Gas Rarefaction Degree Research Articles

State-specific slip boundary conditions in non-equilibrium gas flows: Theoretical models and their assessment

In this study, we develop and assess a new approach to modeling slip boundary conditions in gas mixtures with coupled state-to-state vibrational-chemical kinetics and surface physical and chemical processes: adsorption, desorption, vibrational energy transitions, and chemical reactions. Expressions for velocity slip, temperature jump, and mass fluxes of species are derived on the basis of the advanced kinetic boundary condition taking into account gain and loss of particles in surface processes; theoretical expressions for the mass fluxes obtained in the frame of various approaches are compared. The developed model is implemented to the fluid-dynamic solver for modeling dynamics and state-to-state air kinetics in the boundary layer near stagnation point. Several test cases corresponding to a various degree of gas rarefaction are considered. Recombination probabilities and effective reaction rates are calculated and compared to recent molecular-dynamic simulations; the proposed model yields the best agreement for the recombination rate coefficient. It is shown that temperature jump significantly affects fluid-dynamic parameters and surface heat flux; the role of heterogeneous reactions on the silica surface is weaker. In the surface heating, there is a competition between these two effects: whereas the temperature jump reduces the wall heat flux, surface reactions cause its increase, but to a lesser extent. It is concluded that the model proposed in this study describes self-consistently detailed vibrational kinetics, rarefaction effects, and surface reactions and can be applied both in continuum and slip flow regimes.

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.

Shock-wave thickness influence to the light diffraction on a plane shock wave

This study is devoted to the numerical analysis of the result of light distribution after passing it through a shock wave, depending on the degree of gas rarefaction. The obtained numerical results allowed reproducing the experimental shadowgraph images obtained in our study. The range of shock wave thickness (from 0 mm to 20 mm) allowed considering the qualitative change in the light distribution on the screen during switching from the regime where the wave nature of light has the greatest influence on the distribution of light to the regime of the geometric optics approach. As a result, the criteria for the applicability of the shadowgraphy technique for the experimental description of the shock wave internal structure were obtained.

Thermal Lattice Boltzmann Method for micro-Poiseuille gas flow

In this paper, the thermal Lattice Boltzmann Method (TLBM) is used to simulate a gas microflow. The simulated gas is confined within an inlet/outlet microchannel and driven by a constant inlet velocity Uin . The bottom wall temperature is taken to be equal zero while the top wall temperature is decreasing from 1 to 0. For consistent results velocity slip and temperature jump boundary conditions at the walls are used. The rarefaction effects given by the Knudsen number, on the velocity and temperature profile are investigated. The results obtained from the TLBM are compared with those obtained using the Finite Difference Method (FDM). In addition to the good agreement observed between the two methods, the results show an interesting sensitivity of velocity and temperature profiles with the gas rarefaction degree.

Rarefied Gas Flow Reflection from a Wall with an Orifice and Gas Outflow into a Vacuum

The unsteady two-dimensional problem of the reflection of a uniform supersonic rarefied-gas flow incident normal on a wall with an orifice and the gas outflow through the orifice (slot or channel) is solved on the basis of the kinetic S-model. The slot influence on the reflection nature, the velocity of the shock wave reflected from the wall, and the formation of a gas jet flowing out into a vacuum is studied as a function of the incident flow velocity, the gas rarefaction degree, and the conditions imposed on the wall. The effect of the wall thickness, that is, the length of the channel, through which the gas flows out, is also investigated. The gas flow rate through the orifice is calculated as the main integral characteristic. The attainment of steady reflection regime in the outflow process is traced. The kinetic equation is solved numerically using the conservative difference method of the second order with respect to all variables.

