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Frameworks for investigation of nonlinear dynamics: Experimental study of the turbulent jet
Physics of Fluids
Abstract
Analysis methods that have been developed in the field of nonlinear dynamics have provided valuable insights into the physics of turbulent flows, although their application to open flows is less well explored. The nonlinear dynamics of a turbulent jet with a low-to-moderate Reynolds number is investigated by using the single-trajectory framework and ensemble framework. We have used Lyapunov exponents to calculate the spectra of scaling indices of the attractor. First, we evaluated the frameworks on two theoretical models, one with a stationary attractor (Lorenz-63) and the other with time-varying characteristics (Lorenz-84). Theoretical studies showed that in dynamical systems with a stable attractor, both frameworks estimated the same largest Lyapunov exponent. The ensemble framework enables us to resolve the unsteady characteristics of a time-varying strange attractor. Second, we applied both frameworks to time-resolved planar velocity fields in a turbulent jet at local Reynolds numbers (Reδ) of 3000 and 5000. Time-resolved particle image velocimetry was utilized to measure streamwise and transverse velocity components. Results support the presence of a low-dimensional attractor in the reconstructed phase space with a chaotic characteristic. Despite considerable changes in the dynamics for the higher Reynolds number case, the system’s fractal dimension did not change significantly. We have used Lagrangian Coherent Structures (LCSs) to study the relationship between changes in the Lyapunov exponent with flow topological features. Results suggest that holes in the stable LCSs provide a path for the entrainment of the coflow, which is shown to be one of the main contributors to high Lyapunov exponents.
... Therefore, temporally evolving jets have been considered in the existing literature 8,29,30 as an approximation of spatially evolving jets, which are realizable in both laboratory experiments and numerical simulations. [31][32][33][34] The temporal jets are useful to investigate fundamental properties of intermittent turbulent flows with mean shear. It has been shown that the transverse profiles of various statistics hardly differ between the spatially and temporally evolving jets as also confirmed below. ...
The relation between shearing motions and the turbulent/non-turbulent interfacial (TNTI) layer is studied with direct numerical simulation of a temporally evolving planar jet. Small-scale shear layers are detected with the triple decomposition of the velocity gradient tensor, which is decomposed into shear, rotation, and elongation tensors. The shear layers are found in the turbulent sublayer more frequently than in the turbulent core region although they hardly appear in the viscous superlayer. The shear layers undergo a biaxial strain with stretching in the shear vorticity direction and compression in the interface normal direction. This compressive strain is related to the non-turbulent fluid, which is relatively advected toward the shear layer. The shear layer thickness in the TNTI layer is well predicted by Burgers vortex layer. The velocity jump of the shear layer is about seven times the Kolmogorov velocity both in the turbulent core region and the TNTI layer. However, the layer thickness normalized by the Kolmogorov scale is about 6 in the turbulent core region and decreases in the TNTI layer, where consequently, the shear Reynolds number becomes small. The shear layers have significant contributions to the enstrophy production in the turbulent sublayer and the viscous enstrophy-diffusion toward the viscous superlayer. The shear layer and the outer edge of the TNTI layer have a curvature radius of about 50 times the Kolmogorov scale. The alignment between the shear layer orientation and the interface normal direction confirms that the shear layers near the interface are mostly parallel to the TNTI layer.
... Several changes (indicated in bold type below) are required in the text on pages 13 and 14 of our previous paper. 1 Page 13: ...
The self-sustained interactions between a flexible film and periodic vortices epitomize the spirit of fish swimming and flag flapping in nature, involving intricate patterns of flow–structure coupling. Here, we comprehensively investigate the multiple coupling states of a film in the cylinder wake mainly with experiments, complemented by theoretical solutions and nonlinear dynamical analyses. Four regimes of film motion states are identified in the parameter space spanned by the reduced velocity and the length ratio. These regimes are (i) keeping stationary, (ii) deflection flutter, (iii) hybrid flutter and (iv) periodic large-amplitude flapping, each governed by a distinct coupling mechanism, involving regular and irregular Kármán vortices, local instability of the elongated shear layers and 2P mode vortex shedding. The film futtering in regimes (ii) and (iii) is substantiated to be chaotic and bears a resemblance to the ‘entraining state’ of fish behind an obstacle in the river. The periodic flapping in regime (iv) manifests itself in an amalgam of standing and travelling waves, and has intrinsic relations to the ‘Kármán gaiting’ of fish in periodic vortices. With the spatiotemporal reconstruction for the periodic flapping, we procure the energy distributions on the film, revealing the energy transfer processes between the film and the large-scale vortices. The findings unequivocally indicate that the flow–structure interaction during the energy-release stage of the film is more intense than that during the energy-extraction stage. Given the similarities, the mathematical and physical methods presented in this work are also applicable to the research on biological undulatory locomotion.
