Physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation for microgas flows

Transcription

1 JOURNAL OF APPLIED PHYSICS 99, Physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation for microgas flows Zhaoli Guo a National Laboratory of Coal Combustion, The Huazhong University of Science and Technology, Wuhan, , People s Republic of China, and Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China T. S. Zhao b and Yong Shi Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China Received 21 September 2005; accepted 8 February 2006; published online 5 April 2006 In this paper, we study systematically the physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation LBE for microgas flows in both the slip and transition regimes. We show that the physical symmetry and the spatial accuracy of the existing LBE models are inadequate for simulating microgas flows in the transition regime. Our analysis further indicates that for a microgas flow, the channel wall confinement exerts a nonlinear effect on the relaxation time, which should be considered in the LBE for modeling microgas flows American Institute of Physics. DOI: / I. INTRODUCTION Gas flows in microdevices have received particular attention over the last decade with the rapid development in microelectromechanical systems MEMS. 1 3 Owing to the small size of the devices, a microgas flow usually posses a relatively large Knudsen number, Kn= /h, where represents the mean free path of the gas and h is the characteristic length of the flow domain. Under such a circumstance, the continuum assumption may break down, and the hydrodynamic equations based on the continuum assumption, such as the Navier-Stokes equations NSEs, will fail to work for such flows. Indeed, many previous theoretical and experimental studies have revealed that the microgas flow behavior in a microdevice may deviate significantly from macrosized flows. 3 Up to date, the most widely used method in the study of microgas flows is the direct simulation Monte Carlo DSMC. 4 However, the DSMC usually suffers from the expensive computational cost and highly statistical noise in simulating low speed flows. Alternatively, the gas kinetic theory has also been used to study microgas flows since the work by Maxwell. 5 It is widely accepted that the Boltzmann equation BE is applicable to gas flows in a wide range of Kn, ranging from the continuum Kn limit to the free molecule flows Kn 10. Actually, it has been demonstrated that the results predicted by the linearized Boltzmann equation are indeed in agreement with DSMC and experimental results for some simple flows. 6,7 Recently, some numerical methods based on the gas kinetic theory have been proposed to study the gas flows at finite Knudsen numbers Among these mesoscopic methods, the lattice Boltzmann equation LBE method has a Electronic mail: zlguo@hust.edu.cn b Author to whom correspondence should be addressed; electronic mail: metzhao@ust.hk received particular attention over the last few years, mainly because of its mesoscopic nature and the distinctive computational features. However, before applying the LBE to microgas flows, it is essential to gain better understanding of the fundamental features of the LBE and particularly, it is important to examine the limitations of the LBE for large Knudsen numbers. It is understood that the BE can be used to describe microgas flows in a wide range of Knudsen numbers. However, this may not be the case for the LBE, as the LBE is a special numerical scheme for solving, or an approximation to the original BE. 38 Therefore, the question whether the LBE can be applied to microgas flows depends on the discretization accuracy and other relevant procedures in deriving the LBE from the BE. The objective of the present study is to investigate systematically the effects of large Knudsen numbers and wall confinements associated with microgas flows on the LBE. Specifically, we shall address three fundamental issues of the LBE in simulating microgas flows, i.e., the physical symmetry, spatial accuracy, and relaxation time. The analysis indicates that the existing LBE models, in