Flows of a Rarefied Gas between Coaxial Circular Cylinders with Nonuniform Surface Properties

Flows of a rarefied gas between coaxial circular cylinders with nonuniform surface properties are studied on the basis of kinetic theory. It is assumed that the outer cylinder is a diffuse reflection boundary and the inner cylinder is a Maxwell-type boundary whose accommodation coefficient varies in the circumferential direction. Three fundamental flows are studied: 1) a flow caused by the rotation of the outer cylinder (Couette flow), 2) a flow induced between the cylinders at rest kept at different temperatures (heat transfer problem), and 3) a flow induced by the circumferential temperature distribution along the cylindrical surfaces (thermal creep flow). The linearized ES-BGK model of the Boltzmann equation is numerically analyzed using a finite difference method. The time-independent behavior of the gas is studied over a wide range of the gas rarefaction degree, the radii ratio, and a parameter characterizing the distribution of the accommodation coefficient. Due to an effect of nonuniform surface properties, a local heat transfer occurs between the gas and the cylindrical surfaces in Couette flow; a local tangential stress arises in the heat transfer problem. However, the total heat transfer between the two cylinders in Couette flow and the total torque acting on the inner cylinder in the heat transfer problem vanish irrespective of the flow parameters. Two nondegenerate reciprocity relations arise due to the effect of nonuniform surface properties. The reciprocity relations among the above-mentioned three flows are numerically confirmed over a wide range of the flow parameters. The force on the inner cylinder, which also arises due to the effect of nonuniform surface properties in Couette flow and the heat transfer problems, is studied.

Pulsatile pressure driven rarefied gas flow in long rectangular ducts

The pulsatile pressure driven fully developed flow of a rarefied gas through an orthogonal duct is investigated, based on the time-dependent linear Bhatnagar, Gross, and Krook equation, by decomposing the flow into its steady and oscillatory parts. The investigation is focused on the oscillatory part, which is characterized by the gas rarefaction and oscillation parameters, the duct aspect ratio, and the accommodation coefficient. As the oscillation frequency is increased, the amplitude of all macroscopic quantities is decreased, while their phase angle lag is increased reaching the limiting value of π/2. As the gas becomes more rarefied, higher frequencies are needed to trigger this behavior. At small and moderate frequencies, there is a critical degree of gas rarefaction, where a maximum flow rate is obtained. As the duct aspect ratio is decreased and tends to zero, the flow rate and mean wall shear stress amplitudes are increased, while their phase angle lags are slightly affected. The accommodation coefficient has a significant effect on the amplitude and a very weak one on the phase angle of the macroscopic quantities. The computation of the inertia and viscous forces clarifies when the flow consists of only one oscillating viscous region or of two regions, namely, the inviscid piston flow in the core and the oscillating Stokes layer at the wall with the velocity overshooting. Finally, the time average oscillatory pumping power is increased as the oscillation frequency is reduced and its maximum value is one half of the corresponding steady one.

Poiseuille, thermal transpiration and Couette flows of a rarefied gas between plane parallel walls with nonuniform surface properties in the transverse direction and their reciprocity relations

Slow flows of a rarefied gas between two plane parallel walls with nonuniform surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary whose accommodation coefficient varies periodically in the direction perpendicular to the flow. The time-independent Poiseuille, thermal transpiration and Couette flows are considered. The flow behavior is numerically studied based on the linearized Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation. The flow field, the mass and heat flow rates in the gas, and the tangential force acting on the wall surface are studied over a wide range of the gas rarefaction degree and the parameters characterizing the distribution of the accommodation coefficient. The locally convex velocity distribution is observed in Couette flow of a highly rarefied gas, similarly to Poiseuille flow and thermal transpiration. The reciprocity relations are numerically confirmed over a wide range of the flow parameters.

The 13-moments method for heat transfer in gas microflows

In this work, the heat transfer within MEMS devices materialised by a cavity is envisaged. The problem is studied using the Grad’s 13-moments and the Navier–Stokes and Fourier field of equations (NSF), with first order of slip and jump boundary conditions, compared to the direct simulation Monte Carlo (DSMC) results. A Maxwelllian gas is assumed to be confined in a cavity at the environmental temperature. The cavity is taken to have a hot chip, an adiabatic one while the two others are taken to the environmental temperature. The 13-moments solutions are in accordance with DSMC predicting, besides the classical hot to cold vortices, two additional counter-rotating ones exchanging energy with the two cold walls, which are not predicted by the continuum-based model. The Knudsen number growth, gas rarefaction degree, affects the vortices by pushing the classical ones towards the centre and hot plate, while strengthening the others.

Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.

Unsteady rarefied gas flow in a microchannel driven by a pressure difference

The kinetic S-model is used to study the unsteady rarefied gas flow through a plane channel between two parallel infinite plates. Initially, the gas is at rest and is separated by the plane x = 0 with different pressure values on opposite sides. The gas deceleration effect of the channel walls is studied depending on the degree of gas rarefaction and the initial pressure drop, assuming that the molecules are diffusely reflected from the boundary. The decay of the shock wave and the disappearance of the uniform flow region behind the shock wave are monitored. Special attention is given to the gas mass flux through the cross section at x = 0, which is computed as a function of time. The asymptotic behavior of the solution at unboundedly increasing time is analyzed. The kinetic equation is solved numerically by applying a conservative finite-difference method of second-order accuracy in space.

Extended Macroscopic Study of Dilute Gas Flow within a Microcavity

The behaviour of monatomic and dilute gas is studied in the slip and early transition regimes using the extended macroscopic theory. The gas is confined within a two-dimensional microcavity where the longitudinal sides are in the opposite motion with constant velocity ±Uw. The microcavity walls are kept at the uniform and reference temperature T0. Thus, the gas flow is transported only by the shear stress induced by the motion of upper and lower walls. From the macroscopic point of view, the regularized 13-moment equations of Grad, R13, are solved numerically. The macroscopic gas proprieties are studied for different values of the so-called Knudsen number (Kn), which gives the gas-rarefaction degree. The results are compared with those obtained using the classical continuum theory of Navier-Stokes and Fourier (NSF).

Rarefied gas deceleration in a channel in the case of expansion into a vacuum

The time-dependent problem of the development of a rarefied flow in a plane channel formed by two parallel infinite plates is considered on the basis of a kinetic model. At the initial moment of time the gas at rest occupies half the channel and borders on a vacuum. The effect of gas deceleration at the channel walls is studied as a function of the gas rarefaction degree under the conditions of the diffuse reflection of molecules from the boundary. Emphasis is placed on the calculations of the time dependence of the integral gas mass flux through the boundary separating the gas from the vacuum at the initial moment. The deceleration of the conditional expansion wave front and the conditional front of gas expansion into the vacuum is traced. The kinetic equation is numerically solved using a first-order difference method.

Rarefied gas flow in a planar channel caused by arbitrary pressure and temperature drops

The paper presents a numerical analysis of the nonlinear rarefied gas flow through a long planar channel of finite length. The solution is constructed for the arbitrary pressure and temperature drops, including flow into vacuum. The obtained results for the flow rates are compared with the linearized solutions in the large range of degree of gas rarefaction. The boundaries of the validity of the linearized approximation are established.

Design of micro distribution systems consisting of long channels with arbitrary cross sections

Gas flows through long micro-channels of various cross sections have been extensively investigated over the years both numerically and experimentally. In various technological applications including microfluidics, these micro-channels are combined together in order to form a micro-channel network. Computational algorithms for solving gas pipe networks in the hydrodynamic regime are well developed. However, corresponding tools for solving networks consisting of micro-channels under any degree of gas rarefaction is very limited. Recently a kinetic algorithm has been developed to simulate gas distribution systems consisting of long circular channels under any vacuum conditions. In the present work this algorithm is generalized and extended into micro-channels of arbitrary cross-section etched by KOH in silicon (triangular and trapezoidal channels with acute angle of 54.74°). Since a kinetic approach is implemented, the analysis is valid and the results are accurate in the whole range of the Knudsen number, while the involved computational effort is very small. This is achieved by successfully integrating the kinetic results for the corresponding single channels into the general solver for designing the gas pipe network. To demonstrate the feasibility of the approach two typical systems consisting of long rectangular and trapezoidal micro-channels are solved.

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.

