Singularities in the Density Gradient

The compressible steady state Navier–Stokes system with viscosity is considered on a plane polygonal domain. It is shown that if a solution satisfies certain reasonable regularity conditions, and if the velocity is non-zero and points into the domain at a vertex of the polygon, then the gradient of the density either becomes infinite or has a jump discontinuity at each point of the streamline emanating from the vertex.     

基于广义波阻抗梯度飞片的准等熵压缩技术

基于变截面杆的波传播特性，设计了一种“针床型”广义波阻抗梯度飞片，即在圆薄片基座上密排叠加许多犹如针尖的小正四棱锥。采用LS-DYNA软件中SPH算法对广义波阻抗梯度飞片高速击靶过程进行了数值计算，结果显示:在飞片击靶过程中，每一个小正四棱锥台可以看作“点”式加载脉冲源，产生一系列具有缓慢上升前沿的近似球面波，球面波相互叠加得到具有缓慢上升前沿的平面加载波形，从而实现对靶板准等熵压缩加载。在数值计算中详细讨论了飞片击靶速度、飞片几何特征参数对准等熵压缩加载特性的影响规律，为广义波阻抗梯度飞片的设计与应用提供指导。基于数值计算结果，采用激光选区烧结金属增材制造技术，制备了一种广义阻抗梯度飞片样品，在一级气炮上进行击靶实验，实测了靶板自由面速度时程曲线，波形呈现了准等熵压缩加载特性，并与计算结果进行了对比，两者基本一致，从而验证了广义波阻抗梯度飞片结构设计的可行性以及数值计算结果的可靠性。     

The effect of viscoelasticity on a rising gas bubble

This paper investigates the role of viscoelasticity on the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. There is much debate in the literature over the cause of the jump discontinuity, which is observed once the bubble exceeds a certain critical volume. In this paper, the employment of some choice modelling assumptions allows insights into the mechanisms of the jump discontinuity which cannot be ascertained experimentally. The ambient fluid is assumed incompressible and the flow irrotational, with viscoelastic effects included through the stress balance on the bubble surface. The governing equations are solved using the boundary element method. Some Newtonian predictions are discussed before investigating the role of viscoelasticity. The model predicts the trademark cusp at the trailing end of a rising bubble to a high resolution. However, the irrotational assumption precludes the prediction of the negative wake. The corresponding absence of the jump discontinuity supports the hypothesis that the negative wake is primarily responsible for the jump discontinuity, as mooted in previous studies.     

Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids

The propagation of a three-dimensional shock wave in an elastic solid is studied. The material is assumed to be a simple elastic solid in which the Cauchy stress depends on the deformation gradient only. It is shown that the growth or decay of a discontinuity ψ depends on (i) an unknown quantity φ^? behind the shock wave, (ii) the two principal curvatures of the shock surface, (iii) the gradient on the shock surface of the shock wave speeds and (iv) the inhomogeneous term which depends on the motion ahead of the shock surface and vanishes when the motion ahead of the shock surface is uniform. If a proper choice is made of the propagation vectorb along which the growth or decay of the discontinuity is measured, the dependence on item (iii) can be avoided. However,b assumes different directions depending on the choice of discontinuity ψ with which one is concerned and the unknown quantity φ^? behind the shock wave on which one chooses to depend. As in the case of one-dimensional shock waves, the growth (or decay) of one discontinuity may not be accompanied by the growth (or decay) of other discontinuities. A universal equation relating the growth or decay of discontinuities in the normal stress, normal velocity and specific volume is also presented.     

平面柔性多体系统正碰撞动力学建模理论研究

针对目前柔性多体系统碰撞动力学建模方法存在的不足,对影响碰撞动力学仿真的主要因素如柔性体建模和碰撞初始条件进行分析,建立起基于变约束的柔性体碰撞动力学方程。首先,为了解决子结构法在处理碰撞界面搜索时面临的难题,引入多体系统柔性体有限元描述方法,推导出凸形柔性体接触点间法向位移约束的二阶导数形式。其次,从碰撞引起的接触界面速度不连续机理出发,结合连续介质力学间断面理论,给出碰撞瞬时由物体本身物理性质决定的接触位置处速度跳跃公式。最后对两弹性圆盘低速碰撞问题进行数值仿真。结果表明本文提出的改进方法符合力学基本原理,仿真结果满足收敛性要求。     

The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

Traveling wave with allowance for equilibrium radiation

Three modes of propagation of a traveling-wave front over a noncold gas with different propagation velocities are found using one thermodynamic model. When the indicated velocity is low, transition from constant values of the gas parameters on both sides of the traveling-wave front proceeds continuously. An increase in the traveling-wave velocity leads to an isothermal jump: the density and velocity of the gas undergo a strong discontinuity whereas the temperature varies continuously. With a further increase in the traveling-wave velocity, the isothermal jump disappears and the flow becomes continuous again. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 15–25, July–August, 2006.     

