Number Kn Research Articles

The unreasonable effectiveness of hydrodynamics in heavy ion collisions

Event-by-event hydrodynamic simulations of AA and pA collisions involve initial energy densities with large spatial gradients. This is associated with the presence of large Knudsen numbers (Kn≈1) at early times, which may lead one to question the validity of the hydrodynamic approach in these rapidly evolving, largely inhomogeneous systems. A new procedure to smooth out the initial energy densities is employed to show that the initial spatial eccentricities, εn, are remarkably robust with respect to variations in the underlying scale of initial energy density spatial gradients, λ. For sNN=2.76 TeV LHC initial conditions generated by the MCKLN code, εn (across centralities) remains nearly constant if the fluctuation scale varies by an order of magnitude, i.e., when λ varies from 0.1 to 1 fm. Given that the local Knudsen number Kn≈1/λ, the robustness of the initial eccentricities with respect to changes in the fluctuation scale suggests that the vn's cannot be used to distinguish between events with large Kn from events where Kn is in the hydrodynamic regime. We use the 2+1 Lagrangian hydrodynamic code v-USPhydro to show that this is indeed the case: anisotropic flow coefficients computed within event-by-event viscous hydrodynamics are only sensitive to long wavelength scales of order 1/ΛQCD≈1 fm and are incredibly robust with respect to variations in the initial local Knudsen number. This robustness can be used to justify the somewhat unreasonable effectiveness of the nearly perfect fluid paradigm in heavy ion collisions.

Transient conjugate heat transfer in a circular microchannel involving rarefaction, viscous dissipation and axial conduction effects

Transient conjugate heat transfer of a hydrodynamically developed and thermally developing laminar flow in a circular microchannel is investigated considering the effects of rarefaction, viscous dissipation and axial conduction. The problem is solved numerically by a finite difference method based on Patankar’s (1980) control volume approach. Momentum and energy equations are solved with the boundary conditions of the first degree velocity slip and temperature jump. The problem depends on four parameters. These are the Peclet number (Pe), the Knudsen number (Kn), the Brinkman number (Br) and the wall thickness ratio (d′). The results obtained showed that heat transfer characteristics are significantly affected by these parameters and that in micro channel flows, the fluid’s rarefaction effects, viscous dissipation and axial conduction should not be ignored.

A forced convection heat transfer correlation of rarefied gases cross-flowing over a circular cylinder

With the rapid development of aeronautics and astronautics, there is increased interest in understanding the behavior of heat transfer of rarefied gases particularly in the environment outside of the atmospheric layer. Currently, empirical correlations are correlated at standard atmospheric pressure, but this may not be suitable for rarefied gases flow, particularly at Knudsen number (Kn)>0.1. In this study, the forced convection heat transfer of rarefied gases cross-flowing over a circular cylinder is studied at Reynolds number of 10−3 to 20 and Knudsen number of 10−3 to 3. The experiment is conducted at the velocity of 0.5–20m/s and the pressure of 100Pa∼100kPa. It is found that the heat transfer coefficient in the continuum flow regime mainly depends on the Reynolds number and the effect of the Knudsen number was quite weak. In the regimes of slip flow and transitional flow, the heat transfer coefficient slightly decreased as the Reynolds number decreased. However, the Knudsen number plays a significant role in heat transfer performance in both the slip flow regime and the transitional flow regime. The effects of Reynolds and Knudsen numbers on heat transfer were measured and analyzed. Based on the experimental data, an empirical correlation of the forced convection heat transfer of rarefied gases cross-flowing over a circular cylinder was developed. The correlation coefficient (R) was greater than 0.99, corresponding to a fitting error of less than ±10.5% based on a 95% confidence level. The study provides theoretical guidance for future engineering design of rarefied gases flow and heat transfer.

Simulation of water impregnation through vertically aligned CNT forests using a molecular dynamics method.

