Wide Range Of Gas Rarefaction Research Articles

Sensitivity analysis of the Cercignani - Lampis accommodation coefficients in prototype rarefied gas flow and heat transfer problems via the Monte Carlo method

In rarefied gas dynamics, the Cercignani-Lampis (CL) scattering kernel, containing two accommodation coefficients (ACs), namely the tangential momentum and normal energy ones, is widely employed to characterize gas-surface interaction, particularly in non-isothermal setups, where both momentum and energy may simultaneously be exchanged. Here, a formal and detailed sensitivity analysis of the effect of the CL ACs on the main output quantities of several prototype problems, namely the cylindrical Poiseuille, thermal creep and thermomolecular pressure difference (TPD) flows, as well as the plane Couette flow and heat transfer (Fourier flow), is performed. In each problem, some uncertainties are randomly introduced in the ACs (input parameters) and via a Monte Carlo propagation analysis, the deduced uncertainty of the corresponding main output quantity is computed. The output uncertainties are compared to each other to determine the flow configuration and the gas rarefaction range, where a high sensitivity of the output quantities with respect to the CL ACs is observed. The flow setups and rarefaction regimes with high sensitivities are the most suitable ones for the estimations of the ACs, since larger modeling and experimental errors may be acceptable. In the Poiseuille and Couette flows, the uncertainties of the flow rate and shear stress respectively are several times larger than the input uncertainty in the tangential momentum AC and much smaller than the uncertainty in the normal energy AC in a wide range of gas rarefaction. In the thermal creep flow, the uncertainty of the flow rate depends on the input ones of both ACs, but, in general, it remains smaller than the input uncertainties. A similar behavior with the thermal creep flow is obtained in the TPD flow. On the contrary, in the Fourier flow, the uncertainty of the heat flux may be about the same or even larger than the input ones of both ACs in a wide range of gas rarefaction. It is deduced that in order to characterize the gas-surface interaction via the CL ACs by matching computations with measurements, it is more suitable to combine the Poiseuille (or Couette) and Fourier configurations, rather than, as it is commonly done, the Poiseuille and thermal creep ones. For example, in order to estimate the normal energy AC within an accuracy of 10 %, experimental uncertainties should be less than 4 % in the thermal creep or TPD flows, while may be about 10 % in the Fourier flow.

Rarefied gas flow into vacuum through linearly diverging and converging channels

Using the direct simulation Monte Carlo direct method, rarefied gas flow into a vacuum through linearly diverging and сonverging сhannels has been studied. The problem has been solved in the complete geometrical setup, namely with including certain pre- and post-channel regions in the geometry under consideration. In a wide range of gas rarefaction, a mass flow rate through the channel, the gas flow diodicity, and flow field both inside the channel and in the regions upstream and downstream have been computed. Computation results are compared to corresponding data for the channel with a constant cross-section. A strong effect of channel geometry and gas rarefaction has been stated. In particular, in the dependence of the mass flow rate on the gas rarefaction, a so-called Knudsen minimum has been revealed; while in the flow field, one-dimensional distribution of gas density in the channel has been found.

Aerothermodynamics of a sphere in a monatomic gas based onab initiointeratomic potentials over a wide range of gas rarefaction: transonic, supersonic and hypersonic flows

Aerothermodynamic characteristics of a sphere such as drag and energy transfer coefficients are calculated for Mach numbers varying from 1 to 10 over a wide range of the gas rarefaction degree spanning the free molecular, transitional and near hydrodynamic flow regimes. The effects of major factors determining the gas flow are studied using the direct simulation Monte Carlo method. To reveal the effect of gas species, the simulations are performed based onab initiointeratomic potentials for helium, neon, argon and krypton, as well as based on the hard sphere model. The impact of the accommodation coefficients is evaluated by applying the Cercignani–Lampis model of gas–surface interaction. The calculations are carried out for several values of the free stream and sphere temperatures. It is found that the effects of gas species on the drag and energy transfer coefficients are approximately 3 % and 6 %, respectively. The accommodation coefficients in the Cercignani–Lampis model strongly affect all aerothermodynamic characteristics. The drag and energy transfer coefficients calculated for different accommodation coefficients vary within 30 % and 200 %, respectively. It is found that the variation of the tangential momentum and normal energy accommodation coefficients can induce an increase of the drag coefficient compared to the case of diffuse gas–surface interaction. In hypersonic flows, the drag coefficient varies within 30 % when the sphere temperature varies from the free stream temperature to the stagnation temperature. The drag and energy transfer coefficients are found to be non-monotonic functions of the free stream temperature.

