Microscale Gas Flows Research Articles

Filter-matrix lattice Boltzmann model for microchannel gas flows

The lattice Boltzmann method has been shown to be successful for microscale gas flows, and it has attracted significant research interest. In this paper, the recently proposed filter-matrix lattice Boltzmann (FMLB) model is first applied to study the microchannel gas flows, in which a Bosanquet-type effective viscosity is used to capture the flow behaviors in the transition regime. A kinetic boundary condition, the combined bounce-back and specular-reflection scheme with the second-order slip scheme, is also designed for the FMLB model. By analyzing a unidirectional flow, the slip velocity and the discrete effects related to the boundary condition are derived within the FMLB model, and a revised scheme is presented to overcome such effects, which have also been validated through numerical simulations. To gain an accurate simulation in a wide range of Knudsen numbers, covering the slip and the entire transition flow regimes, a set of slip coefficients with an introduced fitting function is adopted in the revised second-order slip boundary condition. The periodic and pressure-driven microchannel flows have been investigated by the present model in this study. The numerical results, including the velocity profile and the mass flow rate, as well as the nonlinear pressure distribution along the channel, agree fairly well with the solutions of the linearized Boltzmann equation, the direct simulation Monte Carlo results, the experimental data, and the previous results of the multiple effective relaxation lattice Boltzmann model. Also, the present results of the velocity profile and the mass flow rate show that the present model with the fitting function can yield improved predictions for the microchannel gas flow with higher Knudsen numbers in the transition flow regime.

Hybrid continuum–molecular modelling of multiscale internal gas flows

We develop and apply an efficient multiscale method for simulating a large class of low-speed internal rarefied gas flows. The method is an extension of the hybrid atomistic–continuum approach proposed by Borg et al. (2013) [28] for the simulation of micro/nano flows of high-aspect ratio. The major new extensions are: (1) incorporation of fluid compressibility; (2) implementation using the direct simulation Monte Carlo (DSMC) method for dilute rarefied gas flows, and (3) application to a broader range of geometries, including periodic, non-periodic, pressure-driven, gravity-driven and shear-driven internal flows. The multiscale method is applied to micro-scale gas flows through a periodic converging–diverging channel (driven by an external acceleration) and a non-periodic channel with a bend (driven by a pressure difference), as well as the flow between two eccentric cylinders (with the inner rotating relative to the outer). In all these cases there exists a wide variation of Knudsen number within the geometries, as well as substantial compressibility despite the Mach number being very low. For validation purposes, our multiscale simulation results are compared to those obtained from full-scale DSMC simulations: very close agreement is obtained in all cases for all flow variables considered. Our multiscale simulation is an order of magnitude more computationally efficient than the full-scale DSMC for the first and second test cases, and two orders of magnitude more efficient for the third case.

Analytical solution of plane Poiseuille flow within Burnett hydrodynamics

In the current paper, low-speed isothermal microscale gas flows have been investigated utilizing the augmented Burnett equations. There has been limited success to analytically solve the Burnett equations till date. We propose an analytical solution to Burnett equations, which is shown to satisfy the full set of augmented Burnett equations up to Kn of 2.2 with an error of 1 %. Detailed validation shows that the solution represents the integral flow parameters accurately up to Kn ~ 2.2 and local field properties up to Kn ~ 0.5. The range over which the proposed Burnett analytical solution is applicable is substantially better than existing analytical solutions, without incorporating any wall scaling functions into constitutive relations and variation of slip coefficients in the boundary conditions. Normalized mass flow rate, friction factor, and axial velocity profile results show very good agreement with the experimental and simulation data. The analytical solution is also able to predict the change in the curvature of streamwise pressure profile.

