Transition of free-surface flow modes in taylor-couette system

This paper presents our numerical and experimental results of the bifurcation found in Taylor-Couette system with a free surface. The lengths of the two concentric cylinders are finite and their axes are parallel to the direction of the gravitational force. When the end walls of the cylinders are fixed and stationary, numerical and experimental studies have shown that the flow has multiple patterns depending on the cylinder lengths and the Reynolds numbers. Experimental studies on flows with free surfaces also gave various flow modes. Our result shows that the measured and predicted time-dependent displacements of the free surface are in favorable agreement. In case of the cylinder length comparable with the gap width between the cylinders, gradual accelerations of the inner cylinder bring the normal mode flows with one, three and five toroidal vortices. The exchanges of stabilities between these flow modes are summarized in a phase diagram.     

New type of magnetorotational instability in cylindrical taylor-couette flow

We study the stability of cylindrical Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields, and show that adding an azimuthal field profoundly alters the previous results for purely axial fields. For small magnetic Prandtl numbers Pm, the critical Reynolds number Re(c) for the onset of the magnetorotational instability becomes independent of Pm, whereas for purely axial fields it scales as Pm-1. For typical liquid metals, Re(c) is then reduced by several orders of magnitude, enough that this new design should succeed in realizing this instability in the laboratory.     

Resistance law for taylor-couette turbulent flow at very high reynolds numbers

New expressions for the resistance law and dimensionless moment of force are derived for a Taylor-Couette turbulent flow starting from the generalized model of local balance for the turbulent energy. In the case of extremely high Reynolds numbers, the formulas derived involve a single empirical (Karman) constant.     

Lagrangian chaos and Eulerian chaos in shear flow dynamics

Shear flow dynamics described by the two-dimensional incompressible Navier-Stokes equations is studied for a one-dimensional equilibrium vorticity profile having two minima. These lead to two linear Kelvin-Helmholtz instabilities; the resulting nonlinear waves corresponding to the two minima have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter lambda specifying the width of the minima in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangian chaos, leading to mixing of vorticity, i.e., momentum transport, between the chains of vortices or cat's eyes. Lagrangian chaos is quantified by plotting the finite time Lyapunov exponents on a grid of initial points, and by the probability distribution of these exponents. For moderate values of lambda, there is Lagrangian chaos everywhere except near the centers of the vortices and near the boundaries, and there are competing effects of homogenization of vorticity and formation of structures associated with secondary resonances. For smaller values of lambda Lagrangian chaos occurs in the regions in the centers of the vortices, and the Eulerian behavior of the flow undergoes bifurcations leading to Eulerian chaos, as measured by the time series of several Galilean invariant quantities. A discussion of Lagrangian chaos and its relation to Eulerian chaos is given.(c) 2001 American Institute of Physics.     

Transition to chaos in lid-driven square cavity flow

To date, there are very few studies on the transition beyond second Hopf bifurcation in a lid-driven square cavity, due to the difficulties in theoretical analysis and numerical simulations. In this paper, we study the characteristics of the third Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme rectently developed by us. We numerically identify the critical Reynolds number of the third Hopf bifurcation located in the interval of (13944.7021,13946.5333) by the method of bisection. Through Fourier analysis, it is discovered that the flow becomes chaotic with a characteristic of period-doubling bifurcation when the Reynolds number is beyond the third bifurcation critical interval. Nonlinear time series analysis further ascertains the flow chaotic behaviors via the phase diagram, Kolmogorov entropy and maximal Lyapunov exponent. The phase diagram changes interestingly from a closed curve with self-intersection to an unclosed curve and the attractor eventually becomes strange when the flow becomes chaotic.     

Edge of chaos in a parallel shear flow

We study the transition between laminar and turbulent states in a Galerkin representation of a parallel shear flow, where a stable laminar flow and a transient turbulent flow state coexist. The regions of initial conditions where the lifetimes show strong fluctuations and a sensitive dependence on initial conditions are separated from the ones with a smooth variation of lifetimes by an object in phase space which we call the "edge of chaos." We describe techniques to identify and follow the edge, and our results indicate that the edge is a surface. For low Reynolds numbers we find that the surface coincides with the stable manifold of a periodic orbit, whereas at higher Reynolds numbers it is the stable set of a higher-dimensional chaotic object.     

Characterization of chaos in a serrated plastic flow model

We report new results on a dynamical model of serrated yielding. These essentially pertain to the full spectrum of Lyapunov exponents of the non-linear (chaotic) model and fractal characterization of the associated strange attractor. The power spectrum of scalar time series extracted from the phase space trajectories decays exponentially with increase of frequency and the decay constant is found proportional to the Kolmogorov-Sinai entropy.     

Transition to chaos in a confined two-dimensional fluid flow

For a two-dimensional fluid in a square domain with no-slip walls, new direct numerical simulations reveal that the transition from steady to chaotic flow occurs through a sequence of various periodic and quasiperiodic flows, similar to the well-known Ruelle-Takens-Newhouse scenario. For all solutions beyond the ground state, the phenomenology is dominated by a domain-filling circulation cell, whereas the associated symmetry is reduced from the full symmetry group of the square to rotational symmetry over an angle pi. The results complement both laboratory experiments in containers with rigid walls and numerical simulations on double-periodic domains.     

