Efficient method for computing rarefied gas flow in a long finite plane channel

The linearized kinetic S-model is used to study the nonisothermal steady rarefied gas flow driven by differences in pressure and temperature in a plane channel between long finite parallel plates joining two tanks of infinite volume. An efficient composite (asymptotic) method is developed: a one-dimensional asymptotic solution corresponding to an infinitely long channel is constructed in the middle part of the computational domain, while a solution of the two-dimensional kinetic equation matched with the middle-part asymptotic solution is constructed near the ends of the channel. The latter solution is found numerically by a high-order accurate conservative method. The basic quantity to be computed is the gas flow rate through the channel. Characteristic flow features are also investigated. The resulting solutions are compared with previously known results.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

Development and tending to steady state of subsonic gas condensation on a plane condensed phase

The kinetic equation for a monatomic gas with a model collision operator (S-model) is used to study the development and tending to steady state of one-dimensional unsteady half-space gas condensation on a plane condensed phase. Initially, the gas is at rest and in equilibrium with the body’s surface and, then, the body temperature suddenly drops to a constant value. The problem is solved using an implicit second-order accurate quasi-monotone scheme. The process of reaching a steady flow regime is of primary interest. The effect of the evaporation (condensation) coefficient on the flow pattern is analyzed.     

Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic <Emphasis Type="Italic">S</Emphasis>-Model

The nonisothermal steady rarefied gas flow driven by a given pressure gradient (Poiseuille flow) or a temperature gradient (thermal creep) in a long channel (pipe) of an arbitrary cross section is studied on the basis of the linearized kinetic S-model. The solution is constructed using a high-order accurate conservative method. The numerical computations are performed for a circular pipe and for a cross section in the form of a regular polygon inscribed in a circle. The basic characteristic of interest is the gas flow rate through the channel. The solutions are compared with previously known results. The flow rates computed for various cross sections are also compared with the corresponding results for a circular pipe.     

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.     

On transient heat conduction in the walls of a rectangular gas channel

In this paper a problem on transient heat conduction in the walls of a gas channel with a rectangular cross-section is solved. The temperature of the gas flow in the channel rises linearly while the temperature of the surrounding open air is constant.The differential equation and its auxiliary conditions are Laplace transformed, the subsidiary equations are solved by a method resembling the two-dimensional relaxation method for steady state heat conduction problems, and the resulting temperatures are obtained by numerical inversion.Numerical results are presented at the end of the paper.     

A combination of energy method and spectral analysis for study of equations of gas motion

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.      

Kinetic analysis of the isothermal flow in a long rectangular microchannel

The linearized kinetic BGK model is used to study the steady Poiseuille flow of a rarefied gas in a long channel of rectangular cross section. The solution is constructed using the finite-volume method based on a TVD scheme. The basic computed characteristic is the mass flow rate through the channel. The effect of the relative width of the cross section is examined, and the difference of the solution from the one-dimensional flow between infinite parallel plates is analyzed. The numerical solution is compared to available results and to the analytical solution of the Navier-Stokes equations with no-slip and slip boundary conditions. The limits of applicability of the hydrodynamic solution are established depending on the degree of rarefaction of the flow and on the ratio of the side lengths of the channel cross section.     

Solutions of the Boltzmann equation with infinite energy: Existence, stability and long time behavior

This paper is devoted to studying a class of solutions to the nonlinear Boltzmann equation having infinite kinetic energy, these solutions have an upper Maxwellian bound with infinite kinetic energy. Firstly, the existence and stability of this kind of solutions are established near vacuum. Secondly, it is proved that this kind of solutions are stable for any initial data, as a consequence, the Boltzmann equation has at most one solution with infinite kinetic energy. Finally, the long time behavior of the solutions is also established.     

Numerical study of unsteady rarefied diatomic gas flows in a plane microchannel

The numerical solution of a kinetic equation for a diatomic gas (nitrogen) is used to study two-dimensional unsteady gas flows in a plane microchannel caused by discontinuous in the initial distributions of macroscopic gas parameters. The plane discontinuity fronts are perpendicular to the walls of the channel. The arising flows are model ones for gas flows in a shock tube and a microchannel. The reflection of an incident shock wave from a flat end face is studied. It is found that the gas piles up at the cold wall, which slows down the shock wave detachment. The numerical results are in qualitative agreement with experimental data.     

Numerical study of the transverse supersonic flow of a diatomic rarefied gas past a plate

The two-dimensional supersonic rarefied gas flow past an infinite plate placed normally to the flow is analyzed. The gas possesses rotational degrees of freedom. The problem is stated for a model kinetic equation and is solved by applying a second-order accurate implicit conservative finite-difference method. The gas parameters correspond to nitrogen. The results are compared with those obtained for a monatomic gas. The influence exerted by the rotational degrees of freedom and the boundary conditions at the plate’s surface on the aerodynamic characteristics of the plate and the flow pattern is illustrated.     

