Finite-time Lyapunov Exponents Research Articles

Finite Time Lyapunov Exponent Analysis of Model Predictive Control and Reinforcement Learning

Finite Time Lyapunov Exponent Analysis of Model Predictive Control and Reinforcement Learning

Efficient Sensor Location for HVAC Systems Using Lagrangian Coherent Structures

Study of thermo‐fluid flow has its significance in energy analysis of a building. Various thermo‐fluid analyses are performed in heating ventilation and air conditioning (HVAC) systems for energy efficient buildings by scientists and engineers to save energy consumption during the life of a building. In order to achieve energy conservation for different HVAC systems, determination of air flow pattern and flow physics are vital parameters. In case of fluid flow postprocessing techniques such as streamlines, vector plots and contours are often employed. These techniques help in understanding the nature of flow and its properties. Each of these postprocessing techniques mentioned are based on Eulerian methods and have certain inherent deficiencies pertaining to the amount of information they can convey about certain aspects of fluid flow. Lagrangian coherent structures (LCS) on the other hand use Lagrangian data for analysis purposes. LCS are generated using finite time Lyapunov exponent fields which in turn depict the rate of expansion or contraction of the trajectories around a certain point. LCS act as the transport barriers across which there is approximately zero mass flux. This property means that LCS can be applied to problems related to separation and reattachment in fluid flow and find virtual boundaries inside flows. In the current study, we focused on the application of LCS for efficient placement of sensors for HVAC systems. We computed LCS using velocity data extracted from the CFD simulations of a 2D room model. Thus, LCS can be used to identify the virtual boundaries in fluid flow. This helps in indication of regions where mixing of particles occurs and also where particles are stagnated. Inlet angles of 0, 15, 22, and 30 degrees are used and analysis shows that manifolds with 30 degree angles provide better ventilation. The outcome of this study can be used to improve the energy efficiency as well as predict the accurate location of HVAC sensor and control units.

Comparative analysis of the secure communication schemes based on the generators of hyperbolic strange attractor and strange nonchaotic attractor

The purpose of this work is to analyse qualitative features of the information transmission process via several communication schemes based on the synchronization of transmitter and receiver, both being complex signal generators. For this purpose generators of the hyperbolic chaos and generators with the strange nonchaotic attractor are employed. Evaluation of advantages and disadvantages of such schemes is made comparing themselves with each other as well as with schemes based on the nonhyperbolic chaotic generators. Methods. The power spectra and the distributions of the largest finite-time Lyapunov exponent are used to confirm the complexity of the dynamics of the generators in use and to verify the wide-bandness, robustness and stochasticity of their signals. Confidentiality of the informational signal transmission is achieved using its nonlinear mixing to the dynamics of the transmitter. The special phase mixing is used since the model generators employed for the research demonstrate nontrivial dynamics for the angular variable — oscillations phase shift. The digital image is used as an information for transmission. Visual control during the transmission process allows to carry out the qualitative analysis of the success of the signal coding and its detecting by the receiver. Results. Successful transmission and decoding of information for all schemes under investigation are demonstrated for the case of identical transmitter and receiver. Parameter detuning of these generators leads to difficulties in separation of the informational signal from the chaotic/complex carrier due to loss of the full synchronization. For the nonhyperbolic chaos detuning of the parameter responsible for the amplitude of the signal leads to the bad quality of the detection while frequency detuning makes detection absolutely impossible. Schemes with the hyperbolic chaos and strange nonchaotic dynamics appear to demonstrate much better results. The information detection is much better in this case because of the robustness of the generalized synchronization. Conclusion. Robust chaotic and complex nonchaotic generators appear to have significant advantages for communication systems comparing to the chaotic generators of nonhyperbolic type.

