A two‐dimensional numerical scheme of dry/wet fronts for the Saint‐Venant system of shallow water equations

We propose a new two‐dimensional numerical scheme to solve the Saint‐Venant system of shallow water equations in the presence of partially flooded cells. Our method is well balanced, positivity preserving, and handles dry states. The latter is ensured by using the draining time step technique in the time integration process, which guarantees non‐negative water depths. Unlike previous schemes, our technique does not generate high velocities at the dry/wet boundaries, which are responsible for small time step sizes and slow simulation runs. We prove that the new scheme preserves ‘lake at rest’ steady states and guarantees the positivity of the computed fluid depth in the partially flooded cells. We test the new scheme, along with another recent scheme from the literature, against the analytical solution for a parabolic basin and show the improved simulation performance of the new scheme for two real‐world scenarios. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability for manifolds of equilibrium states of generalized Hamiltonian systems

For a generalized Hamiltonian system, stability for the manifolds of equilibrium states is presented based on Lyapunov’s stability theories. Equilibrium equations, perturbation equations and first approximate equations of the system are given. A theorem for the stability of manifolds of equilibrium states of general autonomous system is used to the generalized Hamiltonian system, and three propositions on the stability of manifolds of equilibrium states of the system are obtained. Two examples are given to illustrate application of the method and results.     

Evaluation of well‐balanced bore‐capturing schemes for 2D wetting and drying processes

We consider numerical solutions of the two‐dimensional non‐linear shallow water equations with a bed slope source term. These equations are well‐suited for the study of many geophysical phenomena, including coastal engineering where wetting and drying processes are commonly observed. To accurately describe the evolution of moving shorelines over strongly varying topography, we first investigate two well‐balanced methods of Godunov‐type, relying on the resolution of non‐homogeneous Riemann problems. But even if these schemes were previously proved to be efficient in many simulations involving occurrences of dry zones, they fail to compute accurately moving shorelines. From this, we investigate a new model, called SURF_WB, especially designed for the simulation of wave transformations over strongly varying topography. This model relies on a recent reconstruction method for the treatment of the bed‐slope source term and is able to handle strong variations of topography and to preserve the steady states at rest. In addition, the use of the recent VFRoe‐ncv Riemann solver leads to a robust treatment of wetting and drying phenomena. An adapted ‘second order’ reconstruction generates accurate bore‐capturing abilities.This scheme is validated against several analytical solutions, involving varying topography, time dependent moving shorelines and convergences toward steady states. This model should have an impact in the prediction of 2D moving shorelines over strongly irregular topography. Copyright © 2006 John Wiley & Sons, Ltd.     

Generalization of Vlasov’s Equations for a Cylindrical Shell to the Case of a Transversely Isotropic Material

Differential equations of the general theory of transversely isotropic cylindrical shells are obtained; in a certain sense, these equations are generalizations of Vlasov’s and Ambartsumyan’s equations. This allowed us on the basis of Novozhilov’s criterion (comparison of variability of the stress state in the principal orthogonal directions) to divide the initial equations according to Goldenweiser into approximate equations of the type of the semi-momentless theory, theory of the edge effect and flexural state, which are also generalizations of equations that describe the elementary stress states of an isotropic shell. Numerical values are found for criteria of matching of approximate equations that describe the elementary stress states in the asymptotic synthesis of the full stress state. Examples of calculations and experimental data for a shell with and without allowance for transverse shear strain are given.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 125–132, July– August, 2005.     

The phase configuration of the waves around an accelerating disturbance in a rotating stratified fluid

Ray theory is extended to consider the case of an accelerating disturbance which is producing waves in a rotating stratified fluid. Starting from the equations of motion, dispersion relations are derived for surface gravity waves, capillary waves, Rossby waves and internal-inertial waves. The wave system is studied in each case for the problem of a body starting impulsively from rest and for a body starting from rest and moving with constant acceleration.     

Shock and acceleration waves with large amplitudes in laminated composite plates

Propagation of shock and acceleration waves with large amplitudes is studied. The geometrical nonlinearity in the von Karman sense is included in deriving the plate equations. The dynamical conditions on the wave fronts are derived from the three-dimensional conditions in a way consistent with the derivation of the plate equations. General equations governing the propagation velocities are obtained. Solutions are presented for the case where the plates are initially at rest. It is found that, in this case, the large amplitude has a substantial effect only on the transverse shear shock wave. Finally, stability of the wave front is discussed.     

Discontinuous solutions of the shallow water equations on a rotatable attracting sphere

The shallow water equations on a rotatable attracting sphere represent a system of hyperbolic equations on a compact manifold. These equations are derived in a spherical coordinate system from the integral laws of mass and total momentum conservation with account for the Coriolis and centrifugal forces. An analysis of the stability of discontinuous solutions with discontinuous waves and contact discontinuities is made using the closing law of total energy conservation, which represents a convex extension of the basic conservation-law system. The classes of stationary, one-dimensional (latitude-dependent only) exact solutions with contact discontinuities and discontinuous waves are constructed. Within the framework of the one-dimensional equations the test problem of wave flows resulting from the simultaneous break of two dams confining a fluid at rest in the vicinities of the poles is numerically modeled.     

