Intermediate Knudsen Numbers Research Articles

Kinetic modelling of rarefied gas flows with radiation

Two kinetic models are proposed for high-temperature rarefied (or non-equilibrium) gas flows with internal degrees of freedom and radiation. One of the models uses the Boltzmann collision operator to model the translational motion of gas molecules, which has the ability to capture the influence of intermolecular potentials, while the other adopts the relaxation time approximations, which has higher computational efficiency. In our kinetic model equations, not only the transport coefficients such as the shear/bulk viscosity and thermal conductivity but also their underlying relaxation processes are recovered. The non-equilibrium dynamics of gas flow and radiation are tightly coupled, where the transport properties of gas molecules and photons are correlatively dependent. The proposed kinetic models are validated by the direct simulation Monte Carlo method in several non-radiative rarefied gas flows (e.g. the normal shock wave, Fourier flow, Couette flow and the creep flow driven by the Maxwell demon), and the experimental data of planar heat transfer and normal shock waves in nitrogen. Then, the rarefied gas flows with strong radiation are studied based on the kinetic models, not only in the above one-dimensional gas flows, but also in the two-dimensional radiative hypersonic flow passing a cylinder. The characteristics of heat transfer in the tightly coupled fields of gas and radiation are systematically investigated, particularly the influence of the non-equilibrium photon transport and their interactions with gas molecules are revealed. It is found that the radiation makes a profound contribution to the total heat transfer in radiative hypersonic flow at an intermediate photon Knudsen number.

A modified lattice Boltzmann model for microcylindrical Couette gas flows

We develop an axisymmetric lattice Boltzmann (LB) model to simulate microcylindrical Couette gas flows (MCGF) in the slip regime and transition regime, respectively. A kinetic boundary scheme in the cylindrical coordinate system is proposed to fulfill the second-order slip boundary condition at the cylindrical wall. To consider the effect of the Knudsen layers for transition flows, local effective Knudsen numbers are introduced into the kinetic boundary scheme and relaxation time. Numerical tests are executed to acquire the velocity distributions of the time-independent and time-dependent MCGF. Comparisons with the analytical solution and direct Monte Carlo data are also implemented. The simulation results demonstrate that the developed LB model can successfully acquire the velocity distribution of the MCGF with an intermediate Knudsen number in the transition regime.

Heat transfer in a dense gas between two parallel plates

Time-dependent heat transfer in a dense gas between two parallel plates, which is initiated by an abrupt change in temperature of one plate, is numerically investigated on the basis of the Enskog equation under the diffuse reflection boundary condition. Numerical computation is carried out by a finite-difference scheme combined with the Fourier spectral method for the efficient computation of the collision term of the Enskog equation. As a result, macroscopic quantities of the gas, such as heat flux and temperature, are obtained for various Knudsen numbers and ratios of the molecular diameter to the distance between plates. Compared to the case of an ideal gas, the heat flux in the stationary state is enhanced due to an effect of the finite size of molecules for not only small but also intermediate Knudsen numbers. The results imply that the finite-size effect also affects the propagation of disturbances in the initial stage, particularly for small Knudsen numbers.

Inversion of the transverse force on a spinning sphere moving in a rarefied gas

The flow around a spinning sphere moving in a rarefied gas is considered in the following situation: (i) the translational velocity of the sphere is small (i.e. the Mach number is small); (ii) the Knudsen number, the ratio of the molecular mean free path to the sphere radius, is of the order of unity (the case with small Knudsen numbers is also discussed); and (iii) the ratio between the equatorial surface velocity and the translational velocity of the sphere is of the order of unity. The behaviour of the gas, particularly the transverse force acting on the sphere, is investigated through an asymptotic analysis of the Boltzmann equation for small Mach numbers. It is shown that the transverse force is expressed as $\boldsymbol{F}_L = {\rm \pi}\rho a^3 (\boldsymbol{\varOmega} \times \boldsymbol{v}) \bar{h}_L$ , where $\rho$ is the density of the surrounding gas, a is the radius of the sphere, $\boldsymbol {\varOmega }$ is its angular velocity, $\boldsymbol {v}$ is its velocity and $\bar {h}_L$ is a numerical factor that depends on the Knudsen number. Then, $\bar {h}_L$ is obtained numerically based on the Bhatnagar–Gross–Krook model of the Boltzmann equation for a wide range of Knudsen number. It is shown that $\bar {h}_L$ varies with the Knudsen number monotonically from 1 (the continuum limit) to $-\tfrac {2}{3}$ (the free molecular limit), vanishing at an intermediate Knudsen number. The present analysis is intended to clarify the transition of the transverse force, which is previously known to have different signs in the continuum and the free molecular limits.