Impact of Yucatan Channel Subsurface Velocity Observations on the Gulf of Mexico State Estimates

Abstract The Gulf of Mexico (GoM) surface circulation variability is dominated by the Loop Current (LC) and the episodically released anticyclonic Loop Current eddies (LCEs). The Yucatan Current feeds the LC through the Yucatan Channel (YC), and its flow structure at the YC is hypothesized to affect the LC evolution critically. This study examines the impact of assimilating YC subsurface velocity observations from a tall mooring array across the YC on the GoM circulation. State estimates and forecasts of the LC circulation were produced using a regional implementation of the Massachusetts Institute of Technology general circulation model (MITgcm) and its adjoint-based four-dimensional variational (4DVAR) assimilation system. The estimates were constrained by combinations of the YC observations and satellite-derived sea surface height (SSH) and sea surface temperature (SST). The results show that assimilation of both moored and satellite data improves the model hindcasts and forecasts for all LC phases. Additionally, one realization of the state estimate that assimilates only moored data matches the LCE detachment timing with that of AVISO SSH. Observations from the moorings close to the Yucatan Peninsula significantly impact the LCE detachment. A finite-time Lyapunov exponent analysis reveals the differences among the assimilation experiments, such as eddylike structures intruding into the GoM through the YC, and its relation to the typical LC sudden growth. Finally, an adjoint sensitivity analysis is used to verify the dynamic link between the LC extension and the intrusion of eddylike structures into the GoM.

Using a Synoptic-Scale Mixing Diagnostic to Explain Global Precipitation Variability from Weekly to Interannual Time Scales

Abstract Precipitation often happens along organized filaments or bands of moisture such as convergence zones. Recent regional studies have shown that these moisture filaments arise from synoptic-scale mixing features known as attracting Lagrangian coherent structures (LCSs). In this study, we present a climatology of synoptic-scale mixing and investigate its covariability with precipitation on temporal scales ranging from weekly to interannual. We characterize mixing with the finite-time Lyapunov exponent (FTLE), a measure of parcel deformation, in ERA5 data between 1980 and 2009. Attracting LCSs are identified as ridges of the FTLE. At the interannual time scale, we compare El Niño and La Niña events and find that composites of precipitation and mixing anomalies share similar spatial patterns. We also compare summer and winter seasons and find that composites of seasonal-mean precipitation and mixing anomalies present similar characteristics, with precipitation being particularly intense (weak) where mixing is strong (weak). In particular, these patterns closely match the typical signatures of the intertropical convergence zone (ITCZ) and monsoon systems and the migrations of extratropical cyclone tracks. At the subseasonal scale, we employ daily composites to investigate the influence of the Madden–Julian oscillation and the North Atlantic Oscillation on the mixing regimes of the Atlantic and east Pacific; our results indicate that these oscillations control the synoptic-scale horizontal mixing and the occurrence of LCSs as to suppress or enhance precipitating systems like the ITCZ and the South Atlantic convergence zone. The results presented in this first climatology of synoptic-scale mixing and LCSs indicate that these are powerful diagnostics to identify circulation mechanisms underlying precipitation variability.

Unsteady Mass Transfer in Bubble Wakes Analyzed by Lagrangian Coherent Structures in a Flat-Bed Reactor

To increase the yield and selectivity in reactive bubbly flows, the gas-liquid interactions have to be understood in depth. In the current fundamental study, flow and concentration data of the wakes of two-dimensional bubbles in an organic solvent are obtained experimentally in a flat-bed reactor. The unsteady mass transport phenomena in these turbulent wakes of two freely rising, two-dimensional bubbles with bubble Reynolds numbers Re=949 and Re=388 are evaluated by analyzing Lagrangian Coherent Structures (LCS). To reveal how LCS govern the transport of dissolved gas in bubble wakes, and therefore affect gas-liquid reactions, LCS in two-dimensional velocity fields are computed and compared with concentration fields of dissolved gas. The analysis of backward Finite Time Lyapunov Exponent (bFTLE) fields reveals coherent fluid dynamic structures for both bubble Reynolds numbers studied. In the higher bubble Reynolds number case, two types of coherent structures are found, which hinder the mixing of the dissolved gas and the liquid bulk. Repelling LCS are found to enclose parcels transported into the vortices, and indicate thus, which fluid parcels can possibly take part in chemical reactions. Due to higher mixing, unveiled by details from the LCS and FTLE analyses, and therefore increased contact area between dissolved gas and fresh liquid, higher yields of reaction products are suggested for the lower bubble Reynolds number case in this two-dimensional study. This is contradicting the rule of thumb that mixing increases for higher bubble Reynolds numbers.