Existence of solutions of Kraiko's problem

Three initial-boundary-value problems for the equations of gas dynamics are formulated. Successive solution of these problems yields a solution of Kraiko's problem of the isentropic transition of an ideal gas from a homogeneous state of rest to another state of rest with higher or lower density. Solutions are constructed for plane, cylindrical, and spherical layers of an ideal gas. The existence of locally analytic solutions is proved. Ural State Academy of Service, Ekaterinburg 620034. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 48–55, May–June, 2000.     

Stability for manifolds of equilibrium states of fractional generalized Hamiltonian systems

For a fractional generalized Hamiltonian system, in terms of Riesz derivatives, stability theory for the manifolds of equilibrium states is presented. The gradient representation and second order gradient representation of a fractional generalized Hamiltonian system are studied, and the conditions under which the system can be considered as a gradient system and a second order gradient system are given, respectively. Then, equilibrium equations, disturbance equations, and first approximate equations of a fractional generalized Hamiltonian system are obtained. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to a fractional generalized Hamiltonian system, and three propositions on the stability of the manifolds of equilibrium states of the system are investigated. As the special cases of this article, the conditions which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given, respectively, and the stability theory for the manifolds of equilibrium states of these systems are obtained. Further, a fractional dynamical system and a fractional Volterra model of the three species groups are given to illustrate the method and results of the application. Finally, by using the method in this paper, we construct a new kind of fractional dynamical model, i.e. the fractional Hénon–Heiles model, and we study its stability of the manifolds of equilibrium states.     

STABILITY FOR THE EQUILIBRIUM STATE OF CHAPLYGIN’S SYSTEMS

The stability for the equitibrium states of Chaplygin’s systems is considered. Theequations of motion of Chaplygin’s systems and the existence conditions of their equilibrium states are given. Some criteria of stability for the equilibrium.states of Chaplygin’s systems are obtained. Two examples are finally given.     

Characteristic properties of the system of equations for an incompressible viscoelastic Maxwell medium

Characteristics of a system of equations that describe three-dimensional motion of an incompressible viscoelastic Maxwell medium with the upper and lower convective derivatives and the rotational Jaumann derivative being used in the rheological relation are calculated. An initial-boundary-value problem is formulated for the system linearized in the vicinity of the state at rest, and its unique solvability is established.     

位势流动和非均匀介质中声波方程的36种形式

声波方程是对大多数声学问题进行数学描述的出发点. 那些得到 广泛应用的经典波动方程及对流波动方程都存在苛刻的适用条件, 即仅适用于描述处于静态或匀速运动状态的定常 均匀介质中的线性无耗散声波. 然而, 很多实际场合并不满足这些严格的适用条件. 本文对经典声波方程和对流声波 方程进行推广, 导出了编号为W1$\sim$W36的36种不同形式的声波方程, 涵盖了处于静止、势流或旋涡流状态下的非均匀 和/或非定常介质中的声波传播问题. 所考虑的声波传播情形包括: (1) 线性波, 即具有小梯度(小振幅)性质; (2)非线性波, 即具有陡峭梯度性质, 包括``波纹'(小振幅大梯度)或者大振幅波. 本文仅考虑非耗散声波, 即排除了由剪切、体积黏度及热传导所引起的耗散. 对具有匀熵或等熵(熵沿流线守恒)性质的均匀介质和非均匀介质中的声传播进行了研究但非等熵(即耗散)情况除外; 另外, 对非定常介质中的 声波问题也进行了分析. 所涉及的介质可以处于静止、匀速运动状态, 或者是非匀速的和/或非定常的平均流动, 包括: (1)低Mach数的势平均流(即不可压缩的平均态), 或高速势平均流(即非均匀可压缩的平均流); ② 变截面管 道中的准一维传播, 包括无平均流的号管和具有低或高Mach数平均流的喷管; 或③平面的、空间的、或轴对称的单 向剪切平均流. 本文没有探讨其他类型的旋涡平均流(将与耗散及其他情形一起留待下一步研究), 例如, 可能与剪切效应相结合的轴对称旋转平均流. 通过对流体力学的一般方程进行消元处理或根据声学变分原理, 导出了36种波动方程, 对一些波动方程还采用这两种方法进行相互校验. 尽管声波方程的36种形式没有涵盖非线性、非均匀与非定常及非匀速运动介质 这3个效应的所有可能的组合情形, 但它们的确包括了孤立状态下的各种效应, 并包括了多种多重效应组合的 情形. 虽然经典波动方程和对流波动方程仅适用于处于静止(或匀速运动)的均匀定常介质中的线性无耗散声波, 但它们在 相关文献中已被广泛采用; 本文给出的36种声波方程提供了它们多种有用的推广形式. 在许多实际应用中, 经典波动方 程和对流波动方程仅是粗略的近似, 声波方程的更一般形式可提供更令人满意的理论模型. 本文每节末尾给出了这些应用 的众多范例. 在这篇评论文章中引用了240篇参考文献.     