their present forms, are incapable of capturing the flow behavior in the Knudsen layer, and particularly, these models are not yet accurate enough for modeling microgas flows in the transition regime. The rest of the paper is organized as follows. First, we discuss briefly the Boltzmann equation in Sec. II. We then study the effect of wall confinement on the relaxation time of gas flows in Sec. III. In Sec. IV, we revisit the LBE and analyze its limitations for large-knudsen number flows. In Sec. V, we present a LBE model with an effective relaxation time that incorporates wall confinement for microgas flows in the slip regime, followed by the numerical test results in Sec. VI. Finally, the conclusions drawn in this work are presented in Sec. VII /2006/99 7 /074903/8/$ , American Institute of Physics

2 Guo, Zhao, and Shi J. Appl. Phys. 99, II. THE BOLTZMANN EQUATION In kinetic theory, the dynamic evolution of a gas flow can be described by the Boltzmann equation, f + f = f, f, t 1 where f x,,t is the single-particle distribution function at time t and position x for particles moving with velocity, and is the collision operator. Hydrodynamic variables, such as the density, velocity u, and temperature T, are defined as the velocity moments of the density distribution function, = fd, u = fd, D 2 RT = C2 fd, 2 2 where D is the spatial dimension, R is the gas constant, and C= u is the peculiar velocity. The collision term in Eq. 1 is rather complicated and is usually simplified in practical calculations. One such a approximation is the singlerelaxation time or Bhatnagar-Gross-Krook BGK model, 26 = 1 f f eq, 3 P 2 = p 2 A 1 us + A 2 DS 2 u S + A 3 S S, where p= RT is the pressure, = p is the shear viscosity; S= u+ u T /2, and D= t +u. The tilde over a tensor denotes its traceless part, e.g., S =S D 1 S. The coefficients A 1, A 2, and A 3 are some constants dependent on the relaxation time. As the Knudsen number becomes large, however, the validity of the Chapman-Enskog method is questionable. Nevertheless, we can still gain some insights from the high order results of the formal Chapman-Enskog expansions. As can be seen later, the pressure tensor of the LBE indeed matches that of the Boltzmann equation at the zeroth Euler and first Navier-Stokes orders, but deviates significantly at the second and higher Burnett and super-burnett orders, in terms of the Chapman-Enskog expansion, which means that the LBE deviates from the original Boltzmann equation for large Knudsen number flows. III. THE RELAXATION TIME FOR MICROFLOWS The relaxation time in the BGK model is a key parameter. In general, relates to the mean free path by = c *, 8 9 where is the relaxation time, and f eq = 2 RT exp 2RT C2 4 where c * is a certain averaged value of the particle velocity. Alternatively, can also be defined in terms of the viscosity as is the Boltzmann-Maxwell distribution function. Multiplying the Boltzmann equation 1 by the summational invariants = 1,,C 2 /2 and integrating over the velocity space, we can obtain the hydrodynamic equations with the pressure tensor P and the heat flux vector q given by P CCfd, q 1 2 C2 Cfd. 5 When the Knudsen number is small, the Boltzmann equation can be solved using the Chapman-Enskog method, 27 in which the distribution function f is expressed as an infinite series of the Knudsen number Kn: f = f 0 +Knf 1 +Kn 2 f 2 +, where f 0 = f eq. In this way, P and q can be approximated at various orders, P=P 0 +KnP 1 +Kn 2 P 2 + and q=q 0 +Knq 1 +Kn 2 q 2 +, where P k CCf k d, q k 1 2 C2 Cf k d. 6 For the BGK model, the pressure tensor and heat flux vector at the zeroth through second orders can be evaluated explicitly, 28 which correspond to the hydrodynamic equations at the Euler, Navier-Stokes, and Burnett levels, respectively. For example, for an isothermal flow the pressure tensor at these orders can be expressed as and P 0 = pi, P 1 = 2 S, 7 = p. 10 The viscosity-based relaxation time defined in Eq. 10 facilitates the incorporation of the effect of the intermolecular potential through the viscosity. From Eqs. 9 and 10, we can obtain the averaged velocity c * as c * = p. For