Finite-time Lyapunov fluctuations and the upper bound of classical and quantum out-of-time-ordered expansion rate exponents.

This Letter demonstrates for chaotic maps [logistic, classical, and quantum standard maps (SMs)] that the exponential growth rate (Λ) of the out-of-time-ordered four-point correlator is equal to the classical Lyapunov exponent (λ) plus fluctuations (Δ^{(fluc)}) of the one-step finite-time Lyapunov exponents (FTLEs). Jensen's inequality provides the upper bound λ≤Λ for the considered systems. Equality is restored with Λ=λ+Δ^{(fluc)}, where Δ^{(fluc)} is quantified by k-higher-order cumulants of the (covariant) FTLEs. Exact expressions for Λ are derived and numerical results using k=20 furnish Δ^{(fluc)}∼ln(sqrt[2]) for all maps (large kicking intensities in the SMs).

How sensitive are Lagrangian coherent structures to uncertainties in data?

Lagrangian coherent structures (LCSs) are time-varying entities which capture the most influential transport features of a flow. These can for example identify groups of particles which have greatest stretching, or which maintain a coherent jet or vortical structure. While many different LCS methods have been developed, the impact of the inevitable measurement uncertainty in realistic Eulerian velocity data has not been studied in detail. This article systematically addresses whether LCS methods are self-consistent in their conclusions under such uncertainty for nine different methods: the finite time Lyapunov exponent, hyperbolic variational LCSs, Lagrangian averaged vorticity deviation, Lagrangian descriptors, stochastic sensitivity, the transfer operator, the dynamic Laplacian operator, fuzzy c-means clustering and coherent structure colouring. The investigations are performed for two different realistic data sets: a computational fluid dynamics simulation of a Kelvin–Helmholtz instability, and oceanographic data of the Gulf Stream region. Using statistics gleaned from stochastic simulations, it is shown that the methods which detect full-dimensional coherent flow regions are significantly more robust than methods which detect lower-dimensional flow barriers. Additional insights into which aspects of each method are self-consistent, and which are not, are provided.

Learning ocean circulation models with reservoir computing

Two elementary models of ocean circulation, the well-known double-gyre stream function model and a single-layer quasi-geostrophic (QG) basin model, are used to generate flow data that sample a range of possible dynamical behavior for particular flow parameters. A reservoir computing (RC) machine learning algorithm then learns these models from the stream function time series. In the case of the QG model, a system of partial differential equations with three physically relevant dimensionless parameters is solved, including Munk- and Stommel-type solutions. The effectiveness of a RC approach to learning these ocean circulation models is evident from its ability to capture the characteristics of these ocean circulation models with limited data including predictive forecasts. Further assessment of the accuracy and usefulness of the RC approach is conducted by evaluating the role of both physical and numerical parameters and by comparison with particle trajectories and with well-established quantitative assessments, including finite-time Lyapunov exponents and proper orthogonal decomposition. The results show the capability of the methods outlined in this article to be applied to key research problems on ocean transport, such as predictive modeling or control.

A Lagrangian analysis of partial cavitation growth and cavitation control mechanism

Partial cavitation has a strong unsteadiness, which will cause serious damage to the hydraulic machinery. The spanwise obstacle is nearly the most efficient method for controlling unsteady cavitation. In this study, numerical simulations of partial cavitating flows around NACA (National Advisory Committee for Aeronautics) 66 hydrofoils in two dimensions (2D) were carried out both with and without obstruction. The obstruction is placed at 0.37c, and its height is 0.1c. Utilizing the finite-time Lyapunov exponent, the Lagrangian coherent structures (LCSs) were developed to investigate the dynamic characteristics of the unsteady flow. By showing the dynamic evolution of the Lagrangian behaviors, the time-dependent LCSs over the two different flows demonstrate the effectiveness of LCSs in explaining the evolution of the vortex during the partial cavitation process. With the use of LCSs, the vortex boundary and reentrant jet can be easily located, and the link between the vortexes can be readily seen. In the meantime, the vortex's origin and destination are shown by the stable and unstable manifolds, respectively. LCSs were then utilized to examine how the obstruction had an impact, and the following conclusions were reached. First, the obstruction can stop a portion of reentrant jets from passing through it. Second, the obstruction can curve the pathway of the reentrant jet, which has passed through it. Third, the obstruction prevents the cavity from flowing downstream. Finally, the obstruction continuously obliterates the expanding cavity across it. Simply said, the Lagrangian analysis based on LCSs provides a better understanding of the vortex dynamics than traditional visualization techniques, which is essential to understanding the great performance of the cavitation-induced unsteady flow.