Mixing of the turbine blade tip leakage and mainstream flows causes considerable aerodynamic loss. Its understanding is crucial to raise a consequential improvement of the turbine performance. In the present paper, a typical high pressure turbine rotor flow is simulated by Detached Eddy Simulation. The complex mixing in the blade tip region is assessed by the dilution index algorithm in the streamwise direction. Influences of typical parameters on mixing are consecutively identified and analyzed. Relating the influences to the flow structures, the mixing mechanism is obtained. It is revealed that the normal effective diffusion coefficient is key in mixing and is correlated to the unsteady tip leakage flow stick vortices. The latter induced by the Kelvin-Helmholtz instability can significantly enhance local mass and energy transfer and hence diffusion. As a result, mixing is strengthened. Furthermore, the tip region is knowingly divided into the juxtaposing near and far fields. The tip region is artificially divided into near field and far field. The former contains the leakage concentrated vortex, leakage jet, mainstream flow and two entrainment zones. Meanwhile, the latter contains the leakage jet, mainstream flow and entrainment zone. It was found that the entrainment zone is mixed to a high-degree whereas the leakage jet is barely mixed.
The present study investigates the effect of exit geometry of the main stage in a model combustor with a dual-swirl layout. In practical combustors, the main stage has much larger airflow than the pilot stage, therefore dominating the aerodynamics of the global flame. In this paper, six different geometries of the main stage exit are designed to cover the common styles in engineering applications. Their effects on thermoacoustic instability are examined experimentally. Sudden transitions from the attached flame to the blurred flame are found in most of the exit geometries, excluding the Outer-rough case. This transition of flame shape triggers the onset of strong thermoacoustic oscillation, leading to a jump of the amplitude of pressure fluctuation. Frequency drop and the acoustic mode shift are also found at the transition point. Flame dynamics are further analyzed using the Dynamic Mode Decomposition method. It is concluded that the Outer-rough case has the best thermoacoustic stability within the tested range. The mechanisms of the observed phenomena are analyzed by measuring the reacting flow fields. It is found that the increased roughness of the outer wall can prevent the flame shape transition and the onset of thermoacoustic instability, possibly due to the weakened vortex structures and strain rate along with the outer shear layer.
We experimentally investigate the pulsating circular jet flow at moderate Reynolds numbers. By applying time-resolved particle image velocimetry in the axial-radial plane, we measure the near-field velocity fields with the jet source temporally modulated by sinusoidal pulsations. As a baseline, the steady jet flow with the same mean Reynolds number is tested. The direct comparisons of the mean and fluctuating velocity fields show that the whole potential core as well as the axisymmetric shear layer is modulated by the pulsation effect. Meanwhile, larger-scale vortices are formed in the shear layer with phase correlation of the pulsation cycle. As a result, the pulsation increases the turbulent mixing in the latter half of the potential core, and it extends the fluid entrainment further in the radial direction. The increased fluid entrainment of the ambient quiescent fluid is clearly identified by the attracting Lagrangian coherent structures as the bounds of the growing vortices within the shear layer. By analyzing the dynamic modes, we find that the low-frequency off-the-axis helical structures, which are dominant in the steady jet flow, are inhibited. The axisymmetric jet column mode and its harmonics along the axis are strengthened by the pulsation effect. Furthermore, the vortex formation mainly takes place particularly in the deceleration phase, whereas a shock-like wave front is formed during the acceleration, indicating the distinct roles of the pulsation phases in the jet instability.
