al. ZAMP
0.9
0.92
0.94
0.96
0.98
1
-5 0 5 10 15
n
x
0.96
0.97
0.98
0.99
1
-5 0 5 10 15
T
x
0
0.08
0.16
0.24
0.32
-5 0 5 10 15 20
u x
x
Fig. 9. (Color online) Distributions of dimensionless density, temperature and bulk velocity [Eqs. (13), (14)] along thex-axis for p1/p0 = 0.9. Circles BGK, crosses S-model; red δ = 0.1, green δ = 10, blue δ = 100
quantitatively the same, i.e., the density smoothly changes from its values in the left container to itsvalue in the right one, the temperature has a minimum near the slit and tends to its equilibrium values inboth containers, and the bulk velocity has a maximum near the slit and tends to zero in both containers.However, for the large value of the rarefaction (δ = 100), the distributions n, ux and T qualitatively
Vol. 63 (2012) Rarefied gas flow through a slit 515
0.99
0.992
0.994
0.996
0.998
1
-5 0 5 10 15
n
x
0.998
0.9985
0.999
0.9995
1
-5 0 5 10 15
T
x
0
0.015
0.03
0.045
0.06
-5 0 5 10 15 20
u x
x
Fig. 10. (Color online) Distributions of dimensionless density, temperature and bulk velocity [Eqs. (13), (14)] along thex-axis for p1/p0 = 0.99. Circles BGK, crosses S-model; red δ = 0.1, green δ = 10, blue δ = 100
differ for each pressure ratio p1/p0. It can be seen that results based on the BGK and S-models are ina good agreement between them. For the pressure ratios p1/p0 = 0.1, 0.5 and 0.7, the distributions arecompared with those obtained in Ref. [15] applying the DSMC method. It can be seen that the densitydistributions obtained here are in a good agreement with the DSMC results in all cases. The temperature
516 I. A. Graur et al. ZAMP
xy
0 5 10 150
1
2
3
4
n
x
y
0 5 10 150
1
2
3
4
T
x
y
0 5 10 150
1
2
3
4
0.10 0.12 0.14 0.16 0.18 0.20 0.24
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2ux
Fig. 11. (Color online) Fields of dimensionless density, temperature and bulk velocity for p1/p0 = 0.1 and δ = 100
and bulk velocity distributions are in agreement with the DSMC data near the slit, while far from theslit (x > 5), there is a discrepancy explained by the different computational domains used here and inthe work [15].
A special attention should be paid to the case p1/p0 = 0.1 and δ = 100 when all quantities n, ux
and T have the sharp variations near the point x = 5. In the case of axisymmetrical flows, such sharpvariations are also observed [3,31] and called Mach discs. For the two-dimensional flow considered here,the variations can be called as Mach belts. More detailed information of this phenomenon can be seenin Fig. 11. A zone of silence is observed past from the slit, with the rapidly decreasing density and tem-perature, and the increasing of the flow velocity (see also the axial distributions of these parameters onFig. 6). This zone of silence is followed by the normal shock wave with a strong decreasing of the flowvelocity. The shock wave is characterized by a sharp increase in density and temperature across it. Thezone of silence and the normal shock wave are confined by the barrel shock, almost rethermalized and ofcomparatively high density, with slightly supersonic flow velocity. Just after the barrel shock, the flow isslow.
Vol. 63 (2012) Rarefied gas flow through a slit 517
ux
y
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5(a)
ux
y
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5(b)
ux
y
0
0
0.1
0.1
0.2
0.2
0.3
0.3
0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5(c)
Fig. 12. Profiles [Eqs. (13), (14)] in the slit section (x = 0): a p1/p0 = 0.1, b p1/p0 = 0.5, c p1/p0 = 0.9; dotted line δ = 0.1,dashed line δ = 1, dash-dotted line δ = 10, solid line δ = 100
518 I. A. Graur et al. ZAMP
The velocity profiles in the slit section are shown in Fig. 12. It can be seen that in the hydrodynamicregime for the small pressure drops, the velocity has a minimum at the slit centre. Near the free-molecularregime, the velocity profile is practically constant.
