梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性，同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应，利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析，在不同的条件下，该方程在相平面上存在同宿轨道或异宿轨道， 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法，对该非线性方程进行 求解，得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解，讨论了这些解存 在的必要条件，这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程，从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

曲率对机翼边界层二次失稳影响

采用非线性抛物化扰动方程(NPSE)计算了层流机翼NLF0415(2)在几个工况下横流涡的非线性发展, 应用Floquet理论分析了横流涡的二次失稳. 较系统地分析了后掠机翼在多种参数下表面曲率对其流动稳定性的影响. Haynes证明曲率对于横流在线性稳定性计算(LST)和非线性抛物化扰动方程计算中(NPSE)都起着非常明显的稳定作用; 然而, 该文计算结果表明, 曲率在二次失稳计算中影响不大.      

弹性应力波理论中的运动非线性问题

本文在线性应变条件下导出了应力波在弹性介质中传播的非线性的运动方程。文中指出方程的非线性性质是由变形运动本身的非线性性质决定的。故这类非线性问题可以称为弹性应力波理论中的运动非线性问题。文中对在有限变形下具有线性应力——应变关系的弹性材料,在一维应变下,对方程进行了简化并给出了一个求解的实例。计算结果表明,这种情况下存在着十分明显的非线性动态响应,在分析和计算中必须加以考虑。     

三维方腔介电液体电对流的数值模拟研究

本文对封闭方腔内介电液体电对流进行了三维数值模拟研究.方腔的6个边界为固壁;4个侧边界为电绝缘边界;上下界面为两个电极.直流电场作用在从底部电极注入的自由电荷上,从而对液体施加库伦体积力并驱动流体流动形成电对流.为了求解这一物理问题,发展了一种二阶精度的有限体积法来求解完整的控制方程,包括Navier-Stokes方程和一组简化的Maxwell方程.考虑到电荷密度方程的强对流占优特性,采用了全逆差递减格式来求解该方程,获得了准确有界的解.通过研究发现,该流动在有限振幅区内的分叉类型为亚临界,即系统存在一个线性和非线性临界值,分别对应流动的开始和终止.由于非线性临界值比线性值小,因此两个临界值之间有一个迟滞回线.与无限大域中的自由对流相比,侧壁施加的额外约束改变了流场结构,使这两个临界值均有所增大.此外,还讨论了电荷密度和速度场的空间分布特征,发现电荷密度分布中存在电荷空白区.最后对更小空间尺寸情况计算结果表明,流动的线性分叉类型为超临界.本文的结果拓展了已有的二维有限空间内电对流的研究,并为三维电对流的线性和弱非线性理论分析提供参考.     

三维方腔介电液体电对流的数值模拟研究

本文对封闭方腔内介电液体电对流进行了三维数值模拟研究.方腔的6个边界为固壁;4个侧边界为电绝缘边界;上下界面为两个电极.直流电场作用在从底部电极注入的自由电荷上,从而对液体施加库伦体积力并驱动流体流动形成电对流.为了求解这一物理问题,发展了一种二阶精度的有限体积法来求解完整的控制方程,包括Navier-Stokes方程和一组简化的Maxwell方程.考虑到电荷密度方程的强对流占优特性,采用了全逆差递减格式来求解该方程,获得了准确有界的解.通过研究发现,该流动在有限振幅区内的分叉类型为亚临界,即系统存在一个线性和非线性临界值,分别对应流动的开始和终止.由于非线性临界值比线性值小,因此两个临界值之间有一个迟滞回线.与无限大域中的自由对流相比,侧壁施加的额外约束改变了流场结构,使这两个临界值均有所增大.此外,还讨论了电荷密度和速度场的空间分布特征,发现电荷密度分布中存在电荷空白区.最后对更小空间尺寸情况计算结果表明,流动的线性分叉类型为超临界.本文的结果拓展了已有的二维有限空间内电对流的研究,并为三维电对流的线性和弱非线性理论分析提供参考.      

