Studies on Typical Stalls of Airfoil at Low Reynolds Number Using Lagrangian Coherent Structures

In this work, we introduce a scalable and efficient GPU-accelerated methodology for volumetric particle advection and finite-time Lyapunov exponent (FTLE) calculation, focusing on the analysis of Lagrangian coherent structures (LCS) in large-scale direct numerical simulation (DNS) datasets across incompressible, supersonic, and hypersonic flow regimes. LCS play a significant role in turbulent boundary layer analysis, and our proposed methodology offers valuable insights into their behavior in various flow conditions. Our novel owning-cell locator method enables efficient constant-time cell search, and the algorithm draws inspiration from classical search algorithms and modern multi-level approaches in numerical linear algebra. The proposed method is implemented for both multi-core CPUs and Nvidia GPUs, demonstrating strong scaling up to 32,768 CPU cores and up to 62 Nvidia V100 GPUs. By decoupling particle advection from other problems, we achieve modularity and extensibility, resulting in consistent parallel efficiency across different architectures. Our methodology was applied to calculate and visualize the FTLE on four turbulent boundary layers at different Reynolds and Mach numbers, revealing that coherent structures grow more isotropic proportional to the Mach number, and their inclination angle varies along the streamwise direction. We also observed increased anisotropy and FTLE organization at lower Reynolds numbers, with structures retaining coherency along both spanwise and streamwise directions. Additionally, we demonstrated the impact of lower temporal frequency sampling by upscaling with an efficient linear upsampler, preserving general trends with only 10% of the required storage. In summary, we present a particle search scheme for particle advection workloads in the context of visualizing LCS via FTLE that exhibits strong scaling performance and efficiency at scale. Our proposed algorithm is applicable across various domains, requiring efficient search algorithms in large, structured domains. While this article focuses on the methodology and its application to LCS, an in-depth study of the physics and compressibility effects in LCS candidates will be explored in a future publication.

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.

Microfluidics offers an effective means to carry out a wide range of transport processes within a controlled microenvironment by drawing on the benefits imparted by increasing surface area to volume ratio at the microscale. Critical to the impact of microfluidics on integrated devices in the fields of bioengineering and biomedicine is the ability to transport fluids and biomolecules effectively particularly at the size scales involved. In this context a bio-inspired pumping mechanism, the valveless impedance pump, was explored for applications in microfluidics ranging from micro total analysis systems to microchannel cooling. Adhering to the basic principles of the impedance pump mechanism, pumps have been constructed at a variety of size scales from a few centimeters to a few hundred microns. The micro impedance pump is valveless, bidirectional, and can be constructed simply from a wide range of materials. Depending on the size of the pump flow rates range from nL/min to mL/min and pressures can be generated that exceed 20 kPa. Another benefit of the impedance pump is the pulsatile flow output which can be used in the context of microfluidic applications to enhance transport at low Reynolds numbers as well as metering in drug delivery. Pulsatile flow was therefore investigated as a method of augmenting transport in microfluidic systems. Micro PIV was used to study the affect of both steady and pulsatile flows on transport at low Reynolds number was examined in microscale rectangular cavities. Ventilation of the cavity contents was examined in terms of the residence time or average time a particle remains in the cavity region. Lagrangian coherent structures (LCS) were applied to empirical velocity fields to determine the impact of unsteadiness on time dependent boundaries to fluid transport present in the flow. Experimental results show that there are both frequencies which are beneficial and detrimental to cavity ventilation as well as certain frequencies which more evenly distribute particles originating in the cavity throughout the freestream.

An integrated simulation of a Drosophila wing–body combination in hovering flight has been carried out in order to analyze the Lagrangian and Eulerian coherent structures. A parallel unstructured finite volume method based on an arbitrary Lagrangian–Eulerian (ALE) formulation has been initially validated for a flapping rectangular plate and then employed to solve the incompressible unsteady Navier–Stokes equations around a Drosophila wing–body combination. A robust mesh deformation algorithm based on indirect radial basis function method is utilized at each time level while avoiding remeshing. The time variation of the three-dimensional Eulerian coherent structures around a flapping Drosophila in hovering flight is analyzed in the near wake with $$\lambda _{2}$$ -criterion. Meanwhile, the Lagrangian coherent structures are investigated using finite-time Lyapunov exponent fields. In addition, the instantaneous velocity vectors and particle traces are presented along with the aerodynamic parameters including the force, moment and power for a wing–body combination. Furthermore, a wing-only configuration is also investigated in order to show the body effects on aerodynamic loads. The numerical simulations are used to gain insight into the near wake topology as well as their correlations with the aerodynamic force generation. The present fully coupled ALE algorithm is shown to be sufficiently robust to deal with large mesh deformations seen in flapping wings and reveals highly detailed near wake topology which is very useful to study physics in biological flights and can also provide an effective tool for designing bio-inspired MAVs and MFIs.

