Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

Effect of viscosity on the propagation of nonlinear Alfvén waves along a tangential magnetohydrodynamic discontinuity in an incompressible fluid

It is proposed to consider the propagation of surface waves along a tangential magnetohydrodynamic discontinuity in the particular case where the fluid velocities on both sides of the interface are equal to zero. In [1] it was shown that waves called surface Alfvén waves may be propagated along the surface separating a semi-infinite region without a field from a region with a uniform magnetic field. The linear theory of surface Alfvén waves in a compressible medium was considered in [2]. In [3] the damping of surface Alfvén waves as a result of viscosity and heat conduction was investigated. The propagation of low-amplitude nonlinear surface Alfvén waves in an incompressible fluid in the absence of dissipative processes is described by the integrodifferential equation obtained in [4]. By means of a numerical solution of this equation it was shown that a perturbation initially in the form of a sinusoidal wave will break. The breaking time was determined. In this paper the equation derived in [4] is extended to the case of a viscous fluid. It is shown that the equation obtained does not have steady-state solutions. The propagation of periodic disturbances is investigated numerically. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–104, November–December, 1986. The author wishes to thank L. S. Fedorov for assisting with the calculations.     

Design method for fluid viscous dampers

A basic design method of doubly acting fluid viscous dampers with double guide bars is presented. The flow of the viscoelastic fluid between two parallel plates, one of which is started suddenly and the other of which is still, is analyzed. According to this solution, the velocity and the shear stress of the fluid at the fringe of the piston are solved approximately. A mathematical model of viscous dampers is derived, and the shock test is carried out. From experimental results, the parameters of the mathematical model are determined. Consequently, a semi-empirical design equation is obtained. Applying this equation to a certain practical damper, the damping material is chosen and the physical dimensions of the damper are determined. Shock tests using this damper are performed. Theoretical results are in good agreement with experimental results, which validates the reliability of the calculated physical dimensions of the specimen damper and the validity of the basic design equation. An erratum to this article can be found at     

On steady periodic waves on the surface of a fluid of finite depth

A solution of Nekrasov’s integral equation is obtained, and the range of its existence in the theory of steady nonlinear waves on the surface of a finite-depth fluid is determined. Relations are derived for calculating the wave profile and propagation velocity as functions of the ratio of the liquid depth to the wavelength. A comparison is made of the velocities obtained using the linear and nonlinear theories of wave propagation.     

Propogation of waves in a linear elastic fluid

The non-classical symmetry method is used to determine particular forms of the arbitrary velocity and forcing terms in a linear wave equation used to model the propogation of waves in a linear elastic fluid. The behaviour of solutions derived using the non-classical symmetry method are discussed. Solutions satisfy a given initial profile and wave velocity. For some solutions the arbitrary forcing terms and wave velocity can be written in terms of the initial wave profile. Relationships between the arbitrary forcing, arbitrary velocity and the solution are derived.     

Effect of damping and wave parameters on offshore structure under random excitation

This paper describes the linearized and nonlinear dynamic response of a tension leg platform (TLP) to random waves and current forces. The forcing term of the equation of motion is inherently nonlinear due to the nonlinear drag force. Two analysis procedures are used: nonlinear time domain analysis and linear frequency domain analysis. For the nonlinear analysis, the random wave particle velocities and accelerations are simulated for a given wave spectrum. The nonlinear equation of motion is then integrated directly to obtain the system response statistics. For the linear frequency domain analysis, the nonlinear drag force is linearized through an introduction of linearization coefficients. The main objective of this paper is to investigate the effect of the structural damping and wave parameters on both nonlinear and linear dynamic response of the TLP by parametric studies. The results of stochastic nonlinear and linear dynamic response of the TLP, with and without the presence of current, are presented and compared.     

激励Stuart-Landau方程的研究--周期解、稳定性及流动控制

解析得出了有外部激励的Stuart-Landau（S-L）方程的频率锁定周期解,对这些解与外部激励振幅和频率的依赖关系做了详细研究,并用周期系统稳定性理论确定了解的稳定性边界.还对S-L方程所描述的流动控制效果进行了研究,发现由于外部激励的作用,稳定的锁频解可能比原来的饱和解能量减少了,外部的控制最多能使扰动能量减少为原来的一半.     

EFFECTS OF VARYING BOTTOM ON NONLINEAR SURFACE WAVES

The resonant flow of an incompressible, inviscid fluid with surface tension on varying bottoms was researched. The effects of different bottoms on the nonlinear surface waves were analyzed. The waterfall plots of the wave were drawn with Matlab according to the numerical simulation of the fKdV equation with the pseudo-spectral method. Prom the waterfall plots, the results are obtained as follows: for the convex bottom, the waves system can be viewed as a combination of the effects of forward-step forcing and backward step forcing, and these two wave systems respectively radiate upstream and downstream without mutual interaction. Nevertheless, the result for the concave bottom is contrary to the convex one. For some combined bottoms, the wave systems can be considered as the combination of positive forcing and negative forcing.     

