Ideal gas flow behind a finite-amplitude shock wave

The equations of one-dimensional (with a plane of symmetry) adiabatic motion of an ideal gas are transformed to a form convenient for studying flows between a moving piston and a shock wave of variable intensity. The solution is found for the equations of a motion containing a shock wave which propagates through a quiescent gas with variable initial density and constant pressure. This solution contains four arbitrary constants and, in a particular case, gives an example of adiabatic shockless compression by a piston of a gas initially at rest.     

Ovsyannikov plane vortex: The equations of the submodel

A submodel of the equations of ideal magnetohydrodynamics is constructed that generalizes the classical motion of an ideal continuous medium with plane waves. It is shown that, in contrast to classical motion, in this submodel the velocity and magnetic-field vectors can change direction in a plane orthogonal to a distinguished spatial direction. The submodel is described by a system of equations with two independent variables and a finite relation specifying the orientation of the vector fields in space. The solutions of the submodel define substantially spatial processes and singularities in the motion of continuous media which cannot be studied in the classical one-dimensional formulation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 27–40, September–October, 2008.     

Is Intersonic Dislocation Motion Possible? Singularity Analysis for an Edge Dislocation Accelerating through the Shear Wave Speed Barrier

The analysis of Markenscoff and Clifton (J Mech Phys Solids 29:253–262, 1981) for a generally nonuniformly moving Volterra edge dislocation, valid both for subsonic and intersonic/supersonic motion, is focused on the instant in which the dislocation accelerates through the shear-wave speed barrier. Mathematically, the roots (of the argument of the step function that defines the intervals of the path of the motion that contribute to the field point) change from a pair of complex conjugate to a double real, splitting into two real ones, and, at the instant of the transition to intersonic motion, the stress analysis is performed at this double root maximum of . The stress at the forming Mach front contains a singularity in the coefficient of the delta function, which can be removed by a ramp-core (delta sequence rather than delta function) model of the core displacement.     

Transformation and some solutions of the equations of motion of an ideal gas

The equations of one-dimensional and plane steady adiabatic motion of an ideal gas are transformed to a new form in which the role of the independent variables are played by the stream function and the function introduced by Martin [1, 2], It is shown that the function retains a constant value on a strong shock wave (and on a strong shock for plane flows). For one-dimensional isentropic motions the resulting transformation permits new exact solutions to be obtained from the exact solutions of the equations of motion. It is shown also that the one-dimensional motions of an ideal gas with the equation of state p=f(t) and the one-dimensional adiabatic motions of a gas for which p=f() are equivalent (t is time, is the stream function). It is shown that if k=s=–1, m and n are arbitrary (m+n0) and =1, the general solution of the system of equations which is fundamental in the theory of one-dimensional adiabatic self-similar motions [3] is found in parametric form with the aid of quadratures. Plane adiabatic motions of an ideal gas having the property that the pressure depends only on a single geometric coordinate are studied.     

基于原子体积场拉普拉斯算子对金属玻璃剪切转变区的预测

剪切转变区(shear transformation zone, STZ)作为金属玻璃塑性事件的一个基本特征单元, 已被研究者们逐渐接受,但STZ产生的机制和来源仍具争议. 本文采用分子模拟方法对 Cumetallic glasses,shear transformation zones,nonaffine displacements,atomic volume,molecular simulation,1)国家自然科学基金(11790292);国家自然科学基金(11672298);国家自然科学基金(11432014);中国科学院战略性先导科技专项(B类)(XDB22040501)2019-12-242020-04-10剪切转变区(shear transformation zone, STZ)作为金属玻璃塑性事件的一个基本特征单元, 已被研究者们逐渐接受,但STZ产生的机制和来源仍具争议. 本文采用分子模拟方法对 Cu$_{64}$Zr$_{36}$金属玻璃在受简单剪切加载时的变形行为展开了研究. 结果表明,体系的初始构型与加载后STZ的产生是相关的. 虽然原子体积场及其梯度可以用来有效表征金属玻璃中局部原子构型的非均匀性, 但它们与STZ产生的区域没有直接明显的对应关系. 基于此, 提出一个新的局域结构参数 $\xi $ 来用于金属玻璃中STZ产生区域的预测,它由两部分构成: 原子体积场的拉普拉斯算子和体积场梯度分量的绝对差值. 原子体积场的拉普拉斯算子为负且绝对值较大时, 体积场梯度向量呈现向内指的分布特征, 代表体系中的局域软区; 而体积场梯度分量的绝对差值则用于遴选体积场梯度不同的分布模式. 进一步地,建立了该结构参数与非仿射位移和剪切局部化三者关系, 发现特定的体积场梯度向量分布模式, 将导致局部剪切增强的非仿射位移场, 从而更容易诱发STZ的形成. 相关性分析表明,该参数与STZ区域平均相关性高于78%, 因此, 该参数能有效用于金属玻璃剪切转变区的预测,且运用拉普拉斯算子的思想有望应用于金属玻璃力学行为的理论分析.     