View Video Presentation: https://doi.org/10.2514/6.2022-1208.vid Non-premixed turbulent lifted jet flames are studied from a Lagrangian point of view to understand flame-turbulence interaction better. High-speed planar particle image velocimetry (PIV) was utilized to measure streamwise and transverse velocity components. The field of view was selected, such that it contains the unsteady lifted flame base. The flame front was identified using the abrupt change in the number density of the PIV particles seeded into the jet stream. Lagrangian Coherent Structures (LCSs), defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields, were utilized to resolve invariant material manifolds. The vorticity field is resolved to investigate its association with LCSs. The Lagrangian framework is more effective than Eulerian analysis in capturing the flame-turbulence interaction. The manifolds' time history represents how eddy structures interact with the flame and highlight their role in the dynamics of the lifted jet flames with a local Reynolds number (Re_δdelta) of 5000. The lifted flame dynamics were investigated using a Lagrangian approach, which differs from an Eulerian approach in that the time-history of the flow is inherent to the analysis technique due to the intrinsic nature of tracking particles. The results suggest that the balance of attracting and repelling LCSs induces a barrier preventing the flame from moving. However, the accumulation of the repelling LCSs at the flame base causes the flame to move downstream, while the accumulation of the attracting LCSs pushes the flame upstream.
Advances in time-resolved three-dimensional Particle Tracking Velocimetry (4D-PTV) techniques have consistently revealed more accurate Lagrangian particle motions. A novel track initialization technique as a complementary part of 4D-PTV, based on local temporal and spatial coherency of neighbor trajectories, is proposed. The proposed Lagrangian Coherent Track Initialization (LCTI) applies physics-based Finite Time Lyapunov Exponent (FTLE) to build four frame coherent tracks. We locally determine Lagrangian coherent structures among neighbor trajectories by using the FTLE boundaries (i.e., ridges) to distinguish the clusters of coherent motions. To evaluate the proposed technique, we created an open-access synthetic Lagrangian and Eulerian dataset of the wake downstream of a smooth cylinder at a Reynolds number equal to 3900 obtained from three-dimensional direct numerical simulation. Performance of the proposed method based on three characteristic parameters, temporal scale, particle concentration (i.e., density), and noise ratio, showed robust behavior in finding true tracks compared to the recent initialization algorithms. Sensitivity of LCTI to the number of untracked and wrong tracks is also discussed. We address the capability of using the proposed method as a function of a 4D-PTV scheme in the Lagrangian particle tracking challenge. We showed that LCTI prevents 4D-PTV divergence in flows with high particle concentrations. Finally, the LCTI behavior was demonstrated in a jet impingement experiment. LCTI was found to be a reliable tracking tool in complex flow motions, with a strength revealed for flows with high velocity and acceleration gradients.
We report the interaction between active non-spherical swimmers and a long-standing flow structure, Lagrangian coherent structures (LCSs), in a weakly turbulent two-dimensional flow. Using a hybrid experimental–numerical model, we show that rod-like swimmers have a much stronger and more robust preferential alignment with attracting LCSs than with repelling LCSs. Tracing the swimmers' Lagrangian trajectories, we reveal that the preferential alignment is the consequence of the competition between the intrinsic mobility of the swimmers and the reorientation ability of the strain rate near the attracting LCSs. The strong preferential alignment with attracting LCSs further leads to a strong accumulation near the attracting LCSs. Moreover, we show the self-similarity of this accumulation, which reduces the intricate interaction to only one control parameter. Our results generically elucidate the interaction between active and non-spherical swimmers with LCSs and, thus, can be widely applied to many natural and engineered fluids.
Teleconnections are striking features of the Earth climate system which appear as statistically correlated climate-related patterns between remote geographical regions of the globe. In a changing climate, however, the strength of teleconnections might change, and an appropriate characterization of these correlations and their change (more appropriate than detrending the time series) is lacking in the literature. Here we present a novel approach, based on the theory of snapshot attractors, corresponding in our context to studying parallel climate realizations. Imagining an ensemble of parallel Earth systems, instead of the single one observed (i.e., the real Earth), the ensemble, after some time, characterizes the appropriate probabilities of all options permitted by the climate dynamics, reflecting the internal variability of the climate. We claim that the relevant quantities for characterizing teleconnections in a changing climate are correlation coefficients taken over the temporally evolving ensemble in any time instant. As a particular example, we consider the teleconnections of the North Atlantic Oscillation (NAO). In a numerical climate model, we demonstrate that this approach provides the only statistically correct characterization, in contrast to commonly used temporal correlations evaluated along single detrended time series. The teleconnections of the NAO are found to survive the climate change, but their strength might be time-dependent.