蜿蜒边界下平面流的线性流动稳定性

为便于数值分析,蜿蜒河流水动力和演变模型中一般隐性假设二次时均流-二次涡的关系与明渠流时均流-明渠湍流的关系相同,但由于高雷诺数下的DNS算力限制和实验尺度限制,这种隐含假设是否成立目前尚无相关湍流研究来支撑.文章试图通过分析明渠湍流和二次湍流发展初期的研究,侧面揭示其湍流结构的异同.通过对曲线正交坐标系下的平面二维NS方程使用双参数摄动的方法,建立了一种求解蜿蜒边界弱非线性层流的摄动解法,并推导得出一个适用于蜿蜒边界的EOS方程以及其特征值问题的解法.蜿蜒边界下弱非线性层流解为一系列蜿蜒谐波分量的叠加,其中线性部分使得两壁产生流速差,非线性部分随着雷诺数增大呈指数增长.水流的扰动增长率特征谱的第一模态与直道流相似,由3条曲线、4个波段合成,但其长波段和短波段的扰动流场与直道流不同,所有短波段的扰动流速近似于KH涡.蜿蜒边界对内部水流扰动有一定的选择性.偏角幅值越大扰动增长越快;蜿蜒波数的影响则为先增后减,有一个使扰动增长最快的蜿蜒波数.扰动流场由一个典型的TS波和一对波包形式的二次涡叠加而成,波包只有纵向流速分量,包络线由蜿蜒波数控制,波包内是与直道扰动波参数相同的TS波.     

受撞击粘弹性基支弹性直梁的动力响应

本研究了半无限线性粘弹性Ｗｉｎｋｌｅｒ地基上的弹性直梁受低速运动物体撞击的动力响应分析问题，在不计撞击中应力波传播的条件下推导出了关于撞击力Ｆ（ｔ）的非线性Ｖｏｌｔｅｒｒａ分方程，得出了梁的横向位移Ｗ（ｘ，ｔ）的一般表达式，作为实例，本对两端简支弹性直梁受圆球对心横向撞击的问题进行了分析计算。     

移动载荷作用下浮冰的非线性动力学响应

本文考虑非线性、惯性和阻尼的影响, 研究了任意深度二维理想流体顶部浮冰的振动. 对相关的拟微分算子进行展开并将非线性项保留至三阶后, 完全非线性问题被简化为仅与自由面上的变量相关的三阶截断模型. 为了验证简化模型的准确性, 重点关注了自由孤立波解. 在不考虑阻尼的情况下, 采用多重尺度方法推导了三阶非线性薛定谔方程(NLS), 利用该方程预测了任意水深下原始欧拉方程中自由波包型孤立波解的存在性及三阶截断模型的准确性. 相比于Dinvay等所提出的二阶模型, 三阶截断模型的优势在于其对应的三阶NLS具有准确的非线性项系数, 能够在最小相速度附近更好地模拟冰层的动力学响应. 进一步地对自由孤立波解进行数值计算, 数值结果表明三阶截断模型在分岔曲线和孤立波波形上均与完全欧拉方程吻合良好, 准确性高于二阶截断模型. 基于三阶截断模型, 探究了匀速局域化载荷作用下的浮冰非线性动力学响应并将时间依赖解与实验测量数据进行比较, 数值计算结果与实验记录吻合良好.      

反应堆三角形通道棒束间二次流动的数值研究

采用非线性Ｋ－ε湍流模式数值模拟三角形通道棒束中的二次流动，并考察其对流动和传热的影响。数值方法采用非正交曲线坐标系下求解控制方程的非交错网格方法。计算结果表明该模式能够较为有效地反映棒束中的二次流动，进一步分析表明二次流动有利于改善棒束中的流动和传热特性。     

沿平板下落薄膜流动的研究综述

沿平板下落薄膜流动的时空演化一直是流体力学中一个相当活跃的研究领域.全面回顾了下落薄膜从长波近似方程到积分边界层方程, 从线性稳定性分析到弱非线性分析, 从首次失稳到二次失稳以及从有限振幅计算到直接数值模拟的发展历程, 总结了下落薄膜已有的理论结果和数值结果.此外,还介绍了沿加热平板下落的薄膜流动的最新研究进展, 概述了其它类型下落薄膜的研究情况.      

