Lagrangian-averaged Vorticity Deviation Research Articles

Lagrangian gradient regression for the detection of coherent structures from sparse trajectory data.

Complex flows are often characterized using the theory of Lagrangian coherent structures (LCS), which leverages the motion of flow-embedded tracers to highlight features of interest. LCS are commonly employed to study fluid mechanical systems where flow tracers are readily observed, but they are broadly applicable to dynamical systems in general. A prevailing class of LCS analyses depends on reliable computation of flow gradients. The finite-time Lyapunov exponent (FTLE), for example, is derived from the Jacobian of the flow map, and the Lagrangian-averaged vorticity deviation (LAVD) relies on velocity gradients. Observational tracer data, however, are typically sparse (e.g. drifters in the ocean), making accurate computation of gradients difficult. While a variety of methods have been developed to address tracer sparsity, they do not provide the same information about the flow as gradient-based approaches. This work proposes a purely Lagrangian method, based on the data-driven machinery of regression, for computing instantaneous and finite-time flow gradients from sparse trajectories. The tool is demonstrated on a common analytical benchmark to provide intuition and demonstrate performance. The method is seen to effectively estimate gradients using data with sparsity representative of observable systems.

Automated cardiac vortex ring identification and characterization based on Recurrent All-Pairs Field Transforms and Lagrangian Averaged Vorticity Deviation

Automated identification of cardiac vortices is a formidable task due to the complex nature of blood flow within the heart chambers. This study proposes a novel approach that algorithmically characterizes the identification criteria of these cardiac vortices based on Lagrangian Averaged Vorticity Deviation (LAVD). For this purpose, the Recurrent All-Pairs Field Transforms (RAFT) is employed to assess the optical flow over the Phase Contrast Magnetic Resonance Imaging (PC-MRI), and to construct a continuous blood flow velocity field and reduce errors that arise from the integral process of LAVD. Additionally, Generalized Hough Transform (GHT) is applied for automated depiction of the structure of cardiac vortices. The effectiveness of this method is demonstrated and validated by the computation of the acquired cardiac flow data. The results of this comprehensive visual and analytical study show that the evolution of cardiac vortices can be effectively described and displayed, and the RAFT framework for optical flow can synthesize the in-between PC-MRIs with high accuracy. This allows cardiologists to acquire a deeper understanding of intracardiac hemodynamics and its impact on cardiac functional performance.

On the evidence of helico-spiralling recirculation within coherent cores of eddies using Lagrangian approach

Oceanic eddies exhibit remarkable coherence and longevity compared to other transient features in the surrounding flow. They possess the ability to transport properties over extensive distances while maintaining their material identity intact. The Lagrangian Coherent Structure (LCS) framework has proven effective in capturing these coherent eddies, where they display a solid-body-like rotation. Although various LCS approaches have been employed to investigate different facets of coherent eddies, a comprehensive understanding of their three-dimensional structures and internal dynamics remains elusive. This study aims to advance our comprehension of coherent eddies’ structural characteristics and delve into the precise nature of their internal dynamics by utilizing the Lagrangian Averaged Vorticity Deviation approach. Two eddies, one cyclonic and the other anti-cyclonic, were chosen from a high-resolution simulation carried out in the Bay of Bengal using the Regional Ocean Modeling System (ROMS). The findings unveil that these eddies have three-dimensional coherent cores resembling gently tapered cones that are broader at the surface and gradually narrow towards the bottom. Intriguingly, the dynamically coherent core of these eddies exhibits simultaneous upwelling and downwelling while maintaining their volumes during advection due to persistent material coherence. The three-dimensional trajectories followed by the fluid parcels inside the coherent core are helical. Their two-dimensional horizontal projections show alternating spiral bands of upwelling and downwelling which are the manifestations of Vortex Rossby Waves. These observations lead to a conceptual framework of a three-dimensional helico-spiralling recirculation pattern within the coherent cores of eddies.

