On problems in the weakly nonlinear theory of hydrodynamic stability and its improvement

There are three main problems in the weakly nonlinear theory of hydrodynamic stability: (1)The radius of convergence with respect to the perturbation parameter is too small and there is no concrete estimation for it. (2)The solution has a special structure, thus in general, it can not satisfy the initial condition posed by many practical problems. (3)When the linear part of its solution does not correspond to a neutral case, there are more than one way in determining the Landau constants, and practically no one knows which is the best way. In this paper, problems(1) and (2)are solved theoretically, and ways for its improvement have been proposed. By comparing the theoretical results with those obtained by numerical simulations, problem(3)has also been clarified. Project supported by the National Natural Science Foundation of China     

桁架结构稳定分析的几何非线性欧拉稳定理论

目前,桁架结构稳定分析有两个理论:经典的特征值理论和几何非线性临界点理论。前者发展较早,在许多结构力学书中都有叙述。但是经过工程应用检验,它过高地估计了结构抗稳定的能力。后者是20年前提出的,提出者认为适用于所有的扁桁架。经过几年的研究,作者连续提出三个稳定理论:线性与非线性欧拉理论;非线性临界点-欧拉理论和两个计算方法,进行桁架结构的线性和非线性截面优化设计。经过理论研究和对前人文献中数值例题的计算和比较,发现特征值理论不符合实际,临界点理论只适用于大扁度的桁架,而不适用于一切扁桁架。研究结果提供了若干有用的结论,给出了各种稳定理论的适用范围,指出了国际文献中若干例题的正确和错误,补充了桁架结构的稳定分析理论,使之更为完整和实用了。     

On the energy equation for a spatially developing instability wave in the theory of hydrodynamic stability

In this article, it is shown that the energy equation for a spatially developing disturbance used in all the literatures dealing with the problem of hydrodynamic stability suffers from a small, but crucial error.     

On the theory of stability of a cylindrical hydrodynamic suspension

The stability of a cylindrical hydrodynamic suspension is investigated with complete account for the dependence of the fluid velocity distribution profile on the radial coordinate in the supporting layer without an a priori representation of the inertial terms in the fluid layer in terms of the velocity averaged over the clearance cross-section. It is shown that the central position of a light inner body in the cylindrical hydrodynamic suspension is asymptotically stable. The suspension remains asymptotically stable for a fairly long time as the light inner body is displaced from the central position along the dynamic equilibrium curve. The central position of a heavy inner body is unstable; however, as it is displaced from the central position under the action of a constant external decentering force a stability domain develops.     

THE RE－EXAMINATION OF DETERMINING THE COEFFICIENT OF THE AMPLITUDE EVOLUTION EQUATION IN THE NONLINEAR THEORY OF THE HYDRODYNAMIC STABILITY

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Nonlinear vibration and stability analyses on the basis of a kinetic stability theory for arbitrarily curved rods

Summary After formulating nonlinear kinetic stability equations for curved and twisted rods, these are used for the solution of static and kinetic lateral buckling problems as well as for linear and nonlinear stability analysis of parametrically excited vibrations of circular arches. Accepted for publication 26 May 1997     

Analysis of hybrid algorithms for the Navier–Stokes equations with respect to hydrodynamic stability theory

Hybrid three‐dimensional algorithms for the numerical integration of the incompressible Navier–Stokes equations are analyzed with respect to hydrodynamic stability in both linear and nonlinear fields. The computational schemes are mixed—spectral and finite differences—and are applied to the case of the channel flow driven by constant pressure gradient; time marching is handled with the fractional step method. Different formulations—fully explicit convective term, partially and fully implicit viscous term combined with uniform, stretched, staggered and non‐staggered meshes, x‐velocity splitted and non‐splitted in average and perturbation component – are analyzed by monitoring the evolution in time of both small and finite amplitude perturbations of the mean flow. The results in the linear field are compared with correspondent solutions of the Orr–Sommerfeld equation; in the nonlinear field, the comparison is made with results obtained by other authors. Copyright © 2002 John Wiley & Sons, Ltd.     

流动稳定性弱非线性理论中的问题及其改进

流动稳定性弱非线性理论存在三个主要问题:(1)其摄动参数收敛半径太小且无具体估值;(2)其解有特定结构,一般不能满足很多实际问题的初始条件;(3)当其线性部分不是中性时,Landau系数的确定法不唯一,不知何者应是最好的方法。本文以平面Poiseuille流为例,从理论上解决了问题(1)和(2),并提出了改进办法。又通过将理论和数值模拟结果相比较的方法,澄清了问题(3)。改进了的理论,将可解决许多实际问题。     

The dynamic stability and nonlinear resonance of a flexible connecting rod: Continuous parameter model

The transverse vibrations of a flexible connecting rod in an otherwise rigid slider-crank mechanism are considered. An analytical approach using the method of multiple scales is adopted and particular emphasis is placed on nonlinear effects which arise from finite deformations. Several nonlinear resonances and instabilities are investigated, and the influences of important system parameters on these resonances are examined in detail.     

