Showing papers on "Knudsen number published in 2006"

Hydrodynamic interactions and Brownian forces in colloidal suspensions: coarse-graining over time and length scales.

Abstract: We describe in detail how to implement a coarse-grained hybrid molecular dynamics and stochastic rotation dynamics simulation technique that captures the combined effects of Brownian and hydrodynamic forces in colloidal suspensions. The importance of carefully tuning the simulation parameters to correctly resolve the multiple time and length scales of this problem is emphasized. We systematically analyze how our coarse-graining scheme resolves dimensionless hydrodynamic numbers such as the Reynolds number Re, which indicates the importance of inertial effects, the Schmidt number Sc, which indicates whether momentum transport is liquidlike or gaslike, the Mach number, which measures compressibility effects, the Knudsen number, which describes the importance of noncontinuum molecular effects, and the Peclet number, which describes the relative effects of convective and diffusive transport. With these dimensionless numbers in the correct regime the many Brownian and hydrodynamic time scales can be telescoped together to maximize computational efficiency while still correctly resolving the physically relevant processes. We also show how to control a number of numerical artifacts, such as finite-size effects and solvent-induced attractive depletion interactions. When all these considerations are properly taken into account, the measured colloidal velocity autocorrelation functions and related self-diffusion and friction coefficients compare quantitatively with theoretical calculations. By contrast, these calculations demonstrate that, notwithstanding its seductive simplicity, the basic Langevin equation does a remarkably poor job of capturing the decay rate of the velocity autocorrelation function in the colloidal regime, strongly underestimating it at short times and strongly overestimating it at long times. Finally, we discuss in detail how to map the parameters of our method onto physical systems and from this extract more general lessons—keeping in mind that there is no such thing as a free lunch—that may be relevant for other coarse-graining schemes such as lattice Boltzmann or dissipative particle dynamics.

Efficient kinetic method for fluid simulation beyond the Navier-Stokes equation.

Abstract: We present a further theoretical extension to the kinetic-theory-based formulation of the lattice Boltzmann method of Shan et al. [J. Fluid Mech. 550, 413 (2006)]. In addition to the higher-order projection of the equilibrium distribution function and a sufficiently accurate Gauss-Hermite quadrature in the original formulation, a regularization procedure is introduced in this paper. This procedure ensures a consistent order of accuracy control over the nonequilibrium contributions in the Galerkin sense. Using this formulation, we construct a specific lattice Boltzmann model that accurately incorporates up to third-order hydrodynamic moments. Numerical evidence demonstrates that the extended model overcomes some major defects existing in conventionally known lattice Boltzmann models, so that fluid flows at finite Knudsen number Kn can be more quantitatively simulated. Results from force-driven Poiseuille flow simulations predict the Knudsen's minimum and the asymptotic behavior of flow flux at large Kn.

Challenges in Modeling Gas-Phase Flow in Microchannels: From Slip to Transition

Abstract: It has long been recognized that the fluid mechanics of gas-phase microflows can differ significantly from the macroscopic world. Non-equilibrium effects such as rarefaction and gas-surface interactions need to be taken into account, and it is well known that the no-slip boundary condition of the Navier-Stokes equations is no longer valid. Following ideas proposed by Maxwell, it is generally accepted that the Navier-Stokes equations can be extended into the slip-flow regime, provided the Knudsen number is less than 10− 1. Improvements in micro-fabrication techniques, however, are now enabling devices to be constructed with sub-micron feature sizes. At this scale, the flow will depart even further from equilibrium and will enter the transition regime. In recent years, there has been considerable success in the implementation of second-order slip-boundary conditions to extend the Navier-Stokes equations into the transition regime. Unfortunately, as yet, no consensus has been reached on the correct form of h...

Capturing Knudsen layer phenomena using a lattice Boltzmann model.