The effect of surface roughness on the Lagrangian coherent structures in turbulent Rayleigh–Bénard convection

We perform direct numerical simulations of turbulent Rayleigh–Bénard (RB) convection in a closed square cell with roughness plates at Rayleigh number fixed at Ra=108 and the Prandtl number fixed at Pr = 1. To gain insight into the effect of surface roughness on material transport in turbulent Rayleigh–Bénard convection, the Lagrangian coherent structures (LCSs) are extracted using the finite-time Lyapunov exponent method in the cases of different roughness heights. First, we find that lobe structures are widely present in RB convection and we elucidate how they play a part in transporting heat from coner-flow rolls to large-scale circulation. Then, we quantify the heat flux along the LCSs, which contributes to 80% of the total flux. This implies that the LCSs play an important role in heat transport regardless of the roughness height. Furthermore, two different mechanisms of heat transport in RB convection induced by roughness heights are explained in the Lagrangian perspective: the decrease in Nu number in the cases of h<hc is caused by the LCSs between the roughness elements which hinders the exchange of material between the fluid in the cavity and the bulk region; whereas, the increase in Nu number in the case of h>hc is produced by the enhanced mixing events of the convection that enhance the contribution of heat transport in the bulk region.

Lagrangian surface signatures reveal upper-ocean vertical displacement conduits near oceanic density fronts

Vertical transport in the ocean plays a critical role in the exchange of freshwater, heat, nutrients, and other biogeochemical tracers. While there are situations where vertical fluxes are important, studying the vertical transport and displacement of material requires analysis over a finite interval of time. One such example is the subduction of fluid from the mixed layer into the pycnocline, which is known to occur near density fronts. Divergence has been used to estimate vertical velocities indicating that surface measurements, where observational data is most widely available, can be used to locate these vertical transport conduits. We evaluate the correlation between surface signatures derived from Eulerian (horizontal divergence, density gradient, and vertical velocity) and Lagrangian (dilation rate and finite time Lyapunov exponent) metrics and vertical displacement conduits. Two submesoscale resolving models of density fronts and a data-assimilative model of the western Mediterranean were analyzed. The Lagrangian surface signatures locate significantly more of the strongest displacement features and the difference in the expected displacements relative to Eulerian ones increases with the length of the time interval considered. Ensemble analysis of forecasts from the Mediterranean model demonstrates that the Lagrangian surface signatures can be used to identify regions of strongest downward vertical displacement even without knowledge of the true ocean state.

Heat transfer enhancement by a pair of asymmetric flexible vortex generators and thermal performance prediction using machine learning algorithms

• Heat transfer enhancement system including two flexible vortex generators is analyzed by using an immersed boundary method. • Effects of the inclination angle, bending rigidity, and the gap distance on the thermal performance are analyzed in detail. • The thermal performance of the present heat transfer system is enhanced up to 122% as compared to baseline flow without flexible vortex generators. Heat transfer should be considered in the design process of various industrial fields such as electric-electronics, plants and refrigeration and air conditioning, etc. Many researchers have introduced vortex generators to promote mixing of fluids, thereby improving heat transfer performance. Convective heat transfer can be enhanced by using vortex generators where, however, the disadvantage of increased mechanical loss due to increased pressure drop is not avoidable. In order to compensate for the penalty, the present study utilizes the characteristics of self-sustained flapping motions of the flexible vortex generator to improve heat transfer without much increasing the pressure drop. The two flexible vortex generators (FVGs) fixed on the upper and lower channel walls, respectively, are introduced in the present study and they take the form of asymmetric configurations by adjusting the inclination angles. The heat transfer performance is observed to depend on the parametric conditions of FVGs such as the inclination angle, bending rigidity, and the gap distance between them. The present system including the asymmetric FVGs shows the improvement of thermal performance by 122% as compared to the baseline flow when the bending rigidity is 0.06, the initial inclination angle is 75°, and the gap distance is 2.4 times the length of FVGs. The attracting Lagrangian coherent structures (LCS) are visualized by calculating the finite-time Lyapunov exponent (FTLE) field to analyze the effects of the vortex structures near the heated channel walls on the fluid mixing. The net energy loss and average heat transfer are predicted by the three machine learning algorithms, i.e ., artificial neural network (ANN), support vector regression (SVR), and random forest (RF), where the inclination angle, bending rigidity, and gap distance of FVGs are selected as input data. The SVR and ANN algorithms show the best performance in predicting the mean energy loss and the heat transfer, respectively, with the R 2 value of above 0.99 and 0.95.