An ionization region model for high-power impulse magnetron sputtering discharges

A time-dependent plasma discharge model has been developed for the ionization region in a high-power impulse magnetron sputtering (HiPIMS) discharge. It provides a flexible modeling tool to explore, e.g., the temporal variations of the ionized fractions of the working gas and the sputtered vapor, the electron density and temperature, and the gas rarefaction and refill processes. A separation is made between aspects that can be followed with a certain precision, based on known data, such as excitation rates, sputtering and secondary emission yield, and aspects that need to be treated as uncertain and defined by assumptions. The input parameters in the model can be changed to fit different specific applications. Examples of such changes are the gas and target material, the electric pulse forms of current and voltage, and the device geometry. A basic version, ionization region model I, using a thermal electron population, singly charged ions, and ion losses by isotropic diffusion is described here. It is fitted to the experimental data from a HiPIMS discharge in argon operated with 100 µs long pulses and a 15 cm diameter aluminum target. Already this basic version gives a close fit to the experimentally observed current waveform, and values of electron density ne, the electron temperature Te, the degree of gas rarefaction, and the degree of ionization of the sputtered metal that are consistent with experimental data. We take some selected examples to illustrate how the model can be used to throw light on the internal workings of these discharges: the effect of varying power efficiency, the gas rarefaction and refill during a HiPIMS pulse, and the mechanisms determining the electron temperature.

Plane Poiseuille flow of a rarefied gas in the presence of strong gravitation

Plane Poiseuille flow of a rarefied gas, which flows horizontally in the presence of strong gravitation, is studied based on the Boltzmann equation. Applying the asymptotic analysis for a small variation in the flow direction [Y. Sone, Molecular Gas Dynamics (Birkhäuser, 2007)], the two-dimensional problem is reduced to a one-dimensional problem, as in the case of a Poiseuille flow in the absence of gravitation, and the solution is obtained in a semianalytical form. The reduced one-dimensional problem is solved numerically for a hard sphere molecular gas over a wide range of the gas-rarefaction degree and the gravitational strength. The presence of gravitation reduces the mass flow rate, and the effect of gravitation is significant for large Knudsen numbers. To verify the validity of the asymptotic solution, a two-dimensional problem of a flow through a long channel is directly solved numerically, and the validity of the asymptotic solution is confirmed.

Numerical Analysis of the Time-Dependent Energy and Momentum Transfers in a Rarefied Gas Between Two Parallel Planes Based on the Linearized Boltzmann Equation

Periodic time-dependent behavior of a rarefied gas between two parallel planes caused by an oscillatory heating of one plane is numerically studied based on the linearized Boltzmann equation. Detailed numerical data of the energy transfer from the heated plane to the unheated plane and the forces of the gas acting on the boundaries are provided for a wide range of the gas rarefaction degree and the oscillation frequency. The flow is characterized by a coupling of heat conduction and sound waves caused by repetitive expansion and contraction of the gas. For a small gas rarefaction degree, the energy transfer is mainly conducted by sound waves, except for very low frequencies, and is strongly affected by the resonance of the waves. For a large gas rarefaction degree, the resonance effects become insignificant and the energy transferred to the unheated plane decreases nearly monotonically as the frequency increases. The force of the gas acting on the heated boundary shows a remarkable minimum with respect to the frequency even in the free molecular limit.

Rarefied gas flows through a curved channel: Application of a diffusion-type equation

Rarefied gas flows through a curved two-dimensional channel, caused by a pressure or a temperature gradient, are investigated numerically by using a macroscopic equation of convection-diffusion type. The equation, which was derived systematically from the Bhatnagar–Gross–Krook model of the Boltzmann equation and diffuse-reflection boundary condition in a previous paper [K. Aoki et al., “A diffusion model for rarefied flows in curved channels,” Multiscale Model. Simul. 6, 1281 (2008)], is valid irrespective of the degree of gas rarefaction when the channel width is much shorter than the scale of variations of physical quantities and curvature along the channel. Attention is also paid to a variant of the Knudsen compressor that can produce a pressure raise by the effect of the change of channel curvature and periodic temperature distributions without any help of moving parts. In the process of analysis, the macroscopic equation is (partially) extended to the case of the ellipsoidal-statistical model of the Boltzmann equation.