Showing papers on "Knudsen number published in 1991"

The classical incompressible navier-stokes limit of the boltzmann equation

Abstract: The convergence hypothesis of Bardos, Golse, and Levermore,1 which leads to the incompressible Navier-Stokes equation as the limit of the scaled Boltzmann equation, is substantiated for the Cauchy problem with initial data small but independent of the Knudsen number e. The uniform (in e) existence of global strong solutions and their strong convergence as e→0 are proved. A necessary and sufficient condition for the uniform convergence up to t=0, which implies the absence of the initial layer, is also established. The proof relies on sharp estimates of the linearized operators, which are obtained by the spectral analysis and the stationary phase method.

Kinetic study of pulsed desorption flows into vacuum.

Abstract: The one-dimensional flow of particles thermally desorbed from a plane surface into vacuum is studied on the basis of a Monte Carlo simulation of the Boltzmann equation. It is assumed that particles desorb during a finite period of time with a fixed temperature. It is shown then that the flow is determined by a single parameter, which is essentially the number of monolayers \ensuremath{\Theta} desorbed, and is inverse to the Knudsen number of the problem. The cases of negligible (\ensuremath{\Theta}\ensuremath{\ll}1) and in part also of very intense (\ensuremath{\Theta}\ensuremath{\gg}1) desorption fluxes can be treated analytically using collision-free flow and ideal gas dynamics, respectively. Here the simulation yields very good agreement with theory, indicating that the code can be used on a wide range of Knudsen numbers. For the gas-dynamical case, a Knudsen layer is formed by the equilibration of the flow in the vicinity of the surface; it is studied in some detail. Simulation of flows with \ensuremath{\Theta}\ensuremath{\lesssim}1 shows that the particle distribution is far from thermal equilibrium everywhere in the flow, and deviates strongly from the analytically accessible cases. For stronger desorption fluxes, the flow is partially equilibrated, such that the velocity component in the direction of the flow reaches thermal equilibrium, although with a smaller temperature than the velocity component perpendicular to the flow. The time integrated velocity spectra of particles measured far away from the surface are discussed for moderate desorption fluxes. They are surprisingly well described by a thermal distribution, even though no Knudsen layer forms in this case.

Asymptotic Theory of a Steady Flow of a Rarefied Gas Past Bodies for Small Knudsen Numbers

Abstract: A survey is made of the asymptotic behavior for small Knudsen numbers of the time-independent solution of the boundary-value problem of the Boltzmann equation over a general domain. Included is the hydrodynamic system (hydrodynamic type equations and their slip boundary conditions) describing the asymptotic behavior, with several new results.

Thermophoresis in Gases at Small Knudsen Numbers

Abstract: (1991). Thermophoresis in Gases at Small Knudsen Numbers. Aerosol Science and Technology: Vol. 15, No. 2, pp. 77-92.

Knudsen diffusivities and properties of structures of unidirectional fibers

Abstract: Effective Knudsen diffusion coefficients are presented for fibrous structures consisting of parallel, nonoverlapping or partially overlapping fibers They are computed by means of a Monte Carlo simulation scheme which is employed to determine the mean square displacement of molecules travelling in the interior of the porous medium for large travel times The results show that structures of paralle, non-overlapping fibers have smaller effective diffusion coefficients parallel to the fibers than structures of parallel, randomly overlapping fibers of the same porosity and fiber radius, but larger in directions perpendicular to the fibers Partially overlapping fiber structures are found to exhibit behavior intermediate to those of the two extreme cases Molecular trajectory computations are also used to obtain results for the structural properties of partially overlapping fiber structures (eg, porosity and internal surface area, accessible porosity and internal surface area, and percolation threshold), which are compared with some results of the literature for the equivalent problem of partially overlapping disks on a plane

Kinetic theory of thermal transpiration and the mechanocaloric effect: Planar flow of a rigid sphere gas with arbitrary accommodation at the surface

Abstract: The planar flow of a rarefied gas subject to pressure and temperature gradients is a problem of fundamental interest in vacuum science and technology. This problem is solved based on the linearized Boltzmann equation with the assumption of a rigid sphere molecular interaction and Maxwell’s gas–surface interaction operator. Comprehensive numerical results for the macroscopic flow profiles, and the integrated flow rates for arbitrary Knudsen numbers are obtained. The results are basic to study of planar flows of rarefied gases.