Sonic discontinuities in a relaxing gas

Following Elcrat [1] the phenomena associated with the sonic discontinuities in non-equilibrium gasdynamics have been studied here. The sonic wave in non-equilibrium gaseous medium propagates with the frozen speed of sound. The magnitude of discontinuities of the first derivatives of flow quantities in the unsteady flow of relaxing gas are shown to satisfy Riccatti equations along the orthogonal trajectories of surface S(t). In order to integrate them in full generality, they are transformed to an equation along the bicharacteristic curve in the characteristic manifold S(t). These equations have been solved completely. The criteria for decay or blow up of sonic discontinuities are given and the particular cases of plane and spherical waves existing in an ideal dissociative gasdynamics have been discussed. In the case of planewave for uniform propagation, it is shown that the dissociating character of the gas is to decrease the critical time. Other cases of shock formation have been studied in detail.     

Nonlinear periodic waves in a gas

A system of equations describing the one-dimensional time-dependent polytropic motion of a gas is considered. In special cases the general solutions of this system of equations are obtained and exact solutions with the initial conditions which are periodic with respect to the spatial variable are found. For an arbitrary polytropic exponent an asymptotic solution, which is uniformly suitable till the onset of the gradient catastrophe, is constructed in the form of expansions in series in a small parameter, namely, the initial wave amplitude. Asymptotic dependences of the time of onset and the location of the gradient catastrophe are obtained. The complex correspondence between the initial system of equations and the system of equations describing the motion of quasi-gas media is given. An example of using this correspondence is considered.     

Propagation of discontinuities against a static background

The solution of the ideal gasdynamic equations describing propagation of a shock wave initiated, for example, by the motion of a piston against an inhomogeneous static background is considered. The solution is constructed in the form of Taylor series in a special time variable which is equal to zero on the shock wave. In the case of weak shock waves divergence of the series serves as the constraint for such an approach. Then the solution is constructed by linearizing the equations about the solution with a weak discontinuity. In the case of a given background the last solution can be always found exactly by solving successively a set of transport equations, all these equations are reduced to linear ordinary differential equations. The presentation begins from the one-dimensional solutions with plane waves and ends by discussion of spatial problems.     

Statistical Analysis of Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

Statistically planar turbulent premixed and partially premixed flames for different initial turbulence intensities are simulated for global equivalence ratios ??>?=?0.7 and ??>?=?1.0 using three-dimensional Direct Numerical Simulations (DNS) with simplified chemistry. For the simulations of partially premixed flames, a random distribution of equivalence ratio following a bimodal distribution of equivalence ratio is introduced in the unburned reactants ahead of the flame. The simulation parameters in all of the cases were chosen such that the combustion situation belongs to the thin reaction zones regime. The DNS data has been used to analyse the behaviour of the dissipation rate transports of both active and passive scalars (i.e. the fuel mass fraction Y _F and the mixture fraction ξ) in the context of Reynolds Averaged Navier–Stokes (RANS) simulations. The behaviours of the unclosed terms of the Favre averaged scalar dissipation rates of fuel mass fraction and mixture fraction (i.e. \(\widetilde {\varepsilon }_Y =\overline {\rho D\nabla Y_F^{\prime \prime } \cdot \nabla Y_F^{\prime \prime } } /\overline{\rho }\) and \(\widetilde {\varepsilon }_\xi =\overline {\rho D\nabla \xi ^{\prime \prime }\cdot \nabla \xi ^{\prime \prime }} /\overline {\rho })\) transport equations have been analysed in detail. In the case of the \(\widetilde {\varepsilon }_Y \) transport, it has been observed that the turbulent transport term of scalar dissipation rate remains small throughout the flame brush whereas the terms due to density variation, scalar–turbulence interaction, reaction rate and molecular dissipation remain the leading order contributors. The term arising due to density variation remains positive throughout the flame brush and the combined contribution of the reaction and molecular dissipation to the \(\widetilde {\varepsilon }_Y \) transport remains negative throughout the flame brush in all cases. However, the behaviour of scalar–turbulence interaction term of the \(\widetilde {\varepsilon }_Y \) transport equation is significantly affected by the relative strengths of turbulent straining and the straining due to chemical heat release. In the case of the \(\widetilde {\varepsilon }_\xi \) transport, the turbulent transport term remains small throughout the flame brush and the density variation term is found to be negligible in all cases, whilst the reaction rate term is exactly zero. The scalar–turbulence interaction term and molecular dissipation term remain the leading order contributors to the \(\widetilde {\varepsilon }_\xi \) transport throughout the flame brush in all cases that have been analysed in the present study. Performances of existing models for the unclosed terms of the transport equations of \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) are assessed with respect to the corresponding quantities obtained from DNS data. Based on this exercise either suitable models have been identified or new models have been proposed for the accurate closure of the unclosed terms of both \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) transport equations in the context of Reynolds Averaged Navier–Stokes (RANS) simulations.     