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。     

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

Gravity wave radiation from unsteady rotational flow in an f-plane shallow water system

Spontaneous gravity wave radiation from an unsteady rotational flow is investigated numerically in an f-plane shallow water system. Unlike the classical Rossby adjustment problem, where free development of an initially unbalanced state is investigated, we consider development of a barotropically unstable zonal flow which is initially balanced but maintained by zonal mean forcing. Gravity waves are continuously radiated from a nearly balanced rotational flow region even when the Froude number is so small that balance dynamics is thought to be a good approximation for the full system. The source of gravity waves is discussed by analogy with the theory of aero-acoustic sound wave radiation (the Lighthill theory). It is shown that the source regions correspond to regions of strong rotational flow. The gradual change of rotational flow causes gravity wave radiation. We propose an approximation for these strong sources on the assumption that the dominant flow in the jet region is non-divergent rotational flow. In addition, we calculate the zonally symmetric component of gravity waves far from the source regions, solving the Lighthill equation. Using scaling analyses for perturbations, these gravity waves can be calculated with only one approximated source term that is related to the latitudinal gradient of the fluid depth and the latitudinal mass flux. In spite of its simplicity, this approximation not only explains the physical cause of gravity wave radiation, but gives an amount of source close to that obtained by classical approximation derived from vortical motion.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

Viscosity effect on free surface waves in fluids

The frequencies and damping coefficients of gravitational-capillary waves are found for a wide range of controlling dimensionless parameters. The transition to the limiting cases of deep water and an ideal fluid is analyzed. In the parameter plane, the boundary between the regions of oscillatory and aperiodic perturbations is determined and the region of weak damping is indicated. The equilibrium state of thin liquid films with account for the Van der Waals forces is considered and the dispersion equation for the capillary-Van der Waals surface waves is obtained. For a suitably chosen frequency scale, this equation is the same as that for gravitational-capillary waves. The physical conditions making it possible to observe capillary and Van der Waals waves in thin fluid layers are estimated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 156–164, July–August, 2000.     

Water wave interaction with a sphere in a two-layer fluid flowing through a channel of finite depth

Using linear water wave theory, we consider a three-dimensional problem involving the interaction of waves with a sphere in a fluid consisting of two layers with the upper layer and lower layer bounded above and below, respectively, by rigid horizontal walls, which are approximations of the free surface and the bottom surface; these walls can be assumed to constitute a channel. The effects of surface tension at the surface of separation is neglected. For such a situation time-harmonic waves propagate with one wave number only, unlike the case when one of the layers is of infinite depth with the waves propagating with two wave numbers. Method of multipole expansions is used to find the particular solutions for the problems of wave radiation and scattering by a submerged sphere placed in either of the upper or lower layer. The added-mass and damping coefficients for heave and sway motions are derived and plotted against various values of the wave number. Similarly the exciting forces due to heave and sway motions are evaluated and presented graphically. The features of the results find good agreement with previously available results from the point of view of physical interpretation.     

Wave propagation through mangrove forests in the presence of a viscoelastic bed

The influence of viscoelastic ocean beds on the characteristics of surface waves passing through mangrove forests is analyzed under the assumption of linearized water wave theory in two dimensions. The trunks of the mangroves are assumed to be in the upper-layer inviscid fluid domain, whilst the roots are inside the viscoelastic bed. The associated equation of motion is obtained by coupling the Voigt’s model for flow within the viscoelastic medium with the equation of motion in the presence of mangroves. The modified dynamic conditions are coupled with the kinematic conditions to obtain the boundary condition at the free surface and the interface of the two fluids consisting of the upper layer inviscid fluid and the viscoelastic fluid bed. To understand the effects of bed viscosity as well as elasticity on energy dissipation, the complex dispersion relation associated with the plane progressive wave is derived and analyzed. Effect of physical parameters associated with mangroves and viscoelastic bed on wave motion in surface and internal modes are computed and analyzed to understand their roles in attenuating wave effects. The present model will be useful in the better understanding of wave propagation through mangroves in the coastal zone having muddy seabed.     

A new car-following model with consideration of anticipation driving behavior

A new anticipation driving car-following (AD-CF) model is presented based on the effect of traffic anticipation in the real world. The model??s linear stability condition was obtained by applying the linear stability theory. Additionally, a modified Korteweg?Cde Vries (mKdV) equation was derived via nonlinear analysis to describe the propagating behavior of traffic density wave near the critical point. Good agreement between the simulation and the analytical results shows that the stability of traffic flow can be enhanced when the driver??s anticipation effects are considered.     

Study on wave dispersion characteristics of piezoelectric sandwich nanoplates considering surface effects

In this study, the wave propagation properties of piezoelectric sandwich nanoplates deposited on an orthotropic viscoelastic foundation are analyzed by considering the surface effects (SEs). The nanoplates are composed of a composite layer reinforced by graphene and two piezoelectric surface layers. Utilizing the modified Halpin-Tsai model, the material parameters of composite layers are obtained. The displacement field is determined by the sinusoidal shear deformation theory (SSDT). The Euler-Lagrange equation is derived by employing Hamilton’s principle and the constitutive equations of piezoelectric layers considering the SEs. Subsequently, the nonlocal strain gradient theory (NSGT) is used to obtain the equations of motion. Next, the effects of scale parameters, graphene distribution, orthotropic viscoelastic foundation, and SEs on the propagation behavior are numerically examined. The results reveal that the wave frequency is a periodic function of the orthotropic angle. Furthermore, the wave frequency increases with the increase in the SEs.

     

填隙幂率流体下两刚性圆球相对错移时的粘性阻力

湿颗粒离散元模型以两球作用时填隙流体定常流动解为基础,其中切向作用是难点,国外仅有Goldman的牛顿流体渐近解.基于Reynolds润滑理论导出了两刚性球切向错动时填隙幂律流体的压力方程,并利用傅立叶级数展开简化,通过数值解法得到相应的压力分布、黏性阻力及阻力矩.该方程的解较之作者先前对速度场附加假定的结果精确,而当幂指数为1时等价于Goldman的牛顿流体渐近解.