The book discusses continuous and discrete systems in systematic and sequential approaches for all aspects of nonlinear dynamics. The unique feature of the book is its mathematical theories on flow bifurcations, oscillatory solutions, symmetry analysis of nonlinear systems and chaos theory. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, and a number of examples worked out in detail and exercises have been included. Chapters 1-8 are devoted to continuous systems, beginning with one-dimensional flows. Symmetry is an inherent character of nonlinear systems, and the Lie invariance principle and its algorithm for finding symmetries of a system are discussed in Chap. 8. Chapters 9-13 focus on discrete systems, chaos and fractals. Conjugacy relationship among maps and its properties are described with proofs. Chaos theory and its connection with fractals, Hamiltonian flows and symmetries of nonlinear systems are among the main focuses of this book. Over the past few decades, there has been an unprecedented interest and advances in nonlinear systems, chaos theory and fractals, which is reflected in undergraduate and postgraduate curricula around the world. The book is useful for courses in dynamical systems and chaos, nonlinear dynamics, etc., for advanced undergraduate and postgraduate students in mathematics, physics and engineering.
The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory and numerical simulations. The stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtained. It is also shown that when the bifurcation parameter exceeds a critical value, the Hopf bifurcation occurs in this model. Then, the conditions of the supercritical and subcritical bifurcation are derived through the normal form theory. Finally, the chaotic behavior of the model is also discussed, the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter are obtained, and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further. Especially, a computer-assisted proof of the chaoticity of the model is presented by a topological horseshoe theory.
In the present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit methods for computing all Lyapunov Characteristic Exponents of a dynamical system.
After the general theory of such exponents, developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B).
The authors argue that the concept of snapshot attractors and of their natural probability distributions are the only available tools by means of which mathematically sound statements can be made about averages, variances, etc., for a given time instant in a changing climate. A basic advantage of the snapshot approach, which relies on the use of an ensemble, is that the natural distribution and thus any statistics based on it are independent of the particular ensemble used, provided it is initiated in the past earlier than a convergence time. To illustrate these concepts, a tutorial presentation is given within the framework of a low-order model in which the temperature contrast parameter over a hemisphere decreases linearly in time. Furthermore, the averages and variances obtained from the snapshot attractor approach are demonstrated to strongly differ from the traditional 30-yr temporal averages and variances taken along single realizations. The authors also claim that internal variability can be quantified by the natural distribution since it characterizes the chaotic motion represented by the snapshot attractor. This experience suggests that snapshot-attractor-based calculations might be appropriate to be evaluated in any large-scale climate model, and that the application of 30-yr temporal averages taken along single realizations should be complemented with this more appealing tool for the characterization of climate changes, which seems to be practically feasible with moderate ensemble sizes.
The theory of nonautonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this book. The focus is on dissipative systems and nonautonomous attractors, in particular the recently introduced concept of pullback attractors. Linearization theory, invariant manifolds, Lyapunov functions, Morse decompositions and bifurcations for nonautonomous systems and set-valued generalizations are also considered as well as applications to numerical approximations, switching systems and synchronization. Parallels with corresponding theories of control and random dynamical systems are briefly sketched. With its clear and systematic exposition, many examples and exercises, as well as its interesting applications, this book can serve as a text at the beginning graduate level. It is also useful for those who wish to begin their own independent research in this rapidly developing area.
The rate of turbulent dissipation and the turbulent Reynolds number along the axes of a turbulent plane jet and three circular jets is measured in order to confirm universal relations between the turbulent dissipation rate and the streamwise position derived from the necessary requirements for self-preservation. Flow velocity was determined by a hot-wire anemometer located in a plane jet with exit Reynolds numbers of 20,400 and 42,800 and circular jets with exit Reynolds numbers of 55,600, 109,000 and 471,000. The universal relation for circular jets is found to be valid in the position/diameter ratio range of 20 to 140 in a large range of jet Reynolds numbers, while that for plane jets is confirmed in the range 30 to 160. For either type of jet, the turbulence Reynolds number is found to be proportional to 2.3 times the square root of the local Reynolds number, which is based on local velocity and length scales.
Evidence is presented for low-dimensional strange attractors in Couette-Taylor flow data. Computations of the largest Lyapunov exponent and metric entropy show that the system displays sensitive dependence on initial conditions. Although the phase space is very high dimensional, analysis of experimental data shows that motion is restricted to an attractor of dimension 5 for Reynolds numbers up to 30 percent above the onset of chaos. The Lyapunov exponent, entropy, and dimension all generally increase with Reynolds number.
