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Highlights • Heat and mass transfer of an oscillatory rarefied gas flow in a square cavity is investigated. • Effects of frequency and speed of the oscillating lid are studied. • The hot to cold heat transfer could be dominant for a highly rarefied oscillatory flow. • The convective heat transfer could be dramatically enhanced by lid oscillation. • The average Nusselt number on the lid is reported in all the flow regimes. Abstract Although effective cooling of micro-electro-mechanical systems (MEMS) with oscillatory components is essential for reliable device operation, the role of oscillation on heat transfer remains poorly understood. In this work, heat and mass transfer of the oscillatory gas flow inside a square cavity is computationally studied by solving the Boltzmann model equation, i.e. the Shakhov model. The oscillation frequency of the lid and rarefaction and nonlinearity of the flow field are systematically investigated. Our results show that, when the oscillation frequency of the lid increases, the usual cold-to-hot heat transfer pattern for highly rarefied flow changes to hot-to-cold, which contradicts the well-known anti-Fourier (i.e. cold-to-hot) heat transfer in a non-oscillatory lid-driven cavity. In addition, the thermal convection will be dramatically enhanced by lid oscillation, which may play a dominant role in the heat transfer. Meanwhile, the average Nusselt number varies non-monotonically with the oscillation frequency, with the maximum occurring at the anti-resonance frequency. Finally, the average Nusselt number on the lid at various oscillation frequencies is found to reduce when the gas becomes more rarefied. These findings may be useful for the thermal design of MEMS. [ABSTRACT FROM AUTHOR]

Abstract The Katrin experiment aims to measure the effective mass of the electron antineutrino from the analysis of electron spectra stemming from the β -decay of molecular tritium with a sensitivity of 200 meV/c2. Therefore, a cumulative amount of about 40 g of gaseous tritium is circulated daily in a windowless source section. An accurate description of the gas flow through this section is of fundamental importance for the neutrino mass measurement as it significantly influences the generation and transport of β -decay electrons through the experimental setup. In this paper we present a comprehensive model consisting of calculations of rarefied gas flow through the different components of the source section ranging from viscous to free molecular flow. By connecting these simulations with a number of experimentally determined operational parameters the gas model can be refreshed regularly according to the measured operating conditions. In this work, measurement and modelling uncertainties are quantified with regard to their implications for the neutrino mass measurement. We find that the magnitude of systematic uncertainties related to the source model is represented by | Δ m ν 2 | = (3.06 ± 0.24) ⋅ 10 − 3 eV 2 / c 4 , and that the gas model is ready to be used in the analysis of upcoming Katrin data. Highlights • Description of gas flow in the KATRIN windowless gaseous tritium source. • Improved gas model allows experimental verification and neutrino mass analysis. • Description of measurements to monitor and determine the amount of gas in the KATRIN source, using an angular-selective electron gun. • Quantification of neutrino mass systematic uncertainty related to the description of gas flow in the KATRIN source. [ABSTRACT FROM AUTHOR]

A backward facing micro-step is a building block for many microfluidic devices. Due to micron sized characteristic dimensions, the gas flow in such a geometry is rarefied in nature. Such rarefied gas flows are widely solved using the Direct Simulation Monte Carlo (DSMC) technique. Flow separation, circulation and re-attachment are some of the basic characteristics of step flows. The objective of this study is to analyze the effect of rarefaction on the flow properties and the separation of the flow. The range of selected Knudsen number (Kn) covers the slip and transition regime from a value of 0.0311–13.25. The pressure ratios employed are 3 and 5. It is observed that the slip velocity continuously increases while the centre-line velocity first decreases, then remains constant and finally increases with increase in Kn. At the step, separation of the flow is seen for Kn < 0.1325 while no such separation is observed in the range of Kn from 0.198 to 13.25. The corresponding Re for these ranges are 6.43 to 0.67 and 0.392 to 0.012 respectively. The re-attachment length decreases with increase in Kn whereas it increases with increase in Re. A stronger pressure force and a weaker diffusion effect leads to flow separation in the slip regime whereas stronger diffusion and weaker pressure force lead to an absence of flow separation in the transition regime. Finally, this work presents for the first time the existence of the Knudsen minimum for such a backward step geometry. [ABSTRACT FROM AUTHOR]