The worm-LBM, an algorithm for a high number of propagation directions on a lattice Boltzmann grid: The case of phonon transport

The lattice Boltzmann method (LBM) is a numerical approach to tackle problems described by a Boltzmann type-equation, where time, space, and velocities are discretized to describe scattering and advection. Even though the LBM executes advection along a lattice direction without numerical error, its usage in the high Knudsen number regime (ballistic) has been hindered by the ray effect problem (for dimensions greater than 1D). This problem has its origin in the low number of available propagation directions on standard LBM lattices. Here, to overcome this limitation, we propose the worm-lattice Boltzmann method (worm-LBM), which allows a high number of lattice directions by alternating in time the basic directions described within the next neighbor schemes. Additionally, to overcome the velocity anisotropy issue, which otherwise clearly manifests itself in the ballistic regime (e.g. the 2 higher grid velocity of the D2Q8 scheme along the diagonal direction compared to the axial one), the time-adaptive scheme (TAS) is proposed. The TAS method makes use of pausing advection on the grid, allowing to impose not only isotropic propagation but also arbitrary direction-dependent grid velocity. Last but not least, we propose a grid-mean free path (grid-MFP) correction to correctly handle the aforementioned velocity issue in the diffusive limit, without affecting the ballistic one. We provide a detailed description of the TAS method and the worm-LBM algorithm, and verify their numerical accuracy by using several transient diffusive-ballistic phonon transport cases, including different initial and boundary conditions. We demonstrate the accuracy of the new worm-LBM to describe problems where a high angular resolution (i.e. a high number of propagation directions) is required, as the in-plane thermal transport problem under adiabatic-diffusive boundary conditions. In this particular case, we show that schemes with a low number of propagation directions (D2Q8) result in an overestimation of the analytical Fuchs-Sondheimer solution for intermediate and high Knudsen numbers, and that schemes with a higher number of propagation directions are required to correctly describe the problem. Overall, the new, very accurate, and efficient worm-LBM algorithm, free of numerical smearing and false scattering, has the potential to be at the forefront of the numerical solvers to tackle the advective part of different equations in a wide field of applications.

Barodiffusion in Slow Flows of a Gas Mixture

Barodiffusion in slow flows of a gas mixture is studied with an approximation using hydrodynamic equations of motion for the individual mixture components. It is shown that consideration of the viscous momentum transfer and the contribution of Knudsen layers for the mixture flowing in a channel has a considerable effect on the value of the barodiffusion factor. The relations are obtained for the mean diffusion fluxes of components and for the total flux of the mixture in a circular cylindrical capillary; these relations are valid for moderately small Knudsen numbers used for calculation of the diffusion baroeffect and separation effect when the gas mixture flows in a set of capillaries connecting two volumes. The modification of the relations for the barodiffusion factor (and for the diffusion slip coefficient cross-linked with it) allows interpreting the sign alteration of these effects observed experimentally for some gas mixtures at intermediate Knudsen numbers.

О бародиффузии при медленных течениях газовой смеси

AbstractBarodiffusion in slow flows of a gas mixture is studied with an approximation using hydrodynamic equations of motion for the individual mixture components. It is shown that consideration of the viscous momentum transfer and the contribution of Knudsen layers for the mixture flowing in a channel has a considerable effect on the value of the barodiffusion factor. The relations are obtained for the mean diffusion fluxes of components and for the total flux of the mixture in a circular cylindrical capillary; these relations are valid for moderately small Knudsen numbers used for calculation of the diffusion baroeffect and separation effect when the gas mixture flows in a set of capillaries connecting two volumes. The modification of the relations for the barodiffusion factor (and for the diffusion slip coefficient cross-linked with it) allows interpreting the sign alteration of these effects observed experimentally for some gas mixtures at intermediate Knudsen numbers.

A simple model for the time evolution of the condensation sink in the atmosphere for intermediate Knudsen numbers

Abstract. Transformation of the mass flux towards the particle from the kinetic regime to the continuum regime is often described by the Fuchs–Sutugin coefficient. Kinetic regime can be obtained as a limiting case when only one term of the expansion of the Fuchs–Sutugin coefficient at small 1∕Kn is considered. Here we take the two first terms into account, and get a mass flux which agrees well with the full mass flux down to Kn∼0.5. This procedure allows an analytical solution of the condensation equation valid for the range of intermediate Knudsen numbers to be obtained. The expansion is further applied to analytically calculate the condensation sink. The formula for the condensation sink is tested against field observations. The relative contribution of different aerosol modes to the condensation sink is discussed. Furthermore, we present a simple model describing the coupled dynamics of the condensing vapour and the condensation sink. The model gives reasonable predictions of condensation sink dynamics during the periods of the aerosol modes' growth by condensation in the atmosphere.