A low-dimensional model of general circulation of the atmosphere is investigated. The differential equations are subject to periodic forcing, where the period is one year. A three-dimensional Poincaré mapping P depends on three control parameters F, G, and ε, the latter being the relative amplitude of the oscillating part of the forcing. This paper provides a coherent inventory of the phenomenology of P PF,G,ε. For ε small, a Hopf-saddle-node bifurcation HSN of fixed points and quasi-periodic Hopf bifurcations of invariant circles occur, persisting from the autonomous case ε = 0. For ε = 0.5, the above bifurcations have disappeared. Different types of strange attractors are found in four regions (chaotic ranges) in {F,G} and the related routes to chaos are discussed.
The fractal dimension, Lyapunov-exponent spectrum, Kolmogorov entropy, and predictability are analyzed for chaotic attractors in the atmosphere by analyzing the time series of daily surface temperature and pressure over several regions of the United States and the North Atlantic Ocean with different climatic signal-to-noise ratios. Though the total number of data points (from about 13 800 to about 36 500) is larger than those used in previous studies, it is still too small to obtain a reliable estimate of the Grassberger-Procaccia correlation dimension because of the limitations discussed by Ruelle. However, it can be shown that this dimension is greater than 8. Also, it is pointed out the most, if not all, of the previous estimates of low fractal dimensions in the atmosphere are spurious. -from Authors
In a low-order chaotic global atmospheric circu-lation model the effects of deterministic chaotic driving are investigated. As a result of driving, peak-over-threshold type extreme events, e.g. cyclonic activity in the model, become more extreme, with increased frequency of recurrence. When the characteristic time of the driving is comparable to that of the undriven system, a resonance effect with amplified variance shows up. For very fast driving we find a reduced enhancement of variance, which is also the case with white noise driving. Snapshot attractors and their natural measures are determined as a function of time, and a resonance effects is also identified. The extreme value statistics of group max-ima is found to follow a Weibull distribution.
In a conceptual model of global atmospheric circulation, the effects of annually periodic driving are investigated. The driven system is represented in terms of snapshot attractors, which may remain fractal at all times. This is due to the transiently chaotic behavior in the regular parameter regimes of the undriven system. The driving with annual periodicity is found to be relatively fast: There is a considerable deviation from the undriven case. Accordingly, the existence of a hysteresis loop is identified, namely, the extremal values of a given variable depend not only on the actual strength of the insolation but also on the sign of its temporal change. This hysteresis is due to a kind of internal memory. In the threshold-dependence of mean return times of various extreme events, a roughly exponential scaling is found. Climate sensitivity parameters are defined, and the measure of certain types of extremal behavior is found to be strongly susceptible to changes in insolation.
The possible chaotic nature of the turbulence of the atmospheric boundary layer in and above a decidious forest is investigated. In particular, this work considers high resolution temperature and three-dimensional wind speed measurements, gathered at six alternative elevations at Camp Borden, Ontario, Canada (Shawet al., 1988). The goal is to determine whether these time series may be described (individually) by sets of deterministic nonlinear differential equations, such that: (i) the data's intrinsic (and seemingly random) irregularities are captured by suitable low-dimensional fractal sets (strange attractors), and (ii) the equation's lack of knowledge of initial conditions translates into unpredictable behavior (chaos). Analysis indicates that indeed all series exhibit chaotic behavior, with strange attractors whose (correlation) dimensions range from 4 to 7. These results support the existence of a low-dimensional chaotic attractor in the lower atmosphere.
The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we focus on the large-scale, wind-driven flow of the mid-latitude oceans which contribute in a crucial way to Earth's climate, and to changes therein. We study the low-frequency variability (LFV) of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones. The natural climate variability induced by the LFV of the ocean circulation is but one of the causes of uncertainties in climate projections. Another major cause of such uncertainties could reside in the structural instability in the topological sense, of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics. We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. As a very first step, we study the effect of noise on the topological classes of the Arnol'd family of circle maps, a paradigmatic model of frequency locking as occurring in the nonlinear interactions between the El Nino-Southern Oscillations (ENSO) and the seasonal cycle. It is shown that the maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification. This result is consistent with stabilizing effects of stochastic parametrization obtained in modeling of ENSO phenomenon via some general circulation models.
We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.
