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A novel modi ﬁ ed lattice Boltzmann method for simulation of gas ﬂ ows in wide range of Knudsen number☆

A. Homayoon

^a

, A.H. Meghdadi Isfahani

^b,

⁎ , E. Shirani

^a

, M. Ashra ﬁ zadeh

^a

aDepartment of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran

bIslamic Azad University, Najafabad branch, Isfahan, Iran

a b s t r a c t a r t i c l e i n f o

Available online 21 March 2011 Keywords:

Lattice Boltzmann method Micro and nanoﬂows Knudsen minimum effect Rarefaction

To cover a wide range of theflow regimes, a new relaxation time formulation by considering the rarefaction effect and the effective dynamic viscosity has been obtained. By using the modified lattice Boltzmann method (LBM), pressure drivenflow through micro and nano channels has been modeled for wide range of Knudsen number, Kn, covering the slip, transition and to some extent the free molecular regimes. The results agree very well with existing empirical and numerical data. The velocity profile was predicted as well as the volumetric flow rate and for thefirst time, the well known Knudsen minimum effect has been captured about Kn = 1.

1. Introduction

Fluidflow plays a major role in micro and nano devices Therefore, it has attracted the interests of the computational fluid dynamic researchers significantly. To categorizeflow regimes, a nondimensional number known as the Knudsen number (Kn) is defined. It is the ratio of the molecular mean free path to aflow geometric characteristic length. For 0.01bKnb0.1,flow can be assumed as continuous, but slip velocities appear on solid walls. For the transition regime (KnN0.1), the continuity assumption and consequently the validity of the Navier Stokes Equations, NSE, is questionable as the size is reduced significantly. In such cases, because of the solid walls effect, thefluid flow behavior depends strongly on the geometry dimensions[1].

Molecular based methods such as Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC) methods[2]maybe used to model micro and nanoﬂows for a wide range of the Knudsen numbers[3,4].

However, the computational cost of these molecular based methods is prohibitively high such that they cannot be used for practicalﬂuidﬂow simulations in micro and nano scales except for free molecular regimes.

A midway approach for theflow simulation in small scales is the lattice Boltzmann method (LBM). Because the LBM is a mesoscopic method which can be considered as a particle based method and at the same time is independent of the actual number of molecules, it requires less computational resources for low Knflows. Moreover the lattice Boltzmann equation (LBE) is a more fundamental equation compared to the NSE, which is valid for all ranges of Knudsen number [5]. Therefore, the LBM can be used to simulatefluid flows in all regimes upon appropriate adjustments[6].

Recently there have been attempts to use the LBM for gaseousflows in slipflow regime[7–13]but only a few papers can be mentioned for the use of LBM in transition regime [14–20]. To this end, two methods are proposed based on the use of higher order LBM [14–17] and the modification of the mean free path[17–20]. The multi-speed or higher order LBM has been developed to increase the order of accuracy in the discretization of velocity phase space. Although Ansumali, et al.[16]have demonstrated that the high order LBMs have improved current capability, Kim et al.[17]showed that this method can predict the rarefaction effects only for Kn=O(0.1)and at large Kn, the mass flow rate cannot be predicted properly by these methods. Additionally, the high-order LBMs with large numbers of discrete velocities are not numerically stable[21].

On the other hand, for high Knﬂows that the mean free path,λ, becomes comparable with the channel dimensions, the wall boundaries reduce the local mean free path. Therefore, by using a geometry dependent local mean free path, Tang et al. [20] captured the nonlinear high order rarefaction phenomena, but this local mean free path is complicated and cannot be used for complex geometries such as porous media.

In this article, by relating the viscosity to the local Kn, a generalized diffusion coefﬁcient is obtained in such a way that wide range of Kn regimes ofﬂow can be simulated more accurately.

2. The LBM

The continuum Boltzmann equation is a fundamental model for rareﬁed gases in the kinetic theory[22,23]. It is an integro-differential equation in which the collective behavior of molecules in a system is used to simulate the continuum mechanics of the system. The Boltzmann equation is written as follows:

∂tf+→ξ:∇

f =Q f ;f′

ð1Þ International Communications in Heat and Mass Transfer 38 (2011) 827–832

☆ Communicated by W.J. Minkowycz.