The Boltzmann equation can be used to model rarefied gas flows in the transition or kinetic regime, i.e. for moderate to large Knudsen numbers. However, standard moment methods like Grad’s approach lack hyperbolicity of the equations. This can lead to instabilities and nonphysical solutions. Based on recent developments in this field we have recently derived a quadrature-based moment method leading to globally hyperbolic and rotationally invariant moment equations. We present a 1D five moment case of the equations and use numerical simulations to compare the new model with standard approaches. The tests are done with dedicated numerical methods to solve the new non-conservative moment equations. These first results using the new method show the accuracy of the new method and its benefits compared with Grad’s method or other existing models like discrete velocity. [ABSTRACT FROM AUTHOR]

The paper is devoted to the development of the numerical approaches to solve model kinetic equations as applied to computing high-speed rarefied gas flows over three-dimensional geometries. The use of a very non-uniform unstructured velocity mesh is proposed, and the influence of the velocity mesh resolution is studied in detail. The test problems include a simple flow over a planar circular cylinder and a three-dimensional flow over a model winged re-entry vehicle. [ABSTRACT FROM AUTHOR]

Micro gas flow in porous medium is widespread in nature and industrial engineering. In the paper, a pore-scale model of gas flow through porous medium constructed by Sierpinski fractal is developed and numerically analyzed to study the gas flow behaviors in porous medium. The velocity distribution inside porous media are presented and analyzed, and the effects of the porous structure and rarefaction effect on the gas permeability are examined and investigated. The results indicate that the rarefaction effect greatly affects the gas flow in porous media, especially in the case of lower porosity. In addition, the pore configuration and tortuosity of the porous medium also affect the gas permeability. This may attribute to the existence of wide long straight channels in the porous structure. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the gas-surface interaction on the time-dependent rarefied gas flow through a short tube into vacuum is investigated. Due to a significant scale separation, the flow is simulated using a hybrid scheme proposed by Vargas et al. [1, 2], in which the pressure change in the upstream chamber is coupled to the flow rate obtained by the Direct Simulation Monte Carlo (DSMC) computations. First, the influence of gas-surface interaction is demonstrated through comparison of DSMC simulation results obtained for different values of accommodation coefficients. The obtained dataset is then used to show how the boundary conditions can affect the time-dependent gas flow, even when applied on the small surface area of the device. It is argued that the domain of possible solutions is bounded by the cases of specular and diffuse scattering. Furthermore, it is investigated how this domain can be affected by measurement uncertainties present in the experimental setup. Next, the analysis of the gas-surface interaction is extended to binary mixtures. In the end, it is discussed how the obtained results can be used in order to improve the efficiency of gas separation mechanisms. [ABSTRACT FROM AUTHOR]

The sound propagation through a binary mixture of rarefied gases in a semi-infinite space is investigated on the basis of the McCormack model to the linearized Boltzmann equation. The source of sound waves is an infinite and flat plate oscillating in the direction normal to its own plane. It is considered a fully established oscillation so that the solution of the kinetic equation depends on time harmonically. The diffuse scattering gas–surface interaction law is assumed as the boundary condition on the oscillatory plate. A realistic intermolecular potential based on experimental data for the transport coefficients of the mixture is employed in the calculations. Two mixtures of noble gases, namely, Helium–Argon and Helium–Xenon, are considered in order to investigate the influence of the molecular mass ratio of species on the problem. A wide range of the oscillation speed parameter, defined as the ratio of intermolecular collision frequency to sound frequency, is considered. The amplitudes and phases of the macrocharacteristics of the gas flow are calculated and the results are compared to those obtained for a single gas. [ABSTRACT FROM AUTHOR]

