Viscous falling film instability around a vertical moving cylinder

The present work discusses both the linear and nonlinear stability conditions of a viscous falling film down the outer surface of a solid vertical cylinder which moves in the direction of its axis with a constant velocity.After studying the linear conditions,a generalized nonlinear kinematic model is then derived to present the physical system.Applying the boundary conditions,analytical solutions are obtained using the long-wave perturbation method.In the first step,the normal mode method is used to characterize the linear behaviors.In the second step,the nonlinear film flow model is solved by using the method of multiple scales,to obtain Ginzburg-Landau equation.The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem,and the results are displayed in many plots showing the stability criteria in various parameter planes.     

Nonlinear Stability Analysis of Thin Viscoelastic Film Flow Traveling Down along a Vertical Cylinder

This paper investigates the weakly nonlinear stability theoryof a thin viscoelastic liquid film flowing down along the outsidesurface of a vertical cylinder. The long-wave perturbation method isemployed to solve for generalized nonlinear kinematic equations withfree film interface. The normal mode approach is first used to computethe linear stability solution for the film flow. The method of multiplescales is then used to obtain the weak nonlinear dynamics of the filmflow for stability analysis. The modeling results indicate that both thesubcritical instability and supercritical stability conditions arepossible to occur in a viscoelastic film flow system. The degree ofinstability in the film flow is further intensified by the lateralcurvature of cylinder. This is somewhat different from that of theplanar flow. The modeling results also indicate that by increasing theviscoelastic effect and decreasing the radius of the cylinder the filmflow can become less stable as traveling down along the verticalcylinder.     

Weakly nonlinear stability analysis of a falling film with countercurrent gas flow

Weakly nonlinear stability analysis of a falling film with countercurrent gas–liquid flow has been investigated. A normal mode approach and the method of multiple scales are employed to carry out the linear and nonlinear stability solutions for the film flow system. The results show that both supercritical stability and subcritical instability are possible for a film flow system when the gas flows in the countercurrent direction. The stability characteristics of the film flow system are strongly influenced by the effects of interfacial shear stress when the gas flows in the countercurrent direction. The effect of countercurrent gas flow in a falling film is to stabilize the film flow system.     

Bifurcation analysis for thermal convection in a rotating porous layer

The linear and weakly nonlinear thermal convection in a rotating porous layer is investigated by constructing a simplified model involving a system of fifth-order nonlinear ordinary differential equations. The flow in the porous medium is described by Lap wood-Brinkman-extended Darcy model with fluid viscosity different from effective viscosity. Conditions for the occurrence of possible bifurcations are obtained. It is established that Hopf bifurcation is possible only at a lower value of the Rayleigh number than that of simple bifurcation. In contrast to the non-rotating case, it is found that the ratio of viscosities as well as the Darcy number plays a dual role on the steady onset and some important observations are made on the stability characteristics of the system. The results obtained from weakly nonlinear theory reveal that, the steady bifurcating solution may be either sub-critical or supercritical depending on the choice of physical parameters. Heat transfer is calculated in terms of Nusselt number.     

Simulation of cross-flow-induced vibration of cylinder arrays by surface vorticity method

The surface vorticity method (SVM), which is a fast and practical grid-free two-dimensional (2-D) method, and a fluid–structure interaction model incorporating the effects of cylinder motions and displacements is used to simulate the vortex-induced vibration of cylinder arrays at sub-critical Reynolds number Re=2.67×10⁴. The SVM is found to be most suitable for simulating a 2-D cylinder row with large-amplitude vibrations where the vorticity field and the fluid forces of the cylinder row change drastically, and the effect of the stream on the transverse direction vibration is very significant. The fluidelastic instability of a flexible cylinder row at small pitch ratio is also investigated, and the critical reduced velocity of the cylinder row at a reduced damping parameter S_G=1.29 is calculated, which is in good agreement with experimental and analytical results of the unsteady model. Vortex-induced vibration of a staggered cylinder array is simulated using different structural parameters. When the cylinders are relatively more flexible, the flow pattern changes dramatically and the fluid–structure interaction has a dominant impact on the flow field. Compared with grid-based methods, the grid-free SVM is a fast and practical method for the simulation of the FIV of cylinder arrays due to vortex shedding at sub-critical Reynolds numbers.     

Finite-amplitude stability analysis of liquid films down a vertical wall with and without interfacial phase change

The generalized kinematic equation for film thickness, taking into account the effect of phase change at the interface, is used to investigate the nonlinear stability of film flow down a vertical wall. The analysis shows that supercritical stability and subcritical instability are both possible for the film flow system. Applications of the result to isothermal, condensate and evaporate film flow show that mass transfer into (away from) the liquid phase will stabilize (destabilize) the film flow. Finally, we find that supercritical filtered waves are always linearly stable with regard to side-band disturbance.     