Diffusion approximation and short-path statistics at low to intermediate Knudsen numbers

In the field of first-return statistics in bounded domains, short paths may be defined as those paths for which the diffusion approximation is inappropriate. However, general integral constraints have been identified that make it possible to address such short-path statistics indirectly by application of the diffusion approximation to long paths in a simple associated first-passage problem. This approach is exact in the zero Knudsen limit (Blanco S. and Fournier R., Phys. Rev. Lett., 97 (2006) 230604). Its generalization to the low to intermediate Knudsen range is addressed here theoretically and the corresponding predictions are compared to both one-dimension analytical solutions and three-dimension numerical experiments. Direct quantitative relations to the solution of the Schwarzschild-Milne problem are identified.

On the description of nanoparticle diffusion in gases

Analyses are presented using both experimental and simulation data for nanoparticle diffusion and mobility in gases. These analyses show that it is important to include the diameter of the bath gas when calculating effective cross sections, if simple dependencies are desired. This can be especially important when analyzing experimental data for nanoparticles where accurate values are difficult to obtain. In particular, we show there is little evidence for a transition from specular to diffuse in the nature of the scattering for nanoparticles. The accuracy of the Stokes–Millikan equation, commonly used to fit diffusion and mobility data, is tested for hard sphere diffusion and found to give good predictions, albeit with different parameters than typically used for experimental data. In addition, a new empirical formula, introduced by us previously to fit simulation data, was tested against experimental data and found to be adequate. One strength of this formula is its explicit inclusion of experimentally determined quantities and its reduced number of adjustable parameters. Its predictions are most accurate for intermediate and large Knudsen numbers, the values found in practice for nanoparticle systems.

Gas motion in a microgap between a stationary plate and a plate oscillating in its normal direction

Unsteady motion of a gas between two parallel plates is investigated in the case where one of the plates starts (harmonic) oscillation in its normal direction. A kinetic–theoretic approach is employed under the condition that the distance between the two plates is comparable to the mean free path of the gas molecules and/or the frequency of oscillation of the plate is comparable to their mean collision frequency. More specifically, the Bhatnagar–Gross–Krook model of the Boltzmann equation is solved numerically for wide ranges of parameters, such as the Knudsen number and the Mach number, with special interest in the fully nonlinear wave motion. As the result, the time evolution of the local flow field and the periodic state attained at later times are obtained accurately. It is shown that, in the periodic state, one-period average of the momentum (or energy) transferred from the oscillating to the stationary plate takes a nonzero value in contrast to the linear theory, and it becomes minimum at an intermediate Knudsen number (for a given oscillation of the plate and for a given distance between the center of the oscillating plate and the stationary plate).

A Kinetic Model for Gas Mixtures Based on a Fokker-Planck Equation

We present a generalized nonlinear Fokker–Planck equation, which describes the dynamics of rarefied monatomic gas mixture flows. The devised kinetic model leads to correct transfer of energy and momentum between gas species and to consistent evolution of molecular stresses and heat fluxes with respect to the generalized Boltzmann equation. Thus, the correct diffusion coefficient together with the mixture viscosity and mixture heat conductivity coefficients are obtained. The strength of the presented model lies on the computational efficiency, which is due to the fact that the resulting stochastic processes are continuous in time. Therefore, unlike in Direct Simulation Monte Carlo (DSMC), here simulated particles do not collide with each other, but move along independent continuous stochastic paths. Another aspect of the new Fokker-Planck model equation is that here the effect of collisions is described via drift and diffusion type processes. Accordingly, a scheme can be derived for which the time step size limitation of the corresponding numerical simulation becomes independent of the Knudsen number. Consequently, this leads to more efficient simulations, especially in low or intermediate Knudsen numbers. Results are presented for helium-argon mixture in a one dimensional geometry. The calculated mixture viscosity is found to be in accordance with experimental data, which reveals the accuracy and relevance of the approach.

CFD by first order PDEs

This research originally was aimed at modeling all flows (except free-molecular) by systems of hyperbolic-relaxation equations (moments of the Boltzmann equation), and developing efficient numerical methods for these. Such systems have many potential numerical advantages, mainly because there are no second or higher derivatives to be approximated. This avoids accuracy problems on adaptive unstructured grids, and the source terms, though often stiff, are only local; the compact stencils facilitate code parallelization. A single code could simulate flows up to intermediate Knudsen numbers, and be hybridized with DSMC where needed. In this project, one major problem arose that we have not yet solved: the accurate representation of shock structures. This makes the methodology currently unsuited for, e.g., re-entry flows. We have validated it for subsonic and transonic flows and are concentrating on applications to MEMS-related flows. We discuss the challenges of our approach, present numerical algorithms and results based on the 10-moment model, and report progress in our latest research topic: formulating accurate solid-boundary conditions.