Gas-surface interaction in rarefied gas flows through long capillaries via the linearized Boltzmann equation with various boundary conditions

Reliable gas–surface interaction modeling is of major importance in consistent simulation of rarefied gas flows with applications in vacuum technology and micro-electromechanical systems. The effectiveness of the most prominent kinetic boundary conditions, namely the ones by Maxwell, Cercignani-Lampis and Epstein, is elaborated by numerically solving the classical plane Poiseuille and thermal creep flows via the linearized Boltzmann equation, subject to these boundary conditions. The flow rates are provided in a wide range of the gas rarefaction for many values of the accommodation coefficients of each scattering kernel, identifying in parallel, the sensitivity of the flow rates in small variations of the free parameters. A comparison with available experimental data is performed to deduce that the Cercignani-Lampis and Epstein scattering kernels, compared to the Maxwell one, are more suitable to characterize coupled momentum and energy accommodation modes. With these boundary conditions, the accommodation coefficients may be extracted by fitting computations with measurements, in the Poiseuille flow in a wide range of gas rarefaction, but in the thermal creep flow, experimental data under rarefied conditions, as high as possible, are recommended. The present work may be useful in determining the measurement conditions and uncertainties, allowing accurate extraction of the accommodation coefficients.

Effect of internal degrees of freedom in rarefied gas problems: Plane Couette flow

Present paper studies the influence of internal degrees of freedom (IDOF) on rarefied gas flow in a classic plane Couette problem. The numerical study is performed using Direct Simulation Monte Carlo method in a wide range of gas rarefaction, plates relative velocities and for different values of accommodation coefficients (ACs) on the walls. It is shown that: (1) in presence of IDOF a noticeable thermal non-equilibrium between translational and internal energy modes is observed, which increases with gas rarefaction and with plates relative velocity; (2) presence of IDOF can noticeably decrease shear stress and heat flux to the walls, which is mainly explained by the drain of energy from over-excited translational mode into internal ones, leading to lower values of translational temperature and, hence, viscosity; (3) the magnitude of IDOF influence is a non-monotonic function of gas rarefaction, taking maximal values in transitional regimes Kn≈1; (4) values of ACs can significantly change the influence of IDOF, both qualitatively and quantitatively; (5) the rate of energy exchange between translational and internal modes has a noticeable influence on the results, but the energy exchange model itself is not principal as long as this rate is reproduced correctly; (6) increasing number of IDOF leads to more noticeable influence on shear stress and heat flux. Therefore the influence of IDOF will be more significant for polyatomic gases.

Gas outflow into vacuum over a forward- and backward-facing step in a wide range of rarefaction

The direct simulation Monte Carlo method has been applied to investigate the outflow of rarefied gas into vacuum over a forward- and backward-facing step. The non-equilibrium effects at the ends of the step channel have been taken into account by considering the geometry of sufficiently large regions directly adjacent to the inlet and outlet of the channel. The mass flow rate through a step channel as well as the gas flow field both inside the step channel and in the upstream and downstream regions have been calculated over a wide range of gas rarefaction. The calculation results have been compared with the corresponding data for a channel of constant cross-section. The system configuration has been found to have a pronounced effect on the simulation findings. The so-called Knudsen minimum has been established in the dependence of the mass flow rate on the gas rarefaction, and a gas circulation zone has been found in the flow field.

Oscillatory pressure-driven rarefied binary gas mixture flow between parallel plates.

The rarefied, oscillatory, pressure-driven binary gas mixture flow between parallel plates is computationally investigated in terms of the mixture molar fraction and molecular mass ratio of the species, in a wide range of gas rarefaction and oscillation frequency. Modeling is based on the McCormack kinetic model. The output quantities are in dimensionless form and include the flow rate, wall shear stress and pumping power of the mixture, as well as the velocity and shear stress distributions and flow rates of the species. The presented results are for He-Xe and Ne-Ar. The heavier species are affected more drastically than the lighter ones from the inertial forces, resulting to large differences between the flow rate amplitudes of the species, which are increased as the flow becomes less rarefied, provided that the oscillation frequency is adequately high. At very high frequencies the ratio of the flow rate amplitudes of the light over the heavy species tends to the inverse of their molecular mass ratio in the whole range of gas rarefaction. The velocity overshooting effect becomes more pronounced as the molecular mass is increased. The mixture flow rate amplitude is larger, while its phase angle is smaller, than the corresponding ones of single gas, and they both vary nonmonotonically with the molar fraction. The effect of the mixture composition on the wall shear stress and pumping power is small. The present work may be useful in the design of gas separation devices, operating at moderate and high frequencies in rarefied and dense atmospheres.