Rarefied gas flow through a slit

Abstract: A rarefied gas flow through a slit in an infinite plane wall, induced by a small pressure difference across the slit, is studied on the basis of the kinetic theory. The system of integral equations for the macroscopic variables (the velocity, density, and temperature of the gas), derived from the linearized Boltzmann–Krook–Welander equation with the diffuse reflection boundary condition, is solved by constructing the Neumann series numerically. The flow velocity, density, and temperature fields of the gas as well as the mass flux through the slit are obtained with good accuracy for various Knudsen numbers. The correction to the free molecular flow result for large but finite Knudsen numbers is also obtained analytically, which gives a continuous transition from the free molecular flow to the present numerical result.

Interparticle and Particle-Surface Gas Dynamic Interactions

Abstract: The problems of the gas dynamic interactions of two spherical particles and a single spherical particle with a plane wall are reviewed here. The resistance and mobility functions that describe the relationship between the forces and torques acting on the spheres with their translational and angular velocities are summarized for continuum and near-continuum (small Knudsen number) flows. For intermediate and large Knudsen number (free-molecule) flows, these functions are only known in their asymptotic limit of infinitely large interparticle or particle-wall separation distances (isolated particles). Potentially useful analytical and computational methods to treat these regimes are also reviewed.

An efficient DSMC algorithm applied to a delta wing

Abstract: A new algorithm for 3D direct simulation Monte Carlo (DSMC) is tested and numerical results are compared with wind tunnel data and results obtained earlier with a more traditional DSMC code. The test case is the flowfield around a delta wing at incidence at Knudsen number of 0.016 and Mach number of 20.2. The results are shown to compare favorably with both experimental and earlier numerical results. The new algorithm is described with special emphasis placed on its distinctive features: Cartesian/unstructured combination grid, special body surface definition, discretization in physical space.

Evaporation and condensation of a rarefied gas between its two parallel plane condensed phases with different temperatures and negative temperature-gradient phenomenon — Numerical analysis of the boltzmann equation for hard-sphere molecules —

Abstract: A rarefied gas between its two parallel plane condensed phases is considered, and its steady behavior, especially the rate of evaporation or condensation on the condensed phases and the negative temperature-gradient phenomenon, is studied numerically on the basis of the linearized Boltzmann equation for hard-sphere molecules under the conventional boundary condition and its generalization The method of analysis is the finite-difference method developed recently by the authors Not only the temperature and density distributions and the mass and energy fluxes in the gas but also the velocity distribution function of the gas molecules is obtained with good accuracy for the whole range of the Knudsen number

Kinetic study of evaporation flows from cylindrical jets

Abstract: The steady flow of particles evaporated from a cylindrical jet into vacuum is studied on the basis of a Monte Carlo simulation of the Boltzmann equation. The cases of very small and very large Knudsen numbers Kn can be treated analytically using ideal gas dynamics and collision‐free flow, respectively. Here the simulation yields very good agreement with theory, indicating that the code can be used on a wide range of Knudsen numbers. For the gas‐dynamical case a Knudsen layer is formed by the equilibration of the flow in the vicinity of the jet; it is studied in some detail. Simulation of a flow at Kn≂1 shows that the radial velocity component attains equilibrium nowhere. This example models a recent experiment on the evaporation from liquid water jets, and gives good agreement with the measured speed ratio.