Light-induced heat and mass transfer in a single-component gas in a capillary

A theoretical analysis is presented of light-induced heat and mass transfer in a single-component gas in a capillary tube at arbitrary Knudsen numbers. Surface and collisional mechanisms of transfer are analyzed, due to differences in accommodation coefficient and collision cross section between excited-and ground-state particles, respectively. Analytical expressions for kinetic coefficients characterizing the gas drift and heat transfer in a capillary tube are obtained in the limits of low and high Knudsen numbers. Numerical computations are performed for intermediate Knudsen numbers. Both drift and heat fluxes are determined as functions of the light beam frequency. In the case of an inhomogeneously broadened absorption line, the light-induced fluxes are found to depend not only on the sign, but also on the amount, of light beam detuning from the absorption line center frequency.

Effect of the droplet-size dependence of saturated-vapor pressure on the dynamics of supersaturated-vapor condensation relaxation

409 In this paper, we analyze the variation of the dropletsize distribution functions, of the fractional droplet composition for a condensation aerosol, and of certain integral characteristics (temperature, supersaturation, and the droplet-number density) that occur in the course of condensation-relaxation process. The consideration is restricted to the period after the rapid creation of the supersaturated state in a vapor‐gas mixture as a result of adiabatic expansion. In order to describe the dynamics of the droplet-size distribution function, we make use of the corresponding kinetic equation. This equation is solved by the direct numerical method. The results are obtained for growing droplets in the cases of large and intermediate Knudsen numbers with and without allowance for the dependence of the saturatedvapor pressure on the droplet radius. The analysis of the results has allowed us to reach a conclusion about how the droplet-size dependence of the saturated-vapor pressure affects the dynamics of the condensationrelaxation process.

A history and state-of-the-art of accommodation coefficients

This paper reviews theory and measurements of transport processes between small particles and the surrounding gas. Evaporation and condensation coefficients and gas uptake coefficients are of particular interest. There has long been a great difference in coefficients reported by different experimentalists, and much of this disagreement is considered in this overview. A brief review of the kinetic theory of gases is provided to describe molecular transport to or from a surface when the mean free path of molecules, ℓ, is large compared with the particle dimensions, that is, when the Knudsen number is large. For a sphere of radius a the Knudsen number, Kn = ℓ / a. The condition Kn >> 1 is called the free molecule regime, and for Kn << 1 continuum theory applies. It is shown that accommodation coefficients cannot be determined by experiments operating in the continuum regime. At intermediate Knudsen numbers (the Knudsen regime) transport theory is more difficult, but results based on solution of the Boltzmann equation describing the evolution of the molecular velocity distribution are reviewed. With the advent of high-speed computers transport theory has been supplemented by molecular dynamics calculations, and these calculations are often at odds with experimental measurements of accommodation coefficients. Examples are provided. Experimental methods surveyed include Knudsen cell methods, jet tensimetry, electrodynamic levitation experiments, expansion cloud chamber measurements, and vibrating orifice aerosol generator (VOAG) techniques. VOAG measurements are particularly useful for studying processes over small times, and theory and results related to VOAG experiments are presented. More recent experimental measurements of condensation coefficients involving the use of molecular beams are reported. There is a growing body of evidence that accommodation coefficients are of order unity in many cases, but coefficients smaller than 0.01 are still reported, particularly for uptake coefficients.

On the Spectral Problems for the Discrete Boltzmann Models

The discrete Boltzmann models are used to study the spectral problems related to the one-dimensional plane wave propagation in monatomic gases which are fundamental in the nonequilibrium statistical thermodynamics. The results show that the 8-velocity model can only describe the propagation of the diffusion mode (entropy wave) in the intermediate Knudsen number regime. The 4- and 6-velocity models, instead, can describe the propagation of sound modes quite well, after comparison with the continuum-mechanical results.

Spectral problems for the discrete velocity models

We use the discrete velocity models to study the spectral problems related to the 1D plane wave propagation in monatomic gases which are fundamental in the rarefied gases dynamics and nonequilibrium statistical thermodynamics. The results show that 6- and 8-velocity models can only capture the propagation of diffusion mode (entropy wave) in the intermediate Knudsen number regime. 4-velocity model instead captures the propagation of sound mode quite well after the comparison with the continuum-mechanic results.

Diffusional deposition of finite size aerosol, particles in model filter at intermediate Knudsen numbers

Diffusional deposition of finite size aerosol, particles in model filter at intermediate Knudsen numbers

15.P.07 Aerosol filtration by fibrous filters at intermediate knudsen numbers

15.P.07 Aerosol filtration by fibrous filters at intermediate knudsen numbers