Showing papers on "Knudsen number published in 2010"

Improved Drag Correlation for Spheres and Application to Shock-Tube Experiments

Abstract: MPIRICALcorrelationsforthequasi-steadydragcoef!cientofa sphere in compressible "ow have been presented by severalauthors (e.g., Henderson [1] and Loth [2]). Such correlations areneeded in numerical simulations of compressible multiphase "owsinvolving spherical particles. In this Note, the accuracy of thecorrelationsofHenderson[1]andLoth[2]areassessedusingthedatacollectedbyBaileyandStarr[3],andanimprovedcorrelationforthedrag coef!cient of a sphere in compressible "ow is developed. Theimproved correlation is validated for shock-particle interaction,using the recent shock-tube experiments of Jourdan et al. [4].

Rarefaction effects on gas viscosity in the Knudsen transition regime

Abstract: The effects of rarefaction on gas viscosity are investigated through the simulation of isothermal, low speed flow in a long straight channel using the Direct Simulation Monte Carlo (DSMC) method. Following convergence to the flow field inside the channel, the effective viscosity is calculated directly from its definition using shear stress calculations in each individual cell assuming that the gas flow is close to a local equilibrium state. Averaging over the cross-sectional area at different positions down the pressure gradient allows the determination of the gas viscosity as a function of the local Knudsen number (Kn) along the channel. Following an extensive investigation of this dependence over a wide range of Kn values, it was conveniently found that a Bosanquet-type of approximation describes very satisfactorily the Knudsen number dependence of the viscosity over the entire transition regime, i.e., from the slip-flow to the free-molecular flow limit. Such a simple functional dependence is expected to facilitate significantly phenomenological descriptions and numerical computations of rarefied flows that rely on the notion of an effective viscosity in the transition regime.

Gas Flow Through Square Arrays of Circular Cylinders with Klinkenberg Effect: a Lattice Boltzmann Study

Abstract: It is well known that, as non-continuum gas flows throughmicroscale porous media, the gas permeability derived fromDarcy law is larger than the absolute permeability, which is caused by the so-called Klinkenberg effect or slippage effect. In this paper, an effective definition of Knudsen number for gas flows through square arrays of circular cylinders and a local boundary condition for non-continuum gas flows are first proposed, and then the multi-relaxation-time lattice Boltzmann equation including discrete effects on boundary condition is used to investigate Klinkenberg effect on gas flow through circular cylinders in square arrays. Numerical results show that the celebrated Klinkenberg equation is only correct for low Knudsen number, and secondorder correction to Klinkenberg equation is necessary with the increase of Knudsen number. Finally, the present numerical results are also compared to some available results, and in general an agreement between them is observed. PACS: 44.05.+e, 47.11.-j, 47.56.+r

Investigation of shock waves in the relativistic Riemann problem: A comparison of viscous fluid dynamics to kinetic theory

Abstract: We solve the relativistic Riemann problem in viscous matter using the relativistic Boltzmann equation and the relativistic causal dissipative fluid-dynamical approach of Israel and Stewart. Comparisons between these two approaches clarify and point out the regime of validity of second-order fluid dynamics in relativistic shock phenomena. The transition from ideal to viscous shocks is demonstrated by varying the shear viscosity to entropy density ratio $\ensuremath{\eta}/s$. We also find that a good agreement between these two approaches requires a Knudsen number $\text{Kn}l1/2$.

Falkner-Skan Flow over a Wedge with Slip Boundary Conditions

Abstract: DOI: 10.2514/1.43316 TheFalkner–Skansolutionforlaminar boundary-layer flowoverawedgeismodifiedtoallow foraslipboundary condition. A modified boundary-layer Knudsen number K is introduced, and the coordinate system is transformed fromone-dimensionaltotwo-dimensionaltoallowforthelossofself-similarityinthe flow.Amarchingschemeisused to solve the boundary-layer equations in the rarefied flow regime. The results of this solution show decreased skin friction, boundary-layer thickness, velocity thickness, and momentum thickness because of the presence of the slip boundary condition. When the energy equation is solved using a temperature-jump boundary condition, the heat transfer increases for slightly rarefied flows, and then decreases as the Knudsen number increases.