In this paper we present a coupled Navier–Stokes/direct simulation Monte Carlo (NS/DSMC) solver for simulating the production of organic light-emitting diodes. In this production process layers of organic molecules are deposited from gas phase on a substrate. Guided in a pipe system at low pressure the precipitating material flows through nozzles into a high-vacuum chamber where the deposition takes place. Typical for this system is a strong decrease of pressure along the flow path with an equivalent increase in Knudsen number. Therefore only a coupled NS/DSMC simulation is possible. The commercial simulation software ANSYS Fluent was used for the whole domain while for the domain with high Knudsen numbers DSMCFoam was applied, which is part of the open source CFD toolbox OpenFOAM. Starting with an NS solution of the whole domain the breakdown parameter was computed defining the beginning of the DSMC subdomain. Data is exchanged using a state based coupling and compared in an overlapping region, where both are valid. An analysis in the number of lattice cells in the NS simulation and in the number of DSMC particles per cell in the DSMC simulation was performed. Furthermore the effect of different numbers of degrees of freedom of the fluid was examined. For the validation of the full simulation, results were compared with the results from two experiments, and very good agreement was achieved. [ABSTRACT FROM AUTHOR]

A numerical analysis of the rarefied gas flow caused by a temperature gradient in the direction tangential to a wall through a planar channel of finite length is carried out based on the nonlinear Shakhov model of the Boltzmann kinetic equation. The Maxwell diffuse and Cercignani–Lampis gas–surface interaction models are provided as a boundary condition at solid walls. An implicit scheme for solution of the S-model kinetic equation was applied and the algorithm has been optimized for the use of massive parallelization in both physical and velocities spaces. The competition between rarefaction, gas–surface scattering model and temperature gradient effects on a gas flow is discussed in terms of both mass flow rate and streamwise heat flux. A comparison with available literature results has been carried out and showed a good agreement. It was found that the temperature gradient as well as the gas–surface scattering dynamics play a significant role for the highly rarefied gas flow, beginning from the transition flow regime. At the same time, under certain gas–surface scattering conditions a change of the value of temperature gradient does not affect the heat flux. For the weakly rarefied flow the temperature gradient effect on the mass flow rate is small for any gas–surface scattering condition, while the heat flux does not depend neither on value of temperature gradient nor choice and details of gas–surface scattering dynamics. [ABSTRACT FROM AUTHOR]

This paper presents experimental and three-dimensional numerical study of early onset of separation with rarefied gas flow through a tube with single sharp 90° bend. Experiments are conducted for nitrogen gas flowing at low pressures in three conventional size tubes. The flow is dynamically similar to gas flow in a microchannel as the Knudsen number range (0.0003 < Kn < 0.0385) covers part of the continuum and the slip flow regime while maintaining the Reynolds number between 0.27 and 418.5. The static pressures along the inner, outer and top walls are measured for different mass flow rates and analyzed to understand the flow behavior. The static pressure measurement indicates adverse pressure gradient near the bend along the inner and outer walls of the tube at much lower value of Reynolds number as compared to conventional flow. The numerical solution of the Navier–Stokes equations with the Maxwell’s slip boundary condition shows good agreement with experimental data and helps bring out the complex flow behavior near the bend. The adverse pressure gradient, velocity profile, flow streamlines and velocity vectors in the bend plane clearly indicates secondary flows near the bend at as low a Reynolds number as unity. The flow acceleration and the presence of secondary flows near the bend causes a larger pressure drop as compared with a straight tube. Empirical correlations for Poiseuille number and additional pressure drop coefficient are proposed as part of this work. It is noted that limited experimental data exists in the literature for such flows; these results should therefore help enhance the fundamental understanding of gas flow in microchannels with bend. [ABSTRACT FROM AUTHOR]