Long Nonlinear Water Waves in a Channel Near a Cut-off Frequency

A theoretical study is made of the free-surface flow induced by a wavemaker, performing torsional oscillations about a vertical axis, in a shallow rectangular channel near a cut-off frequency. Exactly at cut-off, linearized water-wave theory predicts a temporally unbounded response due to a resonance phenomenon. It is shown, through a perturbation analysis using characteristic variables, that the nonlinear response is governed by a forced Kadomtsev—Petviashvili (KP) equation with periodic boundary conditions across the channel. This nonlinear initial-boundary-value problem is investigated analytically and numerically. When surface-tension effects are negligible, the nonlinear response reaches a steady state and exhibits jump phenomena. On the other hand, in the high-surface-tension regime, no steady state is obtained. These results are discussed in connection with similar forced wave phenomena studied previously in a deepwater channel and related laboratory experiments.     

Implicit numerical method for computing three-dimensional rarefied gas flows on unstructured meshes

The method based on the numerical solution of a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The basic idea behind the method is the use of a second-order accurate TVD scheme on hybrid unstructured meshes in physical space and a fast implicit time discretization method without iterations at the upper level. The performance of the method is illustrated by computing test examples of three-dimensional rarefied gas flows in variously shaped channels in a wide range of Knudsen numbers.     

KINETIC FLUX VECTOR SPLITTING FOR THE EULER EQUATIONS WITH GENERAL PRESSURE LAWS

This paper attempts to develop kinetic flux vector splitting(KFVS)for the Euler equa-tions with general pressure laws.It is well known that the gas distribution function forthe local equilibrium state plays an important role in the construction of the gas-kineticschemes.To recover the Euler equations with a general equation of state(EOS),a newlocal equilibrium distribution is introduced with two parameters of temperature approx-imation decided uniquely by macroscopic variables.Utilizing the well-known connectionthat the Euler equations of motion are the moments of the Boltzmann equation wheneverthe velocity distribution function is a local equilibrium state,a class of high resolutionMUSCL-type KFVS schemes are presented to approximate the Euler equations of gas dy-namics with a general EOS.The schemes are finally applied to several test problems for ageneral EOS.In comparison with the exact solutions,our schemes give correct location andmore accurate resolution of discontinuities.The extension of our idea to multidimensionalcase is natural.     

Simulation of rarefied gas flows on the basis of the Boltzmann kinetic equation solved by applying a conservative projection method

Flows of a simple rarefied gas and gas mixtures are computed on the basis of the Boltzmann kinetic equation, which is solved by applying various versions of the conservative projection method, namely, a two-point method for a simple gas and gas mixtures with a small difference between the molecular masses and a multipoint method in the case of a large mass difference. Examples of steady and unsteady flows are computed in a wide range of Mach and Knudsen numbers.     

负压抽液分装的数学模型

本文介绍了负压抽液分装的数学模型 .由于考虑到所讨论液态物质的渗透性及污染性大等特点 ,采用负压抽液的原理 ,根据能量守恒理论确定抽液所需的负压 (真空度 ) ;根据克拉珀龙方程、分子流运动论的 Knudsen公式等理论 ,确定达到额定压力 (真空度 )所需的时间 (即装置的起动时间 ) .据此设计了一套含有不完全真空保护系统的分装装置 ,用这套装置进行的实验和实际操作 ,其真空度、时间数据与确定值 (计算值 )基本相符 .表明所建立的数学模型正确 ,据此设计的这套装置已成功用于液态物质的分装 .     

Remarks on the asymptotic behavior of scalar auxiliary variable (SAV) schemes for gradient-like flows

We introduce a time semi-discretization of a damped wave equation by a SAV scheme with second order accuracy. The energy dissipation law is shown to hold without any restriction on the time step. We prove that any sequence generated by the scheme converges to a steady state (up to a subsequence). We notice that the steady state equation associated to the SAV scheme is a modified version of the steady state equation associated to the damped wave equation. We show that a similar result holds for a SAV fully discrete version of the Cahn-Hilliard equation and we compare numerically the two steady state equations.     

Implicit multiblock method for solving a kinetic equation on unstructured meshes

A parallel multiblock implementation of a second-order accurate implicit numerical method based on solving a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The performance of the method is illustrated by computing test examples of gas flows in a circular pipe in a wide range of Knudsen numbers. The convergence rate and scalability of the method are analyzed depending on the number of blocks in the spatial grid.     

Computation of rarefied diatomic gas flows through a plane microchannel

A numerical method based on a model kinetic equation was developed for computing diatomic rarefied gas flows in two dimensions. Nitrogen flows through a plane microchannel were computed, and the gas flow rate was constructed as a function of the Knudsen number for various channel lengths.