6. Conclusion
The flow of rarefied gas through a thin slit at an arbitrary pressure ratio was calculated applying the dis-crete velocity method to BGK and S-model equations. The flow rate and flow field are reported for wideranges of the gas rarefaction and pressure ratio. The results obtained from the two model kinetic equa-tions were compared between them. It was shown that both model equations provide the same flow ratewithin the numerical uncertainty. The distributions of macroscopic quantities along the x axis obtainedfrom the kinetic equations are in a good agreement between them.
The present results for small pressure differences were compared with whose obtained from the linear-ized BGK model [1]. It has been shown that the linear theory works well in a wider range of the pressuredrop than that indicated earlier in Ref. [1].
For three values of the pressure ratio, the present results were compared with the corresponding dataof Ref. [15]. The maximum discrepancy of the flow rate was 4% in the transitional regime that exceedsthe accuracy of the numerical data.
The streamlines are qualitatively different for each pressure ratio in the hydrodynamic regime (δ =100), while they are similar in the transitional regime (δ = 1).
The formation of Mach belt structures past the slip was observed for near continuum regime and thelarge pressure difference.
Acknowledgments
The present work has been done in the frame of the European Community’s Seventh Framework Pro-gramme (ITN—FP7/2007-2013) under grant agreement n 215504. One of the authors (F.Sh.) acknowl-edges Programme Action en Region de Cooperation Universitaire et Scientifique (ARCUS, France) forthe support of his short-term visit to Provence University. The authors thank the IDRIS (Institute forDevelopment and Resources in Intensive Scientific computing) for the supercomputing resources underthe projet number i2010022168. We thank Yann Jobic for his help in the parallelization of the numericalcode.
References
1. Sharipov, F.: Rarefied gas flow through a slit: influence of the gas-surface interaction. Phys. Fluids 8(1), 262–268 (1996)2. Sharipov, F.: Rarefied gas flow into vacuum through a thin orifice. Influence of the boundary conditions. AIAA
J. 40(10), 2006–2008 (2002)3. Sharipov, F.: Numerical simulation of rarefied gas flow through a thin orifice. J. Fluid Mech. 518, 35–60 (2004)4. Liepmann, H.W.: Gas kinetics and gas dynamics of orifice flow. J. Fluid Mech. 10, 65–79 (1961)5. Willis, D.R.: Mass flow through a circular orifice and a two-dimensional slit at high Knudsen numbers. J. Fluid Mech.
21, 21–31 (1965)6. Stewart, J.D.: Mass flow rate for nearly-free molecular slit flow. J. Fluid Mech. 35(3), 599–608 (1969)7. Akinshin, V.D., Seleznev, V.D., Sharipov, F.M.: Non-isothermal rarefied gas flow through a narrow slit. Izvestia Aka-
demii Nauk SSSR, Mekhanika Zhidkosti i Gaza 4, 171–175 (1990) [in Russian]. Translated in Fluid Dyn. 25(4), 642–645(1990)
8. Hasegawa, M., Sone, Y.: Rarefied gas flow through a slit. Phys. Fluids A 3(3), 466–477 (1991)9. Chung, C.H., Jeng, D.R., Witt, K.J.De , Keith, T.G. Jr..: Numerical simulation of rarefied gas flow through a slit.