Detection of materially coherent eddies from satellite altimetry in the Bay of Bengal

Mesoscale eddies are a well-recognized feature of the global oceans, being of dynamical relevance to various ocean processes and of significance to transport of various species. In the last few decades, advancements in satellite observations of the sea surface have revealed mesoscale and sub-mesoscale eddy features in many parts of the ocean. An objective definition and detection of eddies has, however, remained a challenge, thus resulting in unreliable estimates of eddy formation, evolution and dissipation. In this study, we present an implementation of a Lagrangian averaged vorticity deviation (LAVD) based method to detect and characterize eddies in the Bay of Bengal (BoB), a freshwater-dominated semi-enclosed basin in the northern Indian Ocean. Systematic sensitivity studies are performed to identify parameter values that result in robust eddy detection in the BoB, and finite-time coherence in material transport associated with the LAVD-based eddies is demonstrated. LAVD-based eddy detection is then used to statistically characterize the Sri Lanka dome (SLD) eddy, an annual mesoscale cyclonic eddy of dynamical and physical importance in the southern BoB. Analysing satellite-based ocean surface currents from 27 years (1993–2019) of data, the inter-annual variability in the SLD formation, its spatio-temporal evolution and dissipation are elucidated. On an average, SLD has a lifetime of 15 weeks and a maximum area of 5 square degrees, with the longest lifetime of 38 weeks in 1996 and a maximum area of 12 square degrees in 1998. Long-lived SLDs are shown to spatially excurse into the BoB, potentially influencing large-scale transport of physical and biogeochemical properties. A potential correlation between large SLD lifetime and large variations in the monthly evolution of Dipole Mode Index and NINO3.4 is highlighted. An SSH-based Eulerian eddy detection method is also implemented to understand its efficacy in identifying materially coherent eddies in the BoB region. The paper concludes with a summary of our results, and a brief discussion of potential applications of LAVD-based eddy detection in the northern Indian Ocean.

Parameter Identification of the Lagrangian-averaged Vorticity Deviation Vortex Detection Method Through the Investigation of Fluid Flow Around Solid Bodies

The main focus of the current paper is the detection of vortices in fluid flow around a circular cylinder and a square cylinder, with an emphasis on the identification of the parameters used for vortex detection. The authors aim to enhance the practicality of an existing vortex detection method (Lagrangian-averaged vorticity deviation) by providing recommendations for the settings of the vortex detection parameters. The simulations were carried out using ANSYS Workbench 2022 R2, encompassing Reynolds numbers between 12 and 140, and angles of incidence from 0° to 45°. The vortex detection was performed using MATLAB R2020b. The paper provides a comprehensive description of the parameters involved in the detection process and their significance, as well as the implementation of the parameter identification. The study results in the determination of the suggested parameter ranges, and a comparative analysis of different vortex detection methods is also presented for the case of the circular cylinder.

Spatiotemporal Characteristics and Volume Transport of Lagrangian Eddies in the Northwest Pacific

Mesoscale eddies play a crucial role in the transport of mass, heat, salt and nutrients, exerting significant influence on ocean circulation patterns, biogeochemical processes and the global climate system. Based on Lagrangian-Averaged Vorticity Deviation (LAVD) method, this study applies 27 years (1993–2019) of geostrophic current velocity data to detect Rotationally Coherent Lagrangian Vortices (RCLVs) in the Northwest Pacific (NWP; 10°N–30°N, 115°E–155°E), with the spatiotemporal characteristics of Eulerian Sea Surface Height Eddies (SSH eddies) and RCLVs being compared. A higher number of SSH eddies and RCLVs can be observed in spring and winter, and their inter-annual variations are similar. SSH eddies show higher generation number and larger radius in the Subtropical Countercurrent region, while RCLVs occur more favorably in the ocean basin. The propagation speed distributions of both eddy types are nearly identical and decrease with increasing latitude. Due to the material coherent transport maintained by RCLVs within a finite time interval, the coherent cores of RCLVs are considerably smaller in scale as compared to those of SSH eddies. The average zonal transports induced by SSH eddies and RCLVs are estimated to be −0.82 Sv and −0.51 Sv (1 Sv = 106 m3/s), respectively. For non-overlapping SSH eddies with RCLVs, approximately 80% of the water within the eddy leaks out during the eddy’s lifespan. In the case of overlapping SSH eddies, the ratio of coherent water inside the eddy decreases with increasing radius, and the leakage rate is around 58%. Finally, an examination of 36 shedding RCLVs events from the Kuroshio near the Luzon Strait, which induce an average zonal transport of −0.14 Sv, reveals that 54% of the water within the shedding RCLVs originates from the Kuroshio.