In dynamical systems, regions containing similar dynamics can be identified and their boundaries visualized using Lagrangian Coherent Structures (LCS). In fluid flows, LCS can be approximated as virtual boundaries dictating the structure of flow. Current study makes use of LCS to simulate two cases for fluid flows; flow features around the turbine blade and a ventilation system. Detection of LCS will present a useful technique of providing an insight into coherent structures on the suction surface and vortex formation downstream of the blade. The performance of the low pressure turbine is significantly affected by the laminar separation bubble that forms on the suction surface and it is hence very important to capture the details of the associated flow features. This information can in turn help in the examination of procedures developed for their control. A T106 LP turbine blade is selected for the purpose of present study. A numerical simulation is performed using Transition SST model. The results of the computational study are validated using experimental and other computational studies. Information from the CFD study is then used to generate LCS. Different flow features are indentified and their evolution with time is observed. LCS help describe the process of vortex formation and are related to the flow topology. LCS associated with the separation bubble are discussed. A second case of ventilation inside a two dimensional domain is considered. LCS are used to depict the effect of ventilation systems with changing inlet air velocity angles. Study of ventilation using LCS show the patterns of fluid flow inside the domain and can be useful in design of HVAC systems by identifying real time location of virtual walls inside the domain.

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.

In dynamical systems, regions containing similar dynamics can be identified and their boundaries visualized using Lagrangian Coherent Structures (LCS). In fluid flows, LCS can be approximated as virtual boundaries dictating the structure of flow. Current study makes use of LCS to simulate two cases for fluid flows; flow features around the turbine blade and a ventilation system. Detection of LCS will present a useful technique of providing an insight into coherent structures on the suction surface and vortex formation downstream of the blade. The performance of the low pressure turbine is significantly affected by the laminar separation bubble that forms on the suction surface and it is hence very important to capture the details of the associated flow features. This information can in turn help in the examination of procedures developed for their control. A T106 LP turbine blade is selected for the purpose of present study. A numerical simulation is performed using Transition SST model. The results of the computational study are validated using experimental and other computational studies. Information from the CFD study is then used to generate LCS. Different flow features are indentified and their evolution with time is observed. LCS help describe the process of vortex formation and are related to the flow topology. LCS associated with the separation bubble are discussed. A second case of ventilation inside a two dimensional domain is considered. LCS are used to depict the effect of ventilation systems with changing inlet air velocity angles. Study of ventilation using LCS show the patterns of fluid flow inside the domain and can be useful in design of HVAC systems by identifying real time location of virtual walls inside the domain.

Lagrangian fronts (LFs) in the ocean are defined as boundaries between surface waters with strongly different values of Lagrangian indicators. They can be accurately detected in a given velocity field by computing different Lagrangian indicators for synthetic tracers. We report here our results on connection of the LFs with fishing grounds and catch locations. Imposing on the altimetry-based Lagrangian maps available catch data on Pacific saury and neon flying squid in the region of the North Western Pacific with one of the richest fisheries in the world, it is shown that the LFs could serve as a new indicator for potential fishing grounds. It is shown statistically that the catch locations are not randomly distributed over the region but concentrated mainly along the strong LFs where productive cold waters of the Oyashio Current, warmer waters of the southern branch of the Soya Current, and waters of warm-core Kuroshio eddies converge. Electronic tagging and tracking of marine animals and seabirds provides a new source of information on their foraging behavior and its relationship with the marine environment. It is discussed how some top marine predators as great frigates, elephant seals, and Mediterranean fin whales could be able to track somehow mesoscale and submesoscale Lagrangian coherent structures (LCSs) and use them in the foraging strategy and to feed on. Possible biophysical reasons for aggregation of food near strong LFs and LCSs are discussed.