The self-similarly expanding Eshelby ellipsoidal inclusion: I. Field solution

The solution of a self-similarly (subsonically) dynamically expanding ellipsoidal inclusion with general spatially uniform transformation strain temporally constant is obtained by the use of the Radon transform and the satisfaction of the zero initial conditions and the radiation condition at infinity. It constitutes the self-similar evolution of the inclusion singularity (jump discontinuity at the inclusion boundary) starting from zero dimension. The field solutions for the displacement gradient and particle velocity are presented. Due to the fact that for a self-similarly expanding subsonic motion the hyperbolic system of the partial differential equations of motion becomes elliptic (as proved in Ni and Markenscoff, 2015), it is shown here explicitly that the solution for the displacement gradient in the interior domain of the expanding ellipsoid is constant, thus extending the Eshelby property to the self-similarly expanding ellipsoids as pointed out by Burridge and Willis (1969). Also, the particle velocity is shown to be zero in the interior domain (lacuna) as the waves emitted by the self-similarly expanding inclusion cancel each other due to the symmetries of geometry and motion.     

An admissibility condition for equilibrium shocks in finite elasticity

The equations governing the equilibrium of a finitely deformed elastic solid are derived from the Principle of Minimum Potential Energy. The possibility of the deformation gradient and the stresses being discontinuous across certain surfaces in the body — “equilibrium shocks” — is allowed for. In addition to the equilibrium equations, natural boundary conditions and traction continuity condition, a supplementary jump condition which is to hold across the surface of discontinuity is derived. This condition is shown to imply that a stable equilibrium shock must necessarily be dissipation-free.     

Higher-order jump conditions for conservation laws

The hyperbolic conservation laws admit discontinuous solutions where the solution variables can have finite jumps in space and time. The jump conditions for conservation laws are expressed in terms of the speed of the discontinuity and the state variables on both sides. An example from the Gas Dynamics is the Rankine–Hugoniot conditions for the shock speed. Here, we provide an expression for the acceleration of the discontinuity in terms of the state variables and their spatial derivatives on both sides. We derive a jump condition for the shock acceleration. Using this general expression, we show how to obtain explicit shock acceleration formulas for nonlinear hyperbolic conservation laws. We start with the Burgers’ equation and check the derived formula with an analytical solution. We next derive formulas for the Shallow Water Equations and the Euler Equations of Gas Dynamics. We will verify our formulas for the Euler Equations using an exact solution for the spherically symmetric blast wave problem. In addition, we discuss the potential use of these formulas for the implementation of shock fitting methods.     

Considerations on shock wave/boundary layer interaction in undular hydraulic jumps in horizontal channels with a very high aspect ratio

It can be seen in the literature that the fundamental factors governing oblique shock wave development, typically in very large channels with straight sidewalls, have not yet been completely understood and remain at the level of indicating its presence and formation. In this study, in addition to an analysis of various properties of hydraulic jump behaviour in very large channels, some aspects of boundary layer development and its detachment from the channel lateral sidewall are also investigated. At the detachment point of the lateral shock waves, it was noted that the displacement thickness experiences a significant increase; this is accompanied by a significantly reduced gradient normal to the channel sidewalls of the flow velocity as well as the occurrence of a strong, sudden adverse pressure gradient. An analysis of the flow velocity distribution and the background turbulence intensity of both the streamwise and spanwise velocity components was also carried out. Furthermore, it is argued that the supersonic flow separation analogy with a supercritical free surface flow can be applied to this case study and that the behaviour of the supercritical flow during separation can be interpreted by the free interaction theory typically used in aerodynamics.     