Microscale boundary conditions of the lattice Boltzmann equation method for simulating microtube flows

The lattice Boltzmann equation (LBE) method has been shown to be a promising tool for microscale gas flows. However, few works focus on the microtube flows, and there still are some fundamental problems for the LBE to such flows. In this paper, a recently proposed axisymmetric LBE with three kinetic boundary conditions, i.e., the combination of bounceback and specular reflection scheme, the combination of the Maxwell and specular-reflection scheme, and the combination of the Maxwell and bounceback scheme, have been investigated in detail. By analyzing the micro-Hagen-Poiseuille flow, we observed the discrete boundary condition effect and provided a revised boundary scheme to overcome such effect near the slip flow regime. Some numerical tests for the micro-Hagen-Poiseuille have been carried out to validate the analysis, and the numerical results of the revised boundary schemes agree well with the analytic solutions which confirmed our theoretical analysis. In addition, we also applied the revised combination of the Maxwell and bounceback scheme to microtube flow with sudden expansion and contraction, the numerical results of the pressure distribution and normalized slip velocity agree well with the theoretical ones.

Multiple temperature kinetic model and its applications to micro-scale gas flows

This paper presents a gas-kinetic scheme to solve the multiple temperature kinetic model (MTKM), which was proposed in J. Comput. Math. 29(6) (2011) 639–660, for the study of non-equilibrium flows. The MTKM is a two-stage particle collision model possessing an intermediate quasi-equilibrium state with a symmetric second-order temperature tensor. A gas-kinetic finite volume scheme is developed for the numerical solution of the MTKM in the continuum and transition flow regimes. The gas-kinetic scheme is designed for the updating of macroscopic variables, which include the conservative flow variables and the multiple temperature field. In order to validate the kinetic model, the gas-kinetic scheme is used in the study of lid-driven cavity flows in both continuum and transition flow regimes. The numerical results predicted by the MTKM are compared with those from the direct simulation Monte Carlo (DSMC) method, the Navier–Stokes equations (NSE), and the early three-temperature kinetic model (TTKM) proposed in Phys. Fluids 19, 016101(2007). It is demonstrated that the MTKM has obvious advantages in comparison with the NSE and the TTKM in capturing the non-equilibrium flow behavior in the transition flow regime. One distinguishable phenomenon captured by the MTKM is that in the transition flow regime the heat flux direction can be from a low temperature to a high temperature region, which violates the Fourier’s law of continuum flows. The MTKM provides a more accurate physical model than the NSE for the non-equilibrium flows.

Boundary element solution of thermal creep flow in microfluidic devices

Flow in rarefied gases can be caused by a tangential temperature gradient along the contour boundaries (tangential heat flux), without the presence of any other external driven force, inducing a fluid motion from colder to hotter regions. This phenomenon is known as thermal creep and has gained importance in recent years in connection with micro-scale gas flow systems. Prediction of the flow field in micro-systems can be obtained by using continuum based models under appropriate boundary conditions accounting for the slip velocity due to tangential shear rate and heat flux. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the velocity field is presented. The tangential heat flux is evaluated in terms of the tangential gradient of the temperature integral representational formulae presenting singularities of the Cauchy type, which are removed by the use of an auxiliary potential field. These formulations are used to evaluate the performance of different microfluidic devices.

Effect of variable thermal conductivity and viscosity on single phase convective heat transfer in slip flow

For a variety of fields in which micro-mechanical systems and electronic components are used, fluid flow and heat transfer at the microscale needs to be understood and modeled with an acceptable reliability. In general, models are prepared by making some extensions to the conventional theories by including the scaling effects that become important for microscale. Some of these effects are; axial conduction, viscous dissipation, and rarefaction. In addition to these effects, temperature variable thermal conductivity and viscosity may become important in microscale gas flows due to the high temperature gradients that may exist in the fluid. For this purpose, simultaneously developing, single phase, laminar and incompressible air flow in a microtube and in the micro gap between parallel plates is numerically analyzed. Navier–Stokes and energy equations are solved and the variation of Nusselt number along the channel is presented in tabular and graphical forms as a function of Knudsen, Peclet, and Brinkman numbers, including temperature variable thermal conductivity and viscosity.