Variational derivation of second-order slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules

Abstract: The objective of the present paper is to provide an analytic expression for the first- and second-order velocity slip coefficients. Therefore, gas flow rates in microchannels have been rigorously evaluated in the near-continuum limit by means of a variational technique which applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator. The diffuse-specular reflection condition of Maxwell’s type has been considered in order to take into account the influence of the accommodation coefficient on the slip parameters. The polynomial form of the Knudsen number obtained for the Poiseuille mass flow rate and the values of the velocity slip coefficients, found on the basis of our variational solution of the linearized Boltzmann equation for hard-sphere molecules, are analyzed in the frame of potential applications of classical continuum numerical tools in simulations of microscale flows.

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

Abstract: 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.

Shock and boundary structure formation by spectral-lagrangian methods for the inhomogeneous boltzmann transport equation *

Abstract: The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for 1D in x times 3D in v in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with difiusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefled state. Finally, the classical inflnite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the ∞ow in the flnal case turns from a supersonic ∞ow to a subsonic ∞ow across the shock.

Evaluation of a lattice Boltzmann method in a complex nanoflow

Abstract: In order to establish a cost-effective strategy to simulate complex flows in continuum to slip and transitional regimes, the present study assesses the performance of a lattice Boltzmann method (LBM) formerly discussed by the present authors' group [Niu et al., Phys. Rev. E 76, 036711 (2007)]. This LBM is based on a diffuse scattering wall boundary condition, a regularization procedure, and an effective relaxation time associated with the Knudsen number. The present assessment is on its regularization procedure and third-order truncated system based on the two-dimensional twenty-one discrete velocity (D2Q21) model for the Cartesian lattices. The test flow cases are force-driven Poiseuille flows, the Couette flows and a flow around a square cylinder situated in a nanochannel. For producing the reference data of the square cylinder flow, the molecular dynamics simulation using Lennard-Jones potential is also performed. Although the flow profiles and the slip velocities of the Poiseuille flows and the Couette flows are more accurately predicted by the third-order truncated system, the general velocity profiles around the square cylinder are also well predicted by the second-order truncated system based on the two-dimensional nine discrete velocity (D2Q9) model. It is also confirmed that without the regularization process, the entire flow field prediction suffers unphysical momentum oscillations around the square cylinder.

Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities

Abstract: The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exist a family of solutions to the Boltzmann equation globally in time for small Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number e tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.

Higher-order Galilean-invariant lattice Boltzmann model for microflows: single-component gas.

Abstract: We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation (without any cubic error). In the first part of this work, we show that the present model can capture two important features of the microflow in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model. Finally, our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature-based procedure for the construction of the lattice Boltzmann model.

Viscous Dissipation and Rarefaction Effects on Laminar Forced Convection in Microchannels

Abstract: Fluid flow in microchannels has some characteristics, which one of them is rarefaction effect related with gas flow. In the present work, hydrodynamically and thermally fully developed laminar forced convection heat transfer of a rarefied gas flow in two micro-geometries is studied, namely, microannulus and parallel plate microchannel. The rarefaction effects are taken into consideration using first-order slip velocity and temperature jump boundary conditions. Viscous heating is also included for either the wall heating or the wall cooling case. Closed form expressions are obtained for dimensionless temperature distribution and Nusselt number. The results demonstrate that for both geometries, as Brinkman number increases, the Nusselt number decreases. However, the effect of viscous heating on the Nusselt number at greater values of Knudsen number becomes insignificant. In the absence of viscous heating, increasing values of Knudsen number lead to smaller values of Nusselt number. Furthermore, it is observed that viscous heating causes singularities in Nusselt number values. Also, asymmetry causes singularities in Nusselt numbers of both microannulus walls and the parallel plate wall having lower heat flux, even in the absence of viscous heating. For parallel plate microchannel, in the absence of viscous heating, Nusselt number of the wall having larger heat flux is an increasing function of the wall heat fluxes ratio.

Comparative study between computational and experimental results for binary rarefied gas flows through long microchannels

Abstract: A comparative study between computational and experimental results for pressure-driven binary gas flows through long microchannels is performed. The theoretical formulation is based on the McCormack kinetic model and the computational results are valid in the whole range of the Knudsen number. Diffusion effects are taken into consideration. The experimental work is based on the Constant Volume Method, and the results are in the slip and transition regime. Using both approaches, the molar flow rates of the He–Ar gas mixture flowing through a rectangular microchannel are estimated for a wide range of pressure drops between the upstream and downstream reservoirs and several mixture concentrations varying from pure He to pure Ar. In all cases, a very good agreement is found, within the margins of the introduced modeling and measurement uncertainties. In addition, computational results for the pressure and concentration distributions along the channel are provided. As far as the authors are aware of, this is the first detailed and complete comparative study between theory and experiment for gaseous flows through long microchannels in the case of binary mixtures.