Stress channelling in extreme couple-stress materials Part II: Localized folding vs faulting of a continuum in single and cross geometries

The antiplane strain Green's functions for an applied concentrated force and moment are obtained for Cosserat elastic solids with extreme anisotropy, which can be tailored to bring the material in a state close to an instability threshold such as failure of ellipticity. It is shown that the wave propagation condition (and not ellipticity) governs the behaviour of the antiplane strain Green's functions. These Green's functions are used as perturbing agents to demonstrate in an extreme material the emergence of localized (single and cross) stress channelling and the emergence of antiplane localized folding (or creasing, or weak elastostatic shock) and faulting (or elastostatic shock) of a Cosserat continuum, phenomena which remain excluded for a Cauchy elastic material. During folding some components of the displacement gradient suffer a finite jump, whereas during faulting the displacement itself displays a finite discontinuity.     

Mechanism of shear attenuation near the interface under combined compression and shear impact loading

We investigate the compressional/shear coupling plastic wave propagation characteristics analytically for ideal elastic–plastic materials in both stress and particle velocity spaces, focusing on the shear wave attenuation near the interface occurring in pressure–shear plate impact experiments. The results show that the shear attenuation is strongly associated with the wave propagation characteristics of the coupling waves. In the stress space, as the shear stress increases, an adjustment of the stress components is observed and the final stress state along the wave path is a combined pure shear- and hydrostatic pressure-state. In the particle velocity space, the wave structures with different loading and maximal transverse particle velocity are obtained. The maximal transverse particle velocity varies with the longitudinal velocity and forms a boundary line. Once the loading transverse velocity exceeds this line, a transverse particle velocity discontinuity occurs at the impact interface. If the bonding strength is sufficiently high, there will be a shear band in the target in the extreme vicinity of the interface.     

A model for pressure loss, film thickness, and entrained fraction for gas-liquid annular flow

A two-component (air-water) annular flow model is presented requiring only flow rates, absolute pressure, temperature, and tube diameter. Film thicknesses (base film and wave height) are calculated from a critical film thickness model. Modeled pressure gradient is weighted by wave intermittency to compute average pressure gradient. Film flow rate and wave velocity are estimated using the universal velocity profile in the waves and a piecewise linear profile in the base film. For vertical flow, mean absolute errors for film thickness, wave velocity, and pressure gradient are 9%, 9%, and 19%, respectively. In horizontal flow, mean absolute errors for pressure gradient, base film thickness, and disturbance wave velocity are 17%, 10%, and 14%, respectively, on par with those from single-behavior models that require additional film thickness or other data as inputs.     

Roughening Instability of Broken Extremals

We derive a new general jump condition on a broken Weierstrass–Erdmann extremal of a vectorial variational problem. Such extremals, containing surfaces of gradient discontinuity, are ubiquitous in shape optimization and in the theory of elastic phase transformations. The new condition, which does not have a one dimensional analog, reflects the stationarity of the singular surface with respect to two-scale variations that are nontrivial generalizations of Weierstrass needles. The over-determinacy of the ensuing free boundary problem suggests that typical stable solutions must involve microstructures or chattering controls.     

Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.     

Model of a strong discontinuity for the equations of spatial long waves propagating in a free-boundary shear flow

The long-wave equations describing three-dimensional shear wave motion of a free-surface ideal fluid are rearranged to a special form and used to describe discontinuous solutions. Relations at the discontinuity front are derived, and stability conditions for the discontinuity are formulated. The problem of determining the flow parameters behind the discontinuity front from known parameters before the front and specified velocity of motion of the front are investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 206–213, July–August, 2008.     

压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题

分析了压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题。利用傅立叶变换,使问题的求解转换为对一对以裂纹表面上的位移差为未知变量的对偶积分方程的求解。为了求解对偶积分方程,把裂纹面上的位移差展开为雅可比多项式形式,进而得到了裂纹长度、入射波波速及入射波频率对裂纹应力强度因子的影响。从数值结果可以看出,压电压磁复合材料中可导通裂纹的反平面问题的动应力奇异性与一般弹性材料中的反平面断裂问题动应力奇异性相同。     

Convection in a porous medium with velocity slip and temperature jump boundary conditions

The onset of convection in a rarefield gas saturating a horizontal layer of a porous medium has been investigated using both Darcy and Brinkman models. It is assumed that due to rarefaction both velocity slip and temperature jump exist at the boundaries. The results show that (i) when the degree of rarefaction increases the critical Rayleigh number as well as the critical wave number for the onset of convection increases, (ii) stabilizing effect of temperature jump is more than that of velocity slip, (iii) Darcy model is seen to be the most stable one when compared to Brinkman model or the pure gaseous layer (i.e. in the absence of porous medium).