The flow rate of water through carbon nanotube (CNT) membranes is considerably large. Hence, CNT membranes can be used in nanofluidic applications. In this work, we performed a molecular dynamics (MD) simulation of the introduction of water into CNTs in the CNT membranes, especially in vertically aligned CNT forests. The results showed that the Knudsen number (Kn) increased with an increasing volume fraction of CNT (VC) and was greater than 10−3 for each VC. Beyond this value, the flow became a slip flow. Further, the permeability increased as VC increased in the actual state calculated by the MD simulation, whereas the permeability in the no-slip state predicted by the Hagen–Poiseuille relationship decreased. Thus, a clear divergence in the permeability trend existed between the states. Finally, the flow enhancement ranged from 0.1 to 23,800, and the results show that water easily permeates as VC increases.

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is the infinite Karlin “balls-in-boxes” scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals n, we prove several functional limit theorems (FLTs) in the Skorohod space D[0,1] endowed with the J1- or M1-topology for the number Kn∗(t) of boxes containing at most [nt] balls, t∈[0,1], and the random distribution function Kn∗(t)/Kn∗(1), as n→∞. The limit processes for Kn∗(t) are of the form (X(1)−X((1−t)−))t∈[0,1], where X is either a Brownian motion, a spectrally negative stable Lévy process, or an inverse stable subordinator. The small value probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for Kn∗(t)/Kn∗(1) is a Lévy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of Kn∗(1). First, for any Karlin occupancy scheme with deterministic probabilities (pk)k≥1, we obtain an approximation, uniformly in t∈[0,1], of the number of boxes with at most [nt] balls by a counting function defined in terms of (pk)k≥1. Second, we prove several FLTs for the number of visits to the interval [0,nt] by a perturbed random walk, as n→∞. If the stick-breaking factor has a beta distribution with parameters θ>0 and 1, the process (Kn∗(t))t∈[0,1] has the same distribution as a similar process defined by the number of cycles of length at most [nt] in a θ-biased random permutation a.k.a. a Ewens permutation with parameter θ. As a consequence, our FLT with Brownian limit forms a generalization of a FLT obtained earlier in the context of Ewens permutations by DeLaurentis and Pittel (1985), Hansen (1990), Donnelly et al. (1991), and Arratia and Tavaré (1992).

Extended application of lattice Boltzmann method to rarefied gas flow in micro-channels

Abstract Simulation of rarefied gas flow in micro-channels is of great interest owing to its diverse applications in many engineering fields. In this study, a multiple-relaxation-time lattice Boltzmann (MRT-LB) model with a general second-order slip boundary condition is presented to investigate the behaviour of gas flow with a wide range of Knudsen number in micro-channels. With the aid of a Bosanquet-type effective viscosity, the effective relaxation time is correlated with local Knudsen number (Kn) to account for the varying degree of rarefaction effect. Unlike previous studies, the derived accommodation coefficient r for the combined bounce-back/diffusive reflection (CBBDR) boundary condition is dependent on the local Kn, which allows more flexibility to simulate the slip velocity along the channel walls. When compared with results of other methods, such as linearised Boltzmann equation, experimental data, direct simulation Monte Carlo (DSMC) and Information Preservation DSMC (IP-DSMC), it is found that the LB model is capable of capturing the flow behaviour, including the velocity profile, flow rate, pressure distribution and Knudsen minimum of rarefied gas with Kn up to 10. The effect of Knudsen layer (KL) on the velocity of gas flow with a wide range of Kn is also discussed. It is found that KL effect is negligible in the continuum flow and y -independent in the free molecular flow, while in the intermediate range, especially in transition flow, KL effect is significant and particular efforts should be made to capture this effect.