J. Thermophys. Heat Transf. 6(1), 27–34 (1992)
Vol. 63 (2012) Rarefied gas flow through a slit 519
10. Cercignani, C., Sharipov, F.: Gaseous mixture slit flow at intermediate Knudsen numbers. Phys. Fluids A 4(6), 1283–1289 (1992)
11. Wadsworth, D.C., Erwin, D.A.: Numerical simulation of rarefied flow through a slit. Part I: Direct simulation MonteCarlo results. Phys. Fluids A 5(1), 235–242 (1993)
12. Sharipov, F.: Non-isothermal rarefied gas flow through a slit. Phys. Fluids 9(6), 1804–1810 (1997)13. Sharipov, F., Kozak, D.: Rarefied gas flow through a thin slit into vacuum simulated by the Monte Carlo method over
the whole range of the Knudsen number. J. Vac Sci. Technol. A 27(3), 479–484 (2009)14. Graur, I., Polikarpov, A.P., Sharipov, F.: Numerical modelling of rarefied gas flow through a slit into vacuum based on
the kinetic equation. Comput. Fluids (2011). doi:10.1016/j.compfluid.2011.05.00115. Sharipov, F., Kozak, D.: Rarefied gas flow through a thin slit at an arbitrary pressure ratio. Eur. J. Mech. B/Fluids
30(5), 543–549 (2011)16. Bhatnagar, P.L., Gross, E.P., Krook, M.A.: A model for collision processes in gases. Phys. Rev. 94, 511–525 (1954)17. Shakhov, E.M.: Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3(5), 95–96 (1968)18. McCormack, F.J.: Construction of linearized kinetic models for gaseous mixture and molecular gases. Phys. Fluids
16, 2095–2105 (1973)19. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press, Oxford (1994)20. Loyalka, S.K., Hamoodi, S.A.: Poiseuille flow of a rarefied gas in a cylindrical tube: solution of linearized Boltzmann
equation. Phys. Fluids A 2(11), 2061–2065 (1990)21. Tcheremissine, F.G.: Conservative evaluation of the Boltzmann collision integral in discrete ordinates approxima-
tion. Comput. Math. Appl. 35, 215–221 (1998)22. Aristov, V.V.: Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer, Dordr-
echt (2001)23. Sharipov, F., Bertoldo, G.: Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular
potential. J. Comput. Phys. 228(9), 3345–3357 (2009)24. Sharipov, F., Bertoldo, G.: Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones
potential over a wide range of the Knudsen number. Phys. Fluids 21, 067101.1–067101.8 (2009)25. Graur, I., Sharipov, F.: Non-isothermal flow of rarefied gas through a long pipe with elliptic cross section. Microfluid.
Nanofluid. 6(2), 267–275 (2008)26. Titarev, V.: Numerical method for computing two-dimensional unsteady rarefied gas flows in arbitrarily shaped
domains. Comput. Math. Math. Phys. 49(7), 1197–1211 (2009)27. Naris, S., Valougeorgis, D.: The driven cavity flow over the whole range of the Knudsen number. Phys. Fluids
17(9), 097106.1–097106.12 (2005)28. Sharipov, F., Seleznev, V.: Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data 27(3), 657–706 (1998)29. Sharipov, F.: Data on the velocity slip and temperature jump on a gas-solid interface. J. Phys. Chem. Ref. Data
40(2), 023101.1–023101.28 (2011)30. Sharipov, F.M., Subbotin, E.A.: On optimization of the discrete velocity method used in rarefied gas dynamics.
Z. Angew. Math. Phys. (ZAMP) 44, 572–577 (1993)31. Graur, I.A., Elizarova, T.G., Ramos, A., Tejeda, G., Fernandez, J.M., Montero, S.: A study of shock waves in expanding
flows on the basis of spectroscopic experiments and quasi-gasdynamic equations. J. Fluid Mech. 504, 239–270 (2004)
Irina A. Graur and Alexey Ph. PolikarpovEcole PolytechniqueUniversitaire de MarseilleUMR CNRS 6595Universite de Provence5 rue Enrico Fermi13453 MarseilleFrancee-mail: [email protected]
Alexey Ph. PolikarpovDepartment of PhysicsUral Federal University620083 EkaterinburgRussiae-mail: [email protected]
