Moment equations in the kinetic theory of gases and wave velocities

We consider the evolution system for N-moments of the Boltzmann equation and we require the compatibility with an entropy law. This implies that the distribution function depends only on a single scalar variable which is a polynomial in . It is then possible to construct the generators such that the system assumes a symmetric hyperbolic form in the main field. For an arbitrary we prove that the systems obtained maximise the entropy density. If we require that the entropy coincides with the usual one of non-degenerate gases, we obtain an exponential function for , which was already found by Dreyer. From these results the behaviour of the characteristic wave velocities for an increasing number of moments is studied and we show that in the classical theory the maximum velocity increases and tends to infinity, while in the relativistic case the wave and shock velocities are bounded by the speed of light. Received June 5, 1997     

Maximum wave velocity in the moments system of a relativistic gas

We consider the system of moments associated with the relativistic Boltzmann-Chernikov equation. Using the particular symmetric form obtained by the closure procedure of Extended Thermodynamics we deduce a lower bound for the maximum velocity of wave propagation in terms of the number of moments for a non-degenerate gas. When the number of moments increases this velocity tends to the speed of light. We also give the lower bound estimate in the limit cases of ultrarelativistic fluids and in the non relativistic approximation. Received September 28, 1998     

Light scattering and extended thermodynamics

The interaction of light and matter leads to the scattering of light and the scattered light carries information about the thermodynamic properties of the matter. The light scattered on dilute gases carries far more information about the gas than is comprised within the Navier-Stokes-Fourier theory of gases. It takes extended thermodynamics of many moments to satisfactorily describe the characteristic features of such light quantitatively.     

Light scattering experiments and extended thermodynamics

Light in a gas is scattered on density fluctuations and the spectrum of the scattered light is influenced by the constitutive properties of the gas. The Navier-Stokes-Fourier theory does not always describe the spectrum of the scattered light in gases satisfactorily; it fails for small densities. Extended thermodynamics of many moments however may be used to predict the scattering spectra of dilute gases correctly. In this paper we compare the results of extended thermodynamics with measurements. Received: January 2, 1997     

水击驻波场中分散相颗粒的运动分析

根据水击在管段内形成的驻波场现象,分析了流体内分散相颗粒受到的驻波作用力;运用李雅普诺夫稳定判据研究了颗粒积聚与分离的机理;考虑到颗粒运动方程的严重刚性而很难进行数值求解,采用相空间和非对称分析方法获得了分散相颗粒的运动轨迹近似解,并进行了实验验证。结果表明:水击驻波场中分散相颗粒的受力方程中惯性项对颗粒初始运动速率的影响不可忽略;在水击驻波波节的±λ/4范围内,分散相颗粒经过一定的时间会发生积聚,其运动速度呈对称分布,最大速度出现在3λ/8位置处;随着分散相颗粒粒径和密度等物性参数以及水击驻波的频率和连续相初始速度的增大,颗粒达到平衡位置的时间呈减小趋势,且连续相的初始速度对颗粒到达波节时间的影响显著。     

Delayed die swell

The experiments reported here establish that there is a general critical condition associated with die swell which we call delayed die swell. This condition is defined by a critical speed which is the area-averaged velocity, the extrusion velocity, at the exit of the pipe when the swell is first delayed. The delayed swell ratio and delay distance first increase for larger, post-critical values of the extrusion velocity; then the increases are terminated either by instabilities or by smoothing. The maximum post-critical velocity at the pipe exit was always greater than the shear wave speed measured on the shear-wave-speed meter. The post critical area averaged velocity at the position of maximum swell before termination was always less than the shear wave speed. There were always points in the region of swelling where the ratio of the local velocity to the shear wave speed, the viscoelastic Mach number, was unity. The swelling of the jet is a nonlinear phenomenon which we suggest is finally terminated either by instability or when the variations of the velocity, vorticity and stress field are reduced to zero by the inward propagation of shear waves from the free surface of the jet. This propagation is generated by discontinuous “initial” data along χ in which the prescribed values of velocity at the boundary change from no-slip in the pipe to no-shear in the jet. The measurements raise the possibility that the delay may be associated with a change of type from supercritical to subcritical flow.     

Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions

We consider a system of N balance laws compatible with an entropy principle and convex entropy density. Using the special symmetric form induced by the main field, we define the concept of principal subsystem associated with the system. We prove that the 2 ^N −2 principal subsystems are also symmetric hyperbolic and satisfy a subentropy law. Moreover we can verify that for each principal subsystem the maximum (minimum) characteristic velocity is not larger (smaller) than the maximum (minimum) characteristic velocity of the full system. These are the subcharacteristic conditions. We present some simple examples in the case of the Euler fluid. Then in the case of dissipative hyperbolic systems we consider an equilibrium principal subsystem and we discuss the consequences in the setting of extended thermodynamics. Finally in the moments approach to the Boltzmann equation we prove, as a consequence of the previous result, that the maximum characteristic velocity evaluated at the equilibrium state does not decrease when the number of moments increases. (Accepted October 6, 1995)     

Wave and shock velocities in relativistic magnetohydrodynamics compared with the speed of light

We prove that the natural thermodynamic restrictions on the constitutive equations in relativistic magnetohydrodynamics (stability of equilibrium state) are necessary and sufficient to guarantee that the normal and the radial velocity of the wave front of disturbances and the shock velocity do not exceed the light speed.     

转动压电陶瓷体表面声波的存在性及波速特征关系

对转动半无限压电陶瓷体的表面声波给出了存在性证明及其条件。结果表明：对应于不转动时的Rayleigh表面声波与Bleustein-Gulyaev表面声波在转动情形是否仍为表面波有赖于压电体的材料参数;如果转动时表面声波存在,则这些表面声波是色散的。最后,以PZT-5H压电陶瓷材料为例定量给出了波速-转动角速度的特征曲线。     

Turbulent Dynamics of a Critically Reflecting Internal Gravity Wave

Results from computational fluid dynamics experiments of internal wave reflection from sloping boundaries are presented. In these experiments the incident wave lies in a plane normal to the slope. When the angle of wave energy propagation is close to the bottom slope the reflection causes wave breakdown into a quasi-periodic, turbulent boundary layer. Boundary layer energetics and vorticity dynamics are examined and indicate the importance of the three-dimensional turbulence. The boundary layer exhibits intermittent turbulence: approximately every 1.2 wave periods the boundary layer mixes energetically for a duration of about one-third of a wave period, and then it restratifies until the next mixing event. Throughout the wave cycle a strong thermal front is observed to move upslope at the phase speed of the incident waves. Simulations demonstrate that the net effects of turbulent mixing are not confined to the boundary layer, but are communicated to the interior stratified fluid by motions induced by buoyancy effects and by the wave field, resulting in progressive weakening of the background density gradient. Transition to turbulence is determined to occur at Reynolds numbers of approximately 1500, based upon the wavelength and maximum current velocity of the oncoming wave train. The boundary layer thickness depends on the Reynolds number for low Richardson numbers, with a characteristic depth of approximately one-half of the vertical wavelength of the oncoming wave. Received 21 May 1997 and accepted 14 October 1997     

Phase and signal velocities of waves in dissipative media. Special relativistic theory

The signal and phase velocities (and their frequency dependence) for all possible plane waves in a relativistic gas (of molecules or photons) with dissipation have been determined from the linearized relativistic 13-moment theory. For each direction in 3-space, and for each frequency, one transverse and two longitudinal waves were found. (In addition, some waves are associated with the mass flow and have the mass flow speed.) Of the longitudinal waves, the fast one is a pressure (sound) wave. It is accompanied by a slow longitudinal thermal dissipation wave and a transverse viscous dissipation wave. The pressure wave has a velocity larger than the Laplace adiabatic speed of sound, while the two dissipation waves have a velocity less than the Laplace speed. All the speeds have been expressed explicitly in terms of quantities associated with the state of equilibrium which existed before passage of the wave. It has also been shown that in the ultrarelativistic limit (extremely high temperatures) all signal speeds remain less than the speed of light in vacuo.The major part of this article was presented at the Seminar on Natural Philosophy at The Johns Hopkins University, Baltimore, on November 24, 1971.     