Gas-dynamic analogy for vortex free-boundary flows

The classical shallow-water equations describing the propagation of long waves in flow without a shear of the horizontal velocity along the vertical coincide with the equations describing the isentropic motion of a polytropic gas for a polytropic exponent γ = 2 (in the theory of fluid wave motion, this fact is called the gas-dynamic analogy). A new mathematical model of long-wave theory is derived that describes shear free-boundary fluid flows. It is shown that in the case of one-dimensional motion, the equations of the new model coincide with the equations describing nonisentropic gas motion with a special choice of the equation of state, and in the multidimensional case, the new system of long-wave equations differs significantly from the gas motion model. In the general case, it is established that the system of equations derived is a hyperbolic system. The velocities of propagation of wave perturbations are found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 8–15, May–June, 2007.     

Characteristic Properties and Exact Solutions of the Kinetic Equation of Bubbly Liquid

The paper considers a kinetic model for the motion of incompressible bubbles in an ideal liquid that takes into account their collective interaction in the case of one spatial variable. Generalized characteristics and a characteristic form of the equations are found. Necessary and sufficient hyperbolicity conditions of the integrodifferential model of rarefied bubbly flow are formulated. Exact solutions of the kinetic equation for the class of traveling waves are derived. A solution of the linearized equation is obtained.     

Asymptotics of the Solutions of the Random Schrödinger Equation

We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying, then the Fourier transform \({\hat\zeta_\epsilon(t,\xi)}\) of the appropriately scaled solution converges point-wise in ξ to a stochastic complex Gaussian limit. On the other hand, when the two-point correlation function decays slowly, we show that the limit of \({\hat\zeta_\epsilon(t,\xi)}\) has the form \({\hat\zeta_0(\xi){\rm exp}(iB_\kappa(t,\xi))}\) where B _κ (t, ξ) is a fractional Brownian motion.     

Statistical Scaling of Geometric Characteristics in Millimeter Scale Natural Porous Media

We analyze statistical scaling of structural attributes of two millimeter scale rock samples, Estaillades limestone and Bentheimer sandstone. The two samples have different connected porosities and pore structures. The pore-space geometry of each sample is reconstructed via X-ray micro-tomography at micrometer resolution. Directional distributions of porosity and specific surface area (SSA), which are key Minkowski functionals (geometric observables) employed to describe the pore-space structure, are calculated from the images, and scaling of associated order- $q$ sample structure functions of absolute incremental values is analyzed. Increments of porosity and SSA tend to be statistically dependent and persistent (tendency for large and small values to alternate mildly) in space. Structure functions scale as powers $\xi (q)$ of directional separation distance or lag, $s$ , over an intermediate range of $s$ , displaying breakdown in power law scaling at large and small lags. Powers $\xi \!\!\left( q \right) $ of porosity and SSA inferred from moment and extended self-similarity (ESS) analyses of limestone and sandstone data tend to be quasi-linear and nonlinear (concave) in $q$ , respectively. We observe an anisotropic behavior for $\xi (q)$ , which appears to be mild for the porosity of the sandstone sample while it is marked for both porosity and SSA of the limestone rock sample. The documented nonlinear scaling behavior is amenable to analysis by viewing the variables as samples from sub-Gaussian random fields subordinated to truncated fractional Brownian motion or fractional Gaussian noise.     