hard-sphere gases, the kinetic theory gives that, 27 = m 2 2, 5m = RT c, where m is the molecular mass and is the molecular diameter, and c = 8RT/ is the mean molecular velocity. Therefore, the averaged velocity c * is c * = 2RT c = RT It is important to recognize that the mean free path and viscosity given by Eq. 12 are only valid far away from the confined walls. In a microgas flow system confined between solid walls, however, the mean free path of the fluid molecules is significantly influenced by the presence of the walls, and thus the relaxation time determined this way is questionable. It is known from the gas kinetic theory that the free path of a gas molecule follows a distribution of the form, 28

3 Guo, Zhao, and Shi J. Appl. Phys. 99, * = Kn. 19 It is worth mentioning that some heuristic effective viscosity models have been proposed in previous studies for microgas flows from different viewpoints, 8,30 32 which can also be expressed in the form of Eq. 19 with different forms of. It is interesting that with these effective viscosity models, the Navier-Stokes equations can even capture the high-knudsen number effects in the transition regime. FIG. 1. Color online The function against the Knudsen number Kn. p r = r exp r, 14 which means that the probability of a given molecule travels a distance r and then has a collision in dr is p r dr. For an unbounded system, the mean free path of all molecules is just N 0 p r dr/n=, where N is the total number of the molecules in the system. When a wall is included in the gas system, however, some molecules will hit the wall and their flight paths will be terminated by the wall. Therefore, the mean free path of one such molecule will be 0 a p r dr, where a is a finite positive valule. As a result, the mean free path of the total molecules in the system, *, will be smaller than the corresponding in an unbounded system with Kn 0. Formally, we can express * as * = Kn. 15 Here Kn is still the conventional Knudsen number without considering the wall effect; is a monotonous function of Kn and satisfying lim Kn 0 Kn =1. 16 Stops 29 has derived the expression of for a gas flow system confined between two infinite parallel walls. The function is very complicated and is difficult for practical applications. For simplicity, we approximate Stops with x = 2 arctan 2x 3/4. 17 As seen from Fig. 1, Eq. 17 well fits the original function over a wide range of Kn. If we assume that the averaged velocity c * is not influenced by the presence of the wall this approximation is reasonable by noticing that c * depends on the system temperature only, the bounded relaxation time * for the bounded system can be determined from Eq. 9, i.e., * = Kn. 18 According to the relation between the viscosity and the relaxation time, Eq. 18 essentially defines an effective viscosity, IV. LATTICE BOLTZMANN EQUATION The lattice Boltzmann equation is a discrete version of the Boltzmann equation 1. The discretization involves three parts, 38 i.e., the microscopic molecular velocity, the spatial gradient, and the time derivative. First, the particle velocity space R D is discretized into a finite discrete velocity set c i :i=0,1,...,b 1. The choice of c i depends on the quadrature in evaluating the moments Eq. 2. In general, the quadrature must be accurate enough so that not only the conservation constraints are preserved exactly, but also some necessary symmetries required by macroscopic hydrodynamic equations are retained. 38 This requirement implies that in the Chapman-Enskog expansion, the moments k f m d 20 should be accurately evaluated for k =0,1,...,K and m =0,1,...,M, where K and M are two non-negative integers dependent on the intended hydrodynamic equations. In other words, certain physical symmetry associated with the discrete velocity set must be ensured in order to obtain the desired hydrodynamic equations. 