Control of cross-flow-induced vibrations of square cylinders using linear and nonlinear delayed feedbacks

We investigate the effectiveness of linear and nonlinear time-delay feedback controls to suppress high amplitude oscillations of an elastically mounted square cylinder undergoing galloping oscillations. A representative model that couples the transverse displacement and the aerodynamic force is used. The quasi-steady approximation is used to model the galloping force. A linear analysis is performed to investigate the effect of linear time-delay controls on the onset speed of galloping and natural frequencies. It is demonstrated that a linear time-delay control can be used to delay the onset speed of galloping. The normal form of the Hopf bifurcation is then derived to characterize the type of the instability (supercritical or subcritical) and to determine the effects of the linear and nonlinear time-delay parameters on their outputs near the bifurcation. The results show that the nonlinear time-delay control can be efficiently implemented to significantly reduce the galloping amplitude and suppress any dangerous behavior by converting any subcritical Hopf bifurcation into a supercritical one.     

A Numerical Method for 2-D Bed Morphology Calculations

A fully coupled two-dimensional sub-critical and/or supercritical, free-surface flow numerical model is developed to calculate bed variations in alluvial channels. Vertically averaged free-surface flow equations in conjunction with sediment transport equation are numerically solved using an explicit finite-volume scheme in integral form. The capabilities of the proposed method are first demonstrated by analyzing supercritical flow in an expansion channel. Thereafter, one and two-dimensional applications referring to aggradation and scouring are reported. For each of these test cases, computed results compare satisfactorily with measurements as well as with other numerical solutions. The method is stable, reliable and accurate, although time consuming, handling a variety of sediment transport equations with rapid changes of sediment transport at the boundaries.     

Kinetic analysis of longitudinal Couette flow between coaxial cylinders

On the basis of a model kinetic equation, the rarefied gas flow between coaxial circular cylinders, of which the outer one is at rest while the inner one travels along its symmetry axis at a constant velocity, is studied. The problem is solved numerically in both the linear and nonlinear formulations by an implicit conservative method of second-order accuracy. The effect of the rarefaction, the cylinder radius ratio, and the inner cylinder velocity on the flow parameters is investigated. The limits of applicability of the linearized kinetic equation are established.     

Cross-flow-induced instability and nonlinear dynamics of cylinder arrays with consideration of initial axial load

The instability and nonlinear dynamics of planar motions of a cylinder array subjected to cross-flow have been studied via a five-mode discretization of the governing partial differential equation, focusing on the effect of initial axial load externally imposed on the cylinder. Theoretical results based on a stability analysis have indicated that, with increasing initial axial load and flow velocity, the system may lose stability either via flutter or via buckling. The boundaries of these two forms of instability are predicted analytically. To explore the post-instability dynamics of the system, a Runge–Kutta scheme is used to solve the nonlinear governing equation of motion. Three typical behaviors, including limit cycle motions of the system, are obtained. It is shown that, for relatively low flow velocity, with increasing initial axial load, just beyond the pitchfork bifurcation the cylinder would settle in a buckled equilibrium position; and for high flow velocity, however, this phenomenon only occurs when the initial axial load becomes sufficiently large.     

Analytical treatment on magnetohydrodynamic (MHD) flow and heat transfer due to a stretching hollow cylinder

This paper studied on magnetohydrodynamics flow and heat transfer outside a stretching cylinder. Momentum and energy equations are reduced using similarity transformation and converted into a system of ordinary differential equations which are solved analytically by the homotopy analysis method. The effects of the parameters involved, namely the magnetic parameter (M), Prandtl number (Pr) and Reynolds number (Re) on the velocity and temperature fields are investigated. The obtained results are valid for the whole solutions' domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.     

进动圆筒内液体流动不稳定的非线性特征

在建立进动充液圆筒内液体偏差流动方程的基础上，结合液体惯性波和轴向二次流动线性解，通过对定常二次流动的线性稳定性分析，提出了函数空间表达的流动不稳定性非线性分岔分析方程. 对非惯性坐标系下液体流动的Navier-Stokes方程进行了数值求解，并对惯性波发生破裂(实验提供的3种主模态下得出的共振破裂现象)时的压力时间序列进行分析，得出了液体流动不稳定的基本非线性特征.     

Solution of shallow-water equations using least-squares finite-element method

A least-squares finite-element method （LSFEM） for the non-conservative shallow-water equations is presented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercritical flows, and flows with smooth and sharp gradient changes. Advantages of the model include： （1） sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; （2） upwind scheme is no needed; （3） a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi （LBB） condition; and （4） the resulting system of equations is symmetric and positive-definite （SPD） which can be solved efficiently with the preconditioned conjugate gradient method. The model is verified with flow over a bump, tide induced flow, and dam-break. Computed results are compared with analytic solutions or other numerical results, and show the model is conservative and accurate. The model is then used to simulate flow past a circular cylinder. Important flow charac-teristics, such as variation of water surface around the cylinder and vortex shedding behind the cylinder are investigated. Computed results compare well with experiment data and other numerical results.     

Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane

A thin film of a power–law fluid flowing down a porous inclined plane is considered. It is assumed that the flow through the porous medium is governed by the modified Darcy’s law together with Beavers–Joseph boundary condition for a general power–law fluid. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium and a slip condition at the bottom is used to incorporate the effects of the permeability of the porous substrate. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. The results show that the substrate porosity in general destabilizes the film flow system and the shear-thinning rheology enhances this destabilizing effect. A weakly nonlinear stability analysis reveals the existence of supercritical stable and subcritical unstable regions in the wave number versus Reynolds number parameter space. The numerical solution of the nonlinear evolution equation in a periodic domain shows that the fully developed nonlinear solutions are either time-dependent modes that oscillate slightly in the amplitude or time independent stable two-dimensional nonlinear waves with large amplitude referred to as ‘permanent waves’. The results show that the shape and the amplitude of the nonlinear waves are strongly influenced by the permeability of the porous medium and the shear-thinning rheology.     

弹性支承滑动轴承不平衡转子系统非线性动力稳定性和分岔的研究

研究弹性支承滑动轴承不平衡转子系统的稳定性及分岔特性。建立了弹性支承-滑动轴承-转子非线性动力系统的力学模型，在油膜力非线性的情况下，应用数值模拟，采用打靶法计算了刚性转子系统的周期解，并与龙格-库塔法计算的结果进行了对比，依据Floquet理论，分析了周期解的稳定性，再结合龙格-库塔法、Poineare映射法作出了系统运动分岔图。最后，讨论了轴的柔性对转子系统运动稳定性的影响。     

Nonlinear dynamics of extensible fluid-conveying pipes,supported at both ends

In this paper, the post-divergence behaviour of extensible fluid-conveying pipes supported at both ends is studied using the weakly nonlinear equations of motion of Semler, Li and Païdoussis. The two coupled nonlinear partial differential equations are discretized via Galerkin's method and the resulting set of ordinary differential equations is solved either by Houbolt's finite difference method or via AUTO. Typically, the pipe is stable at its original static equilibrium position up to the flow velocity where it loses stability by static divergence via a supercritical pitchfork bifurcation. The amplitude of the resultant buckling increases with increasing flow, but no secondary instability occurs beyond the pitchfork bifurcation. The effects of the system parameters on pipe behaviour as well as the possibility of a subcritical pitchfork bifurcation have also been studied.     

非定常短轴承油膜力公式的变分修正

本文采用了变分方法对非定常短轴承的油膜压力分布公式进行了修正，既保留了短轴承公式的简洁形式，又使其适用于大长径比轴承。得出了具有足够精度、适合轴颈大扰动情况下的有限长圆柱轴承非定常油膜力的解析公式。与差分充零算法相比，短轴承公式的结果在轴承长径比为0．6时，误差已经超过百分之二百，而本方法计算结果的误差小于百分之五。因此采用本方法既提高了短轴承油膜力公式的计算精度，又保持了油膜力公式的简洁形式，不失为进行转子-轴承系统非线性动力分析的一种有效方法。     

Estimating Nonlinearity Using Volterra Kernels in Feedback with Linear Models

Aeroelastic dynamics must be accurately known to ensure safe and efficient flight testing. Unfortunately, most models of aircraft systems typically describe only the linear dynamics. These models are inadequate for predicting behaviors, such as limit cycle oscillations, resulting from nonlinearities. This paper presents an approach to augment a linear model by identifying associated nonlinear operators. Essentially, the difference between a flight data measurement and a simulated measurement indicates the unmodeled dynamics. Volterra kernels are computed to represent the difference in measurement and, consequently, represent the unmodeled dynamics. The approach is applied to a nonlinear pitch–plunge system for which only a linear model is assumed available. The method is able to characterize errors due to incorrect parameters in the linear model and errors due to unmodeled nonlinearities of the dynamics.     

A time–frequency technique for the stability analysis of impulse responses from nonlinear aeroelastic systems

A time–frequency method is proposed for the analysis of response time histories from nonlinear aeroelastic systems. The approach is based on a time-varying curve-fit of the short time Fourier transform of the impulse response. It is shown that the method can be used in order to obtain a clear picture of the sub-critical stability of a number of aeroelastic systems with a variety of structural and aerodynamic nonlinearities. Additionally, frequency and amplitude information can be obtained for both the linear and nonlinear signatures of the response signals in the sub- and post-critical regions. Finally, it is shown that, given certain types of nonlinear functions, sub-critical damping trends can be extrapolated to predict bifurcation airspeeds.     

Stability of conducting liquid flowing down an inclined plane at moderate Reynolds number in the presence of constant electromagnetic field

The stability of a conducting viscous film flowing down an inclined plane at moderate Reynolds number in the presence of electromagnetic field is investigated under induction-free approximation. Using momentum integral method a non-linear evolution equation for the development of the free surface is derived. The linear stability analysis of the evolution equation shows that the magnetic field stabilizes the flow whereas the electric field stabilizes or destabilizes the flow depending on its orientation with the flow. The weakly non-linear study reveals that both the supercritical stability and subcritical instability are possible for this type of thin film flow. The influence of magnetic field on the different zones is very significant, while the impact of electric field is very feeble in comparison.