非局部平面应变和平面应力问题界定及其精确性讨论

在非局部弹性理论框架下对平面应变和平面应力状态重新界定.首先,分别在其相应简化假设下推导控制方程,并与经典局部情况进行比较.然后,引入变形协调条件对两类非局部平面问题的精确性进行讨论.其中,对于非局部平面应力状态,通过应变协调方程的Fourier变换形式来进行研究,使问题得以简化.通过以上分析,最终得到一些有价值的结论.     

Dynamics of formation and dripping of drops of deformation-rate-thinning and -thickening liquids from capillary tubes

Dynamics of formation of drops of non-Newtonian liquids from capillary tubes is studied computationally. The rheology of the drop liquids is described by a constitutive relation that accounts for both deformation-rate-thinning and -thickening. The analysis is expedited by reducing the original system of three-dimensional but axisymmetric equations to a system of one-dimensional slender-jet equations. The slender-jet equations are solved by a method of lines using a finite element method for spatial discretization and an adaptive finite difference method for time integration. The simulations follow the formation in time of thousands of drops in sequence, including any satellites that may be produced upon the breakup of a thin thread connecting an about-to-form primary drop to the rest of the liquid attached to the tube. Rate-thickening is shown to produce bead-on-string patterns, which are typically attributed to viscoelastic effects, along the thin threads as they near pinch-off. Rate-thinning, on the other hand, is demonstrated to reduce the length of such thin threads. Simulations are used to identify conditions that may lead to minimization and/or elimination of unwanted satellites. Analysis of dripping or leaky faucets of non-Newtonian liquids reveals rich nonlinear dynamical behavior. As with Newtonian liquids, simple periodic or P-1, where P stands for period, dripping at low flow rates gives way to more complex responses as flow rate is increased. In addition to P-1, P-2, and P-4 responses seen in recent computational analyses of dripping faucets of Newtonian liquids, the new non-Newtonian simulations have also uncovered difficult-to-find P-3 responses as well as chaotic states. Rate-thinning and low viscosities are shown to enhance the complexity of observed responses. Rate-thickening, on the other hand, lowers the critical value of the flow rate for the onset of complexity but narrows the range of flow rates over which the dynamics is complex. The possibility of hysteresis is demonstrated and the effect of fluid rheology on the value of the flow rate for transition from dripping to jetting is determined.     

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.     

Numerical Investigation of the Coalescence of Viscous-Liquid Drops

The coalescence of two identical viscous-liquid drops in weightlessness under the action of surface tension is considered. Inside the drops, the flow is described by the Navier-Stokes and Poisson (for the pressure) equations. On the liquid free surface in contact with the medium at rest, the dynamic and kinematic conditions are used. The problem is solved by a finite-difference method. The coalescence of the drops and the subsequent oscillations of the resultant drop are investigated. The results are compared with the known data.     

Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids

In this paper, based on three-dimensional linear generalized thermoelasticity, an exact analysis of free vibration of a simply supported homogeneous isotropic, thermally conducting, cylindrical panel with voids initially at uniform temperature and undeformed state has been presented. Three displacement potential functions are introduced for solving the equations of motion, heat conduction and volume fraction field. The purely transverse wave gets decoupled from rest of motion and is not affected by thermal and volume fraction (voids) fields. After expanding the displacement potentials, volume fraction and temperature functions with orthogonal series, the equations of the considered vibration problem are reduced to five-second order coupled ordinary differential equations whose formal solution can be expressed by using Bessel functions with complex arguments. The corresponding results for thermoelastic panel without voids, elastic panel with and without voids have been deduced as special cases from the present analysis. In order to illustrate the analytical results, the numerical solutions of various relations and equations have been obtained to compute the lowest frequency as function of different cylindrical panel parameters. The computer simulated results have been presented graphically.     

基于MUSTA格式的全非线性Boussinesq波浪传播数值模型

建立了求解二维全非线性布氏（Boussinesq）水波方程的有限差分/有限体积混合数值格式. 针对守恒形式的控制方程,采用有限体积方法并结合 MUSTA格式计算数值通量, 剩余项则采用有限差分方法求解, 采用具有总变差减小（totalvariation diminishing, TVD）性质的三阶龙格-库塔法进行时间积分.该格式具备间断捕捉、程序实现简单、数值稳定性强、海岸动边界以及波浪破碎处理方便和可调参数少等优点.利用典型算例对数值模型进行了验证,计算结果与实验数据吻合较好.      

Deformations of helical spring and cuboid into hollow cylinders

We study finite inhomogeneous deformations of a helical spring with a rectangular cross-section and a long cuboid. Two surfaces of the spring or the cuboid are joined to obtain a hollow cylinder. When body forces are absent the equilibrium equations reduce to ordinary differential equations. The stress-strain states are the same in each cross-section. The proposed deformations correspond to an inflation, an extension and a torsion of the obtained hollow cylinders. If the obtained cylinders are free of external applied loads, then they have residual stresses.