New equations for nonlinear acoustics in a low Mach number and weakly heterogeneous atmosphere

Various scalar equations are proposed, modeling the pressure field in the linear and nonlinear acoustical regimes. They are derived by assuming a flow with a small Mach number and a smaller medium heterogeneity. Such assumptions are well satisfied in the atmospheric boundary layer. Further simplifications can be obtained when less intense turbulent fluctuations are superimposed to a sheared mean flow. In the linear regime, a hierarchy of equations with increasing orders of precision is established. A new equation is found where all terms quadratic with respect to the ambient flow are retained, either related to sound convection by the flow, or to the flow inhomogeneity. Numerical solutions indicate that it is more precise than the equations in the literature for small Mach numbers, but less robust for larger negative Mach numbers. Two generalizations of Lilley’s equation incorporate the effects of turbulent fluctuations. Nonlinear terms are of different origins, either thermodynamical, inertial, or related to the flow shear. For a locally plane wave, they simplify into a single term which appears as the classical Westervelt quadratic nonlinearity convected by the flow. Consequently, all linear equations can easily be generalized to nonlinear ones, such as a new Lilley’s equation augmented with acoustical nonlinearities and turbulent flow fluctuations.     

New Approaches to the Propagation of Nonlinear Transients in Porous Media

Many problems in regional groundwater flow require the characterization and forecasting of variables, such as hydraulic heads, hydraulic gradients, and pore velocities. These variables describe hydraulic transients propagating in an aquifer, such as a river flood wave induced through an adjacent aquifer. The characterization of aquifer variables is usually accomplished via the solution of a transient differential equation subject to time-dependent boundary conditions. Modeling nonlinear wave propagation in porous media is traditionally approached via numerical solutions of governing differential equations. Temporal or spatial numerical discretization schemes permit a simplification of the equations. However, they may generate instability, and require a numerical linearization of true nonlinear problems. Traditional analytical solutions are continuous in space and time, and render a more stable solution, but they are usually applicable to linear problems and require regular domain shapes. The method of decomposition of Adomian is an approximate analytical series to solve linear or nonlinear differential equations. It has the advantages of both analytical and numerical procedures. An important limitation is that a decomposition expansion in a given coordinate explicitly uses the boundary conditions in such axis only, but not necessarily those on the others. In this article we present improvements of the method consisting of a combination of a partial decomposition expansion in each coordinate in conjunction with successive approximation that permits the consideration of boundary conditions imposed on all of the axes of a transient multidimensional problem; transient modeling of irregularly-shaped aquifer domains; and nonlinear transient analysis of groundwater flow equations. The method yields simple solutions of dependent variables that are continuous in space and time, which easily permit the derivation of heads, gradients, seepage velocities and fluxes, thus minimizing instability. It could be valuable in preliminary analysis prior to more elaborate numerical analysis. Verification was done by comparing decomposition solutions with exact analytical solutions when available, and with controlled experiments, with reasonable agreement. The effect of linearization of mildly nonlinear saturated groundwater equations is to underestimate the magnitude of the hydraulic heads in some portions of the aquifer. In some problems, such as unsaturated infiltration, linearization yields incorrect results.     

Analysis of efficient preconditioned defect correction methods for nonlinear water waves

Robust computational procedures for the solution of non‐hydrostatic, free surface, irrotational and inviscid free‐surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable prediction of free‐surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow equations. We present new detailed fundamental analysis using finite‐amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high‐order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow models. Our study is particularly relevant for fast and efficient simulation of non‐breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization parameters for different PDC strategies, we study systematically limits of accuracy, convergence rate, algorithmic and numerical efficiency and scalability of the most efficient known PDC methods. These strategies are of interest, because they enable generalization of geometric multigrid methods to high‐order accurate discretizations and enable significant improvement in numerical efficiency while incuring minimal storage requirements. We demonstrate robustness using such PDC methods for practical ranges of interest for coastal and maritime engineering, that is, from shallow to deep water, and report details of numerical experiments that can be used for benchmarking purposes. Copyright © 2014 John Wiley & Sons, Ltd.     