Qualitative and quantitative analysis of interaction between cavitation patterns and vortices of a pitching hydrofoil from Lagrangian viewpoint

The evolution of transient flow structures and mass transport in cavitating flow around a pitching hydrofoil is investigated qualitatively and quantitatively, and the interaction between cavitation patterns and vortices is elucidated from Lagrangian viewpoint. First, turbulence effects are estimated by the density-corrected k–ω model to account for the local compressibility of the multiphase flow at Reynolds number Re=6.4×105. Then, the formation and evolution of vorticity structures during the whole pitching cycle are analyzed using Lagrangian averaged vorticity deviation method. By comparing the flow structures and hydrodynamic properties at varying angles of attack, the cavitating flow is divided into two distinct stages, namely multi-scale cloud cavitation phase from α+=10° to α−=8°, and traveling sheet cavitation phase from α−=8° to α+=10°. Specifically in cloud cavitation, the formation of the cavitation pattern is closely related to the development of the main vortex. Furthermore, the quantitative analysis method based on Lagrangian flow network is developed to deeply analyze the transport and mixing processes. Importantly, the coherence ratio and the mixing parameter are proposed as transport indicators to precisely quantify the spatial connectivity behavior. Finally, the correlations between vapor fraction, codelength, global coherence ratio and global mixing parameter are evaluated. As the conclusion, it is shown that Lagrangian methods are powerful tool for both qualitative and quantitative analysis, and the results obtained could provide a key and important understanding of the flow structure and changing mechanism between cavitation and vortices in marine hydro and propulsion systems.

A global Lagrangian eddy dataset based on satellite altimetry

Abstract. The methods used to identify coherent ocean eddies are either Eulerian or Lagrangian in nature, and nearly all existing eddy datasets are based on the Eulerian method. In this study, millions of Lagrangian particles are advected by satellite-derived surface geostrophic velocities over the period of 1993–2019. Using the method of Lagrangian-averaged vorticity deviation (LAVD), we present a global Lagrangian eddy dataset (GLED v1.0, Liu and Abernathey, 2022, https://doi.org/10.5281/zenodo.7349753). This open-source dataset contains not only the general features (eddy center position, equivalent radius, rotation property, etc.) of eddies with lifetimes of 30, 90, and 180 d, but also the trajectories of particles trapped by coherent eddies over the lifetime. We present the statistical features of Lagrangian eddies and compare them with those of the most widely used sea surface height (SSH) eddies, focusing on generation sites, size, and propagation speed. A remarkable feature is that Lagrangian eddies are generally smaller than SSH eddies, with a radius ratio of about 0.5. Also, the validation using Argo floats indicates that coherent eddies from GLED v1.0 exist in the real ocean and have the ability to transport water parcels. Our eddy dataset provides an additional option for oceanographers to understand the interaction between coherent eddies and other physical or biochemical processes in the Earth system.

Influence of Caribbean eddies on the Loop current system evolution

The Loop Current (LC) system dynamics are an essential component of the processes influencing circulation and transport in the Gulf of Mexico (GoM). The LC evolution is influenced by various factors, including the rich eddy field of the region and the flow exchange through the Yucatan Strait with the neighboring Caribbean Sea. These factors contribute to the complexity of the LC and, as a result, to the limitations in the predictability of the system. The focus of this study is to further elucidate the evolution of the LC, by quantifying the influence of coherent eddy fluxes originating in the Caribbean Sea. This is achieved by employing the Lagrangian-Averaged Vorticity Deviation (LAVD) method, an objective metric to evaluate eddy coherence in the Caribbean Sea that allows, for the first time, to quantify at different depths the evolution of coherent Caribbean eddies through the Yucatan Channel towards the GoM. The physical connectivity between the Caribbean Sea and the GoM is addressed using Lagrangian techniques to analyze processes that take place south of the Yucatan Channel and help quantify their strong relationship with the GoM eddy field. Coherent anticyclonic vorticity fluxes, as well as the net coherent anticyclonic volume transport between the Caribbean Sea and the GoM are associated with Loop Current Eddy (LCE) detachments through direct connectivity between the coherent Caribbean anticyclones and the forming LCE. The findings have important implications for understanding and predicting the LC system and the physical connectivity processes between the GoM and the Caribbean Sea.

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.

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.