Rarefaction Effect on Gas Flow in Microchannels with Various Aspect Ratios

AbstractThis study investigated the effect of rarefaction on microchannel gas flow by measuring pressure profiles in microchannels with various aspect ratios. Pressure-sensitive paint (PSP) was applied in rectangular microchannels to obtain the global flow field by using detailed pressure data. The effect of rarefaction on the microchannel gas flow was clearly observed in the microchannels through the pressure data obtained using PSP measurements. A nonlinear pressure distribution was observed inside the microchannels, and this distribution decreased as the Knudsen number (Kn) increased because of the rarefaction effect. The dimensionless pressure deviation from the linear assumption dropped from 0.25 to 0 when the outlet Kn number increased to 0.066 in the 100-μm-wide microchannel, and the dimensionless location of the maximum deviation moved upstream because of the gaseous slip at the wall. The nonlinear pressure distribution also decreased in the 50-μm-wide microchannel as the outlet Kn number increased; however, the peak of the maximum deviation could no longer be identified because of the characteristic of the narrow channel.

Effect of the sliding friction on heat transfer in high-speed rarefied gas flow simulations in CFD

Different nature of the heat transfer computation between the CFD and DSMC methods in rarefied gas flow simulations leads to the difference of their heat transfer results. This paper will revisit a formulation of heat transfer in CFD that includes the sliding friction part for a planar surface. The sliding friction was first introduced by Maslen, and used to be omitted in calculating the CFD heat transfer. This formulation will be extended to compute the heat transfer over the curved surfaces such as NACA0012 micro-airfoil and the sharp 25–55-deg. biconic. Two these configurations 1) NACA0012 micro-airfoil (M = 2) with various Kn = 0.026, 0.1 and 0.26 with the angles-of-attack from 0-deg. to 20-deg., and a sharp 25–55-deg. biconic (M = 15.6) are simulated by the CFD method, which solves the Navier-Stokes equations within the OpenFOAM framework. Moreover, the flat plat cases with various wall temperatures (Tw = 77 K and 294 K) are also undertaken to investigate the significance of the sliding friction on the heat transfer. The CFD simulation results of the heat transfer involving the sliding friction give good agreements with the DSMC data, especially the NACA0012 micro-airfoil cases with high Knudsen numbers Kn = 0.1 and 0.26.

Direct-Simulation Monte Carlo Investigation of a Berea Porous Structure

SummaryShale-gas and tight gas reservoirs consist of porous structures with pore diameter in the range of 1 to 200 nm. At these scales, the pore diameter becomes comparable to the gas mean free path. Flows in these structures fail often in the transition and slip flow regimes. Standard continuum fluid methods such as the Navier-Stokes-Fourier (NSF) set of equations fail to describe flows of these regimes. We present a direct-simulation monte carlo (DSMC) study of a 3D porous structure in an unlimited parallel simulation. The 3D geometry was obtained with microcomputed-tomography (micro-CT). The gas considered is CH4 (100%), and the gas intermolecular-collision model for the simulation is the variable hard sphere (VHS). Simulations were carried out for three different Knudsen (Kn) numbers within the transition and slip flow regimes. The results demonstrate some of the significant differences that appear in gas-flow properties depending on the Kn number and the flow regime. In addition, the velocity profile appears to depend on the Kn number. At the inlet of the porous structures, more-uniform velocity profile occurs for the three Kn numbers. At the outlet, the velocity profile varies depending on the Kn number. For Kn ≈ 0.037, a parabolic shape is observed for the velocity profile, whereas a more-uniform shape is observed for Kn ≈ 3.7.

Conjugate heat transfer in rarefied gas in enclosed cavities

The heat transfer in buoyancy driven rarefied gas in a conjugate cavity has been investigated in this paper. The governing equations along with the slip flow and temperature jump boundary conditions are solved using finite-volume technique. The simulations are carried by varying cavity aspect ratios, conductivity ratios and cavity tilt angles. The Nusselt number values are calculated for different Knudsen, Rayleigh and Prandtl numbers. It was found that Kn number is the most influential parameter on the heat transfer; as it increases the heat transfer in the cavity decreases. The effect of all other investigated factors was found and documented; however, it is found that, in general, at high Kn Numbers their effect on the heat transfer is marginal. It is also found that cavity walls with good conducting materials will have minor effect on heat transfer while insulation material will have moderate effect on heat transfer. Two correlations for average Nusselt number are proposed: one for horizontal cavities; Nu=0.149kr0.0101Ra0.135Kn−0.393Pr0.178 and the second correlation equation introduces the effect of the cavity tilt angle along with Ra and Kn; Nu = 0.158Ra0.130, Kn−0.396, cos(θ)0.216.