39 Generally, in order to have a correct transport equation for the moment of order k, one must consider its flux, i.e., the moment of order k For instance, if we aim to solve the isothermal Navier-Stokes equations, we must take K=1 and M =1 at least; however, for isothermal Burnett or super-burnett equations, it is necessary to take K 1 and M 2 orm 3. After the velocity space is discretized, the Boltzmann equation with the BGK model leads to the discrete velocity model DVM, f i t + c i f i = i = 1 f i f eq i, 21 where f i x,t =W i f x,c i,t and f eq i =W i f eq c i, with W i being the weight coefficient of the quadrature. It is noted that the nonpolynomial form of the equilibrium distribution function was also used in previous studies. 40 Integrating the DVBE 21 along the characteristic line with a time step t, we obtain f i x + c i t,t + t f i x,t = 0 t i x + c i t,t + t dt. 22 In the standard LBE, the time step t is related to the underlying lattice so that the particles can hop from nodes to nodes on the lattice. In the LBE corresponding to the Navier-Stokes equations, it is necessary and adequate to use the trapezoi-

4 Guo, Zhao, and Shi J. Appl. Phys. 99, dal rule in the evaluation of the integration on the right hand side of Eq. 22, i.e., 0 t i x + c i t,t + t dt = t 2 i x,t + i x + c i t,t + t. 23 However, such a treatment makes Eq. 22 implicit in time. Fortunately, the implicitness can be removed by introducing a transformation, 41 f i= f i t i /2. With this transformation, Eq. 22 can be expressed equivalently as f i x + c i t,t + t f i x,t = t + t /2 f i x,t f i eq x,t, 24 which is just the standard LBE. It is noted that the discrete collision operator i is mass, momentum, and energy conservative, and therefore, the macroscopic variables in the LBE can be computed as = i f i, u = c i f i, i 2 D 2 RT = C i i 2 f i. 25 For isothermal flows, T is a constant and does not need to be computed. For simplicity, hereafter we will omit the overbar from the transformed distribution function. We now address some fundamental issues about the LBE as follows. a b c d Remark I. Up to date, most of the existing LBE models have been constructed for the Navier-Stokes equations. It should be recognized that the discrete velocity set c i :i=0,1,,b 1 in these models can ensure the accurate evaluation of the moments for f 0 and f 1, but not for f 2 or other higher order parts of the distribution function. In other words, the physical symmetry of the discrete velocities, or the accuracy of the velocity discretization, is not adequate such that the DVM 21 matches the Boltzmann equation 1 at the higher than Navier-Stokes orders. Remark II. The scheme 22 or 24 is time accurate in the discretization of the streaming derivation on the left hand of Eq. 21, but the trapezoidal rule used in the evaluation of the collision term is only of second order accuracy, which in fact assumes that i is linear in the cell x,x+c i t. Therefore, the LBE is of second order accuracy in space. This fact is also evident from an alternative viewpoint: the LBE 24 can in fact be viewed as a finite-difference scheme of Eq. 21, by discretizing the temporal derivative with the explicit Euler scheme and the spatial gradient with the first order upwind scheme. 39 Fortunately, the first order numerical error can be absorbed into the physical viscosity, so that the resultant LBE becomes a second order scheme. The second order accuracy of the LBE indicates that the third order and higher order spatial derivatives of the macroscopical variables, which appears in the Burnett or super-burnett equations, cannot be captured by the LBE. Therefore, in addition to the insufficient accuracy or symmetry in the velocity discretization, the accuracy in the spatial discretization of the available LBE models is also inadequate for describing the large-knudsen number flows. Remark III. Actually, the formal high order dynamics of the LBE has been analyzed. 