<p>Lagrangian particle tracking and associated diagnostics may be used to examine advective pathways of material and to identify coherent structures in the flow.  Lagrangian coherent structures are material transport barriers and act to separate different flow regimes.</p><p>The drift of the International Multidisciplinary Observatory for the Study of Arctic Climate (MOSAiC) expedition onboard R/V Polarstern began in October 2019 and will continue for the full year.   Our study has the goals to (i) characterise advective pathways and (ii) examine potential predictability of the MOSAiC drift.  Eddies, jets and boundary currents feature large spatiotemporally varying velocity gradients.  Since operational ocean forecasts have a limited time horizon (~weeks), we focused on hindcast to examine typical sea ice/ocean circulation scenarios for 2005-15.  We applied off-line ARIANE particle tracking in an eddying 1/12 deg. global NEMO sea ice-ocean model to estimate the most likely drift pathways. </p><p>Over 10,000 trajectories were initialised in October each year, started at the best estimated MOSAiC location, advected for one year and analysed for key coherent drift structures.  The advection and deformation of the initial particle cluster provided information about MOSAiC drift predictability, but also elucidated transport processes of the biogeochemical tracers, such as nutrients and carbon, and spread of pollution and microplastics. We analysed observations from a newly curated dataset of the Arctic to examine various watermass properties, their origin, fate and connectivity.</p><p>The MOSAiC surface drift trajectories depend on release time and location, but to leading-order, they are governed by the interannual variability of the wind and of the underlying ocean circulation.  Mesoscale flow deformation is linked to a spreading of the cluster of particles and is associated with reduced potential predictability of separation of particles within the cluster (~ 450 km after 12 months).  Gyre-scale flow affects the ensemble drift path over long times and influences whether particular coherent structures are encountered by the particles, their location and strength (in terms of velocity magnitude and gradient).  Saddle-type structures play a major role in bifurcation of particle trajectories.  In the examples studied, saddles north of Nares Strait, near Northern Greenland and Northern Iceland, topologically associated with streamline connectivity between gyres, coastal boundary currents and inflow/outflow at the Arctic gateways, were significant.  On seasonal-interannual scales, the position and strength of the Beaufort Gyre, as well as an anomalous cyclonic gyre in the eastern basin, affected both the ensemble drift path and the coherent flow structures.</p><p>The variability of ensemble drift path, cluster deformation and coherent flow structures across the full Arctic basin were often very different from the climatological advective behaviour of Trans-Polar Drift.  For estimation of advective pathways and sea ice drift it is important to consider the varying flow from gyre-scale to mesoscale, where velocity gradients are large, and to identify robust Lagrangian measures for steady features.</p><p>The study is supported from NE/R012865/1 (APEAR), part of the Changing Arctic Ocean programme, jointly funded by the UKRI Natural Environment Research Council (NERC) and the German Federal Ministry of Education and Research (BMBF).</p><div> </div>

Paddling motion of a free-swimming jellyfish and Lagrangian coherent structure analysis

The lobe dynamics andmass transport between separation bubble and main flow in flow over airfoil are studied in detail, using Lagrangian coherent structures (LCSs), in order to understand the nature of evolution of the separation bubble. For this problem, the transient flow over NACA0012 airfoil with low Reynolds number is simulated numerically by characteristic based split (CBS) scheme, in combination with dual time stepping. Then, LCSs and lobe dynamics are introduced and developed to investigate themass transport between separation bubble and main flow, from viewpoint of nonlinear dynamics. The results show that stable manifolds and unstable manifolds could be tangled with each other as time evolution, and the lobes are formed periodically to induce mass transport between main flow and separation bubble, with dynamic behaviors. Moreover, the evolution of the separation bubble depends essentially on the mass transport which is induced by lobes, ensuing energy and momentum transfers. As the results, it can be drawn that the dynamics of flow separation could be studied using LCSs and lobe dynamics, and could be controlled feasibly if an appropriate control is applied to the upstream boundary layer with high momentum.

The method of using Finite Time Liapunov Exponents (FTLE) to extract Lagrangian Coherent Structures (LCS) in aperiodic flows, as originally developed by Haller, is applied to geophysical flows, and flows in the phase space of second order dynamical systems. In this approach, the LCS are identified as surfaces of greatest separation that parse the flow into regions with different dynamical behavior. In this way, the LCS reveal the underlying skeleton of turbulence. The time-dependence of the LCS provides insight into the mechanisms by which fluid is transported from one region to another. Of especial interest in this study, is the utility with which the FTLE-LCS method can be used to reveal homoclinic and horseshoe dynamics in aperiodic flows. The FTLE-LCS method is applied to turbulent flow in hurricanes and reveals LCS that delineate sharp boundaries to a storm. Moreover, intersections of the LCS define lobes that mediate transport into and out of a storm through the action of homoclinic lobe dynamics. Using FTLE-LCS, the same homoclinic structures are seen to be a dominant transport mechanism in the Global Ocean, and provide insights into the role of mesoscale eddies in enhancing lateral mixing. Beyond geophysical flows, we also study transport in the phase space of a coupled oscillator model for biomolecules. Before we can analyze transport in this model, we first introduce an appropriate model reduction that captures the relevant statistics of the full system. In the reduced model, we see that transport is again mediated by the process of horseshoe dynamics in a perturbed homoclinic tangle. We also consider some theoretical aspects of FTLE-LCS, including the relationship between LCS and stable/unstable manifolds, the invariance of LCS, and the possibility of an evolution equation describing the motion of the LCS. A parallelized software for computing FTLE is also introduced.