SLA-Based Orthogonal Parallel Detection of Global Rotationally Coherent Lagrangian Vortices

Abstract In this paper, we present a highly effective orthogonal parallel algorithm for identifying global rotationally coherent Lagrangian vortices (RCLVs) in heterogeneous systems and a long-time-scale global sea level anomaly (SLA)-based RCLVs product. First, a many-core parallel computing method is used to accelerate the Lagrangian-averaged vorticity deviation (LAVD) computing process. The computation is approximately 8000 times faster than that of a previous method. Second, the global LAVD field is divided into several regions. These regions are searched with a multiprocess CPU parallel pool to identify simultaneously RCLVs. All the identified RCLVs in these regions are merged seamlessly into a global eddy map. The algorithm improves the global RCLV identification efficiency, making the proposed method approximately 20 times faster than a single-threaded method. The LAVD many-core computing method and the RCLV multiprocess parallel method are orthogonally combined. The resulting algorithm is at least 500 times faster than previous nonparallel methods, making the computing of global RCLVs feasible. Third, the advection of Lagrangian particles in RCLVs and Eulerian eddies is analyzed to demonstrate the material coherence of RCLVs and the reliability of our algorithm. Finally, a global RCLVs product from 1993 to 2019 containing 52 567 eddies is produced with a 90-day time interval. This is the first time that a long-time-scale global Lagrangian eddy product has been presented.

Lagrangian Study of Several Long-Lived Agulhas Rings

AbstractBy using a Lagrangian-averaged vorticity deviation (LAVD)-based vortex detection scheme, rotationally coherent Lagrangian vortices in the South Atlantic Ocean are detected. These vortices act as agents for water transport and can stay coherent in a limited time scale. Our study starts from the life cycle of several long-lived Agulhas rings detected with the LAVD-based vortex detection method. The life cycle of those long-lived Agulhas rings can be separated into two distinct stages: the growing stage and the decaying stage. It is found that at the growing stage, the ambient water spins in and provides effective shielding for the coherent core. The rate of change of material belt width with respect to the detection time scale at the end of the growing stage can represent the decay rate of coherence. We further find a linear relationship between the mean strain rate and the mean square root of kinetic energy (KE1/2). Mean finite-time Lyapunov exponents (FTLE) increase monotonically with the mean strain rate or mean KE1/2. The long existence of the Agulhas rings can be partly attributed to the energetic boundaries around the rings. The ratio of the boundary kinetic energy to the spatial mean kinetic energy (KE/MKE) is also found to be a contributing factor that can influence the lifetime of Agulhas rings. In the retroflection area, the short-lived Agulhas rings might be attributed to the low KE/MKE in this area.

Global Oceanic Mass Transport by Coherent Eddies

Abstract Mesoscale eddies are one of the most prominent processes in the world’s ocean. The eddy-induced transport of water mass, heat, and energy has a great impact on the ocean and atmosphere. The study of global mass transport by mesoscale eddies is important. However, most existing studies have used Eulerian eddy detection methods. Compared with Lagrangian methods, Eulerian methods fail to distinguish the coherent transport from the incoherent transport induced by eddies. Using a Lagrangian-averaged vorticity deviation (LAVD)-based coherent eddy detection method, this study identifies global coherent mesoscale eddies in the upper 1000 m of the ocean. Based on the eddy dataset, the eddy-induced coherent mass transport is calculated. Compared with Eulerian estimates, the Lagrangian results shown in this study are one order of magnitude smaller. This means that roughly only about 10% of eddy-induced global water mass transport is coherent. The cumulative eddy-induced coherent transport across each latitude or longitude is only around 1 Sv (1 Sv ≡ 106 m3 s−1), which is much less than the transport induced by wind-driven gyres and thermohaline circulation.

New Approach for Analyzing Dynamical Processes on the Surface of Photospheric Vortex Tubes

Abstract The majority of studies on multi-scale vortex motions employ a two-dimensional geometry by using a variety of observational and numerical data. This approach limits the understanding the nature of physical processes responsible for vortex dynamics. Here, we develop a new methodology to extract essential information from the boundary surface of vortex tubes. 3D high-resolution magneto-convection MURaM numerical data has been used to analyze photospheric intergranular velocity vortices. The Lagrangian averaged vorticity deviation technique was applied to define the centers of vortex structures and their boundary surfaces based on the advection of fluid elements. These surfaces were mapped onto a constructed envelope grid that allows the study of the key plasma parameters as functions of space and time. Quantities that help in understanding the dynamics of the plasma, e.g., Lorentz force, pressure force, and plasma-β were also determined. Our results suggest that, while density and pressure have a rather global behavior, the other physical quantities undergo local changes, with their magnitude and orientation changing in space and time. At the surface, the mixing in the horizontal direction is not efficient, leading to appearance of localized regions with higher/colder temperatures. In addition, the analysis of the MHD Poynting flux confirms that the majority of the energy is directed in the horizontal direction. Our findings also indicate that the pressure and magnetic forces that drive the dynamics of the plasma on vortex surfaces are unbalanced and therefore the vortices do not rotate as a rigid body.