42 For the isothermal LBE models, the formal stress tensor at the Burnett order depends only on the second order spatial derivatives, 2 P LBE = B, 26 where B is a parameter dependent on the relaxation time. It is clearly seen that the second order stress tensor of the LBE are quite different from that of the continuous Boltzmann equation See Eq. 8 : the spatial changes in the flow velocity disappear; particularly, for nearly incompressible flows this is, in fact, the 2 focus of the LBE method, P LBE 0, which means that the second order contributions are completely lost. In 2 fact, P LBE given by Eq. 27 is nothing but a truncation error in the LBE for solving the Navier-Stokes equations, and has no physical significance. This fact again confirms the statement that the LBE method in its present form, is not suitable for simulating the gas flows with relative large Knudsen numbers in theory. Remark IV. In the region of a gas flow close to the solid surface there always exists a so-called Knudsen layer, whose thickness is of order. In this region, the quasithermodynamic-equilibrium assumption, upon which the Navier-Stokes equations depend, does not hold any more because there are insufficient molecularmolecular and molecular-surface collisions over this very small scale. Therefore, the Navier-Stokes equations become inappropriate within the Knudsen layer anymore. Since all of the existing LBE models are, in fact, a numerical solver for the Navier-Stokes equations, it is unrealistic to expect the LBE to capture the Knudsen layer. This situation will be more serious for microgas flows, in which the Knudsen layer usually takes up a large portion of the whole flow domain. This fact, again, indicates that the existing LBE models in their present form are inappropriate for microgas flows. V. BOUNDARY CONDITION AND RELAXATION TIME IN THE LBE FOR MICROFLOWS As analyzed above, most of the previous LBE models, if not all, are designed for the Navier-Stokes equations, and the contributions from the high order dynamics cannot be described correctly. However, it is generally agreed that the Navier-Stokes equations can still be used to describe the bulk flow in the slip regime where Kn is in the range of , provided that an appropriate slip boundary condition is specified. Based on this belief, the LBE has recently been extended to simulate gas flows in the slip regime In order to extend the LBE to microflows in the slip regime, two fundamental issues must be addressed, namely, how to specify the slip boundary condition, and how to relate

5 Guo, Zhao, and Shi J. Appl. Phys. 99, the relaxation time with the Knudsen number Kn. In previous studies, some different approaches have been proposed for this purpose. A. Boundary conditions There are two ways to implement the slip boundary condition in the LBE. One rule is to use the combination of the no-slip bounce-back scheme and the free-slip specularreflection sheme, 14 that is f i x w = 1 f sr i x w + f bb i x w, c i u w n 0, 27 where 0 1 is the combination coefficient, f bb sr i and f i are the distribution functions at the wall node x w determined by the bounce-back and specular-reflection rules, respectively, f bb i x w = f i x w, f sr i x w = f i x w,t, 28 with c i being the opposite velocity of c i c i = c i and c i the specular velocity of c i c i =c i 2 c i u w n n, n is the unit normal vectors. The degree of the slip velocity of this rule depends on the coefficient. Notice that the pure bounce-back and specular-reflection schemes have ever been used in simulations of microgas flows. 10,11 The other way to specify the slip boundary condition in LBE is to use the discrete version of Maxwell s diffuse boundary condition for the continuous Boltzmann equation, 6 which can be expressed as f i x w = 1 f sr i x w + Af eq i x w ; w,u w, 29 for those distribution functions with c i u w n 0, where is the accommodation coefficient, and the parameter A is given by A = cj u w n 0 c j u w n f j x w ck u w n 0 c k u w n f k eq x w ; w,u w. 30 The diffuse rule for the slip boundary condition was first obtained by Ansumali et al. 18 for complete diffusing walls =1, and some other versions were proposed later. 