This article reviews rarefied gas flow computations based on nonlinear model Boltzmann equations using deterministic high-order gas-kinetic unified algorithms (GKUA) in phase space. The nonlinear Boltzmann model equations considered include the BGK model, the Shakhov model, the Ellipsoidal Statistical model and the Morse model. Several high-order gas-kinetic unified algorithms, which combine the discrete velocity ordinate method in velocity space and the compact high-order finite-difference schemes in physical space, are developed. The parallel strategies implemented with the accompanying algorithms are of equal importance. Accurate computations of rarefied gas flow problems using various kinetic models over wide ranges of Mach numbers 1.2–20 and Knudsen numbers 0.0001–5 are reported. The effects of different high resolution schemes on the flow resolution under the same discrete velocity ordinate method are studied. A conservative discrete velocity ordinate method to ensure the kinetic compatibility condition is also implemented. The present algorithms are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the steady normal shock wave structures for different Mach numbers, the two-dimensional flows past a circular cylinder and a NACA 0012 airfoil to verify the present methodology and to simulate gas transport phenomena covering various flow regimes. Illustrations of large scale parallel computations of three-dimensional hypersonic rarefied flows over the reusable sphere–cone satellite and the re-entry spacecraft using almost the largest computer systems available in China are also reported. The present computed results are compared with the theoretical prediction from gas dynamics, related DSMC results, slip N–S solutions and experimental data, and good agreement can be found. The numerical experience indicates that although the direct model Boltzmann equation solver in phase space can be computationally expensive, nevertheless, the present GKUAs for kinetic model Boltzmann equations in conjunction with current available high-performance parallel computer power can provide a vital engineering tool for analyzing rarefied gas flows covering the whole range of flow regimes in aerospace engineering applications. [ABSTRACT FROM AUTHOR]

The NACA 0012 airfoil and the sharp 25–55-deg biconic cases are usually considered in evaluating the accuracy of new forms of velocity slip and temperature jump wall boundary conditions in the Navier–Stokes–Fourier equations. These cases have been investigated by many researchers in previous works with the first order Maxwell slip and Smoluchowski jump boundary conditions. Recently, we proposed a new form of the second order temperature jump for simulating rarefied gas flows and obtained accurate results for low-speed microflows. The current work is an extension of our previous CFD work where we applied both the second order velocity slip condition and the second order temperature jump boundary condition in new forms to simulate rarefied external gas flows for these cases. The second order boundary conditions are implemented into the rhoCentralFoam solver in the OpenFOAM software. The NACA 0012 airfoil case is investigated with angles-of-attack of 0 and 10 degrees and Mach numbers of 0.8 and 2. The sharp 25–55-deg biconic case is simulated at Mach of 15.6. The simulation results obtained for the NACA 0012 airfoil show that the combination of the second order slip/jump boundary conditions gives suitable agreements with DSMC for surface gas temperature and drag coefficients in comparison with those of the first order conditions. However, the results of the biconic case figure out that the second order conditions are not superior to the first order conditions. [ABSTRACT FROM AUTHOR]

Molecular distillation process is a type of vapors transport process at high vacuum in the distillation space between evaporator and condenser. To get a better understanding of vapor flow characteristics in molecular distillation, vapor phase behavior is simulated using the direct simulation Monte Carlo (DSMC) method. The variable hard sphere (VHS) model is adopted to calculate molecular collisions. Molecular rotation and translation have both been taken into account while dealing with molecular energy distribution after collisions. Flow characteristics obtained for velocity distribution and number density are used to calculate the collision frequency and mean free path in the distillation space. The effects of evaporating temperature, condensing temperature and distillation gap on the vapors flow characteristics are discussed. It is found that for gas molecules by increasing the evaporating temperature, the number density and velocity, collision frequency in distillation space increase, while the mean free path decreases. As the condensing temperature increases, the number density and collision frequency in the distillation space increases, while velocity and mean free path decreases. It is also observed that much narrower distillation gap results in steeper variation tendency of flow characteristics. [ABSTRACT FROM AUTHOR]

Transient flow of rarefied gas through an orifice caused by various pressure ratios between the reservoirs is investigated for a wide range of the gas rarefaction, varying from the free molecular to continuum regime. The problem is studied on the basis of the numerical solution of unsteady S-model kinetic equation. It is found that the mass flow rate takes from 2.35 to 30.37 characteristic times, which is defined by orifice radius over the most probable molecular speed, to reach its steady state value. The time of steady flow establishment and the steady state distribution of the flow parameters are compared with previously reported data obtained by the Direct Simulation Monte Carlo (DSMC) method. A simple fitting expression is proposed for the approximation of the mass flow rate evolution in time. [ABSTRACT FROM AUTHOR]