Motion of a vertical wall fixed on springs under the action of surface waves

A two-dimensional unsteady hydroelastic problem of interaction between surface waves and a moving vertical wall fixed on springs is studied. An analytical solution of the problem is constructed using a linear approximation, and a numerical solution within the framework of a nonlinear model of a potential fluid flow is found by a complex boundary element method. By means of analysis of the linear and nonlinear solutions, it is found that the linear solution can be used to predict some important characteristics of the wall motion and the fluid flow in the case of moderate wave amplitudes.     

Instability in gravity-driven flow over uneven permeable surfaces

We develop a theoretical model for inclined free-surface flow over a porous surface exhibiting periodic undulations. The effect of bottom permeability is incorporated by imposing a slip condition that accounts for the nonplanar geometry of the fluid–porous medium interface. Under the assumption of shallow flow, equations of motion accounting for inertial effects are obtained by retaining in the Navier-Stokes equations terms that are up to second-order with respect to a small shallowness parameter. The explicit dependence on the cross-stream coordinate is eliminated from these equations by means of a weighted residual procedure. A linear stability analysis of the steady flow is performed in connection with Floquet–Bloch theory. The results predict that bottom permeability has a destabilizing influence on the flow. A physical explanation has been proposed which involves examining how permeability affects the steady-state flow. Conclusions are drawn regarding the combined effect of the surface tension of the fluid and the parameters describing the bottom surface including permeability, inclination and the amplitude and wavelength of the undulations that generate the bottom topography. A numerical scheme for solving the fully nonlinear governing equations is also outlined. The instability of particular steady flows is determined by conducting nonlinear simulations of the temporal evolution of the flow and comparisons are made with the predictions from the linear analysis. Comparisons with existing experimental data are also included.     

Integral boundary layer relations in the theory of wave flows for capillary liquid films

A generalized method of deriving the model equations is considered for wave flow regimes in falling liquid films. The viscous liquid equations are used on the basis of integral boundary layer relations with weight functions. A family of systems of evolution differential equations is proposed. The integer parameter n of these systems specifies the number of a weight function. The case n = 0 corresponds to the classical IBL (Integral Boundary Layer) model. The case n ≥ 1 corresponds to its modifications called the WIBL (Weighted Integral Boundary Layer) models. The numerical results obtained in the linear and nonlinear approximations for n = 0, 1, 2 are discussed. The numerical solutions to the original hydrodynamic differential equations are compared with experimental data. This comparison leads us to the following conclusions: as a rule, the most accurate solutions are obtained for n = 0 in the case of film flows on vertical and inclined solid surfaces and the accuracy of solutions decreases with increasing n. Hence, the classical IBL model has an advantage over the WIBL models.     

Explosive discharge of a gas-saturated liquid from channels and tanks

A mathematical model for the discharge of a gas-saturated liquid from cylindrical channels is developed. Two limiting cases of linear and quadratic, relations between the flow friction force and the flow velocity are considered. It is established that the process of evacuation, from a semi-infinite channel consists of two stages. In the initial stage, the flow drag can be ignored, and the process of discharge is described by a Riemann wave solution. For the next stage, in which inertia is insignificant, nonlinear equations are obtained and self-similar solutions are constructed for them. The problem of flow through a slot in a tank of finite volume is solved. It is shown that the discharge proceeds either in a gas-dynamic choking regime or in a subsonic regime, depending on the conditions inside the tank and at the outlet. Examples of numerical calculations are given. Institute of Mechanics, Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000 Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 64–73, January–February, 1999.     

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.     

Waves in a solid medium with liquid-filled pores

The dynamic nonlinear theory of deformation of a two-phase medium, a solid with pores filled with a liquid, is developed. The variational principle is used to derive nonlinear equations that take into account the motions of the solid and liquid phases and the porosity variations. All types of nonlinearity, including nonlinear friction, are also taken into account. Formulas for the velocities of the linear and nonlinear waves and the absorption coefficient are derived. The one- and three-dimensional cases are considered. In the three-dimensional case, an equation describing the wave profile evolution is obtained as well as a nonlinear Schrödinger equation. Their solutions are analyzed; soliton-type solutions and solutions for narrow beams are obtained.