15,16 It is noted that the combination rule 27 and the diffusive rule 29 are quite different in general, although they take similar forms and are the same as = =0. For example, as = =1, the combination rule 27 reduces to the bounce-back rule, which yields the no-slip boundary condition; while the diffusive rule still gives a slip velocity. In this work we prefer to use the diffusive rule since it has a clear physical significance in the kinetic theory. B. The relaxation time and Knudsen number In continuum flows, the relaxation time in the LBE can be determined from the viscosity directly as = / RT, and thus can be given in terms of the Reynolds number of the flow. On the other hand, in microflows the basic characteristic nondimensional parameter is the Knudsen number. Therefore, in order to apply LBE to microflows, one must relate the relaxation time to the Knudsen number. From Eq. 18, we can see that the relaxation time for a bounded system is * = c * = Kn c * Kn H, 31 where c * is given by Eq. 13. It is clear that the wall effect is taken into account in this definition, and owing to the property of the function Eq. 16, this bounded relaxation time satisfies one basic consistency requirement: in the continuum limit Kn 0, the relation between the dynamic viscosity and the relaxation time, = p, is kept exactly. It should be pointed out that in all of the previous applications of LBE for microgas flows, the wall confinement effect on the relaxation time is completely ignored 1, i.e., the relaxation time is related to the Knudsen number as = Kn c H, 32 where c is a certain mean velocity. In the literature, the choice of c is rather diverse. For example, in some works, 13 c takes the mean molecular velocity, c =c = 8RT/, just as in the elementary kinetic theory. 28 Instead, some authors 11,15,17 use the root-mean-square rms velocity c 2 = 3RT. Also, some authors 13,23 choose the viscositydependent average velocity, c =c * = p/. Whatever c is used, however, the basic consistency requirement, i.e., = p as Kn 0, should be fulfilled. Apparently, neither the mean molecular velocity c nor the rms velocity c 2 satisfies this consistency condition. For clarity, we will call the LBE with the unbounded relaxation time given by Eq. 32 as LBE UBRT, while the LBE with the bounded relaxation time * given by Eq. 31 as LBE BRT. VI. NUMERICAL RESULTS In this section we apply the present LBE BRT and previous LBE UBRT to simulate the Poiseuille flow of a hardsphere gas in a long microchannel, and compare the results with the DSMC and the information preservation IP results reported in Ref. 43. The LBE employed here is the isothermal two-dimensional nine discrete velocity D2Q9 model see Ref. 38. The ratio of the length L to the height H of the channel is 100, and the ratio of the pressure at the inlet p i to that at the exit takes 1.4 and 2 in our simulations. Three cases are studied according to the Knudsen number at the exit, i.e., Kn e =0.0194, 0.194, and This problem has also been used to test the validity of the LBE for microgas flows. 43 In computations, a regular lattice is used, which can produce grid-independent results. The two walls are assumed to be fully diffusing =1. The linear extrapolation method 44 is used to specify the pressure boundary condition at the inlet and exit. In Fig. 2, we show the streamwise x direction and spanwise y direction velocities u and predicted by LBE- BRT with Kn=0.194 and p i / p e =2.0. The velocities are normalized by the maximum streamwise velocity u * at the exit. Figure 2 a clearly shows the velocity slip at the walls. It is also seen that the slip increases along the channel. Further-

6 Guo, Zhao, and Shi J. Appl. Phys. 99, FIG. 2. Color online Streamwise a and spanwise b velocities prediced by LBE BRT with Kn=0.194 and p i / p e =2.0. more, unlike in the continuum case, the spanwise velocity is distinct and its magnitude increases along the channel. The distribution of v also shows that the flow migrates from the channel centerline toward the wall as it progresses down the channel. Similar phenomena are also observed for other cases. These observations are in agreement with previous analytical results for microflows in a long channel qualitatively. 