Shale gas permeability upscaling from the pore-scale

The effective permeability of large shale samples provides useful insights into shale gas production. However, its determination can only be achieved through upscaling, since the direct pore-scale simulation of gas flows in large rock samples is not feasible due to the high computational cost and absence of pore connectivity in sample images. Although the Brinkman formulation is widely used in the permeability upscaling of conventional rocks, how to choose the effective viscosity in this coarse-scale model is not clear. Moreover, its application in shale rocks, where the rarefaction effects are important so that the conventional Navier–Stokes equations are inadequate, is rare, and its accuracy has not been assessed. This study aims to address the above two problems, by comparing the Brinkman solutions of several two-dimensional and three-dimensional random porous media containing fractures with the fine-scale solutions of the Stokes and Boltzmann equations (for continuum and rarefied gas flows, respectively). It is found that the use of the fluid viscosity in the Brinkman model, instead of the controversial effective viscosity, leads to accurate results for the cases considered. Additionally, the macroscopic quantity in rarefied gas flows in shale rocks is well predicted by the Brinkman model for a wide range of gas rarefaction, since the error is found to be less than 7%, while neglecting the rarefaction effects leads to significant underestimation of effective permeability (up to 90% in the cases studied). Although heterogeneity and anisotropy of the porous medium increase the error of the effective permeability derived from the Brinkman model, generally speaking, the effective permeability extracted from this coarse-scale model compares favorably to its fine-scale counterpart.

Rarefied gas flow into vacuum through a channel with sudden contraction or expansion

Based on the direct simulation Monte Carlo method, we have simulated a flow of rarefied gas into vacuum through a channel with sudden contraction or expansion in cross section. The nonequilibrium effects at the input and output of the channel have been taken into account by including certain pre- and post-channel regions into the geometry under consideration. In a wide range of gas rarefaction, the mass flow rate through the channel and the flow field both inside the channel and in the upstream and downstream regions have been calculated. The results have been compared with the characteristics of a channel with the constant cross section. A significant effect of the channel geometry has been established. The Knudsen minimum has been found in the mass flow rate as a function of gas rarefaction, while a compression zone and a circulation zone have been discovered in the flow field.

Rarefied gas flow through a rough channel into a vacuum

Rarefied gas flow through a rough channel of finite length into a vacuum was investigated using the direct simulation Monte Carlo method. The non-equilibrium effects at the input and output of the channel were considered by including certain pre-and post-channel regions into the geometry under consideration. The mass flow rate through a short and long channel was computed in a wide range of gas rarefaction from the free molecular regime to the hydrodynamic near. It is shown that the noticeable effect of surface roughness on flow rate is manifested in the free molecular and transition regimes. The analysis is provided for the flow field inside the channel as well as in the upstream and downstream regions. The results obtained are consistent with the theoretical, numerical, and experimental data available in the open literature.

INVESTIGATION OF COUPLED EFFECTS OF TEMPERATURE AND RELATIVE HUMIDITY ON DYNAMIC PERFORMANCE OF MEMS CANTILEVER RESONATORS IN GAS RAREFACTION

The modified molecular gas lubrication (MMGL) equation with the effective viscosity of moist air is utilized to solve for the squeeze film damping (SFD) problem on the dynamic performance of MEMS cantilever resonators. Thus, the coupled effects of temperature and relative humidity are discussed on the Q-factors of MEMS cantilever resonators in a wide range of gas rarefaction (pressure, p and accommodation coefficients, ACs) and resonant mode of vibration. The results showed that the Q-factor of moist air decreases more significantly as temperature and relative humidity increase at higher gas rarefaction (lower p, and ACs) conditions.