Abstract: In recent years, lattice Boltzmann methods have been increasingly used to simulate rarefied gas flows in microscale and nanoscale devices This is partly due to the fact that the method is computationally efficient, particularly when compared to solution techniques such as the direct simulation Monte Carlo approach However, lattice Boltzmann models developed for rarefied gas flows have difficulty in capturing the nonlinear relationship between the shear stress and strain rate within the Knudsen layer As a consequence, these models are equivalent to slip-flow solutions of the Navier-Stokes equations In this paper, we propose an effective mean-free path to address the Knudsen layer effect, so that the capabilities of lattice Boltzmann methods can be extended beyond the slip-flow regime The model has been applied to rarefied shear-driven and pressure-driven flows between parallel plates at Knudsen numbers between 001 and 1 Our results show that the proposed approach significantly improves the near-wall accuracy of the lattice Boltzmann method and provides a computationally economic solution technique over a wide range of Knudsen numbers

Mass flow rate measurements in gas micro flows

Abstract: The main objective of this experimental investigation on the gas flow slip regime is to measure the mass flow rate in isothermal steady flows through cylindrical micro tubes. Two technical procedures devoted to mass flow rate measurements are compared, and the measured values are also compared with the results yielded by different approximated analytical solutions of the gas dynamics continuum equations. Satisfactory results are obtained and the way is clearly opened to measuring mass flow rates for higher Knudsen numbers, over all the micro flow transitional regime.

Lattice Boltzmann simulations of apparent slip in hydrophobic microchannels

Abstract: Various experiments have found a boundary slip in hydrophobic microchannel flows, but a consistent understanding of the results is still lacking While Molecular Dynamics (MD) simulations cannot reach the low shear rates and large system sizes of the experiments, it is often impossible to resolve the needed details with macroscopic approaches We model the interaction between hydrophobic channel walls and a fluid by means of a multi-phase lattice Boltzmann model Our mesoscopic approach overcomes the limitations of MD simulations and can reach the small flow velocities of known experiments We reproduce results from experiments at small Knudsen numbers and other simulations, namely an increase of slip with increasing liquid-solid interactions, the slip being independent of the flow velocity, and a decreasing slip with increasing bulk pressure Within our model we develop a semi-analytic approximation of the dependence of the slip on the pressure

Hybrid Particle-Continuum Simulations of Nonequilibrium Hypersonic Blunt-Body Flowfields

Abstract: A modular particle-continuum (MPC) numerical method is presented which solves the Navier-Stokes (NS) equations in regions of near-equilibrium and uses the direct simulation Monte Carlo (DSMC) method where the flow is in non-equilibrium. The MPC method is designed specifically for steady-state, hypersonic, nonequilibrium flows and couples existing, state-of-the-art DSMC and NS solvers into a single modular code. The MPC method is tested for 2D flow of N2 at various Mach numbers over a cylinder where the global Knudsen number is 0.01. For these conditions, NS simulations significantly over-predict the local shear-stress, and also over-predict the peak heating rate by 5-10% when compared with full DSMC simulations. DSMC also predicts faster wake closure and 10-15% higher temperatures in the immediate wake region. The MPC code is able to accurately reproduce DSMC flow field results, local velocity distributions, and surface properties up to 2.8 times faster than full DSMC simulations. The computational time saved by the MPC method is directly proportional to the fraction of the flow field which is in near-equilibrium. It is found that particle simulation of the shock interior is not necessary for accurate prediction of surface properties, however particle simulation of the boundary layer and near-wake region is.

Solution to the Boltzmann kinetic equation for high-speed flows

Abstract: The Boltzmann kinetic equation is solved by a finite-difference method on a fixed coordinate-velocity grid. The projection method is applied that was developed previously by the author for evaluating the Boltzmann collision integral. The method ensures that the mass, momentum, and energy conservation laws are strictly satisfied and that the collision integral vanishes in thermodynamic equilibrium. The last property prevents the emergence of the numerical error when the collision integral of the principal part of the solution is evaluated outside Knudsen layers or shock waves, which considerably improves the accuracy and efficiency of the method. The differential part is approximated by a second-order accurate explicit conservative scheme. The resulting system of difference equations is solved by applying symmetric splitting into collision relaxation and free molecular flow. The steady-state solution is found by the relaxation method.

