Monatomic gas as a singular limit of polyatomic gas in molecular extended thermodynamics with many moments

The difference in the theoretical structure between monatomic and polyatomic gases in highly nonequilibrium states is discussed from the viewpoint of molecular extended thermodynamics (MET) of rarefied gases, which is free from the local equilibrium assumption. The MET theories of these two types of gases are based on the moment balance equations with different hierarchy structures due to whether the internal degrees of freedom of a molecule are incorporated in their distribution functions or not. In particular, the number of balance equations in the MET theory of polyatomic gases is greater than the number in the corresponding theory of monatomic gases. The closure procedure for the system of balance equations of polyatomic gases obtained in a recent paper (Arima et al., 2014) is adopted. We prove that the solutions for polyatomic gases converge, in the limit where the degrees of freedom of a molecule D$D$ tend to 3, to the ones for monatomic gases provided that we impose appropriate initial conditions compatible with monatomic gases. Thus a MET theory of rarefied monatomic gases can be identified as a singular limit of the corresponding MET theory of rarefied polyatomic gases. As illustrative examples, the asymptotic behaviors when D→3$D\to 3$ in the dispersion relation of ultrasonic waves and in the shock wave structure are shown.     

The role of the dynamic pressure in stationary heat conduction of a rarefied polyatomic gas

The effect of the dynamic pressure (non-equilibrium pressure) on stationary heat conduction in a rarefied polyatomic gas at rest is elucidated by the theory of extended thermodynamics. It is shown that this effect is observable in a non-polytropic gas. Numerical studies are presented for a para-hydrogen gas as a typical example.     

Maximum entropy principle for rarefied polyatomic gases

The aim of this paper is to show that the procedure of maximum entropy principle for the closure of the moments equations for rarefied monatomic gases can be extended also to polyatomic gases. The main difference with respect to the usual procedure is the existence of two hierarchies of macroscopic equations for moments of suitable distribution function, in which the internal energy of a molecule is taken into account. The field equations for 14 moments of the distribution function, which include dynamic pressure, are derived. The entropy and the entropy flux are shown to be a generalization of the ones for classical Grad’s distribution. The results are in perfect agreement with the recent macroscopic approach of extended thermodynamics for real gases.     

气体传热对多层绝热性能影响的试验研究

文中通过建立的能进行夹层气体置换的稳态量热器试验系统,试验分析了夹层气体传热对多层绝热材料有效热导率的影响,重点对置换气体种类、气体压强、材料层数及冷热边界温度对多层材料的影响进行试验研究。试验表明在10—60层/cm层密度范围,真空度低于100Pa时,Kn数属于自由分子状态区域和中间压强区域,此时材料的有效热导率随残留气体热适应系数的增大而减小,并随着真空度的降低而增大,当残留气体为空气时,为保证多层材料的绝热性能,应尽量维持真空度不低于10-2Pa。同时,分析表明为有效降低低真空下稀薄气体传热对多层绝热性能的影响,可以采用综合热适应系数较低的气体置换夹层中的空气,以减少低真空多层绝热材料的有效热导率,改善绝热性能。     

层间稀薄气体传热对多层绝热材料性能的影响分析

通过建立的热量传递模型,分析了不同的气体稀薄程度(Knudsen数)时,气体传热对多层绝热材料有效热导率和各层温度分布的影响。分析表明:由多层绝热材料真空度变化引起的稀薄气体传热量波动较大,在10—60层/cm层密度范围,真空度低于100Pa时,Kn数属于自由分子状态区域和中间压强区域,此时材料的有效热导率随残留气体热适应系数的增大而减小,并随着真空度的降低而增大;当残留气体为空气时,为保证多层材料的绝热性能,尽量维持真空度不低于10-2Pa。同时分析表明,为有效降低低真空下稀薄气体传热对多层绝热性能的影响,可以采用综合热适应系数较低的气体置换夹层中的空气,以减少低真空多层绝热材料的有效热导率,改善绝热性能。     

Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments

Molecular extended thermodynamics of rarefied polyatomic gases is characterized by two hierarchies of equations for moments of a suitable distribution function in which the internal degrees of freedom of a molecule is taken into account. On the basis of physical relevance the truncation orders of the two hierarchies are proven to be not independent on each other, and the closure procedures based on the maximum entropy principle (MEP) and on the entropy principle (EP) are proven to be equivalent.     

