Intermittency

Abstract: Preface.Color Plates.1 Introduction.2 Experiments and Simple Models.2.1 Experimental Detection of Deterministic Chaos.2.2 The Periodically Kicked Rotator.3 Piecewise Linear Maps and Deterministic Chaos.3.1 The Bernoulli Shift.3.2 Characterization of Chaotic Motion.3.3 Deterministic Diffusion.4 Universal Behavior of Quadratic Maps.4.1 Parameter Dependence of the Iterates.4.2 Pitchfork Bifurcation and the Doubling Transformation.4.3 Self-Similarity, Universal Power Spectrum, and the Influence of External Noise.4.4 Behavior of the Logistic Map for r ≤ r.4.5 Parallels between Period Doubling and Phase Transitions.4.6 Experimental Support for the Bifurcation Route.5 The Intermittency Route to Chaos.5.1 Mechanisms for Intermittency.5.2 Renormalization-Group Treatment of Intermittency.5.3 Intermittency and 1/f-Noise.5.4 Experimental Observation of the Intermittency Route.6 Strange Attractors in Dissipative Dynamical Systems.6.1 Introduction and Definition of Strange Attractors.6.2 The Kolmogorov Entropy.6.3 Characterization of the Attractor by a Measured Signal.6.4 Pictures of Strange Attractors and Fractal Boundaries.7 The Transition from Quasiperiodicity to Chaos.7.1 Strange Attractors and the Onset of Turbulence.7.2 Universal Properties of the Transition from Quasiperiodicity to Chaos.7.3 Experiments and Circle Maps.7.4 Routes to Chaos.8 Regular and Irregular Motion in Conservative Systems.8.1 Coexistence of Regular and Irregular Motion.8.2 Strongly Irregular Motion and Ergodicity.9 Chaos in Quantum Systems?9.1 The Quantum Cat Map.9.2 A Quantum Particle in a Stadium.9.3 The Kicked Quantum Rotator.10 Controlling Chaos.10.1 Stabilization of Unstable Orbits.10.2 The OGY Method.10.3 Time-Delayed Feedback Control.10.4 Parametric Resonance from Unstable Periodic Orbits.11 Synchronization of Chaotic Systems.11.1 Identical Systems with Symmetric Coupling.11.2 Master-Slave Configurations.11.3 Generalized Synchronization.11.4 Phase Synchronization of Chaotic Systems.12 Spatiotemporal Chaos.12.1 Models for Space-Time Chaos.12.2 Characterization of Space-Time Chaos.12.3 Nonlinear Nonequilibrium Space-Time Dynamics.Outlook.Appendix.A Derivation of the Lorenz Model.B Stability Analysis and the Onset of Convection and Turbulence in the Lorenz Model.C The Schwarzian Derivative.D Renormalization of the One-Dimensional Ising Model.E Decimation and Path Integrals for External Noise.F Shannon's Measure of Information.F.1 Information Capacity of a Store.F.2 Information Gain.G Period Doubling for the Conservative H-enon Map.H Unstable Periodic Orbits.Remarks and References.Index.

Abstract: The axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers. It was established that the jet was truly self-preserving some 70 diameters downstream of the nozzle and most of the measurements were made in excess of this distance. The quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity. The r.m.s. values of the various velocity fluctuations differ from those measured previously as a result of lack of self-preservation and insufficient frequency range in the instrumentation of the previous investigations. It appears that Taylor's hypothesis is not applicable to this flow, but the use of convection velocity of the appropriate scale for the transformation from temporal to spatial quantities appears appropriate. The energy balance was calculated from the various measured quantities and the result is quite different from the recent measurements of Sami (1967), which were obtained twenty diameters downstream from the nozzle. In light of these measurements some previous hypotheses about the turbulent structure and the transport phenomena are discussed. Some of the quantities were obtained by two or more different methods, and their relative merits and accuracy are assessed.

Abstract: We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier–Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.

Abstract: Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

Abstract: The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e., to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in nonequilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.