Entropic lattice Boltzmann method for microflows

Abstract: A new method for the computation of flows at the micrometer scale is presented. It is based on the recently introduced minimal entropic kinetic models. Both the thermal and isothermal families of minimal models are presented, and the simplest isothermal entropic lattice Bhatnagar–Gross–Krook (ELBGK) is studied in detail in order to quantify its relevance for microflow simulations. ELBGK is equipped with boundary conditions which are derived from molecular models (diffusive wall). A map of three-dimensional kinetic equations onto two-dimensional models is established which enables two-dimensional simulations of quasi-two-dimensional flows. The ELBGK model is studied extensively in the simulation of the two-dimensional Poiseuille channel flow. Results are compared to known analytical and numerical studies of this flow in the setting of the Bhatnagar–Gross–Krook model. The ELBGK is in quantitative agreement with analytical results in the domain of weak rarefaction (characterized by Knudsen number Kn , the ratio of mean free path to the hydrodynamic scale), up to Kn ∼ 0.01 , which is the domain of many practical microflows. Moreover, the results qualitatively agree throughout the entire Knudsen number range, demonstrating Knudsen's minimum for the mass flow rate at moderate values of Kn , as well as the logarithmic scaling at large Kn . The present results indicate that ELBM can complement or even replace computationally expensive microscopic simulation techniques such as kinetic Monte Carlo and/or molecular dynamics for low Mach and low Knudsen number hydrodynamics pertinent to microflows.

Mesoscopic modelling of heterogeneous boundary conditions for microchannel flows

Abstract: We present a mesoscopic model of the fluid–wall interactions for flows in microchannel geometries. We define a suitable implementation of the boundary conditions for a discrete version of the Boltzmann equations describing a wall-bounded single-phase fluid. We distinguish different slippage properties on the surface by introducing a slip function, defining the local degree of slip for hydrodynamical fields at the boundaries. The slip function plays the role of a renormalizing factor which incorporates, with some degree of arbitrariness, the microscopic effects on the mesoscopic description. We discuss the mesoscopic slip properties in terms of slip length, slip velocity, pressure drop reduction (drag reduction), and mass flow rate in microchannels as a function of the degree of slippage and of its spatial distribution and localization, the latter parameter mimicking the degree of roughness of the ultra-hydrophobic material in real experiments. We also discuss the increment of the slip length in the transition regime, i.e. at ${O}(1)$ Knudsen numbers. Finally, we compare our results with molecular dynamics investigations of the dependence of the slip length on the mean channel pressure and local slip properties and with the experimental dependence of the pressure drop reduction on the percentage of hydrophobic material deposited on the surface.

Recovery of full rotational invariance in lattice Boltzmann formulations for high Knudsen number flows

Abstract: Kinetic representation of fluid dynamics is essential for describing flows far beyond the Navier–Stokes regime. Existing lattice Boltzmann models suffer from discrete rotation artifacts at high Knudsen number ( Kn ) . In this paper, we present a new approach that directly addresses this issue and makes possible quantitative simulations of flows at high Kn .

Slip effects on the peristaltic flow of a non-Newtonian Maxwellian fluid

Abstract: In real systems there is always a certain amount of slip, which, however, is hard to detect experimentally because of the required space resolution. In this paper, we analyze the effect of slip boundary conditions on the dynamics of fluids in porous media by studying the flow of a Newtonian and non-Newtonian Maxwellian fluid in an axisymmetric cylindrical tube (pore), in which the flow is induced by traveling transversal waves on the tube wall. Like in peristaltic pumping, the traveling transversal waves induce a net flow of the liquid inside the pore. The viscosity as well as the compressibility of the liquid is taken into account. This problem has numerous applications in various branches of science, including stimulation of fluid flow in porous media under the effect of elastic waves and studies of blood flow dynamics in living creatures. The Navier-Stokes equations for an axisymmetric cylindrical pore are solved by means of a perturbation analysis, in which the ratio of the wave amplitude to the radius of the pore is small parameter. In the second order approximation, a net flow induced by the traveling wave is calculated for various values of the compressibility of the liquid, relaxation time and Knudsen number. The calculations disclose that the compressibility of the liquid, Knudsen number of slip flow and non-Newtonian effects in presence of peristaltic transport have a strong influence of the net flow rate. The effects of all parameters of the problem are numerically discussed and graphically explained.

Analytic damping model for an MEM perforation cell

Abstract: The concept of the perforation cell is specified for compact modelling of perforated gas dampers with micromechanical dimensions. Both, analytic expressions and FEM simulations, are used to derive its flow resistance. An extensive set of FEM simulations is performed to characterize the flow resistance of the cell, and to derive approximations for different flow regions by fitting simple functions to them. Sinusoidal small-amplitude velocities are assumed, and micromechanical dimensions are considered with rare gas effects in the slip flow regime (Knudsen number <0.1). The model is capable of modelling all practical combinations of the perforation cell dimensions in a wide range of perforation ratios (1,...,90%). Its validity is verified with a Navier–Stokes solver, and it is shown to be accurate (relative error <4.5%) in the continuum and slip flow regimes. Estimates for cut-off frequencies due to inertial and compressibility effects are specified in a way that the maximum operation frequency of the model can be easily tested. Using a harmonic FEM solver, these estimates are verified. The perforation cell model is also applied to estimate the damping in a perforated rectangular damper (4,...,64 square holes). The damping predicted by the simple model is in moderate agreement with that obtained with 3D FEM simulations.