The application of the Lattice Boltzmann method to the one-dimensional modeling of pulse waves in elastic vessels

The one-dimensional nonlinear equations for the blood flow motion in distensible vessels are considered using the kinetic approach. It is shown that the Lattice Boltzmann (LB) model for non-ideal gas is asymptotically equivalent to the blood flow equations for compliant vessels at the limit of low Knudsen numbers. The equations of state for non-ideal gas are transformed to the pressure-luminal area response. This property allows to model arbitrary pressure-luminal area relations. Several test problems are considered: the propagation of a sole nonlinear wave in an elastic vessel, the propagation of a pulse wave in a vessel with varying mechanical properties (artery stiffening) and in an artery bifurcation, in the last problem Resistor–Capacitor–Resistor (RCR) boundary conditions are considered. The comparison with the previous results shows a good precision.     

基于θ1方法的多体动力学数值算法研究

将结构动力学领域的\theta_1方法拓展到数值求解多体系统运动方程------微分--代数方 程(DAEs), 分别求解指标-3 DAEs形式的运动方程和指标-2超定DAEs (ODAEs)形式的运动方程. 通过数值算例验证了方法的有效性, 并得到\theta _1 方法中参数\theta _1的选取与数值耗散量之间的关系. 数值算例还说明对于同 一个多体系统, 采用指标-3的DAEs 描述时存在速度违约, 用指标-2的ODAEs描述时, 从计算机精度上讲, 位置和速度约束方程 同时满足, 并且\theta_1方法在求解非保守系统DAEs和ODAEs形式的运动方程时 都具有2阶精度. 最后\theta_1 方法与其他直接积分法求解DAEs和ODAEs形式运 动方程的CPU时间进行了比较.     

Energy separation of gases with low and high Prandtl numbers

The process of energy separation of a gas with the Prandtl number which is not equal to unity is investigated. The gas flows through a heat exchanger consisting of two coaxial axisymmetric pipes with sub- and supersonic velocities. Heat exchange between the gas streams takes place as a result of the fact that the recovery factor is not equal to unity. The flow is described by one-dimensional gas dynamic equations for averaged quantities.     

Partially invariant solutions in gas dynamics and implicit equations

This paper studies a nonbarochronic, regular, partially invariant solution (submodel) of rank one and defect two to the equations of gas dynamics which describes spatial unsteady gas motion. The equations of gas dynamics are reduced to an implicit ordinary differential equation of the first order for an auxiliary function and to an integrable system. A complete classification of the irregular singular points of the key equation according to a parameter characterizing the gas flow is given, and transformations of the irregular singular points with variation in the parameter are obtained. Qualitative properties of the solution are investigated and physically interpreted in terms of gas motion. It is shown that there are two modes of motion, one of which is supersonic, and in the second modes, a continuous transition through the speed of sound is possible.     

Gas dynamics of a selectively excited gas

In order to describe the motion of a selectively excited gas, model kinetic equations are proposed in the present study, and they are used to construct the equations of gas dynamics. There is consideration of the one-dimensional problem of heat transmission between plates across a selectively excited gas. Discontinuities are found in the temperature on the boundaries of the phases. It is shown that a selectively excited gas may be used to transmit heat from a cold body to a hot one. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 151–158, September–October, 1988.     

壁湍流相干结构尺度及边界层内的SL标度律

利用高分辨率、高帧率PIV系统对湍流边界层中相干结构的多种空间尺度和边界层内SL 标度律在不同尺度下的具体表达形式进行了实验研究. 实验在两个动量损失厚度雷诺数 (Re_{\theta}=628.5和Re_{\theta}=1032.9)下测量平板湍流边界层中缓冲 层、对数区和外区的二维瞬时速度场. 应用 小波分析以及传统的统计学方法,在垂直于平板和平行于平板的平面内考察平板湍流边界层 中存在的相干结构的流向和展向尺度,并与已知的相干结构尺度实验结果进行了对比分析. 利用在动量损失厚度雷诺数628.5下测得的数据,对多种脉动结构(脉动速度结构等) 的空间关系及其标度律进行了研究. 第2项工作直接利用湍流边界层空间速度分布,对多种 流场尺度结构内部的She-Leveque(简称SL)标度律及自相似律进行了验证. 结果表明,各 单一流场尺度结构内部,流向脉动速度{\pmb u}'、法向脉动速度{\pmb v}'及 脉动涡分量\d {\pmb v}'/\d {\pmb x}的统计结构量均存在明显的标度律,标度 指数的形式与自相似律和SL标度律均非常吻合,只是常数随流场尺度的不同而不同, 且呈现一定的规律性. 但对于结构量的五阶矩随距离l的研究表明,自相似律和SL 标度律成立的范围并不完全一致,同时标度律成立的范围大小与流场尺度有明显关系.