Three-dimensional green water velocity on a model structure

Flow kinematics of green water due to plunging breaking waves impinging on a simplified, 3D model structure was investigated in the laboratory. Two breaking wave conditions were tested: one with waves impinging on the vertical wall of the model at still water level, and the other with waves impinging on the horizontal deck surface. The bubble image velocimetry (BIV) technique was used to measure flow velocities. Measurements were taken on both vertical and horizontal planes. Evolution of green water flow kinematics in time and space was revealed and was found to be quite different between the two wave conditions, even though the incoming waves are essentially identical. The time history of maximum velocity is demonstrated and compared. In both cases, the maximum velocity occurs near the green water front and beneath the free surface. The maximum horizontal velocity for the deck impinging case is 1.44C with C being the wave phase speed, which is greater than 1.24C for the wall impingement case. The overall turbulence level is about 0.3 of the corresponding maximum velocity in each wave condition. The results were also compared with 2D experimental results to examine the 3D effect. It was found that the magnitude of the maximum vertical velocity during the runup process is 1.7C in the 3D model study and 2.9C in the 2D model study, whereas the maximum horizontal velocity on the deck is similar, 1.2C in both 3D and 2D model studies.     

Wave development on a thinning liquid film

Experimental results are presented for the growth of surface waves on a liquid film that thins as it flows under gravity over the surface of an upright circular cone. The characteristics of the mean film are calculated on the assumption of quasi-parallel flow, and the actual mean thickness found to relate very closely to that found on this basis. The development of the film was found to fall into three phases: the entry zone in which the velocity profile of the film becomes established where no waves are visible, a region of wave growth in which amplitude, wave speed, and wave length all grow, and a final region in which amplitude and wave speed decline as the film thins further although wave length continues to grow. An empirical relationship is presented which expresses the wave number at any point on the cone in terms of the flow rate and a parameter based on the local Reynolds and Weber numbers and cone angle. It was found that for a given flow rate the maximum wave amplitude was reached at a value of wave number of 0·048.     

Experimental investigation of a maximum entropy assumption for acceleration terms within a poly‐disperse moment framework

Moment transport methods are being developed to model poly‐dispersed multiphase flows by transporting statistical moments of the particle size–velocity joint probability density function (JPDF). A common feature of these methods is the requirement to reproduce or approximate the form of the JPDF from the transported moments for calculation of body force terms and other source terms. This paper examines the application of a maximum entropy technique against phase Doppler anemometry data sets from an electrostatically charged kerosene spray and also an automotive pressure swirl atomizer. An assessment of which moments are required to reproduce the JPDFs using a maximum entropy assumption to a sufficient level of accuracy is made. It is found that it is possible to reproduce the JPDFs to a high level of accuracy using a large number of moments; however, this incurs large computational overheads. If the moments to be transported are chosen on the basis of physical reasoning (such as the relationship between size and velocity due to drag) it is possible to reduce the number of moments to those which would be conserved via balance equations. This permits an approximation to the JPDF commensurate with the closure level of the moment transport method and thus the closure model method is naturally scalable with the degree of information from available conservation equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Klein-Gordon波动方程多波传播的非线性特性

基于双波初值问题,讨论非线性对多波传播的影响。通过选取合适的多重尺度,对Klein-Gordon波动方程进行变形,得到方程的解的多尺度展式首项近似和三波传播时速度相互影响的定量关系,揭示了多波传播的非线性特性;最后,应用Mathematica对波动方程进行数值仿真。研究结果表明,另外多个波的存在会使波的传播速度（相速）超过独自传播时的速度（相速）。     