Automated vortex identification based on Lagrangian averaged vorticity deviation in analysis of blood flow in the atrium from phase contrast MRI

ObjectiveTo present and validate a method for automated identification of the Lagrangian vortices and Eulerian vortices for analyzing flow within the right atrium (RA), from phase contrast magnetic resonance imaging (PC-MRI) data. MethodologyOur proposed algorithm characterizes the trajectory integral associated with vorticity deviation and the spatial mean of vortex rings, for the Lagrangian averaged vorticity deviation (LAVD) based identification and tracking of vortex rings within the heart chamber. For this purpose, the optical flow concept was adopted to interpolate the time frames between larger discrete frames, to minimize the error caused by constructing a continuous velocity field for the integral process of LAVD. Then the Hough transform was used to automatically extract the vortex regions of interest. The computed flow data within the RA of the participants’ hearts was then used to validate the performance of our proposed method. ResultsIn the paper, illustrations are provided for derived evolution of Euler vortices and Lagrangian vortices of a healthy subject. The visualization results have shown that our proposed method can accurately identify the Euler vortices and Lagrangian vortices, in the context of measuring the vorticity and vortex volume of the vortices within the RA chamber. Then the employment of Hough transform-based automated vortex extraction has improved the robustness and scalability of the LAVD in identifying cardiac vortices. The analytical results have demonstrated that the introduction of the Horn-Schunck optical flow can more accurately synthesize the intermediate PC-MRI to construct a continuous velocity field, compared with other interpolation methods. ConclusionA novel analytical framework has been developed to accurately identify the flow vortices in the RA chamber based on Horn-Schunck optical flow and Hough transform. From the obtained analytical study results, the development and changes of dominant vortices within this cardiac chamber during the cardiac cycle can be acquired. This can provide to cardiologists a deeper understanding of the hemodynamics within the heart chambers.

A Hybrid Approach for Cardiac Blood Flow Vortex Ring Identification Based on Optical Flow and Lagrangian Averaged Vorticity Deviation.

Objective: The measurement of cardiac blood flow vortex characteristics can help to facilitate the analysis of blood flow dynamics that regulates heart function. However, the complexity of cardiac flow along with other physical limitations makes it difficult to adequately identify the dominant vortices in a heart chamber, which play a significant role in regulating the heart function. Although the existing vortex quantification methods can achieve this goal, there are still some shortcomings: such as low precision, and ignoring the center of the vortex without the description of vortex deformation processes. To address these problems, an optical flow Lagrangian averaged vorticity deviation (Optical flow-LAVD) method is proposed.Methodology: We examined the flow within the right atrium (RA) of the participants’ hearts, by using a single set of scans pertaining to a slice at two-chamber short-axis orientation. Toward adequate extraction of the vortex ring characteristics, a novel approach driven by the Lagrangian averaged vorticity deviation (LAVD) was implemented and applied to characterize the trajectory integral associated with vorticity deviation and the spatial mean of rings, by using phase-contrast magnetic resonance imaging (PC-MRI) datasets as a case study. To interpolate the time frames between every larger discrete frame and minimize the error caused by constructing a continuous velocity field for the integral process of LAVD, we implemented the optical flow as an interpolator and introduced the backward warping as an intermediate frame synthesis basis, which is then used to generate higher quality continuous velocity fields.Results: Our analytical study results showed that the proposed Optical flow-LAVD method can accurately identify vortex ring and continuous velocity fields, based on optical flow information, for yielding high reconstruction outcomes. Compared with the linear interpolation and phased-based frame interpolation methods, our proposed algorithm can generate more accurate synthesized PC-MRI.Conclusion: This study has developed a novel Optical flow-LAVD model to accurately identify cardiac vortex rings, and minimize the associated errors caused by the construction of a continuous velocity field. Our paper presents a superior vortex characteristics detection method that may potentially aid the understanding of medical experts on the dynamics of blood flow within the heart.

Lagrangian-averaged vorticity deviation of spiraling blood flow in the heart during isovolumic contraction and ejection phases.