⁎ Corresponding author.

E-mail address:amir_meghdadi@pmc.iaun.ac.ir(A.H.M. Isfahani).

doi:10.1016/j.icheatmasstransfer.2011.03.007

Contents lists available atScienceDirect

International Communications in Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

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wheref t;→x;→ξ is the probability offinding a molecule at position→x with velocity of→ξat time t, andQ(f,f′)Q(f,f′) is a nonlinear collision integral term withfandf′being the pre and post collision probability distribution functions, respectively. Due to the complicated nature of the collision integral, direct solution of the Boltzmann equation is very difficult except forKn→∞, whereQ(f,f′) vanishes[24]. In order to solve the Boltzmann equation, Bhatnagar, Gross and Krook[25]proposed a simplified model for the collision term known as the BGK model:

Q f ;f′

= 1 λf−f^eq

ð2Þ where λ is the relaxation time, f^eq is the equilibrium distribution function approximated as Maxwellian form given by Eq.(3):

f^eq= ρ 2πRT

ð Þ³² exp −→ξ−→u2

2RT 0 B@

1

CA ð3Þ

whereRis the gas constant andρ,→uandTare density, velocity and temperature, respectively. The density and velocity are computed using the moments of the distribution function as below:

ρ=∫fd→ξ

ρ→u=∫f→ξd→ξ: ð4Þ

In the LBM, the BGK–Boltzmann equation is discretized in momentum and physical spaces as well as in time. The evolution of the discrete particle distribution functionfi→x;t

is determined using the following equation:

f_i→x+→c_iΔt;t+Δt

−fi→x;t

=1

2 Q_i→x+→c_iΔt;t+Δt

+Q_i→x;t

: ð5Þ In the present work, the nine velocity 2D model (D2Q9) (Fig. 1), is used to discretize momentum space for the present calculations[26].

In this model, the discrete velocityﬁeld→ci= cix;ciy

is:

→c_i= 0;0 ð Þ; i= 0

cos i−1 2 π

; sin i−1 2 π

c; i= 1;…;4 ffiffiffi2

p cos 2i−9 4 π

; sin 2i−9 4 π

c; i= 5;…;8 8>

>>

><

>>

:

ð6Þ

where c=Δx/Δt. Δx and Δt are lattice spacing and time step, respectively. Using BGK collision operator, (Eq.(5)) yields to:

f_i→x+→c_iΔt;t+Δt

−f_i→x;t

=−1 τ f_i→x;t

−f_i^eq→x;t

h i

ð7Þ where τ=λ/Δt is the nondimensional relaxation time and fieqis the discrete equilibrium distribution function. This function can be

calculated through a second order Taylor series expansion of the Maxwell distribution function, Eq.(3), followed by a discretization of the result on the D2Q9 lattice[27], which results in:

f_i^eq→x;t

=ρ:w_i 1 + →u:→c_i RT +

→u:→c_i 2

2ð ÞRT ² →u:→u 2RT 0

B@

1 CA

w₀= 4

9;w_i_{= 1}_;₂_;₃_;₄=1

9;w_{i= 5}_;₆_;₇_;₈= 1 36:

ð8Þ

Theﬂow parameters are then calculated in terms of the particle distribution functionfi→x;t

, by:

ρ=∑⁸

i= 0

fi ; →u=1 ρ∑⁸

i= 0

→cifi

P=ρc²_s ; c_s²=RT= c² 3

ð9Þ

wherecsis the lattice speed of sound andPis the pressure. In the standard LBM, the kinematic viscosityυis shown to be proportional to the relaxation time, as given below and is a constant value.