Molecular velocity distribution at large Knudsen numbers

Abstract: It is commonly assumed that the distribution of molecular velocities follows a Maxwellian distribution also in the range of high and ultrahigh vacuum, i.e., at large Knudsen numbers. This distribution is theoretically derived from statistical analysis of an ensemble of molecules with frequent molecule–molecule collisions. Such a situation prevails at higher pressures, i.e., at small Knudsen numbers. However, at smaller pressures the molecule–wall collisions prevail, and due to the different nature of these collisions, the statistical arguments fail. As a consequence, the molecular velocity distribution cannot be predicted a priori at such pressures. In the present paper the applicability of the Maxwellian distribution in vacuum metrology at large Knudsen numbers, i.e., in the molecular regime is investigated. For this purpose, the pressures generated in primary standards of the static and dynamic expansion type are calculated for an arbitrary velocity distribution after expansion. The pressure generated by static expansion is proportional to the expectation value 〈v2z〉, that generated by dynamic expansion proportional to the ratio of expectation values 〈v2z〉/〈‖vz‖〉 (vz denotes the one‐dimensional velocity component). Available experimental calibration data obtained by static and dynamic expansion of common gases are compared to each other and with direct pressure measurements with a liquid column manometer. These data provide a decisive test for the expectation values 〈v2z〉 and 〈‖vz‖〉 in the molecular range. Excellent agreement within the uncertainty of less than 1% between experimental values and values calculated assuming a Maxwellian distribution is found. Thus one may conclude that in common vacuum metrology the Maxwellian distribution can be applied for deriving pressures also in the molecular regime with high accuracy.

Effects of Fiber Orientation and Overlapping on Knudsen, Transition, and Ordinary Regime Diffusion in Fibrous Substrates

Abstract: We prescnt effective diffusion coefficients of gases in porous media whose structure can be represented as an assemblage of cylindrical fibers, such as the media used as substrates in chemical vapor infiltration. Structures consisting of non-, partially, or freely overlapping fibers of various orientation distributions are considered, and effective diffusion coefficients are computed by means of a Monte Carlo simulation scheme. In order to be able to examine the interrelation of ordinary, transition, and Knudsen diffusivities and tortuosities, computations are carried out over the whole diffusion regime, i.e., from bulk to Knudsen. Our simulation results are compared with variational bounds and experimcntal values of tortuosity of fibrous beds reported by other investigators.

The behaviour of a single, horizontally discharged, buoyant flow in a non-turbulent coflowing ambient fluid

Abstract: An integral model of the behaviour of a single horizontally discharged buoyant flow in a coflowing ambient fluid is presented. The usual integral equations for the conservation of mass, momentum and buoyancy flux are used. However, the model differs from those presented previously by using the spread assumption rather than the more usual entrainment assumption and by explicitly forcing the initially gaussian velocity distribution to become a thermal distribution. The position and the form of this transition is determined from laboratory data (Brown [1984], Knudsen [1988]) and field data (Lee and Neville-Jones [1987]). The performance of the model is compared with both laboratory and field data and it reproduces the data for a wide range of cases. A model of this type is essential in gaining an understanding of the range of behaviour of merging plumes (Cheng, Davidson and Wood) and is a preliminary to the understanding of the more general case where the buoyant discharge is ejected at an angle to the flow.

Simulation of molecular transport in pores and pore networks

Abstract: Previous work on the reformulation and refinement of the general theoretical treatment of the transport of simple gases in porous adsorbents in the Knudsen regime and the Henry-law adsorption region is reviewed and developed further by the use of model pores with realistic (‘structured pore wall’) adsorption potentials. Previous conclusions regarding the effects of gas adsorbability and temperature based on simplified (‘structureless pore wall’) adsorption potentials are confirmed and amplified; while the combined effect of gas adsorbability and molecular size is illustrated. The chief factors controlling the transport behaviour of pore networks with randomly varying pore width are noted and examples of their effect given.

Long-time behavior of the one-particle distribution function for the Knudsen gas in a convex domain

Abstract: In this paper some insight into the behavior of the one-particle distribution function describing the Knudsen gas in a convex-plane domain is given. The influence of different kinds of boundary conditions is studied, namely of those following ergodic or chaotic reflection laws and of those with a stable periodic orbit. By applying numerical and theoretical methods it is shown that, under appropriate boundary conditions and after a sufficiently long period of time, there appear enclaves free of particles in the configuration space, and moreover the collection of limiting velocities reached by every particle is finite. On the basis of this result we prove that the one-particle distribution function becomes nonanalytic after a sufficiently long period of time, independent of the initial conditions. On the other hand, it turns out that in our model ergodicity or chaotic properties of the reflection law imply ergodicity or chaotic behavior of the Knudsen gas, respectively. From this result some related properties of the one-particle distribution function of the corresponding Knudsen gas for long periods of time are deduced.