45 However, a more detailed analysis of the results reveals some serious problems of LBE in simulating microflows. In Figs. 3 5, the streamwise velocity at the exit and the pressure along the channel are presented for the three cases. Note that here the pressure deviation from the linear pressure distribution p l rather than the pressure itself is shown so that the difference can be seen more clearly. For Kn= and p i / p e =1.4, the flow is in the slip flow regime, and we can observe from Fig. 3 that overall both the velocity profile and the pressure distribution are in good agreement with those of the DSMC and IP methods for both the LBE BRT and the LBE UBRT. However, a small discrepancy in the velocity is still observed near the two walls due to the Knudsen layer effect. The Knudsen layer effect becomes more significant for larger Kn. In Fig. 4, the velocity profile at the exit and the pressure distribution obtained by the LBE URT and LBE- FIG. 3. Color online Streamwise velocity at the exit a and pressure along the channel centerline b. p i / p e =1.4, Kn= DSMC and IP data are taken from Ref. 43. BURT are presented for Kn= It is known from the kinetic theory that the thickness of the Knudsen layer near a solid surface is about It is seen from Fig. 4 a that in the two Knudsen layers near the walls 0 x/h 1.4Kn and 1 1.4Kn x/h 1, the velocity profiles predicted by both LBE UBRT and LBE BRT deviate significantly from the DSMC and IP results, which confirms our statement in the preceding section that LBE is insufficient to capture the flow behavior within the Knudsen layer. However, we can observe that the use of the bounded relaxation time does make slight improvements on the results as compared with the DSMC and IP results. As Kn further increases up to Kn=0.388, the flow falls into the transition flow regime, and the two Knudsen layers will overlap in the central region. It is observed from Fig. 5 that both the velocity profile and the pressure distribution predicted by the both LBE models deviate from the those of the DSMC and IP methods. The portion of the Knudsen layers is so large that the velocities obtained by both the LBE UBRT and LBE BRT deviate from the DSMC and IP results in the whole region. Particularly, it is observed that

7 Guo, Zhao, and Shi J. Appl. Phys. 99, FIG. 4. Color online Same as Fig. 3 except for Kn=0.194 and p i / p e =2. the pressure deviation predicted by the LBE UBRT drifts around the null line, which is contradictive qualitatively to the positive prediction of the DSMC and IP methods. On the other hand, the pressure deviation predicted by the LBE- BRT still agree qualitatively with the DSMC and IP results. These observations indicate that the standard LBE is unfeasible for simulating microgas flows in transition regime; the LBE BRT can yield some improvements, but is still inadequate in this regime although the wall confinement effect is taken into account. VII. SUMMARY We have studied systematically the feasibility of the LBE method for simulating microgas flows. We show that the symmetry of discrete velocity set, or the accuracy in the velocity discretization, is inadequate for the LBE to match the original Boltzmann equation at the second and higher order levels in terms of the Chapman-Enskog analysis. Our analysis also indicates that since the mesoscopic LBE is of second order accuracy in space, it corresponds reasonably to the macroscopic Navier-Stokes equations. However, the LBE is inconsistent with the Burnett/super-Burnett equations FIG. 5. Color online Same as Fig. 4 except for Kn= where the third and higher order spatial changes are involved. Therefore, fundamentally, the LBE in the present form is unsuitable for simulating microgas flows in the transition regime. Even in the slip regime, cautions must also be taken in implementing the LBE. Unlike in the simulation of continuum flows, we must determine the relaxation time from the Knudsen number when we apply the LBE to microgas flows. Although several methods have been proposed in the literature, the consistency requirement for the relaxation time has rarely been considered. Furthermore, in previous studies, the wall effect on the relaxation time has never been considered. The model proposed in this work LBE BRT not only satisfies the consistency requirement but also takes into account the influence of the wall confinement. Numerical tests show that this treatment can improve the accuracy of the LBE for simulating microgas flows with a relative large Knudsen number. In summary, owing to some inherent natures of the LBE method, the existing LBE models, in their present forms, are inadequate for microflows in the transition regime. The mean-free-path correction presented in this paper can remedy