The Effect of Slip Velocity and Temperature Jump on the Hydrodynamic and Thermal Behaviors of MHD Forced Convection Flows in Horizontal Microchannels

The effect of velocity slip and temperature jump on the hydrodynamic and thermal behaviors of MHD flows in the case of forced convection over flat plate, and in the case of forced convection between two parallel flat plates has been studied. Four mathematical models that represent these cases have been developed. The continuum model of fluid has been used with Knudsen (Kn) number regime 0.001 < Kn < 0.1, with the Maxwell slip velocity being applied along with the Smoluchowski temperature jump boundary on the solid–fluid surface interface. COMSOL software multiphysics was used to solve those four models numerically. In addition, the case of the forced convection between two parallel flat plates has also been solved analytically using MATLAB 7 software. It was found that the applied transverse magnetic produces Lorentz force that acts as an external body force and tends to retard the flow velocity; this retardation was found to be directly proportional with both the magnetic field number (N) and the Kn number. It was noticed that when Kn = 0 the reduction in the velocity for the same value of magnetic influence number (N = 1) is about 10 %, while for Kn = 0.1 the reduction is about 15 % in the case of forced flow between two flat plates. This shows that Kn and the magnetic influence number are both affecting the flow velocity at the center by decreasing its value. Also, it was found that an increase in both the magnetic field applied and Kn number will decrease the skin friction factor, decrease the Nusselt number, and decrease the thickness of the velocity boundary layer.

Prediction of Chemical Nonequilibrium Flow Structure at Different Free-Stream Knudsen Numbers

AbstractA numerical study is conducted to investigate the effects of the inflow Knudsen number (Kn) on hypersonic chemical nonequilibrium rarefied flows using the unstructured direct simulation Monte-Carlo (DSMC) method, and an empirical formula for fast prediction of flow-field structure for different values of Kn is proposed and verified. First, the flow around a vertical plate is analyzed at a free-stream velocity of 7,500 m/s using a perfect-gas model with inflow Kn values ranging from 0.035 to 13.36. Through analysis of the flow-field characteristics with varying Kn values, a linear expansion law of flow-field versus the square root of Kn is discovered and an empirical formula based on least-squares fitting is developed for fast prediction of the flow structure at different values of Kn. Further, the effects of chemical nonequilibrium on hypersonic rarefied flow-field structure are investigated to verify the applicability of the empirical formula. Finally, cases of the cylinder flow are simulated in...

DSMC–LBM mapping scheme for rarefied and non-rarefied gas flows

We present the formulation of a kinetic mapping scheme between the Direct Simulation Monte Carlo (DSMC) and the Lattice Boltzmann Method (LBM) which is at the basis of the hybrid model used to couple the two methods in view of efficiently and accurately simulate isothermal flows characterized by variable rarefaction effects. Owing to the kinetic nature of the LBM, the procedure we propose ensures to accurately couple DSMC and LBM at a larger Kn number than usually done in traditional hybrid DSMC—Navier–Stokes equation models. We show the main steps of the mapping algorithm and illustrate details of the implementation. Good agreement is found between the moments of the single particle distribution function as obtained from the mapping scheme and from independent LBM or DSMC simulations at the grid nodes where the coupling is imposed. We also show results on the application of the hybrid scheme based on a simpler mapping scheme for plane Poiseuille flow at finite Kn number. Potential gains in the computational efficiency assured by the application of the coupling scheme are estimated for the same flow.