In a rarefied gas the temperature field and the gas motion are closely related, and the temperature field can cause, in a confined flow geometry, a steady flow without the help of external forces. This is due to the creep of the fluid along the walls induced by a wall temperature gradient, as a consequence of the molecular transfer of momentum to the wall. Experiments performed in microgravity conditions in two small cells, with mono- and bi-atomic carrier gases (Ar, N2), and a thermal gradient between upper and bottom horizontal plates, allowed the measurement of the width of the thermal creep and the induced velocity of the gas near the vertical cell wall, due to thermal gradient. In addition, experiments demonstrated the existence of the thermal creep flow in the cells, even with Knudsen number as low as 10-5. The work evidences experimentally, and for the first time, the thermal creep flow in small cells, due to wall temperature gradient, with Knudsen number as low as 10-5. In view of these results, the no-slip boundary conditions of the Navier-Stokes law in the "continuum" regime can be inadequate in non-isothermal flow geometry and in microgravity conditions. [ABSTRACT FROM AUTHOR]

Abstract: The method, proposed previously by other authors, is applied here to calculate a mass flow rate of rarefied gas through a long rectangular channel of variable cross-section aspect ratio. The gas flow through this channel is generated by the pressure and/or temperature gradient. The method is based on the results obtained previously on the basis of the kinetic equation and it requires very modest computational efforts. As a demonstration of its application some examples of the mass flow rate calculations for the isothermal and non-isothermal flows through the channels with variable rectangular cross sections are given. The analytical expressions in the case of the hydrodynamic and free molecular flow regimes are proposed. [Copyright &y& Elsevier]

Abstract: A method to take into account the influence of the inlet/outlet ends on rarefied gas flows through moderately long capillaries proposed previously [Vacuum 97, 26 (2013)] is applied to gas flows through circular tubes. The method is based on the concept of effective tube length, representing a sum of its real length and an increment. To calculate the length increment, a flow field near the inlet/outlet of tube is calculated on the basis of the linearized kinetic equation. It is shown that the value of the length increment is independent of the tube length, but depends only on the rarefaction parameter so that it can be used to obtain accurate predictions of the mass flow rate and the axial pressure distribution without great computational effort. Comparisons with results obtained by considering the complete flow domain have shown the efficiency of the end correction concept. [Copyright &y& Elsevier]

Abstract: A transient gas flow through a rectangular slit is simulated on the basis of unsteady S-model kinetic equation. The flow into vacuum and at two pressure ratios between the reservoirs are considered for a wide range of the rarefaction parameter lying from the free molecular to the near continuum regime. The time of steady flow establishment is obtained and it is found that this time depends on the pressure ratio between the reservoirs and on the gas rarefaction. The time evolution of the macroscopic flow parameters is analyzed in detail. The present results are compared with the results on the orifice flow previously obtained by other authors. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the study of a rarefied gas flow through a finite length conical pipe into vacuum. The problem is solved in the complete geometrical setup, which included not only the pipe, but also high- and low-pressure reservoirs. The analysis is based on the direct numerical solution of the Boltzmann kinetic equation with the S-model collision integral in three space dimensions. The method of the solution is based on the recent implicit total variation diminishing (TVD) method on unstructured spatial meshes. It is conservative with respect to the collision integral and work across all flow regimes. The results are provided for various ratios of the outlet and inlet diameters, pipe's lengths and Knudsen numbers. The computed flow rates are compared with the case of the circular pipe of constant radius as well as an approximate method for very long pipes. The influence of the pipe geometry on the flow field is also examined. The presented results can be used as a benchmark for calculations by other methods and codes. [Copyright &y& Elsevier]

Abstract: We present the results of an experimental study concerning the gas flow through a packed bed of spheres located between two concentric cylinders, in non isothermal conditions. This problem is prototypal for more complex geometries which can be found in Micro Electro-Mechanical Systems (MEMS) where the heat transfer is a crucial issue. In these applications the characteristic lengths are such that the gas flow regime could fall in almost rarefied regime where the features of heat and mass flows can be markedly different with respect to the continuum regime. We investigated a wide range of regimes, spanning from the compressible continuum flow to rarefied conditions with a variation of the peculiar Knudsen number of about three orders of magnitude, i.e. from Kn =10−3, which corresponds to almost continuum regime, to Kn ≈1 which falls in the transitional region. The heat transfer through the porous medium have been studied by the analysis of the temperature field and the heat flux as a function of the flow rate, the boundary conditions and the porous materials. [Copyright &y& Elsevier]