8 Guo, Zhao, and Shi J. Appl. Phys. 99, the inability to some extent, but more fundamental models which can include correct high order hydrodynamics are still desired. ACKNOWLEDGMENTS One of the authors Z.G. is grateful to Professor K. Xu and Professor L.-S. Luo for illuminative discussions. The author also thanks Professor V. Sofonea for bringing Ref. 15 to his attention and sharing Ref. 16 before its publication. This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China Project No. HKUST6101/04E. 1 C. M. Ho and Y. C. Tai, Annu. Rev. Fluid Mech. 30, D. J. Beebe, G. A. Mensing, and G. M. Walker, Annu. Rev. Biomed. Eng. 4, G. E. Karniadakis and A. Beskok, Micro Flows, Fundamentals and Simulation Springer, New York, G. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows Clarendon, Oxford, J. C. Maxwell, Philos. Trans. R. Soc. London, Ser. A 170, C. Cercignani, Mathematical Methods in Kinetic Theory Plenum, New York, Y. Sone, Kinetic Theory and Fluid Dynamics Birkhäuser, Boston, K. Xu, Phys. Fluids 14, L K. Xu, Phys. Fluids 15, ; K. Xu and Z. Li, J. Fluid Mech. 513, ; T. Ohwada and K. Xu, J. Comput. Phys. 201, X. Nie, G. D. Doolen, and S. Chen, J. Stat. Phys. 107, C. Y. Lim, C. Shu, X. D. Niu, and Y. T. Chew, Phys. Fluids 14, A. Agrawal, L. Djenidi, and R. A. Antonia, J. Fluid Mech. 530, G. Tang, W. Tao, and Y. He, Int. J. Mod. Phys. C 15, ; Phys. Fluids 17, S. Succi, Phys. Rev. Lett. 89, V. Sofonea and R. F. Sekerka, J. Comput. Phys. 207, V. Sofonea and R. F. Sekerka, Phys. Rev. E 71, T. Lee and C. L. Lin, Phys. Rev. E 71, S. Ansumali and I. V. Karlin, Phys. Rev. E 66, ; S.Ansumali, I. V. Karlin, C. E. Frouzakis, and K. B. Boulouchos, arxiv.org/abs/cond-mat/ S. Ansumali, C. E. Frouzakis, I. V. Karlin, and I. G. Kevrekidis, arxiv.org/abs/cond-mat/ X. Niu, C. Shu, and Y. T. Chew, Europhys. Lett. 67, F. Toschi and S. Succi, Europhys. Lett. 69, M. Sbragaglia and S. Succi, Phys. Fluids 17, Y. Zhang, R. Qin, and D. R. Emerson, Phys. Rev. E 71, M. Teixeira, AIP Conf. Proc. 762, W. T. Kim, M. S. Jhon, Y. Zhou, I. Staroselsky, and H. Chen, J. Appl. Phys. 97, 10P P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, S. Chapman and T. G. Cowling, The Mathematical Theory of Non- Uniform Gases, 3rd ed. Cambridge University Press, Cambridge, L. C. Woods, An Introduction to the Kinetic Theory of Gases and Magnetoplasmas Oxford University Press, Oxford, D. W. Stops, J. Phys. D 3, A. Beskok and G. E. Karniadaki, Microscale Thermophys. Eng. 3, ; P. Bahukudumbi, J. H. Park, and A. Beskok, ibid. 7, ; J. H. Park, P. Bahukudumbi, and A. Beskok, Phys. Fluids 16, T. Veijola and M. Turowski, J. Microelectromech. Syst. 10, Y. Sun and W. K. Chan, J. Vac. Sci. Technol. A 22, R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222, Y. H. Qian, S. Succi, and S. A. Orszag, in Recent Advances in Lattice Boltzmann Computing, Annual Reviews of Computational Physics Vol. III edited by Dietrich Stauffer World Scientific, Singapore 1996, pp S. Chen and G. D. Doolen, Annu. Rev. Fluid Mech. 30, L.-S. Luo, in Proceedings of Applied Computational Fluid Dynamics, Beijing, 2000, edited by J.-H. Wu unpublished, pp S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond Oxford University Press, Oxford, X. He and L.-S. Luo, Phys. Rev. E 55, R ; 56, N. Cao, S. Chen, S. Jin, and D. Martínez, Phys. Rev. E 55, R S. Ansumali and I. V. Karlin, J. Stat. Phys. 107, X. He, X. Shan, and G. Doolen, Phys. Rev. E 57, R ; X.He,S. Chen, and G. Doolen, J. Comput. Phys. 146, Y. H. Qian and Y. Zhou, Phys. Rev. E 61, ; Y.H.QianandS. Chen, ibid. 61, C. Shen, D. B. Tian, C. Xie, and J. Fan, Microscale Thermophys. Eng. 8, S. Chen, D. Martinez, and R. Mei, Phys. Fluids 8, E. B. Arkilic, M. A. Schmidt, and K. S. Breuer, J. Microelectromech. Syst. 6, D. A. Lockerby, J. M. Reese, and M. A. Gallis, Phys. Fluids 17,