Influences of relative humidity on the quality factors of MEMS cantilever resonators in gas rarefaction

In this paper, the effect of relative humidity of moist air is discussed on the quality factor (Q factor) of micro-electro-mechanical systems (MEMS) cantilever resonators in wide range of gas rarefaction (ambient pressure and accommodation coefficients, ACs). The modified molecular gas lubrication (MMGL) equation is used to model the squeeze film damping problem of MEMS cantilever resonators. Dynamic viscosity and Poiseuille flow rate are used to modify the MMGL equation to consider the coupled effects of relative humidity and gas rarefaction. Thermoelastic damping and anchor loss, which are dominant damping mechanisms of MEMS cantilever resonators, are also included to calculate total Q factor. Thus, the influences of relative humidity are discussed on the Q factors of MEMS cantilever resonators in wide range of gas rarefaction and dimension of cantilever. The results showed that the Q factor decreases as relative humidity increases in wide range of gas rarefaction (pressure, and ACs) and dimension of cantilever (length, width, and thickness). The influences of relative humidity on the Q factor become more significantly in larger length, larger width, smaller thickness of cantilever, and higher gas rarefaction (lower pressure and ACs). Whereas, the influences of relative humidity on the Q factor reduce or are neglected in smaller length, larger thickness of cantilever and lower gas rarefaction (higher pressure and ACs).

Limits of one dimensional modeling of rarefied Couette Poiseuille clearance flow in vacuum pumps

Clearance flows are the main loss mechanism in dry running positive displacement vacuum pumps. In order to calculate the operation of those pumps a detailed knowledge of the clearance mass flow rates is crucial. The dimensions of such pumps and the large pressure range of the operating points require a wide range of gas rarefaction to be taken into account. The clearance flow can be described by a combined Couette Poiseuille flow due to the pressure gradient between two chambers and the rotation of the rotary pistons. A clearance of variable cross section is investigated and in order to determine the mass flow rate, the DSMC (direct simulation Monte Carlo) method is used. Results in the slip and transition flow regime are compared to experimental findings and a good agreement can be found. In addition, a one dimensional theory is presented based on a linear superposition of Poiseuille and Couette flow rates. The aim of the present work is to determine the limits of the one dimensional theory concerning the geometry of such clearances. For this, the boundary conditions circumferential speed, inlet and outlet pressure are varied.

Measurement and simulation of rarefied Couette Poiseuille flow with variable cross section

Clearance flows are the main loss mechanism in dry running positive displacement vacuum pumps. In order to calculate the operation of those pumps, a detailed knowledge of the clearance mass flow rates is crucial. The dimensions of such pumps and the large pressure range of the operating points require a wide range of gas rarefaction to be taken into account. The clearance flow can be described by a combined Couette Poiseuille flow due to the pressure gradient between two chambers and the rotation of the rotary pistons. These clearance flows are studied experimentally and theoretically in the present work. Therefore, a suitable experimental setup is described together with the requirements of sensors and the necessity of a low leakage. A theoretical approach is presented, and the results are compared to experimental investigations varying the pressure ratio and the circumferential speed of a clearance boundary in a wide range of the gas rarefaction.

On the apparent permeability of porous media in rarefied gas flows

The apparent gas permeability of a porous medium is an important parameter in the prediction of unconventional gas production, which was first investigated systematically by Klinkenberg in 1941 and found to increase with the reciprocal mean gas pressure (or equivalently, the Knudsen number). Although the underlying rarefaction effects are well known, the reason that the correction factor in Klinkenberg’s famous equation decreases when the Knudsen number increases has not been fully understood. Most of the studies idealize the porous medium as a bundle of straight cylindrical tubes; however, according to the gas kinetic theory, this only results in an increase of the correction factor with the Knudsen number, which clearly contradicts Klinkenberg’s experimental observations. Here, by solving the Bhatnagar–Gross–Krook equation in simplified (but not simple) porous media, we identify, for the first time, two key factors that can explain Klinkenberg’s experimental results: the tortuous flow path and the non-unitary tangential momentum accommodation coefficient for the gas–surface interaction. Moreover, we find that Klinkenberg’s results can only be observed when the ratio between the apparent and intrinsic permeabilities is ${\lesssim}30$; at large ratios (or Knudsen numbers) the correction factor increases with the Knudsen number. Our numerical results could also serve as benchmarking cases to assess the accuracy of macroscopic models and/or numerical schemes for the modelling/simulation of rarefied gas flows in complex geometries over a wide range of gas rarefaction. Specifically, we point out that the Navier–Stokes equations with the first-order velocity-slip boundary condition are often misused to predict the apparent gas permeability of the porous medium; that is, any nonlinear dependence of the apparent gas permeability with the Knudsen number, predicted from the Navier–Stokes equations, is not reliable. Worse still, for some types of gas–surface interactions, even the ‘filtered’ linear dependence of the apparent gas permeability with the Knudsen number is of no practical use since, compared to the numerical solution of the Bhatnagar–Gross–Krook equation, it is only accurate when the ratio between the apparent and intrinsic permeabilities is ${\lesssim}1.5$.