Turbulence transition and the edge of chaos in pipe flow

The linear stability of pipe flow implies that only perturbations of sufficient strength will trigger the transition to turbulence. In order to determine this threshold in perturbation amplitude we study the edge of chaos which separates perturbations that decay towards the laminar profile and perturbations that trigger turbulence. Using the lifetime as an indicator and methods developed in Skufca et al., Phys. Rev. Lett. 96, 174101 (2006), we show that superimposed on an overall 1/Re scaling predicted and studied previously there are small, nonmonotonic variations reflecting folds in the edge of chaos. By tracing the motion in the edge we find that it is formed by the stable manifold of a unique flow field that is dominated by a pair of downstream vortices, asymmetrically placed towards the wall. The flow field that generates the edge of chaos shows intrinsic chaotic dynamics.     

Viscous heating and the stability of newtonian and viscoelastic taylor-couette flows

The effects of viscous heating on the stability of Taylor-Couette flow were investigated through flow visualization experiments for Newtonian and viscoelastic fluids. For highly viscous Newtonian fluids, viscous heating drives a transition to a new, oscillatory mode of instability at a critical Reynolds number significantly below that at which the inertial transition is observed in isothermal flows. The effects of viscous heating may explain the discrepancies between the observed and predicted critical conditions and the symmetry of the disturbance flow for viscoelastic instabilities.     

Application of scattering chaos to particle transport in a hydrodynamical flow

The dynamics of a passive particle in a hydrodynamical flow behind a cylinder is investigated. The velocity field has been determined both by a numerical simulation of the Navier-Stokes flow and by an analytically defined model flow. To analyze the Lagrangian dynamics, we apply methods coming from chaotic scattering: periodic orbits, time delay function, decay statistics. The asymptotic delay time statistics are dominated by the influence of the boundary conditions on the wall and exhibit algebraic decay. The short time behavior is exponential and represents hyperbolic effects.     

Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow

The steady incompressible flow in a unit sphere introduced by Bajer and Moffatt [J. Fluid Mech. 212, 337 (1990)] is discussed. The velocity field of this flow differs by a small perturbation from an integrable field whose streamlines are almost all closed. The unperturbed flow has two stationary saddle points (poles of the sphere) and a two-dimensional separatrix passing through them. The entire interior of the unit sphere becomes the domain of streamline chaos for an arbitrarily small perturbation. This phenomenon is explained by the nonconservation of a certain adiabatic invariant that undergoes a jump when a streamline crosses a small neighborhood of the separatrix of the unperturbed flow. An asymptotic formula is obtained for the jump in the adiabatic invariant. The accumulation of such jumps in the course of repeated crossings of the separatrix results in the complete breaking of adiabatic invariance and streamline chaos. (c) 1996 American Institute of Physics.     

Characterization of the spatial complex behavior and transition to chaos in flow systems

We introduce a “spatial” Lyapunov exponent to characterize the complex behavior of non-chaotic but convectively unstable flow sytems. This complexity is of spatial type and is due to sensitivity to the boundary conditions. We show that there exists a relation between the spatial-complexity index we define and the comoving Lyapunov exponents. In such systems the transition to chaos, i.e., the occurrence of a positive Lyapunov exponent, can manifest itself in two different ways. In the first case (from neither chaotic nor spatially complex behavior to chaos) one observes the typical scenario; i.e., as the system size grows up the spectrum of the Lyapunov exponents gives rise to a density. In the second case (when the chaos develops from a convectively unstable situation) one observes only a finite number of positive Lyapunov exponents.     

Correlation dimension of the flow and spatial development of dynamical chaos in a boundary layer

The relation between the spatial development of turbulence in a real hydrodynamical system (a boundary layer on a plate in a wind tunnel) and the increase of dimension downstream is established experimentally.     

A transition from a doubly periodic flow to chaos via successive frequency lockings

In a four dimensional dynamical system depending on a parameter a transition from doubly periodic to aperiodic behaviour occurs through a sequence of frequency lockings with rotation numbers _n =n/2n+1. The aperiodic motion sets in when the unstable hyperbolic orbits associated with the final locked regime (rotation number =1/2) collide with the strange attractor present at higher values of the parameter.Partially supported from CRRNSM (Comitato Regionale per le Ricerche Nucleari e di Struttura della Materia)     

摇摆条件下自然循环系统流量混沌脉动的检验与预测

利用替代数据法检验了摇摆条件下自然循环系统不规则复合型脉动的混沌特性, 并在此基础上进行混沌预测. 关联维数、最大Lyapunov指数等几何不变量计算结果表明不规则复合型脉动具有混沌特性, 但是由于计算结果受实验时间序列长度的限制和噪声的影响, 可能会出现错误的判断结果. 为了避免出现误判, 在提取流量脉动的非线性特征的同时, 需要用替代数据法进一步检验混沌特性是否来自于确定性的非线性系统. 本文用迭代的幅度调节Fourier 算法进行混沌检验, 在此基础上用加权一阶局域法进行混沌脉动的预测. 计算结果表明: 不规则复合型脉动是来自于确定性系统的混沌脉动, 加权一阶局域法对流量脉动进行混沌预测效果较好, 并提出动态预测方法. 关键词： 混沌时间序列 替代数据法 实时预测 两相流动不稳定性