Thin Film Deposition Using Rarefied Gas Jet

The rarefied gas jet of Aluminium is studied at Mach number Ma = (U_j / \\sqrt{kb T_j / m}) in the range 0.01 < Ma < 2, and Knudsen number Kn = (1 / (\\sqrt{2} \\pi d^2 n_d H) in the range 0.01 < Kn < 15, using two-dimensional (2D) Direct Simulation Monte Carlo (DSMC) simulations [1, 2, 3, 4, 5, 6], to understand the flow phenomena and deposition mechanisms in a physical vapor deposition (PVD) process. Here, H is the characteristic dimension, U_j and T_j are the jet velocity and temperature, n_d is the number density of the jet, d is the molecular diameter and kb is the Boltzmann constant. In 2D DSMC simulations we find that the jet velocities both before and after the substrate reduces as the Knudsen number is decreased (increasing the number density). As Knudsen number decreases, the gas jet became much more concentrated on the centerline, and the gas depleted region behind the substrate (flat plate) decreases due to the decrease in the streamwise velocity and hence increased residence time. The decreased Knudsen number (smaller mean free path) confined the gas jet to the centerline by reducing the rate of lateral diffusion. Increasing the jet Mach number also leads to a narrowing of the gas jet. This results from the higher jet velocity, and hence higher momentum of the gas jet along the streamwise direction, which reduces the transverse scattering effects of collisions. We examine the jet streamlines in the vicinity of the substrate by determining the average trajectories along the simulation cells. The variation of the local fluxes along the streamwise direction away from the jet are studied. The qualitative nature of the local flux at high Mach number (Ma = 2) is similar to those in the incompressible limit (Ma = 0.01). These include the initial fast decay, then slow variation, and finally rapid decay near the substrate. However, there are important differences. The amplitudes of the local flux increase as the Mach number increases. The flux decreases by an order of magnitude before reaching to the substrate. There is significant velocity and temperature slip [1, 2] at the solid surfaces of the substrate. In a compressible rarefied jet flow, we determine the local flux deposited on the substrate surface with the thermal accommodation coefficient equal to 1. The gas atom concentration in the jet slowly decreases due to the infrequent scattering, and deposition occurs due to the reduced diffusion distance. The gas flux profile at the front and rear surfaces of the substrate is determined, and show significant dependence upon the Knudsen number. An important finding is that the capture width (cross-section of the gas jet deposited on the substrate) is symmetric around the centerline of the substrate, and decreases with increased Mach number due to an increase in the momentum of the gas molecules. DSMC simulation results reveals that at low Knudsen number ((Kn = 0.01); shorter mean free paths), the atoms experience more collisions, which direct them toward the substrate. However, the atoms also move with lower momentum at low Mach number, which allows scattering collisions to rapidly direct the atoms to the substrate. At high Knudsen number ((Kn = 15); longer mean free paths), the atoms travel greater distances without depositing onto the substrate. Changing the Mach number influences the flux profile by controlling the momentum of the incident gas atoms. At high Mach number (Ma = 2), the gas atoms have large momentums, which increases the distance atoms traveled before depositing onto the substrate. A very small Mach number (Ma = 0.01) resulted in atoms depositing quickly near the centerline of the substrate.

Simulation of collisional effects on divertor pumping in JT-60SA

In this work, the exhausted neutral gas flow is modeled for two cases of a simplified JT-60SA sub-divertor geometry and compared via three different approaches, namely (i) a collisionless approach based on the ProVac3D code, (ii) the DSMC approach based on the DIVGAS code that can be run with and without consideration of particle collisions, and (iii) the NEUT2D approach which has been extensively used in the past for the JT-60 design. In a first case study, the transmission probability was calculated by the 3 approaches and very good agreement is found between NEUT2D-ProVac3D whereas discrepancies between DIVGAS and NEUT2D are found and further analyzed. In the second case, the assessment of collisions is done by means of DIVGAS. Simulations showed that the flow is found in the transitional regime with Kn numbers between 0.1 and 0.4. The DIVGAS collisionless case yielded lower values of temperature than the collisional one by factors of 0.5–0.8 in regions near the inlets of the sub-divertor whereas in regions near the pump and the chevron, the temperature difference is marginal. Moreover, a relative percentage difference of 16–40% in pressure values was found between collisionless and collisional approaches.