Impact of computational physics on multi-scale CFD and related numerical algorithms

Development of accurate computational methods for the constitutive relation that plays a role as the bridge between microscopic and macroscopic physics becomes a key issue in a continuum approach for describing rarefied and micro-scale gas flows. The mathematical form of the constitutive relation dictates the resulting computational methods and related algorithms. It is, therefore, vital to develop proper computational models on the basis of a correct understanding of the multi-scale physics inherent in non-equilibrium gases. In this study the computational issue is discussed by considering two benchmark multi-scale problems: the compression-dominated shock structure and velocity shear-dominated gas flows. Special emphasis is placed on efficient CFD algorithms within the finite volume formulation. In addition, the verification and validation issue of the multi-scale methods is discussed and a simple verification method based on basic physical laws is proposed.

Predicting microscale gas flows and rarefaction effects through extended Navier–Stokes–Fourier equations from phoretic transport considerations

We test an extended continuum-based approach for analyzing micro-scale gas flows over a wide range of Knudsen number and Mach number. In this approach, additional terms are invoked in the constitutive relations of Navier–Stokes–Fourier equations, which originate from the considerations of phoretic motion as triggered by strong local gradients of density and/or temperature. Such augmented considerations are shown to implicitly take care of the complexities in the flow physics in a thermo-physically consistent sense, so that no special boundary treatment becomes necessary to address phenomenon such as Knudsen paradox. The transition regime gas flows, which are otherwise to be addressed through computationally intensive molecular simulations, become well tractable within the extended quasi-continuum framework without necessitating the use of any fitting parameters. Rigorous comparisons with direct simulation Monte Carlo (DSMC) computations and experimental results support this conjecture for cases of isothermal pressure driven gas flows and high Mach number shock wave flows through rectangular microchannels.

Effects of rarefaction in microflows between coaxial cylinders

Microscale gas flows between two rotating coaxial circular cylinders of infinite length with different temperatures are investigated. Navier-Stokes-Fourier (NSF) and regularized 13-moment (R13) equations in their linear form are used to independently analyze velocity and temperature fields in shear-driven rotary flows, i.e., cylindrical Couette flows. Knudsen boundary layers, which present non-Newtonian stress and non-Fourier heat flow, are predicted as the dominant rarefaction effects in the linear theory. We show that the R13 system yields more accurate results for this boundary value problem by predicting the Knudsen boundary layers, which are not accessible for NSF equations. Furthermore, a set of second-order boundary conditions for velocity slip and temperature jump are derived for the NSF system. It is shown that the proposed boundary conditions effectively improve the classical hydrodynamics. The accuracy of NSF and R13 equations is discussed based on their comparison with available direct simulation Monte Carlo data.

Experimental Analyses of High Knudsen Number Flows

In late years, the necessity of high Knudsen number flows, such as highly rarefied gas flows with large mean free path and gaseous flows around nanodevices with small characteristic length have increased significantly, especially in fields of researches on space technology, surface science, fabrication of thin film for semi-conductor and so on. In this paper, we describe mainly the optical diagnostic techniques for the high Knudsen number flows, such as laser induced fluorescence (LIF), resonantly enhanced multiphoton ionization (REMPI) and pressure sensitive molecular film (PSMF), and our experimental results obtained by the use of the techniques, i.e., applications of LIF to visualization of flow field structures including complicated shock wave system and to a measurement of rotational temperature, establishment of a REMPI system and its application to detection of rotational nonequilibrium in highly rarefied gas flows, and development of the suitable PSMF for low density and micro-scale gas flows.

Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects

The constitutive relations of gases in a thermal nonequilibrium (rarefied and microscale) can be derived by applying the moment method to the Boltzmann equation. In this work, a model constitutive relation determined on the basis of the moment method is developed and applied to some challenging problems in which classical hydrodynamic theories including the Navier–Stokes–Fourier theory are shown to predict qualitatively wrong results. Analysis of coupled nonlinear constitutive models enables the fundamentals of gas flows in thermal nonequilibrium to be identified: namely, nonlinear, asymmetric, and coupled relations between stresses and the shear rate; and effect of the bulk viscosity. In addition, the new theory explains the central minimum of the temperature profile in a force-driven Poiseuille gas flow, which is a well-known problem that renders the classical hydrodynamic theory a global failure.

Lattice Boltzmann equation for microscale gas flows of binary mixtures

Modeling and simulating gas flows in and around microdevices are a challenging task in both science and engineering. In practical applications, a gas is usually a mixture made of different components. In this paper we propose a lattice Boltzmann equation (LBE) model for microscale flows of a binary mixture based on a recently developed LBE model for continuum mixtures [P. Asinari and L.-S. Luo, J. Comput. Phys. 227, 3878 (2008)]. A consistent boundary condition for gas-solid interactions is proposed and analyzed. The LBE is validated and compared with theoretical results or other reported data. The results show that the model can serve as a potential method for flows of binary mixture in the microscale.

Application of a droplet cluster to visualize microscale gas and liquid flows

The possibility of using microdroplets, which form a dissipative droplet-cluster structure, as tracers localized in the boundary layer of a gas adjoining a liquid-gas interface is experimentally demonstrated. Examples of the visualization of boundary layer flow fields generated by both processes in the gaseous phase and liquid convection are considered.

Microscale Flow Through Channels With a Right-Angled Bend: Effect of Fillet Radius

Microscale gas flow through channels with a right-angled bend has been numerically analyzed to study the effect of the fillet radius on flow characteristics. The flow is assumed to be incompressible, laminar, and hydrodynamically developing. The fillet radius has been varied from zero, representing a sharp corner, to 0.6 times the height of the channel. The Knudsen number has been varied from zero, representing no-slip at the boundary, to 0.1, which is the limiting case for the slip-flow regime. A low Reynolds number of value 1 has been considered in the present study, which makes the flow to be within the incompressible slip-flow regime. The flow characteristics in terms of velocity profiles, velocity vectors, and the pressure ratio between the inlet and outlet of the channel have been presented for several cases. Results show that for the case of the fillet radius equal to zero, the flow separation occurs after the bend and due to this, the exit velocity profile changes significantly. The highest pressure ratio between the inlet and the outlet is required to maintain a specific mass flow rate for this case. The cases with a nonzero fillet radius exhibit exit velocity profiles identical to that of a straight channel. The pressure ratio decreases when the fillet radius and the Knudsen number are increased.

Analysis of lattice Boltzmann equation for microscale gas flows: Relaxation times, boundary conditions and the Knudsen layer

In this work, we apply the lattice Boltzmann equation (LBE) with multiple relaxation times (MRTs) to simulate the Poiseuille flow in the slip flow regime. We analyse in detail the discrete diffusive and combined bounce-back-specular-reflection boundary conditions for the LBE, the discrete effects at the boundary, and determinations of the relaxation times for the Poiseuille flow in the slip flow regime. In particular, we implement second-order slip boundary conditions with the MRT–LBE model and validate our numerical results for the slip Poiseuille flow with the Knudsen number Kn ≤ 0.2 by using the analytic solution. Our analysis shows that the lattice Bhatnagar–Gross–Krook (BGK) model cannot yield correct results for the Poiseuille flow in the slip-flow regime. We also discuss the possibilities of extending the LBE for the Knudsen layer in micro flows.