Lyapunov exponents measure the average exponential growth rate of typical linear perturbations in a chaotic system, and the inverse of the largest exponent is a measure of the time horizon over which the evolution of the system can be predicted. Here, Lyapunov exponents are determined in forced homogeneous isotropic turbulence for a range of Reynolds numbers. Results show that the maximum exponent increases with Reynolds number faster than the inverse Kolmogorov time scale, suggesting that the instability processes may be acting on length and time scales smaller than Kolmogorov scales. Analysis of the linear disturbance used to compute the Lyapunov exponent, and its instantaneous growth, show that the instabilities do, as expected, act on the smallest eddies, and that at any time, there are many sites of local instabilities.
This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time. Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with
the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence.
The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.
A time series resulting from a single initial condition is shown to be insufficient for quantifying the internal variability in a climate model, and thus one is unable to make meaningful climate projections based on it. The authors argue that the natural distribution, obtained from an ensemble of trajectories differing solely in their initial conditions, of the snapshot attractor corresponding to a particular forcing scenario should be determined in order to quantify internal variability and to characterize any instantaneous state of the system in the future. Furthermore, as a simple measure of internal variability of any particular variable of the model, the authors suggest using its instantaneous ensemble standard deviation. These points are illustrated with the intermediate-complexity climate model Planet Simulator forced by a CO2 scenario, with a 40-member ensemble. In particular, the leveling off of the time dependence of any ensemble average is shown to provide a much clearer indication of reaching a steady state than any property of single time series. Shifts in ensemble averages are indicative of climate changes. The dynamical character of such changes is illustrated by hysteresis-like curves obtained by plotting the ensemble average surface temperature versus the CO2 concentration. The internal variability is found to be the most pronounced on small geographical scales. The traditionally used 30-yr temporal averages are shown to be considerably different from the corresponding ensemble averages. Finally, the North Atlantic Oscillation (NAO) index, related to the teleconnection paradigm, is also investigated. It is found that the NAO time series strongly differs in any individual realization from each other and from the ensemble average, and climatic trends can be extracted only from the latter.
The global attractor, whose well established definition we recall below, is an object that captures the asymptotic behaviour of autonomous systems. The aim of this chapter is to introduce the ‘pullback attractor’, which seems to be the correct generalisation of this concept for use with non-autonomous processes. We pay particular attention to how this non-autonomous definition relates to the autonomous one.
Typical fluid particle trajectories are sensitive to changes in their initial conditions. This makes the assessment of flow models and observations from individual tracer samples unreliable. Behind complex and sensitive tracer patterns, however, there exists a robust skeleton of material surfaces, Lagrangian coherent structures (LCSs), shaping those patterns. Free from the uncertainties of single trajectories, LCSs frame, quantify, and even forecast key aspects of material transport. Several diagnostic quantities have been proposed to visualize LCSs. More recent mathematical approaches identify LCSs precisely through their impact on fluid deformation. This review focuses on the latter developments, illustrating their applications to geophysical fluid dynamics.
This paper examines whether hyperbolic Lagrangian structures-such as stable and unstable manifolds-found in model velocity data represent reliable predictions for mixing in the true fluid velocity field. The error between the model and the true velocity field may result from velocity interpolation, extrapolation, measurement imprecisions, or any other deterministic source. We find that even large velocity errors lead to reliable predictions on Lagrangian coherent structures, as long as the errors remain small in a special time-weighted norm. More specifically, we show how model predictions from the Okubo-Weiss criterion or from finite-time Lyapunov exponents can be validated. We also estimate how close the true Lagrangian coherent structures are to those predicted by models.
The results of several works concerning estimates of correlation dimension have recently been criticized on the basis of a new upper bound on such estimates (i.e. 2 log10 N, where N is the number of data). It is shown here that this bound is not valid for correlation dimension estimates in general, and furthermore it is too weak to be useful for regimes where it does hold. This casts the indicated criticism into question.
We study the transport of particles in a general, two-dimensional, incompressible flow in the presence of a transient eddy, i.e., a bounded set of closed streamlines with a finite time of existence. Using quantities obtained from Eulerian observations, we provide explicit conditions for the existence of a hyperbolic structure in the flow, which induces mixing between the eddy and its environment. Our results can be used directly to study finite-time transport in numerically or experimentally generated vector fields with general time-dependence.