Pressure drop and thrust predictions for transonic micronozzle flows

In this paper, the expansion of xenon, argon, krypton, and neon gases through a Laval nozzle is studied experimentally and numerically. The pressurized gases are accelerated through the nozzle into a vacuum chamber in an attempt to simulate the operating conditions of a cold-gas thruster for attitude control of a micro-satellite. The gases are evaluated at several mass flow rates ranging between 0.178 mg/s and 3.568 mg/s. The Re numbers are low (8–256) and the estimated values of Kn number lie between 0.33 and 0.02 (transition and slip-flow regime). Direct Simulation Monte Carlo (DSMC) and continuum-based simulations with a no-slip boundary condition are performed. The DSMC and the experimental results show good agreement in the range Kn &amp;gt; 0.1, while the Navier-Stokes results describe the experimental data more accurately for Kn &amp;lt; 0.05. Comparison between the experimental and Navier-Stokes results shows high deviations at the lower mass flow rates and higher Kn numbers. A relation describing the deviation of the pressure drop through the nozzle as a function of Kn is obtained. For gases with small collision cross sections, the experimental pressure results deviate more strongly from the no-slip assumption. From the analysis of the developed function, it is possible to correct the pressure results for the studied gases, both in the slip-flow and transition regimes, with four gas-independent accommodation coefficients. The thrust delivered by the cold-gas thruster and the specific impulse is determined based on the numerical results. Furthermore, an increase of the thickness of the viscous boundary layer through the diffuser of the micronozzle is observed. This results in a shock-less decrease of the Mach number and the flow velocity, which penalizes thrust efficiency. The negative effect of the viscous boundary layer on thrust efficiency can be lowered through higher values of Re and a reduction of the diffuser length.

Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid

In this study, effects of nonlocal elasticity and slip condition on free vibration and flutter instability analysis of viscoelastic cantilever carbon nanotubes (CNTs) conveying fluid are investigated. Nonlocal Euler–Bernoulli beam theory is employed to establish the governing equations of motion for the vibrational behavior of CNT. The material property of the CNT is simulated by Kelvin–Voigt viscoelastic constitutive relation. The slip boundary conditions of CNT conveying fluid are considered based on Knudsen number (Kn). The equation of motion and associated boundary conditions are derived by using the extended form of Hamilton’s principle. The extended Galerkin method is then employed to discretize the equation of motion and boundary conditions which are then solved to calculate the eigenvalues and critical flutter speeds. Detailed results are demonstrated for the dependence of internal moving fluid, nonlocal parameter, structural damping coefficients and Knudsen number on the dimensionless eigenvalues and flutter boundaries of the cantilever CNT. It can be concluded that the structural damping coefficients and Knudsen number does not have a significant effect on the eigenvalues. However, the results show that the structural damping coefficients, Knudsen number and nonlocal parameter have significant effect on the flutter boundaries of the cantilever CNTs.

AN EMPIRICAL METHOD FOR FAST PREDICTION OF RAREFIED FLOW FIELD AROUND A VERTICAL PLATE

Numerical study is conducted to investigate the effects of free-stream Knudsen (Kn) number on rarefied flow field around a vertical plate employing an unstructured DSMC method, and an empirical method for fast prediction of flow-field structure at different Kn numbers in a given inflow velocity is proposed. First, the flow at a velocity 7500m/s is simulated using a perfect-gas model with free-stream Kn changing from 0.035 to 13.36. The flow-field characteristics in these cases with varying Kn numbers are analyzed and a linear-expansion phenomenon as a function of the square of Kn is discovered. An empirical method is proposed for fast flow-field prediction at different Kn based on the least-square-fitting method. Further, the effects of chemical reactions on flow field are investigated to verify the applicability of the empirical method in the real gas conditions. Three of the cases in perfect-gas flow are simulated again by introducing five-species air chemical module. The flow properties with and without chemical reactions are compared. In the end, the variation of chemical-reaction flow field as a function of Kn is analyzed and it is shown that the empirical method are also suitable when considering chemical reactions.