υ=c_s²δtðτ−0:5Þ ð10Þ

3. LBM for high Knudsen numberﬂows

From the Direct Simulation Monte Carlo, DSMC, method and the linearized Boltzmann equation [28], it is evident that the velocity profiles of flow in a channel in the transition and free molecular regimes remain approximately parabolic. But the velocity profile obtained from continuum based relations does not predict theflow rate properly [29]. This is because of the fact that the dynamic viscosity which is related to the diffusion of momentum due to the intermolecular collisions must be modified to consider the diffusion of momentum due to the intermolecular collisions and the collision of molecules with the walls. In the transition regime, because of the rarefaction, intermolecular collisions and molecule–wall collisions have the same order and in the free molecular regime, the molecule– wall collisions are the dominant phenomenon.

From the gas dynamics theorem, it is found that the mean free path of molecules is directly proportional to the dynamic viscosity and is inversely proportional to the gas pressure, e.g. λ=μ=p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

πRT=2

p ,

whereμis the viscosity, p is the gas pressure, R is the gas constant, and T is temperature. Thus reduction of the mean free path can reduce the gas dynamic viscosity[30]. Therefore, a general form of the effective viscosity as a function of the Knudsen number has been proposed[29], which is valid for a wide range ofﬂow regimes.

μeff = μ

1 +f Knð Þ ð11Þ

Several models have been proposed for the rarefaction coefﬁcientf (Kn)[31], which are based on numerical and analytical data and cover the entire range of the Knudsen number. Polard and Present[32]

proposed a rather simple model for generalized diffusion coefﬁcient which setsf(Kn) =αKn. Karniadakis and Beskok[29]suggested the value,α= 2.2 in their effort to match their numerical results for the massﬂow rate in a channel with the corresponding DSMC results.

To model the rarefaction effect in the transition regime, kinematic viscosity is modiﬁed as given below:

υeff= μeff

ρeff

=υ 1

1 +f Knð Þ

ð Þ ρ

ρeff

!

: ð12Þ

One should note thatυand ρin Eq.(12) are the macroscopic kinematic viscosity and density of theﬂuid, whereas the quantities Fig. 1.Conﬁguration of the lattice and discrete velocity vectors, D2Q9 model.

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with theeffsubscript represent the effective values at the particular Knudsen number. Considering the effective viscosity and density, we have deﬁned a new effective relaxation time in the LBM as follows:

τeff = 0:5 + 1 1 +f knð Þ

ð Þ ρ

ρeff

! τ−0:5

ð Þ ð13Þ

where τ= 0:5 + ffiffiffiffiffiffiffiffiffiffi 6=π

p KnH is the nondimensional relaxation time andHis a characteristic length (e.g. height of the micro channel)[33].

4. Boundary conditions in LBM

In the present work, the slip reﬂection boundary condition (SRBC) [34]which is a combination of the bounce back and specular boundary conditions is used to predict the slip velocity on solid walls. For example, at the lower boundaries, the unknown particle distribution functions f2,f5andf6, can be calculated as follows:

f₅ðx;0Þ f₂ðx;0Þ f₆ðx;0Þ 2 4

3 5=

s 0 1−s

0 1 0

1−s 0 s 2

4

3

5 f₇ðx+δx;δyÞ f₄ðx;δyÞ f₈ðx−δx;δyÞ 2

4

3

5 ð14Þ

wheresis the slip-reﬂection coefﬁcient on the solid wall. In Eq.(14), one obtains the bounce back boundary condition whens= 1 and specular boundary condition whens= 0. It can be shown that this boundary

condition is related to the relaxation time and slip-reﬂection coefﬁcient as follows[6,34,35]:

U_s−U_w ð Þ∝1−s

s τ ∂U

∂n ð15Þ

whereUsis slip velocity on solid wall,Uwis the solid wall velocity andnis the normal direction to the wall. Considering the fact that in standard LBM, the relaxation time is directly related to the Knudsen number, the proposed boundary condition is equivalent to the ﬁrst order slip boundary condition[21].

τ∝Kn⇒ ðU_s−U_wÞ∝1−s s Kn ∂U

∂n ð16Þ

In this paper, the relaxation time forﬂows with high Knudsen number is modiﬁed as given in Eq.(13). Therefore, the slip velocity on solid walls can be stated as a function of Knudsen number.