A continuum model of gas flows with localized density variations

We discuss the kinetic representation of gases and the derivation of macroscopic equations governing the thermomechanical behavior of a dilute gas viewed at the macroscopic level as a continuous medium. We introduce an approach to kinetic theory where spatial distributions of the molecules are incorporated through a mean-free-volume argument. The new kinetic equation derived contains an extra term involving the evolution of this volume, which we attribute to changes in the thermodynamic properties of the medium. Our kinetic equation leads to a macroscopic set of continuum equations in which the gradients of thermodynamic properties, in particular density gradients, impact on diffusive fluxes. New transport terms bearing both convective and diffusive natures arise and are interpreted as purely macroscopic expansion or compression. Our new model is useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluid flows.     

Higher-order quadrature-based moment methods for kinetic equations

Kinetic equations containing terms for spatial transport, body forces, and particle–particle collisions occur in many applications (e.g., rarefied gases, dilute granular gases, fluid-particle flows). The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows sufficiently far away from the Maxwellian limit. In previous work, a quadrature-based third-order moment closure was derived for approximating solutions to the kinetic equation for arbitrary Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the non-negative weights and velocity abscissas. Here, a robust inversion procedure is proposed for three-component velocity moments up to ninth order. By reconstructing the velocity distribution function, the spatial fluxes in the moment equations are treated using a kinetic-based finite-volume solver. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the kinetic equation, the mass, momentum and energy are conserved for arbitrary Knudsen and Mach numbers. The computational algorithm is tested for the Riemann shock problem and, for increasing Knudsen numbers (i.e. larger deviations from the Maxwellian limit), the accuracy of the moment closure is shown to be determined by the discrete representation of the spatial fluxes.     

Mesoscopic approach to inviscid gas dynamics with thermal lag

A phenomenological model for thermal relaxation and wave propagation in ideal polyatomic gases is developed by introducing a dynamical non‐equilibrium temperature. The system of equations governing the evolution of the gas is derived and the speeds of propagation of thermo‐mechanical disturbances together with the Rankine‐Hugoniot jump conditions for shock waves are calculated. The hyperbolic theories of heat propagation in incompressible fluids and rigid solids are recovered as particular cases. For rigid solids, the well posedness of the Cauchy problem is proved by a classical method.     

Mayer and Virial Series at Low Temperature

We analyze the Mayer and virial series (pressure as a function of the activity resp. the density) for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series’ radii of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results are consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson model.     

Collisional Relaxation of Luminescence Anisotropy in Rarefied Gases in the Condensed Phase

The orientational relaxation of optically induced anisotropy in rarefied gases and at a damped rotation has been investigated. It has been found that the anisotropy relaxation in rarefied gases is described by a reduced kinetic equation depending only on free rotation integrals. The behavior of the integral anisotropy of luminescence for free symmetric and asymmetric top molecules has been elucidated. The law of luminescence depolarization has been obtained for asymmetric top molecules in the Gordon J-diffusion model. It represents the sum of two Stern–Volmer-type dependences, whose relative contribution is determined by the orientation of the dipole moments of transitions with absorption and emission of light in the molecular coordinate system and by the principal moments of inertia of the molecular top. It has been established that in the limit of a strongly damped rotation, kinetic equations of the general form reduce to equations of rotational diffusion. A number of modified diffusion equations correctly describing the contribution of inertial effects to the orientational relaxation of anisotropy have been obtained.     

A fast iterative model for discrete velocity calculations on triangular grids

A fast synthetic type iterative model is proposed to speed up the slow convergence of discrete velocity algorithms for solving linear kinetic equations on triangular lattices. The efficiency of the scheme is verified both theoretically by a discrete Fourier stability analysis and computationally by solving a rarefied gas flow problem. The stability analysis of the discrete kinetic equations yields the spectral radius of the typical and the proposed iterative algorithms and reveal the drastically improved performance of the latter one for any grid resolution. This is the first time that stability analysis of the full discrete kinetic equations related to rarefied gas theory is formulated, providing the detailed dependency of the iteration scheme on the discretization parameters in the phase space. The corresponding characteristics of the model deduced by solving numerically the rarefied gas flow through a duct with triangular cross section are in complete agreement with the theoretical findings. The proposed approach may open a way for fast computation of rarefied gas flows on complex geometries in the whole range of gas rarefaction including the hydrodynamic regime.     