微差爆破振动波速度峰值-位移分布特征的延时控制

为了通过振动波传播规律研究微差爆破延时控制,改善爆破振动和提高爆炸能利用,选取普通雷管和澳瑞凯高精度雷管进行不同段别延期起爆对比实验,试爆过程对振动波测振。基于振动波函数优化理论基础,对实测数据和波状谱处理分析,总结了不同微差时间下振动波传播规律及速度峰值、主频、频带能量、总能量等变化特征,根据该特征发现振动波速度图谱和该速度积分所得位移图谱中两者最大值对应时间点相同。依据此速度峰值-振动位移分布特征,对某实测简单振动波进行高斯多峰拟合,结果表明:该波段能量最大化时间点为60 ms左右,振动最小的时间点为25 ms左右。     

氢燃料缓燃向爆震转捩过程中波与火焰的匹配特性

为了解氢燃料爆震过程中压力波与火焰之间相互匹配的特性,在60mm60mm2000mm 方 爆震管内,用氢气和空气混合物进行了单爆震性能研究。在爆震转捩区内布置压力传感器与离子探针,用来 监控压力波和火焰的信号,同时利用高速摄影仪集中拍摄转捩区域。根据压力波和火焰面在爆震管不同时刻 的强度特性、速度特性及位置特性来分析爆震过程中波与火焰匹配的规律。结果表明:压力波和火焰的强度 呈现为相互正反馈匹配性质;缓燃向爆震转捩(DDT)过程中,压力波和火焰的速度表现为相互交替的变化过 程,且缓燃阶段中火焰速度的增幅大于压力波速度的增幅;当火焰面追赶上激波时,产生过爆,火焰面会临时 位于激波前面;在过爆衰减为正常爆震波的过程中,激波在火焰前面。     

Tsunamis and Barge Canals

Russell’s velocity formula was at the center of the controversy over the existence of the solitary wave; but today the topic is rarely mentioned. It is an immediate corollary of modern bifurcation theory; and it is fundamental to modelling waves in deep water. A tsunami 60 cm high in an ocean 4 km deep is 377 km long, travels with a velocity of 713 km/hr, and carries a transverse energy density of 2 × 10⁹ joules/meter, yet the maximum speed of the surface current is only 2.97 cm/s.     

川藏公路地质环境与整治改建方案的思考

川藏公路由于地质环境复杂、建设标准低、后遗病害多,抗灾能力差,泥石流、滑坡、山崩、雪害、水毁等自然灾害频繁发生,公路阻车断道严重。国家投入巨资进行整治改建,并取得了明显的效果,但由于自然环境特殊、影响因素复杂,许多特大型、大型工程地质病害问题还没有可行、可靠的解决方案。本文通过分析川藏公路沿线的地质环境和灾害特点,总结历年整治改建和经验的教训,提出川藏公路建设的途径、可能达到的目标和应采用的原则。     

驱动气体的密度梯度对弹丸发射速度的影响

为进一步提升轻气炮的发射能力,提出采用梯度气体替代单一氢气或氦气作为驱动气体的方法,通过对等直径发射器进行分析,建立了弹丸在梯度气体驱动下的加速运动模型,对比了氖-氦梯度气体驱动与单一氦气驱动的发射能力差异,分析了梯度气体参数对发射性能的影响。结果表明,与单一氦气驱动相比,氖-氦梯度气体驱动能够提升0.4～1.4 km/s的发射速度或降低0.2～0.9 GPa的发射过载;气体的密度和活塞的运动速度对发射速度和过载的影响最大,气体压力和多方气体指数的影响次之;梯度气体中,高密度气体应选择多方气体指数和密度较高的气体（如氖气、氩气等）;梯度气体界面位置（高密度气体占比）对发射速度的影响不大,但高密度气体占比少有利于降低弹底压力。