Sound propagation through a rarefied gas in rectangular channels.

Sound propagation through a rarefied gas inside a two-dimensional cavity is investigated on the basis of the linearized Boltzmann equation, where one of the cavity walls oscillates harmonically in the direction normal to its own surface and is considered as a sound source. An analytical solution at high oscillation frequencies is obtained and detailed numerical results for a wide range of gas rarefaction are presented. The influence of both the aspect ratio of the cavity and the oscillation frequency on the average gas pressure exerted on the oscillating plate is studied. It is found that, at large values of the aspect ratio, the average pressure oscillates when the sound frequency varies, due to the sound resonance and antiresonance along the oscillation direction of the plate. However, at small values of the aspect ratio, the average pressure is a monotonically decreasing function of the sound frequency, which cannot be observed in the corresponding one-dimensional counterpart. This is explained by the sound interference in the direction parallel to the oscillating plate. The influence of both the cavity aspect ratio and oscillation frequency on the sound speed is also investigated: Again it is found that a different aspect ratio leads to the different behavior of the sound speed as a function of the oscillation frequency.

Rarefied gas flow through a slit and channel of finite length due to a large pressure difference. A benchmark problem

The rarefied gas flow through a slit and channel of finite length is presented as a benchmark problem. Computation of gas flow rate and flow field through a slit and channel of finite length has been carried out in case of a large pressure difference in a wide range of gas rarefaction. We used the direct simulation Monte Carlo method based on the majorant frequency scheme. Input and output parameters in computation correspond to parameters of the benchmark problem. A comparison of results in solving this problem obtained by various methods has been made and recommendations for future research prospective have been proposed.

Oscillatory rarefied gas flow inside rectangular cavities

AbstractTwo-dimensional oscillatory lid-driven cavity flow of a rarefied gas at arbitrary oscillation frequency is investigated using the linearized Boltzmann equation. An analytical solution at high oscillation frequencies is obtained, and detailed numerical results for a wide range of gas rarefaction are presented. The influence of both the aspect ratio of the cavity and the oscillating frequency on the damping force exerted on the moving lid is studied. Surprisingly, it is found that, over a certain frequency range, the damping is smaller than that in an oscillatory Couette flow. This reduction in damping is due to the anti-resonance of the rarefied gas. A scaling law between the anti-resonant frequency and the aspect ratio is established, which would enable the control of the damping through choosing an appropriate cavity geometry.

Rarefied gas mixture flow between plates of arbitrary length due to small pressure difference

The linearized flow of a binary gas mixture between two parallel plates of any length driven by a small pressure difference is numerically solved based on the McCormack kinetic model. In this case the flow is not fully developed and the simulated flow field includes, in addition to the channel, efficiently large upstream and downstream regions significantly increasing the required computational effort. In the present work results are provided for one gas mixture, namely He-Ne with a specific concentration of 0.5 in a wide range of gas rarefaction. The ratio of the length over the height of the channel is taken to be equal to 0, 1 and 5 corresponding to flow through channels of very short (slit), short and moderate lengths respectively. Results are provided for the two kinetic coefficients related to Poiseuille and barodiffusion flow rates and the velocity profiles of each species. The effect of the length to height ratio and of the gas rarefaction on the flow quantities is investigated deducing that as the ratio in increased the separation phenomenon becomes more dominant, while a Knudsen minimum is observed at the ratio equal to 5.

Pressure-driven flow of rarefied gas through a slit at a various pressure ratios

This paper numerically studies a pressure-driven flow of rarefied gas through a slit using a direct simulation Monte Carlo method. A gas mass flow rate through a slit and a gas flow field were calculated. The calculations were carried out for various values of the pressure ratio over a wide range of gas rarefaction. The dependency of the mass flow rate and the flow field characteristics on the pressure ratio value and gas rarefaction degree was determined. In addition, the paper presents a discussion on the existence of the shock waves within a flow field in a hydrodynamic regime of the gas flow.