Normal solutions of the Boltzmann equation for highly nonequilibrium Fourier flow and Couette flow

Abstract: The state of a single-species monatomic gas from near-equilibrium to highly nonequilibrium conditions is investigated using analytical and numerical methods. Normal solutions of the Boltzmann equation for Fourier flow (uniform heat flux) and Couette flow (uniform shear stress) are found in terms of the heat-flux and shear-stress Knudsen numbers. Analytical solutions are found for inverse-power-law molecules from hard sphere through Maxwell at small Knudsen numbers using Chapman-Enskog (CE) theory and for Maxwell molecules at finite Knudsen numbers using a moment-hierarchy (MH) method. Corresponding numerical solutions are obtained using the direct simulation Monte Carlo (DSMC) method of Bird. The thermal conductivity, the viscosity, and the Sonine-polynomial coefficients of the velocity distribution function from DSMC agree with CE results at small Knudsen numbers and with MH results at finite Knudsen numbers. Subtle differences between inverse-power-law, variable-soft-sphere, and variable-hard-sphere representations of Maxwell molecules are observed. The MH and DSMC results both indicate that the effective thermal conductivity and the effective viscosity for Maxwell molecules are independent of the heat-flux Knudsen number, and additional DSMC simulations indicate that these transport properties for hard-sphere molecules decrease slightly as the heat-flux Knudsen number is increased. Similarly, the MH and DSMC results indicate that the effective thermal conductivity and the effective viscosity for Maxwell molecules decrease as the shear-stress Knudsen number is increased, and additional DSMC simulations indicate the same behavior for hard-sphere molecules. These results provide strong evidence that the DSMC method can be used to determine the state of a gas under highly nonequilibrium conditions.

On Phase Transition in Compressible Flows: Modelling and Validation

Abstract: A physical model for compressible flows with phase transition is described, in which all the processes of phase transition, i.e. nucleation, droplet growth, droplet evaporation and de-nucleation, are incorporated. The model is focused on dilute mixtures of vapour and droplets in a carrier gas with typical maximum liquid mass fraction smaller than 0.02. The new model is based on a reinterpretation of Hill's method of moments of the droplet size distribution function. Starting from the general dynamic equation, it is emphasized that nucleation or de-nucleation correspond to the rates at which droplets enter or leave droplet size space, respectively. Nucleation and de-nucleation have to be treated differently in agreement with their differences in physical nature. Attention is given to the droplet growth model that takes into account Knudsen effects and temperature differences between droplets and gas. The new phase transition model is then combined with the Euler equations and results in a new numerical method: ASCE2D. The numerical method is first applied to the problem of shock/expansion wave formation in a closed shock tube with humid nitrogen as a driver gas. Nucleation and droplet growth are induced by the expansion wave, and in turn affect the structure of the expansion wave. When the main shock, reflected from the end wall of the low-pressure section, passes the condensation zone, evaporation and de-nucleation occur. As a second example, the problem of the flow of humid nitrogen in a pulse-expansion wave tube, designed to study nucleation and droplet growth in monodisperse clouds, is investigated experimentally and numerically.

Prediction of Darcy-Forchheimer drag for micro-porous structures of complex geometry using the lattice Boltzmann method

Abstract: The micro flows through two-dimensional and three-dimensional granular and fibrous porous media at various Knudsen numbers are studied by using the lattice Boltzmann method. For the granular materials, the results for the medium of rounded inclusions agree well with the existing empirical and numerical correlations between the permeability and the Reynolds number. However, the agreement becomes poor for the medium of sharp-cornered inclusions. A new correlation for the Darcy–Forchheimer drag for various inclusion shapes and arrangements is then proposed. For the fibrous materials, the current results are also compared with existing experimental and numerical data. They are in good agreement. The calculations are further carried out for these porous media in the slip-flow regime. The effect of rarefaction on the permeability in different porous media is discussed. A new correlation between the permeability, the porosity and the Knudsen number for both granular and fibrous porous media is presented.