Rarefied Gas Flow Around a 3D-Deltawing

Abstract: This paper presents numerical results for hypersonic flows around a 3D-deltawing at a low Knudsen number. The underlying body geometry as well as the physical parameters correspond to testcase 7.2.1. of the workshop. The numerical method used for the calculations is the Finite-Pointset-Method (FPM) developed at the University of Kaiserslautern since 1987. The paper gives a short introduction to the method and then follows the required output formats of the workshop. Further calculations can be found in [3],[4].

A simple substitute for Knudsen cells using an existing pendant-drop type electron beam evaporator

Abstract: A simple and elegant technique is outlined in which a specially designed crucible may be used for noble metal evaporation in a pendant‐drop type electron beam evaporator with virtually no modifications.

Some problems of the statistical description of hydrodynamic motion and autowave processes

Abstract: The main aim of this report is to expose some ideas, methods and results concerning the statistical description of autowave processes in any active medium and of the hydrodynamic motion in liquids on the basis of the kinetic equations for the joint distribution functions of the internal and space variables of these media. For transition to hydrodynamic equations, “physical Knudsen parameter” is used instead of the Knudsen parameter. The role of self-diffusion is considered. Three examples of active media of small macroscopic elements are presented: a medium of bistable elements, a medium of Van der Pol oscillators and a medium with chemical reactions. In all cases the generalized Fisher-Kolmogorov-Petrovski-Piscunov (FKPP) equations are obtained from the kinetic equations. For the description of kinetic fluctuations, the corresponding Langevin equations are considered.

Numerical analysis of rarefied slit flows. II - Navier-Stokes simulations

Abstract: The model problem of pressure-driven flow of a rarefied monatomic gas through a two-dimensional slit is analyzed via full Navier-Stokes numerical simulation. Parametric solutions are generated for slit-height based Knudsen number ranging from continuum to transitional flow and for reservoir pressure ratios leading to subsonic and supersonic flow. The change in the structure of the flowfield near the slit as a function of pressure ratio and Knudsen number are quantified from a purely continuum standpoint. The choice of numerical domain size, boundary conditions and treatment of the slit are also discussed. As expected, comparison with a Direct Simulation Monte Carlo solution for a highly rarefied case shows large differences in the predicted mass flow. The cause of these differences can be quantified through detailed comparison of the local flowfield properties. For the larger pressure ratio cases qualitative trends with increasing rarefaction are discussed, including the change in the sonic line shape in the slit and the in total mass flow.

Force and Heat Transfer on a Delta Wing in Rarefied Flow

Abstract: For testing numerical methods in the rearefied regime a broad range in parameter variation, especially in the Knudsen number is advisable Therefore the drag, lift total heat transfer and recovery temperatureof a small delta wing (5mm length) was tested in a sonic orifice free jet flow In this report the experimental technique and the main results will be given Details are given in Ref1 where also different correlation parameters are examined by comparing experimental and theoretical results, especially of the workshops in Antibes 1990 and 1991

Hypersonic rarefied flow about a delta wing - Direct simulation and comparison with experiment

Abstract: Three-dimensional simulations of hypersonic rarefied flow about a delta wing are made using the direct simulation Monte Carlo (DSMC) method of Bird, and the results of the computations are compared with recent experimental data obtained in a vacuum wind tunnel at the DLR in Gottingen, Germany. The present study considers Mach 8.89 nitrogen flow for a range of conditions that include Knudsen numbers of 0.016 to 3.505 for an incidence angle of 30 deg, and angles of incidence of 15 to 60 deg for a constant Knudsen number of 0.389. The calculations provide details concerning the flowfield structure and surface quantities. Comparisons between the calculations and the available experimental measurements are made for aerodynamic and overall heat-transfer coefficients and recovery temperature. The agreement between the measured and calculated data are very good, well within the estimated measurement uncertainty. Comparisons are also made with modified Newtonian and free-molecule theories.

Light-induced slip

Abstract: We derive the slip boundary conditions for the velocity and temperature of a one- component gas irradiated by resonant light. In contrast to standard gas kinetics the slip now already arises in zeroth order in the Knudsen number. This zeroth-order slip is therefore induced by the resonant light. Explicit expressions are found in the case of monochromatic excitation, after defining a set of accomodation coefficients that characterize the wall collision kernel. We calculate the light-induced particle flow, pressure difference and heat flow, which arise from the zeroth-order slip.