Applying dynamical systems techniques to real ocean drifters

Abstract. This paper presents the first comprehensive comparison of several different dynamical-systems-based measures of stirring and Lagrangian coherence, computed from real ocean drifters. Seven commonly used methods (finite-time Lyapunov exponent (FTLE), trajectory path length, trajectory correlation dimension, trajectory encounter volume, Lagrangian-averaged vorticity deviation, dilation, and spectral clustering) were applied to 144 surface drifters in the Gulf of Mexico in order to map out the dominant Lagrangian coherent structures. Among the detected structures were regions of hyperbolic nature resembling stable manifolds from classical examples, divergent and convergent zones, and groups of drifters that moved more coherently and stayed closer together than the rest of the drifters. Many methods highlighted the same structures, but there were differences too. Overall, five out of seven methods provided useful information about the geometry of transport within the domain spanned by the drifters, whereas the path length and correlation dimension methods were less useful than others.

Periodic Lagrangian Coherent Structures around a tidal inlet

We present an extensive experimental campaign dedicated to the identification of coherent trajectory patterns owing to flow occurring in tidal environments, characterized by a tidal inlet and a channel with lateral tidal flats. Single and multiple harmonics tides are here reproduced on a large-scale physical model. The study of the large scale macro-vortices, generated by vortex shedding during the flood phase from the inlet barrier, is performed employing the Lagrangian Average Vorticity Deviation (LAVD). The presence of large-scale vortices with a complex dynamics within a tidal period suggested a deeper understanding on the possible environmental implications in terms of transport connections or barriers. Finite Time Lyapunov Exponents are employed in order to recognize stable and unstable manifolds within the flow that are defined as preferred paths along which particles are repelled (forward integration) or attracted (backward).

Lagrangian coherent structures in flow past a backward-facing step

This paper investigates flow past a backward-facing step (BFS) in a duct at Reynolds number $Re = 5080$ based on step height, mean inflow velocity and fluid kinematic viscosity. The flow configuration matches a combustion experiment conducted by Pitz and Daily in 1983. High-resolution velocity fields are obtained in OpenFOAM by direct numerical simulation, and the flow field analysed by Lagrangian approaches. Trajectories of fluid particles in areas of interest are obtained by high-order numerical integration, and used to compute finite-time Lyapunov exponents (FTLEs) and polar rotation angles. Lagrangian coherent structures (LCSs) are extracted using geodesic theory, including hyperbolic LCSs and elliptic LCSs. We use complementary qualitative and quantitative LCS analyses to uncover the underlying flow structures. Notably, we find that a flow pathway in which fluid particles rarely diverge from adjacent particles is opened and closed by FTLE ridges determined by the periodic shedding of vortices from the BFS. Two dominant vortices with significant Lagrangian coherence, generated respectively by the separated boundary layer and shear layer, are self-sustaining and of comparable strength. Hyperbolic repelling LCSs act as transport barriers between the pathway and cores of the coherent vortices, thus playing a major part in the fluid entrainment process. Interactions between these different geometric regions partitioned by LCSs lead to intrinsic complexity in the BFS flow.