This theoretical paper discusses recent advances in the fluid dynamics of insect and micro air vehicle (MAV) flight and considers theoretical analyses necessary for their future development. The main purpose is to propose a new conceptual framework and, within this framework, two analytic approaches to aerodynamic modelling of an insect-like flapping wing in hover in the context of MAVs. The motion involved is periodic and is composed of two half-cycles (downstroke and upstroke) which, in hover, are mirror images of each other. The downstroke begins with the wing in the uppermost and rearmost position and then sweeps forward while pitching up and plunging down. At the end of the half-cycle, the wing flips, so that the leading edge points backwards and the wing's lower surface becomes its upper side. The upstroke then follows by mirroring the downstroke kinematics and executing them in the opposite direction. Phenomenologically, the interpretation of the flow dynamics involved, and adopted here, is based on recent experimental evidence obtained by biologists from insect flight and related mechanical models. It is assumed that the flow is incompressible, has low Reynolds number and is laminar, and that two factors dominate: (i) forces generated by the bound leading-edge vortex, which models flow separation; and (ii) forces due to the attached part of the flow generated by the periodic pitching, plunging and sweeping. The first of these resembles the analogous phenomenon observed on sharp-edged delta wings and is treated as such. The second contribution is similar to the unsteady aerodynamics of attached flow on helicopter rotor blades and is interpreted accordingly. Theoretically, the fluid dynamic description is based on: (i) the superposition of the unsteady contributions of wing pitching, plunging and sweeping; and (ii) adding corrections due to the bound leading-edge vortex and wake distortion. Viscosity is accounted for indirectly by imposing the Kutta condition on the trailing edge and including the influence of the vortical structure on the leading edge. Mathematically, two analytic approaches are proposed. The first derives all the quantities of interest from the notion of circulation and leads to tractable integral equations. This is an application of the von Kármán-Sears unsteady wing theory and its nonlinear extensions due to McCune and Tavares; the latter can account for the bound leading-edge vortex and wake distortion. The second approach uses the velocity potential as the central concept and leads to relatively simple ordinary differential equations. It is a combination of two techniques: (i) unsteady aerodynamic modelling of attached flow on helicopter rotor blades; and (ii) Polhamus's leading-edge suction analogy. The first of these involves both frequency-domain (Theodorsen style) and time-domain (indicial) methods, including the effects of wing sweeping and returning wake. The second is a nonlinear correction accounting for the bound leading-edge vortex. Connections of the proposed framework with control engineering and aeroelasticity are pointed out.

In this study, mass transport between separation bubbles and the flow around a two-dimensional airfoil are numerically investigated using Lagrangian Coherent Structures (LCSs). Finite Time Lyapunov Exponent (FTLE) technique is used for the computation to identify invariant manifolds and LCSs. Moreover, the Characteristic Base Split (CBS) scheme combined with dual time stepping technique is applied to simulate such transient flow at low Reynolds number. We then investigate the evolution of vortex structures during the transport process with the aid of LCSs. To explore the vortex formation at the surface of the airfoil, the dynamics of separatrix is also taken into account which is formed by the combination of stable (unstable) manifold. The Lagrangian analysis gives a detailed understanding of vortex dynamics and separation bubbles which plays a significant role to explore the performance of the unsteady flow generated by an airfoil. Transport process and flow separation phenomena are studied extensively to analyze the flow pattern by the Lagrangian point of view.

This work explores the utility of the finite-time Lyapunov exponent (FTLE) field forrevealing flow structures in low Reynolds number biological locomotion. Previous studies ofhigh Reynolds number unsteady flows have demonstrated that ridges of the FTLEfield coincide with transport barriers within the flow, which are not shown by amore classical quantity such as vorticity. In low Reynolds number locomotion (O(1)–O(100)), in which viscous diffusion rapidly smears the vorticity in the wake, the FTLEfield has the potential to add new insight to locomotion mechanics. The targetof study is an articulated two-dimensional model for jellyfish-like locomotion,with swimming Reynolds number of order 1. The self-propulsion of the model isnumerically simulated with a viscous vortex particle method, using kinematicsadapted from previous experimental measurements on a live medusan swimmer. Theroles of the ridges of the computed forward- and backward-time FTLE fields areclarified by tracking clusters of particles both backward and forward in time. It isshown that a series of ridges in front of the jellyfish in the forward-time FTLEfield transport slender fingers of fluid toward the lip of the bell orifice, which arepulled once per contraction cycle into the wake of the jellyfish, where the fluidremains partitioned. A strong ridge in the backward-time FTLE field reveals apersistent barrier between fluid inside and outside the subumbrellar cavity. Thesystem is also analyzed in a body-fixed frame subject to a steady free stream,and the FTLE field is used to highlight differences in these frames of reference.