Simulation of micro- and nano-scale flows via the lattice Boltzmann method

Abstract: We use the lattice Boltzmann method (LBM) for analysis of high and moderate Knudsen number phenomena. Simulation results are presented for microscale Couette and Poiseuille flows. The slip velocity, nonlinear pressure drop, and mass flow rate are compared with previous numerical results and/or experimental data. The Knudsen minimum is successfully predicted for the first time within the LBM framework. These results validate the usage of the LBM based commercial, arbitrary geometry code PowerFLOW for simulating nanoscale problems.

Evaluation of a Particle Method for the Ellipsoidal Statistical Bhatnagar-Gross-Krook Equation

Abstract: A particle method is presented for the numerical simulation of rarefied gas flows, based on the ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) model of the Boltzmann equation The simulation procedure includes consideration of rotational nonequilibrium, and enforces exact momentum and energy conservation for a mixture involving monatomic and diatomic species This method is applied to the simulation of a nozzle flow of the type associated with cold-gas spacecraft thrusters, and flowfield characteristics are compared with experimental data as well as results from direct simulation Monte Carlo (DSMC) and Navier-Stokes simulations of the same flow The ES-BGK method is shown to allow for a relatively high degree of accuracy in transitional flow regimes, while avoiding the intermolecular collision calculations which typically make the DSMC simulation of low Knudsen number flows prohibitively expensive I Introduction n te im the design and performance analysis of low-thrust spacecraft propulsion systems, various numerical simulation chniques may be employed to determine efficiency, thrust characteristics, or the potential for plume pingement and contamination on spacecraft surfaces A particular challenge in the simulation of small thrusters involving a chemically inert neutral gas, such as electro-thermal or cold-gas thrusters, is the accurate consideration of a wide range of Knudsen number regimes In a typical thruster of this type, gas is expelled through a convergentdivergent (Laval) nozzle into a near vacuum, with subsonic near-equilibrium flow in the convergent section of the nozzle As the gas accelerates through the divergent section beyond the nozzle throat, a subsonic boundary layer grows along the wall, while a supersonic core-flow region exists around the nozzle centerline The gas density continues to decrease with downstream distance through the nozzle, and rarefaction effects become more prominent within both the supersonic and subsonic regions Here the gas velocity distribution begins to diverge significantly from the equilibrium limit, and thermal energy is increasingly distributed non-uniformly among the translational and internal degrees of freedom Beyond the nozzle exit plane, these nonequilibrium effects continue to increase as the gas rapidly expands and thermal energy is converted into energy associated with bulk motion of the exhaust flow Rotational freezing occurs due to the large gradients and low collision frequency, and the flow approaches the free molecular limit within a short distance of the nozzle exit, particularly at points far from the nozzle centerline I The simulation of highly rarefied flows, as described above for the divergent nozzle region and plume, is typically performed using the direct simulation Monte Carlo (DSMC) method of Bird 1 This method approximates a numerical solution to the Boltzmann equation – the governing equation for dilute gas flows based on a statistical representation of molecular velocities – by decoupling in time the collision and advection terms in the equation A large number of particles, each representing a large number of atoms or molecules, are tracked through a computational grid, and are sorted into cells according to their location During each time step, some fraction of the particles in a cell collide with each other, and probabilistic techniques are used for calculations of individual collisions All particles are then moved through the grid according to assigned velocities, and particles are created or removed at inflow and outflow boundaries Finally, macroscopic quantities are sampled by averaging various particle properties in each cell, and the process is then repeated at the next time step The DSMC method has been shown to provide accurate solutions for highly rarefied nozzle and plume flows, 2

Measurements and computations of mass flow and momentum flux through short tubes in rarefied gases

Abstract: Gas flows through orifices and short tubes have been extensively studied from the 1960s through the 1980s for both fundamental and practical reasons. These flows are a basic and often important element of various modern gas driven instruments. Recent advances in micro- and nanoscale technologies have paved the way for a generation of miniaturized devices in various application areas, from clinical analyses to biochemical detection to aerospace propulsion. The latter is the main area of interest of this study, where rarefied gas flow into a vacuum through short tubes with thickness-to-diameter ratios varying from 0.015 to 1.2 is investigated both experimentally and numerically with kinetic and continuum approaches. Helium and nitrogen gases are used in the range of Reynolds numbers from 0.02 to 770 (based on the tube diameter), corresponding to Knudsen numbers from 40 down to about 0.001. Propulsion properties of relatively thin and thick tubes are examined. Good agreement between experimental and numerica...