τeff∝ Kn 1 +f Knð Þ

⇒ ðU_s−U_wÞ∝1−s s

Kn 1 +f Knð Þ

∂U

∂n

ð17Þ

Several slip boundary conditions have been used in solving the NSE, among which, the ﬁrst and second order Maxwell boundary

Fig. 2.Normalized velocity distribution across the micro/nano channel (α= 2.2).

A. Homayoon et al. / International Communications in Heat and Mass Transfer 38 (2011) 827–832

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conditions as well as the general slip velocity model given in Eq.(18) can be mentioned.

U_s−U_w= 2−σv

σv

Kn 1−B Knð ÞKn

∂U

∂n ð18Þ

whereσvis the tangential momentum accommodation coefficient and B(Kn)is a semi empirical function of Knudsen number which is determined by curvefitting of results obtained using other numerical methods for flows with Knudsen numbers from zero to infinity [29,36]. Although Eq.(18)isfirst order in velocity gradient, but by Fig. 2(continued).

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using an appropriate form forB(Kn) and taking a Taylor series for the Knudsen function, one can show that it is equivalent to higher order boundary conditions[36]. By comparing Eqs.(17) and (18), it can be realized that both forms are equivalent if an appreciate function is used forf(Kn). The other issue that can be found by comparing Eqs.(17) and (18)is the relationship between slip reflection coefficient and the tangential momentum accommodation coefficient as σv= 2s[36]. Hence, by using the new relaxation time, the form of the slip boundary condition used in the proposed LBM will be equivalent to the higher order boundary conditions used elsewhere, and consequently, higher Knudsen numberflows can be simulated.

5. Results and discussion

Fully developed pressure driven flows in micro channels with different Knudsen numbers are simulated as a benchmark case for validating present work. InFig. 2the nondimensional velocity profiles normalized by local average velocity which is obtained in the transition flow regime for K= 0.1 to 10 is plotted across the channel, where K is ffiffiffi

pπ

=2

Kn. The corresponding linearized Boltzmann solution is also included[28]. It is seen that the LBM velocity proﬁle and the linearized Boltzmann solution agree quiet well through the Knudsen numbers up to 10. However, a little discrepancy appears very close to the walls. It is because of the fact that the intermolecular forces between ﬂuid molecules and solid walls become important near the walls, in the transition regime[37].

Fig. 3shows the normalizedflow rate through micro/nano channel as a function of the Knudsen number. It can be seen that by increasing the Knudsen number, the flowrate decreases initially and has a minimum value forKn≈1, then it increases[38,39]. This phenomenon is called Knudsen minimum effect[40]. Although from the kinetic theory of gases, there is no analytical formula for flow rate in the transition regime, there are two asymptotes for normalizedflow rate which is evident in Eq.(19) [23].

Q₀= 6knð Þ⁻¹+s+ 2s ²−1

Kn Knb1 Q_∞= 1= ffiffiffi

pπ

ln Knð ÞKn→∞ ð19Þ

wheres= 1.015.

It can be seen from theﬁgure that our results are in a good agreement with those of Eq.(19)as well as the linearized Boltzmann method [39]. InFig. 3, the results of the standard LBM, are also presented.

These results agree very well forflows with Kn less than 0.1, which corresponds approximately to the slipflow regime, but the predicted flow rate is overestimated in the early transitionflow regime for KnN0.4 and diverges from results of the linearized Boltzmann method[41,42].

However, the present results by modifying LBM are accurately com- patible with linearized Boltzmann method in the entire transitionﬂow regime. As shown inFig. 3, the Knudsen minimum effect is captured for Kn≈1 by using our new model. Noticeably, this interesting agreement is achieved without incorporating any kind of complex and adjustable slip models.

6. Conclusion

The new LBM is capable of simulating theflow for a wide range of Knudsen numbers including the transition regime. It is shown that the proposed model by modifying the relaxation time in LBM, is able to predict theflow features in micro and nano scales for wide range of Kn, accurately. Nondimensional velocity distribution andflowrates are in good agreement with the existing analytical data and the well known Knudsen minimum effect in micro and nano channels are achieved forKn≈1. These results are obtained without incorporating any kind of adjustable slip models.

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