Extended Macroscopic Study of Dilute Gas Flow within a Microcavity

The behaviour of monatomic and dilute gas is studied in the slip and early transition regimes using the extended macroscopic theory. The gas is confined within a two-dimensional microcavity where the longitudinal sides are in the opposite motion with constant velocity ±Uw. The microcavity walls are kept at the uniform and reference temperature T0. Thus, the gas flow is transported only by the shear stress induced by the motion of upper and lower walls. From the macroscopic point of view, the regularized 13-moment equations of Grad, R13, are solved numerically. The macroscopic gas proprieties are studied for different values of the so-called Knudsen number (Kn), which gives the gas-rarefaction degree. The results are compared with those obtained using the classical continuum theory of Navier-Stokes and Fourier (NSF).

Simulation of micro flow in the transition regime using effective-viscosity-based multi-relaxation-time lattice Boltzmann model

With the rapid development of micro-electro-mechanical systems (MEMS), microscale rarefied gas flows have received considerable attention in the past decades. Recently, the lattice Boltzmann method (LBM) emerges as a promising way to study the flow in MEMS for its kinetic nature and distinctive computational features. Various LBM models have been used to simulate the microscale and nanoscale flow, among which the two-dimensional and nine-velocities (D2Q9)-based LBM is most widely accepted due to its extremely simplicity and high efficiency. However, the D2Q9-based LBM encounters great difficulties in the transition regime due to the rarefaction effects on mean free path and gas viscosity. An effective way to improve the capability of the existing LBM model is to incorporate an effective viscosity into the relaxation time, which can improve the accuracy of LBM model while keeping the simplicity and efficiency of LBM. However, the existing D2Q9-based LBM models with effective viscosity cannot give satisfactory predictions of the none-equilibrium phenomenon at moderate or high Knudsen (Kn) number both in accuracy and efficiency. To solve the above problem, in this study, an effective mean free path function proposed by Dongari et al. (Dongari N, Zhang Y H, Reese J M 2011 J. Fluids Eng. 133 071101) via modular dynamics mean is introduced into the D2Q9 multi-relaxation-time lattice Boltzmann model (MRT-LBM) to account for the effect of Knudsen layer in transition flow regime, and the viscosity in the MRT-LBM model is modified correspondingly. The combination of the bounce-back and specular reflection boundary condition is used to deal with the velocity slip, and the relaxation time and the reflection coefficient are properly set to eliminate the numerical artifact on the boundaries as the kinetic boundary condition is used. Micro Couette flow at Kn=0.1-6.77, and periodic Poiseuille flow at Kn=0.1128-2.2568, respectively, are numerically investigated by using the proposed MRT-LBM model, and the numerical results, including the non-dimensional velocity profile and the mass flow rate, are verified by the direct simulation Monte~Carlo (DSMC) data, the linearized Boltzmann solutions and the existing LBM model. The calculation results demonstrate that in transition regime, with the increase of Knudsen number, the dimensionless slip velocity at the wall significantly increases. It is shown that the velocity profiles predicted by the present MRT-LBM model agree well with the DSMC data and linearized Boltzmann solutions up to Kn=4.5 in Couette flow, which is much more accurate than that obtained from the existing LBM model. And the present LBM model gives at least the same order of accuracy in the prediction of velocity profile and mass flow rate as the existing LBM model in periodic Poiseuille flow. What is more, the Knudsen minimum phenomenon of flow in the microchannel is successfully captured at around Kn=1. The results demonstrate that the proposed model can enhance the ability of LBM in capturing the non-equilibrium phenomenon in micro flow in the transition regime both in accuracy and efficiency.