The scalar concentration and velocity field of an axisymmetric turbulent jet in a coflow are measured using laser-induced fluorescence and laser Doppler anemometry techniques. Both the time-averaged and turbulence data suggest the use of the top-hat average velocity as the characteristics velocity in this shear flow. The visual boundary and the 25% maximum concentration (0.25C(m)) contour are found to correspond closely to the edge of the top-hat profile, where r = B = √2b(g) (B and b(g) are the nominal half-width of the top-hat and Gaussian profiles, respectively). This jet edge, defined by the 25%C(m) contour, also matches with the 50% contour of the intermittency factor. An integral model of the coflowing jet has been formulated using a Lagrangian jet spreading hypothesis for turbulent closure. The model predictions are in good agreement with the present concentration, velocity data obtained in the near field (strong jet), and the velocity data of a recent study in the far field (weak jet).
Previous investigations of riddling have focused on the case where the dynamical invariant set in the symmetric invariant manifold of the system is a chaotic attractor. A situation expected to arise commonly in physical systems, however, is that the dynamics in the invariant manifold is in a periodic window. We argue and demonstrate that riddling can be more pervasive in this case because it can occur regardless of whether the chaotic set in the invariant manifold is transversly stable or unstable. Scaling behavior associated with this type of riddling is analyzed and is supported by numerical experiments.
Hyperbolicity is an important concept in dynamical system theory; however, we know little about the hyperbolicity of concrete physical systems including fluid motions governed by the Navier-Stokes equations. Here, we study numerically the hyperbolicity of the Navier-Stokes equation on a two-dimensional torus (Kolmogorov flows) using the method of covariant Lyapunov vectors developed by Ginelli et al. [Phys. Rev. Lett. 99, 130601 (2007)]. We calculate the angle between the local stable and unstable manifolds along an orbit of chaotic solution to evaluate the hyperbolicity. We find that the attractor of chaotic Kolmogorov flows is hyperbolic at small Reynolds numbers, but that smaller angles between the local stable and unstable manifolds are observed at larger Reynolds numbers, and the attractor appears to be nonhyperbolic at a certain Reynolds numbers. Also, we observed some relations between these hyperbolic properties and physical properties such as time correlation of the vorticity and the energy dissipation rate.
Recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby (the basin is “riddled” with holes). Here we consider the dynamics near the transition to this situation as a parameter is varied. Using a simple analyzable model, we obtain the characteristic behaviors near this transition. Numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered.
a b s t r a c t We develop a mathematical theory that clarifies the relationship between observable Lagrangian Coherent Structures (LCSs) and invariants of the Cauchy–Green strain tensor field. Motivated by physical observations of trajectory patterns, we define hyperbolic LCSs as material surfaces (i.e., codimension-one invariant manifolds in the extended phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. We also define weak LCSs (WLCSs) as stationary solutions of the above variational problem. Solving these variational problems, we obtain computable sufficient and necessary criteria for WLCSs and LCSs that link them rigorously to the Cauchy–Green strain tensor field. We also prove a condition for the robustness of an LCS under perturbations such as numerical errors or data imperfection. On several examples, we show how these results resolve earlier inconsistencies in the theory of LCS. Finally, we introduce the notion of a Constrained LCS (CLCS) that extremizes normal repulsion or attraction under constraints. This construct allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.
This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. A high-resolution numerical study of two highly idealized models of fundamental interest for climate dynamics allows one to obtain a good approximation of their global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to be random Sinai-Ruelle-Bowen (SRB) measures. The first of the two models is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño–Southern Oscillation (ENSO).
Some early ideas concerning the general circulation of the atmosphere are reviewed. A model of the general circulation, consisting of three ordinary differential equations, is introduced. For different intensities of the axially symmetric and asymmetric thermal forcing, the equations may possess one or two stable steady-state solutions, one or two stable periodic solutions, or irregular (aperiodic) solutions. Qualitative reasoning which has been applied to the real atmosphere may sometimes be applied to the model, and checked for soundness by comparing the conclusions with numerical solutions. The implications of irregularity for the atmosphere and for atmospheric science are discussed.
It is shown that by a smallC
2 (resp.C
) perturbation of a quasiperiodic flow on the 3-torus (resp. them-torus,m>3), one can produce strange AxiomA attractors. Ancillary results and physical interpretation are also discussed.