Computations of gas microflows using pressure correction method with Langmuir slip model

A Langmuir slip model combined with continuum-based compressible Navier–Stokes equations is proposed and implemented for the purpose of analyzing complex microscale gas flows. For our model, an efficient compressible pressure correction algorithm based on an unstructured grid is developed and modified to be applicable to low Reynolds number slip flows in microgeometries. Gaseous slip flows in a uniform microchannel and compressible flow at backward-facing step are computed for the assessment of the adequacy of the method. Separated flow in a T-shaped micro-manifold is also simulated for the Reynolds number ranging from 10 to 60. In the uniform microchannel flow, the pressure increases nonlinearly in Langmuir slip model as the Knudsen number increases, while it drops nonlinearly in Maxwell slip model. The results from Langmuir slip model have been found to be more compatible with physics. From all the simulation cases, nonlinear behavior owing to both compressibility and rarefaction clearly appears in terms of streamwise velocity, pressure profiles and even reattachment length in the separation-associated flows. These results show that the suggested pressure correction method along with the Langmuir slip model may effectively simulate complex microscale gas flows, thereby offering a sound theoretical and numerical basis and an inexpensive computation procedure.

Numerical Modeling Based on the Generalized Hydrodynamic Equations for a Microscale Gas Flow

Numerical Modeling Based on the Generalized Hydrodynamic Equations for a Microscale Gas Flow Hyung-Il Choi, Joo-Sung Maeng and Kwan-Soo Lee School of Mechanical Engineering, Hanyang University, Seoul 133-791 (Received 4 October 2007) In a microscale ow with a high Knudsen number, numerical models based on Navier-Stokes equations may yield erroneous results because the models assume local thermodynamic equilibrium. In this study, a new numerical model based on the generalized hydrodynamic equations (GH) is proposed for high Knudsen number ows. The model uses a pressure-based method adequate for low-speed and compressible ows. The nonlinear steady-state non-conserved ux evolutions are preliminarily calculated with the aid of Newton's approximation before solving the equations for the macroscopic observable quantities. The model is tested for non-uniform microscale ows in a microchannel and over a backward-facing step. The results show that the numerical model can be used to predict ows ranging from the continuum to the transition regions. PACS numbers: 47.11.Df, 47.61.Fg, 46.15.-x

Discrete effects on boundary conditions for the lattice Boltzmann equation in simulating microscale gas flows

The lattice Boltzmann equation (LBE) has shown its promise in the simulation of microscale gas flows. One of the critical issues with this advanced method is to specify suitable slip boundary conditions to ensure simulation accuracy. In this paper we study two widely used kinetic boundary conditions in the LBE: the combination of the bounce-back and specular-reflection scheme and the discrete Maxwell's scheme. We show that (i) both schemes are virtually equivalent in principle, and (ii) there exist discrete effects in both schemes. A strategy is then proposed to adjust the parameters in the two kinetic boundary conditions such that an accurate slip boundary condition can be implemented. The numerical results demonstrate that the corrected boundary conditions are robust and reliable.

Kinetic lattice Boltzmann method for microscale gas flows: Issues on boundary condition, relaxation time, and regularization

It is well known that the Navier-Stokes equations cannot adequately describe gas flows in the transition and free-molecular regimes. In these regimes, the Boltzmann equation (BE) of kinetic theory is invoked to govern the flows. However, this equation cannot be solved easily, either by analytical techniques or by numerical methods. Hence, in order to efficiently maneuver around this equation for modeling microscale gas flows, a kinetic lattice Boltzmann method (LBM) has been introduced in recent years. This method is regarded as a numerical approach for solving the BE in discrete velocity space with Gauss-Hermite quadrature. In this paper, a systematic description of the kinetic LBM, including the lattice Boltzmann equation, the diffuse-scattering boundary condition for gas-surface interactions, and definition of the relaxation time, is provided. To capture the nonlinear effects due to the high-order moments and wall boundaries, an effective relaxation time and a modified regularization procedure of the nonequilibrium part of the distribution function are further presented based on previous work [Guo et al., J. Appl. Phys. 99, 074903 (2006); Shan et al., J. Fluid Mech. 550, 413 (2006)]. The capability of the kinetic LBM of simulating microscale gas flows is illustrated based on the numerical investigations of micro Couette and force-driven Poiseuille flows.