Theoretical and Numerical Studies of Noncontinuum Gas-Phase Heat Conduction in Micro/Nano Devices

Abstract: This article presents a comprehensive study of various modeling techniques for noncontinuum gas-phase heat conduction encountered in micro/nano devices over a broad range of Knudsen number. A new slip model is proposed for slip flows and an analytical approach is developed for collisionless steady-state heat conduction inside a fully diffuse enclosure. Excellent agreements with direct simulation Monte Carlo (DSMC) simulations have been achieved for both of them. For problems in the transition regime and/or with partially thermal accommodated walls, the DSMC method is employed. Some noncontinuum phenomena such as the steady gas flows induced by the nonuniform temperature field are observed.

Origin of Knudsen forces on heated microbeams.

Abstract: The presented work probes the fundamentals of Knudsen forces. Using the direct simulation Monte Carlo (DSMC) method, the flows induced by temperature inhomogeneity within a representative configuration and the Knudsen force acting on a heated microbeam are captured as functions of Knudsen number in the entire flow regime. Both flow strength and Knudsen force peak in the transition regime and negative Knudsen force absent in experimental data is observed. The mechanisms of the thermally induced flows and Knudsen forces are studied. It has been found that thermal edge flow is the main driven source for the formation of the Knudsen force on microbeams and domain configuration plays an important role in the process.

Corrected second-order slip boundary condition for fluid flows in nanochannels.

Abstract: A corrected second-order slip boundary condition is proposed to solve the Navier-Stokes equations for fluid flows confined in parallel-plate nanochannels. Compared with the classical second-order slip boundary condition proposed by Beskok and Karniadakis, the corrected slip boundary condition is not only dependent on the Knudsen number and the tangential momentum accommodation coefficient, but also dependent on the relative position of the slip surface in the Knudsen layer. For the fluid flows in slip-flow regime with the Knudsen number less than 0.3, Couette cell is investigated using molecular-dynamics simulations to verify Newtonian flow behaviors by examining the constitutive relationship between shear stress and strain rate. By comparing the velocity profiles of Poiseuille flows predicted from the Navier-Stokes equations with the corrected slip boundary condition with that from molecular-dynamics simulations, it is found that the flow behaviors in our models can be effectively captured.

Quadrature-Based Moment Model for Moderately Dense Polydisperse Gas-Particle Flows

Abstract: A quadrature-based moment model is derived for moderately dense polydisperse gas-particle flows starting from the inelastic Boltzmann-Enskog kinetic equation including terms for particle acceleration (e.g., gravity and fluid drag). The derivation is carried out for the joint number density function, f(t,x,m,u), of particle mass and velocity, and thus, the model can describe the transport of polydisperse particles with size and density differences. The transport equations for the integer moments of the velocity distribution function are derived in exact form for all values of the coefficient of restitution for particle-particle collisions. For particular limiting cases, the moment model is shown to be consistent with hydrodynamic models for gas-particle flows. However, the moment model is more general than the hydrodynamic models because its derivation does not require that the particle Knudsen number (and Mach number) be small.

Network numerical analysis of optically thick hydromagnetic slip flow from a porous spinning disk with radiation flux, variable thermophysical properties, and surface injection effects

Abstract: The steady, incompressible, laminar Newtonian magnetohydrodynamic slip flow with heat transfer from an impulsively started, spinning porous disk is investigated when strong injection (blowing) and significant thermal radiation heat transfer are present. The properties of the fluid, i.e., density, viscosity, and thermal conductivity, are assumed to vary with temperature. Using appropriate transformations, the axisymmetric flow conservation equations for mass, momentum, and energy in a cylindrical polar coordinate system (r, ϕ, z) are normalized to yield a series of highly nonlinear, coupled ordinary differential equations that are solved under appropriate boundary conditions with the network simulation method (NSM). Comparisons are made with an earlier study for the case of Prandtl number = 0.64 with suction present and found to be in excellent agreement. The effects of the radiation-conduction parameter (Nr), hydromagnetic parameter (Nm), slip factor (γ, which is related to Knudsen number), uniform inject...