Theoretical analyses on hydrodynamic instability in narrow deep river with variable curvature

Bending river is the most common river type in nature, and it is also a typical example for river evolution. The transform of the flow pattern can affect the development of the riverbed form. In return, the variation in the riverbed form can affect the hydrodynamic characteristics of the flow, thereby leading to the continuous evolution of the bending river. Based on this, a theoretical model for the bending river is established. The hydrodynamic instability characteristics of the laminar flow in the channel with a variable curvature, a typical model as the meandering river, are studied, and the variations of some parameters such as the curvature, the wave number, and the wave frequency are also discussed.     

The nonlinear stability of plane poiseulle flow for non-newtonian power series fluid

Non-Newtonan fluid is a kind of fluid whose components of stresstensor aren’t theliear funtions of compoents of the strain rate tensor.Non-Newtonianfluid is beingprocessed in many kinds of modern industry,Stability of flows for Non-Newtonianfluid is of important applicatuib,In this article we calculate subcritical thrdshold of flow which oecurs in polymer-processing when the melting substance is driven throughtwo parallel fixed boundaries.     

Unconditional nonlinear exponential stability of the motionless conduction-diffusion solution

Nonlinear stability of the motionless state of a heterogeneous fluid with constant temperature-gradient and concentration-gradient is studied for both cases of stress-free and rigid boundary conditions. By introducing new energy functionals we have shown that for τ=P _C /P _T ≤1, the motionless state is always stable and for τ≤1, the sufficient and necessary conditions for stability coincide, whereP _C ,P _T ,C andR are the Schmidt number, Prandtl number, Rayleigh number for solute and heat respectively. Moreover, the criteria guarantees the exponential stability. The Project supported by SRF for ROCS, China Postdoctoral Science Foundation and the National Basic Research Project “Nonlinear Science”     

Studies on nonlinear stability of three-dimensional H-type disturbance

The three-dimensional H-type nonlinear evolution process for the problem of boundary layer stability is studied by using a newly developed method called parabolic stability equations (PSE). The key initial conditions for sub-harmonic disturbances are obtained by means of the secondaryinstability theory. The initial solutions of two-dimensional harmonic waves are expressed in Landau expansions. The numerical techniques developed in this paper, including the higher order spectrum method and the more effective algebraic mapping for dealing with the problem of an infinite region, increase the numerical accuracy and the rate of convergence greatly. With the predictor-corrector approach in the marching procedure, the normalization, which is very important for PSE method, is satisfied and the stability of the numerical calculation can be assured. The effects of different pressure gradients, including the favorable and adverse pressure gradients of the basic flow, on the “H-type“ evolution are studied in detail. The results of the three-dimensional nonlinear “H-type“ evolution are given accurately and show good agreement with the data of the experiment and the results of the DNS from the curves of the amplitude variation, disturbance velocity profile and the evolution of velocity.     

Analysis and calculation of the nonlinear stability of the rotational composite shell

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Free-parameter perturbation-method solutions of the nonlinear stability of shallow spherical shells

The free-parameter perturbation method is applied to solve the problems of nonlinear stability of spherical shallow shells under uniform load. As a modified perturbation method, the free-parameter perturbation method enables researchers to obtain all characteristic relations without choosing the certain perturbation parameter. Some examples were discussed to study the variety regulations of deflections and stress of shells in the process of buckling, and the results were compared with those of other researchers.     

Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium

Convection induced by the selective absorption of radiation is investigated, for the case of an internal heat source that is modelled quadratically with respect to concentration. The growth rate for the linearised system is shown to be real, and a linear instability analysis is performed. To establish conditional and unconditional nonlinear stability results, both the Darcy and Forchheimer models are employed to describe fluid flow. Due to the presence of significant regions of potential subcritical instabilities, the results indicate that linear theory may only be accurate enough to predict the onset of convective motion when the model for the internal heat source is predominantly linear.Received: 6 May 2003, Accepted: 9 August 2003, Published online: 12 December 2003     

A nonlinear composite plate theory

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.     

Dynamics of electroconvective structures in a weakly conducting liquid

The finite-amplitude evolution of electroconvective structures in a weakly conducting liquid with an electroconductive charge formation mechanism is examined. The liquid is in the electrostatic field of a horizontally placed capacitor and is heated from below, and the electric charge time constant is much shorter than the characteristic hydrodynamic time. The interaction between the electroconductive convection and thermogravitational convection is considered. The evolution of the supercritical structures is investigated by direct numerical simulation using the finite-difference method. The bifurcations leading to the formation of stationary and wave liquid flows are analyzed. Nonlinear modes of stationary convection and traveling waves with different space-time patterns are identified and investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 20–27, May–June, 2008.     

On the nonlinear stability behaviour of distorted plane Couette flow

This paper discusses the nonlinear stability behaviour of distorted plane Couette flowto 2-dimensional disturbances,and compares it with that of distorted plane Poiseuille flow.The results show that plane Couette flow is more unstable than plane Poiseuille flow tofinite-amplitude disturbances.     

On the nonlinear stability of symmetrically laminated cylindrically orthotropic shallow spherical shells

In this paper,the large deflection theory of symmetrically laminated cylindricallyorthotropic shallow spherical shells is established.Based on this theory,applying themodified iteration method,the analytic solution for critical buckling loads of the shells withrigidly clamped edges under actions of uniform pressure has been obtained.