In this paper, the attractors of turbulent flows in phase space are reconstructed by the time delay technique using observed data of atmospheric boundary-layer turbulence, which include high resolution temperature, humidity andthree-dimensional wind speed measurements in Gansu province and Beijing, China. The correlation dimensions and largest Lyapunov exponents have been computed. The results indicate that all the largest Lyapunov exponents in different conditions of time, site and atmospheric stability are greater than zero. This means that the atmospheric boundary-layer turbulence system is really chaotic and has appropriate low-dimensional strange attractors whose dimension numbers range from 3 to 7 and vary with different variables (dynamical variables or non-dynamical variables) and atmospheric stability. Turbulent kinetic energy is first applied to reconstruct the attractor of turbulence, and is found to be feasible.
We exhibit random strange attractors with random Sinai-Bowen-Ruelle measures for the composition of independent random diffeomorphisms.
It is shown how the existence of low-dimensional chaotic dynamical systems describing turbulent fluid flow might be determined experimentally. Techniques are outlined for reconstructing phase-space pictures from the observation of a single coordinate of any dissipative dynamical system, and for determining the dimensionality of the system's attractor. These techniques are applied to a well-known simple three-dimensional chaotic dynamical system.
Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.
Three theories of the liftoff of a turbulent jet flame were assessed using cinema-particle imaging velocimetry movies recorded at 8000 images/s. The images visualize the time histories of the eddies, the flame motion, the turbulence intensity, and streamline divergence. The first theory assumes that the flame base has a propagation speed that is controlled by the turbulence intensity. Results conflict with this idea; measured propagation speeds remains close to the laminar burning velocity and are not correlated with the turbulence levels. Even when the turbulence intensity increases by a factor of 3, there is no increase in the propagation speed. The second theory assumes that large eddies stabilize the flame; results also conflict with this idea since there is no significant correlation between propagation speed and the passage of large eddies. The data do support the “edge flame” concept. Even though the turbulence level and the mean velocity in the undisturbed jet are large (at jet Reynolds numbers of 4300 and 8500), the edge flame creates its own local low-velocity, low-turbulence-level region due to streamline divergence caused by heat release. The edge flame has two propagation velocities. The actual velocity of the flame base with respect to the disturbed local flow is found to be nearly equal to the laminar burning velocity; however, the effective propagation velocity of the entire edge flame with respect to the upstream (undisturbed) flow exceeds the laminar burning velocity. A simple model is proposed which simulates the divergence of the streamlines by considering the potential flow over a source. It predicts the well-established empirical formula for liftoff height, and it agrees with experiment in that the controlling factor is streamline divergence, and not turbulence intensity or large eddy passage. The results apply only to jet flames for Re<8500; for other geometries the role of turbulence could be larger.
We study the three standard methods for reconstructing a state space from a time series; delays, derivatives, and principal components. We derive a closed-form solution to principal component analysis in the limit of small window widths. This solution explains the relationship between delays, derivatives, and principal components, it shows how the singular spectrum scales with dimension and delay time, and it explains why the eigenvectors resemble the Legendre polynomials Most importantly, the solution allows us to derive a guideline for choosing a good window width. Unlike previous suggestions, this guideline is based on first principles and simple quantities. We argue that discrete Legendre polynomials provide a quick and not-so-dirty substitute for principal component analysis, and that they are a good practical method for state space reconstruction.
The combined influences of boundary effects at large scales and nonzero nearest neighbor separations at small scales are used to compute intrinsic limits on the minimum size of a data set required for calculation of scaling exponents. A lower bound on the number of points required for a reliable estimation of the correlation exponent is given in terms of the dimension of the object and the desired accuracy. A method of estimating the correlation integral computed from a finite sample of a white noise signal is given.
We introduce a Lagrangian definition for the boundaries of coherent structures in two-dimensional turbulence. The boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors. Such material lines are responsible for stretching and folding in the mixing of passive tracers. We derive an analytic criterion that can be used to extract coherent structures with high precision from numerical or experimental data sets. The criterion provides a rigorous link between the Lagrangian concept of hyperbolicity, the Okubo–Weiss criterion, and vortex boundaries. We apply the results to simulations of two-dimensional barotropic turbulence.
We prove analytic criteria for the existence of finite-time attracting and repelling material surfaces and lines in three-dimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories. An alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite-time Lyapunov exponent field computed directly from particle paths. Both approaches provide effective tools for extracting distinguished Lagrangian structures from three-dimensional velocity data. We illustrate the results on steady and unsteady ABC-type flows.