Manifestation of strange nonchaotic attractors in extended systems: a study through out-of-time-ordered correlators

We study the spatial spread of out-of-time-ordered correlators (OTOCs) in coupled map lattices (CMLs) of quasiperiodically forced nonlinear maps. We use instantaneous speed (IS) and finite-time Lyapunov exponents (FTLEs) to investigate the role of strange non-chaotic attractors (SNAs) on the spatial spread of the OTOC. We find that these CMLs exhibit a characteristic on and off type of spread of the OTOC for SNA. Further, we provide a broad spectrum of the various dynamical regimes in a two-parameter phase diagram using IS and FTLEs. We substantiate our results by confirming the presence of SNA using established tools and measures, namely the distribution of finite-time Lyapunov exponents, phase sensitivity, spectrum of partial Fourier sums, and $0-1$ test.

Elucidating coherent structures, transport barriers, and entrainment in turbulent fountains in stratified media

We analyze the flow organization of turbulent fountains in stratified media under different conditions, using three-dimensional finite-time Lyapunov exponents. The dominant Lagrangian coherent structures responsible for the transport barriers present a similar shape in the different configurations studied. After proposing a criterion for delimiting the boundary surface of the uprising fountain, we quantify the entrainment and re-entrainment rates under fully developed flow conditions using the proper coefficients. Finally, our analysis was applied to the selective inverted sink, a technological application of turbulent fountains, identifying turbulence as the primary mechanism favoring the device's efficiency.

Chaos and thermalization in a classical chain of dipoles.

We explore the connection between chaos, thermalization, and ergodicity in a linear chain of N interacting dipoles. Starting from the ground state, and considering chains of different numbers of dipoles, we introduce single site excitations with excess energy ΔK. The time evolution of the chaoticity of the system and the energy localization along the chain is analyzed by computing, up to a very long time, the statistical average of the finite-time Lyapunov exponent λ(t) and the participation ratio Π(t). For small ΔK, the evolution of λ(t) and Π(t) indicates that the system becomes chaotic at approximately the same time as Π(t) reaches a steady state. For the largest considered values of ΔK the system becomes chaotic at an extremely early stage in comparison with the energy relaxation times. We find that this fact is due to the presence of chaotic breathers that keep the system far from equipartition and ergodicity. Finally, we show numerically and analytically that the asymptotic values attained by the participation ratio Π(t) fairly correspond to thermal equilibrium.

Lagrangian Identification of Coherent Structures and Mass Transport in a Buoyant Jet Diffusion Flame

ABSTRACT Using Lagrangian coherent structures (LCSs), the mass transport process of large-scale coherent structures in a buoyant jet diffusion flame is studied numerically to gain a key understanding of combustion instability. First, the reacting flow fields of the jet diffusion flame are numerically simulated for two cases, a stable case without considering the buoyancy effect and an unstable case that includes the buoyancy effect corresponding to combustion instability. In particular, large-scale Kelvin-Helmholtz instability (KHI) vortex structures outside the flame surface are captured in the unstable case. Then, the ridges in finite-time Lyapunov exponent (FTLE) fields that sketch the distinct regions in the reacting flow field are extracted as LCSs. The mass transport process is studied by comparing the distribution of LCSs with reaction fields. The flame surface is found to coincide with the attracting LCSs, which separate the fuel and air on opposite sides of the flame surface. In contrast, the reaction products, including species and reaction heat, are confined in the boundaries consisting of repelling LCSs on the inner and outer sides of the flame surface. Finally, the evolution of LCSs is tracked to analyze the generation of KHI vortices and species transport in the jet diffusion flame. With the aid of LCSs, it is found that there exist some saddle-type flow features separating the distinct flow structures into several regions, and the jet fuel and airflow enter into the separated regions from different open boundaries. Importantly, the attracting LCSs in the separated regions then bulge outwards, forming KHI vortices as the attracting LCSs further stretch and fold. Furthermore, the vortices continue moving upwards while species flow from repelling LCSs to attracting LCSs, leading to mixing. These results show that attracting and repelling LCSs can act as the flame surface and the mass and energy transport boundary of the reaction products, respectively. In summary, the work presented can provide a new method in combustion controlling to estimate the location of the flame surface and the transport boundaries of species from the velocity field, by using LCSs.