Relativistic Rational Extended Thermodynamics of Polyatomic Gases with a New Hierarchy of Moments

A relativistic version of the rational extended thermodynamics of polyatomic gases based on a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal mode is proposed. The moment equations associated with the Boltzmann–Chernikov equation are derived, and the system for the first 15 equations is closed by the procedure of the maximum entropy principle and by using an appropriate BGK model for the collisional term. The entropy principle with a convex entropy density is proved in a neighborhood of equilibrium state, and, as a consequence, the system is symmetric hyperbolic and the Cauchy problem is well-posed. The ultra-relativistic and classical limits are also studied. The theories with 14 and 6 moments are deduced as principal subsystems. Particularly interesting is the subsystem with 6 fields in which the dissipation is only due to the dynamical pressure. This simplified model can be very useful when bulk viscosity is dominant and might be important in cosmological problems. Using the Maxwellian iteration, we obtain the parabolic limit, and the heat conductivity, shear viscosity, and bulk viscosity are deduced and plotted.     

Kinetic theory analysis of a binary mixture reacting on a surface

A mixture of two rarefied gases is considered between two parallel planes. On one side of the domain evaporation/condensation conditions are imposed, while on the other side accommodation at the temperature of the wall and chemical equilibrium conditions are considered. The small Knudsen number asymptotics of this problem is performed at the formal level, and fluid-dynamic equations are derived and then solved numerically. We discuss the possible appearance of a ghost effect in this situation.     

Is Brenner＇s Modification to the Classical Navier-Stokes Equations Able to Describe Sound Propagation in Gases？

We analyse the problem concerning the propagation of sound waves in gases by using the modified hydrodynamic theory proposed recently by Brenner for single-component fluids. The modifications introduced by Brenner are based on his proposal that the translational momentum in fluid motion is not given by the mass flux. Comparison of the sound propagation results derived from Brenner＇s theory with available experimental data for monatomic gases shows that this modified continuum theory is unable to describe the acoustic measurements not even in the low-frequency limit, a result that from our point of view makes Brenner＇s proposal questionable.     

Fluctuating hydrodynamics for a rarefied gas based on extended thermodynamics

We develop a theory of fluctuating hydrodynamics based on extended thermodynamics through studying the 13-variable theory for a monatomic rarefied gas as a representative case. After analyzing the relationship between the present theory and the Landau-Lifshitz theory, we discuss the hierarchy structure of the hydrodynamic fluctuations.     

Rarefaction effects in thermally-driven microflows

Rarefied gas flow in a parallel-plate micro-channel is considered, where a streamwise constant temperature gradient is applied in the channel walls. An analytical approach to the problem is conducted based on linearized and semi-linearized forms of the regularized 13-moment equations (R13 equations), which are a set of macroscopic transport equations for rarefied gases at the super-Burnett order. Typical nonequilibrium effects at the boundary, i.e., velocity slip, temperature jump, and formation of Knudsen boundary layers are investigated. Nonlinear contributions lead to temperature, density, and normal stress profiles across the channel which are not reported elsewhere in literature.     

Time-reversal symmetry relation for nonequilibrium flows ruled by the fluctuating Boltzmann equation

A time-reversal symmetry relation is established for out-of-equilibrium dilute or rarefied gases described by the fluctuating Boltzmann equation. The relation is obtained from the associated coarse-grained master equation ruling the random numbers of particles in cells of given position and velocity in the one-particle phase space. The symmetry relation concerns the fluctuating particle and energy currents of the gas flowing between reservoirs or thermalizing surfaces at given particle densities or temperatures.     

The effect of surface roughness on rarefied gas flows by lattice Boltzmann method

This paper studies the roughness effect combining with effects of rarefaction and compressibility by a lattice Boltzmann model for rarefied gas flows at high Knudsen numbers. By discussing the effect of the tangential momentum accommodation coefficient on the rough boundary condition, the lattice Boltzmann simulations of nitrogen and helium flows are performed in a two-dimensional microchannel with rough boundaries. The surface roughness effects in the microchannel on the velocity field, the mass flow rate and the friction coefficient are studied and analysed. Numerical results for the two gases in micro scale show different characteristics from macroscopic flows and demonstrate the feasibility of the lattice Boltzmann model in rarefied gas dynamics.     

大尺度浅水波方程中相互调制的非线性波

基于一般的浅水波方程, 根据大尺度正压大气的特点, 得到无量纲的控制大尺度大气的动力学非线性方程组. 利用多尺度法, 由无量纲的动力学方程组导出了扰动位势的非线性控制方程. 采用椭圆方程构造该扰动位势控制方程的解, 获得了扰动位势和速度的多周期波与冲击波(爆炸波) 并存的解析解. 扰动位势的解表明经向和纬向具有不同周期和波长的周期波, 且都受纬向孤波的调制; 速度的解表明大尺度大气流动存在气旋和反气旋周期性分布的现象. 关键词： 浅水波方程 大尺度正压大气 解析解 非线性波