Slow gas microflow past a sphere: Analytical solution based on moment equations

Abstract: The regularized 13-moment equations are solved analytically for the microflow of a gas past a sphere in the case of low Mach numbers. The result is given in fully explicit expressions and shows nontrivial behavior for all fluid fields including stress, heat flux, and temperature. Various aspects of the flow such as temperature polarization and total force are reproduced correctly for moderate Knudsen number. The analytical solution allows studying the rise of Knudsen layers and their interaction and coupling to the fluid variables in the bulk. Additionally, based on the regularized 13-moment equations system, hybrid boundary conditions are given for the standard Stokes equations in order to enable them to predict nonequilibrium effects in the flow past a sphere.

Modeling pure gas permeation in nanoporous materials and membranes.

Abstract: The low-pressure transport of simple fluids in nanopores and in disordered nanoporous networks is analyzed, using a recent oscillator model theory from the author's laboratory, considering the trajectories of molecules moving in the potential energy field of the fluid-pore wall interaction. The scaling behavior of the single-pore theory is discussed, and it is shown that the Knudsen model provides an upper bound to the diffusivity scaled with the pore radius. The single-pore theory is shown to apply well to ordered materials and successfully interprets recent literature data on the variation of permeability with diffusant molecular size for a DDR zeolite membrane. A peak in permeability is seen at a pore-size-dependent molecular size because of the opposing effects of equilibrium and transport. Application to disordered pore networks is also presented on the basis of a hybrid correlated random walk effective medium theory imbedding the oscillator model at the single-pore level, and a rigorous expression for the tortuosity is derived from the theory. A rich variety of behavior is predicted for the tortuosity, which can increase or decrease with increasing extent of pore size nonuniformity as well as with changes in temperature because the diffusing species preferentially flows through more conducting pores. Weakly adsorbing gases such as helium are seen to have a higher tortuosity than more strongly adsorbing ones. The predicted values of tortuosity are shown to be in line with those obtained from the interpretation of recent experimental mesoporous membrane transport data and are in the range of 5-10 whereas those extracted using the Knudsen model are unrealistically high, in the range of 10-20.

Heat transfer through rarefied gases between coaxial cylindrical surfaces with arbitrary temperature difference

Abstract: The problem of nonlinear heat transfer through a rarefied gas confined between concentric cylinders maintained at different temperatures is investigated. The formulation is based on the nonlinear Shakhov kinetic model subject to Cercignani–Lampis boundary conditions, while molecular interaction is modelled by the inverse power law. The detailed behaviour of the radial heat flow, density, temperature and pressure distributions in terms of the normalized temperature difference between the cylindrical walls, the ratio of the two cylindrical radii and the gas rarefaction is investigated and certain interesting characteristics are revealed. The study includes small, moderate and large temperature differences and various radius ratios and is extended in the whole range of the Knudsen number. It is verified that the type of molecular interaction plays an important role when the heat transfer configuration becomes strongly nonlinear, while the influence of the gas–surface scattering law has similar effects both in linear and nonlinear conditions. By comparing linear and nonlinear results corresponding to the same conditions, it is concluded that linearized analysis can capture the correct behaviour of the heat flow configuration not only for infinitesimally small but also for finite temperature differences and that its range of applicability is wider than expected.

Rarefaction effects in thermally-driven microflows

Abstract: Rarefied gas flow in a parallel-plate micro-channel is considered, where a streamwise constant temperature gradient is applied in the channel walls. An analytical approach to the problem is conducted based on linearized and semi-linearized forms of the regularized 13-moment equations (R13 equations), which are a set of macroscopic transport equations for rarefied gases at the super-Burnett order. Typical nonequilibrium effects at the boundary, i.e., velocity slip, temperature jump, and formation of Knudsen boundary layers are investigated. Nonlinear contributions lead to temperature, density, and normal stress profiles across the channel which are not reported elsewhere in literature.

Combination of a Knudsen effusion cell with a quartz crystal microbalance: in situ measurement of molecular evaporation rates with a fully functional deposition source.

Abstract: We describe a straightforward, reliable, and inexpensive design of a Knudsen type molecular effusion cell capable of measuring molecular evaporation rates in situ. This is accomplished by means of a quartz crystal microbalance integrated into the shutter of the effusion cell. The presented layout facilitates both the measurement of effusion rates under ultrahigh vacuum conditions without the need for a separate experimental setup and the growth of surface supported molecular layers and nanostructures. As an important prerequisite for reproducible deposition of molecular films with defined coverages ranging from submonolayers up to multilayers, the Knudsen cell features a stable deposition rate for crucible temperatures between 50 and 500 °C. Experimental determination of deposition rates for different crucible temperatures allows to approximate sublimation enthalpies of the evaporant based on the Clausius–Clapeyron equation.