The formation of vortex rings in the left ventricular (LV) blood flow is a mechanism for optimized blood transport from the mitral valve inlet to aortic valve outlet, and the vorticity is an important measure of a well-functioning LV. However, due to lack of quantitative methods, the process of defining the boundary of a vortex in the LV and identifying the dominant vortex components has not been studied previously. The Lagrangian-averaged vorticity deviation (LAVD) can enable us to compute the trajectory integral of the normed difference of the vorticity from its spatial mean. Therefore, in this work, we have employed LAVD to identify the Lagrangian vortices and Eulerian vortices for measuring the vortex volume and vorticity in the LV blood flow. We found that during the LV ejection period, the positive (counterclockwise) and negative (clockwise) vorticity of patients are consistently stronger than those of the healthy groups, and the counterclockwise vortex volume of healthy groups (0.84+0.26ml) is greater than that of patients (0.55+0.28ml) during the pre-ejection period. Then, during the middle ejection phase, the counterclockwise vortex ring volume of patients (1.89+0.36ml) exceeds that of healthy groups (1.38+0.43ml). Finally, during the end-ejection period, the counterclockwise vortex ring volume of healthy subjects (0.61+0.17ml) is the same as that of patients (0.60+0.19ml). The results presented in this paper can provide new insights into the blood flow patterns within the LV. It can accurately indicate the role of vortices and vorticity values in intra-LV flow, and portray how cardiomyopathy (and its distorted contractile mechanism) can affect intra-LV flow patterns and mitigate adequate LV outflow.

Study on dynamics of vortices in dynamic stall of a pitching airfoil using Lagrangian coherent structures

The evolution of vortex structure and mass transport process during dynamic stall of the two-dimensional pitching NACA0012 airfoil are studied in detail, using Lagrangian coherent structures (LCSs), and the nature of dynamic stall is given from viewpoint of nonlinear dynamics. First, the numerical method with SST k−ω turbulence model is used to simulate the flow field around the pitching oscillation airfoil. Then, the Lagrangian-averaged vorticity deviation (LAVD) method is used to analyze the formation and evolution of vortex structure during dynamic stall. Further, the circulation development process of the primary leading edge vortex (LEV) is studied, with comparison between LAVD method with Q criterion. Finally, the dynamic behaviors, such as the mass transport and evolution of flow structure during the process of dynamic stall, are analyzed in depth by using LCSs, and the fluctuation of lift coefficient and the dynamic stall are investigated from the viewpoint of nonlinear dynamics. The results show that the evolution of vortex structure and mass transport are closely related to the fluctuation of lift coefficient during the dynamic stall, and the low-pressure region formed by vortices and the complex mass transport and mixing process on the suction surface of the airfoil are the main reasons for the high lift during the dynamic stall process. In particular, the primary LEV, which can be accurately identified and tracked by LAVD method, plays a very important role in lift enhancement, and the primary LEV and the secondary LEV form and grow in turn by the intermittent feeding process of shear layer. Additionally, the fluid particles in front of the leading edge of the airfoil enter the LEV through a special mass transport channel, which is textured by LCSs and opened and closed intermittently. More importantly, the saddle point of LCSs is an important component of vortex boundary, and its dynamic behaviors represent the behaviors of vortex. The first separation of saddle point from the shear layer at leading edge indicate the separation of primary LEV from the suction surface of the airfoil, and the occurrence of dynamic stall. The second separation of saddle point from the shear layer at leading edge indicate that primary LEV begins to leave the upper region of the airfoil, corresponding to a sharp drop in lift coefficient. Generally, compared to the traditional visualization techniques, the Lagrangian analysis based on LCSs and LAVD can provide a deeper insight into the dynamics of the vortex in flow field around pitching oscillation airfoil and the dynamic mechanism of dynamic stall.

Lagrangian chaotic saddles and objective vortices in solar plasmas

We report observational evidence of Lagrangian chaotic saddles in plasmas, given by the intersections of finite-time unstable and stable manifolds, using an ≈22h sequence of spacecraft images of the horizontal velocity field of solar photosphere. A set of 29 persistent objective vortices with lifetimes varying from 28.5 to 298.3min are detected by computing the Lagrangian averaged vorticity deviation. The unstable manifold of the Lagrangian chaotic saddles computed for ≈11h exhibits twisted folding motions indicative of recurring vortices in a magnetic mixed-polarity region. We show that the persistent objective vortices are formed in the gap regions of Lagrangian chaotic saddles at supergranular junctions.