Lagrangian Vortex Research Articles

Wall-bounded flow simulation on vortex dynamics

Vortical flow animation has attracted considerable attention within the realm of computer graphics. Given that boundaries are the source of vorticity, we introduce a novel approach for the wall-bounded flow simulation in the vortex dynamics framework. We enhance the traditional Lagrangian vortex particle method in terms of boundary treatment and viscosity computation for boundary layer to furnish support for the simulation of the vortex shedding phenomenon behind the moving solids. We extend the boundary treatment strategy based on the idea of vortex generation with a reasonable placement algorithm for the generated vortex element to better satisfy the impermeability of solids. Furthermore, we modify the particle strength exchange method at solid boundaries to capture momentum transfer of moving solids. We demonstrate the efficacy of our approach by simulating a series of wall-bounded flows, such as the wake behind a delta wing and the vortex shedding behind the rotating sphere.

LES on wind turbines by comparison of Vortex Particle Method and Finite Volume Method codes

The aim of this paper is to compare two different computational methods that analyse the flow around wind turbine using Large-Eddy-Simulations (LES). The first method uses a three-dimensional unsteady Lagrangian Vortex Particle method (VP) associated to a lifting-line (LL) approach providing radial loads, integrated thrust and power coefficients, circulations and angle of attack across the turbine blades. The second method solves the incompressible Navier-Stokes equations with the classical Finite Volume (FV) method coupled to the Actuator Line (AL) method to model the rotor blades. Both methods are compared by means of a benchmark consisting in a single full-scale NREL5MW wind turbine and the results are thus compared to Martínez-Tossas et al (2018). The comparisons involve loads, angle of attack and velocity along the blade. Wakes downstream of the turbine are analysed via the evolution of flow fields as mean velocity and vorticity. Results are found to be in good agreement with the verification case. A comparison is also performed on the evaluation of the numerical cost, precision and efficiency of both computational methods.

Influence of wakes interaction and upstream turbulence on three tidal turbines behaviour

The current study presents numerical results on three tidal turbine models (two in front, one downstream) interacting in a turbulent upstream flow. The numerical results come from a lifting-line (LL) embedded in a Lagrangian vortex particle (VP) solver: Dorothy LL-VP. The objective is to assess the extent to which this numerical tool is suited to reproduce accurately wakes interaction as well as fluctuating loads perceived by the downstream turbine. To this aim, the numerical set-up reproduces an experimental campaign led at IFREMER’s wave and current flume tank. The downstream turbine is placed at different positions to change the wakes interaction. Two upstream turbulence intensities (TI) experimentally tested are reproduced numerically using the synthetic eddy method (SEM). Favourable comparisons are obtained between numerical and experimental wakes, including velocity profiles. Preliminary results suggest that the downstream turbine performance decrease is numerically well captured. More investigations are needed on the loads fluctuations with longer computation time, and adding an angular velocity controller as well as hub modelling to Dorothy LL-VP.

Enhancing transport barriers with swimming micro-organisms in chaotic flows

We investigate the effects of bacterial activity on the mixing and transport properties of a passive scalar in time-periodic flows in experiments and in a simple model. We focus on the interactions between swimming Escherichia coli and the Lagrangian coherent structures (LCSs) of the flow, which are computed from experimentally measured velocity fields. Experiments show that such interactions are non-trivial and can lead to transport barriers through which the scalar flux is significantly reduced. Using the Poincaré map, we show that these transport barriers coincide with the outermost members of elliptic LCSs known as Lagrangian vortex boundaries. Numerical simulations further show that elliptic LCSs can repel elongated swimmers and lead to swimmer depletion within Lagrangian coherent vortices. A simple mechanism shows that such depletion is due to the preferential alignment of elongated swimmers with the tangents of elliptic LCSs. Our results provide insights into understanding the transport of micro-organisms in complex flows with dynamical topological features from a Lagrangian viewpoint.

Lagrangian dynamics and regularity of the spin Euler equation

We derive the spin Euler equation for ideal flows by applying the spherical Clebsch mapping. This equation is based on the spin vector, a unit vector field encoding vortex lines, instead of the velocity. The spin Euler equation enables a feasible Lagrangian study of fluid dynamics, as the isosurface of a spin-vector component is a vortex surface and material surface in ideal flows. We establish a non-blowup criterion for the spin Euler equation, suggesting that the Laplacian of the spin vector must diverge if the solution forms a singularity at some finite time. The direct numerical simulations (DNS) of three ideal flows – the vortex knot, the vortex link and the modified Taylor–Green flow – are conducted by solving the spin Euler equation. The evolution of the Lagrangian vortex surface illustrates that the regions with large vorticity are rapidly stretched into spiral sheets. The DNS result exhibits a pronounced double-exponential growth of the maximum norm of Laplacian of the spin vector, showing no evidence of the finite-time singularity formation if the double-exponential growth holds at later times. Moreover, the present criterion with Lagrangian nature appears to be more sensitive than the Beale–Kato–Majda criterion in detecting the flows that are incapable of producing finite-time singularities.

Development and validation of a lifting‐line code associated with the vortex particle method software Dorothy

AbstractThis paper presents a lifting‐line implementation in the framework of a Lagrangian vortex particle method (LL‐VP). The novelty of the present implementation lies in the fluid particles properties definition and in the particles shedding process. In spite of mimicking a panel method, the LL‐VP needs some peculiar treatments described in the paper. The present implementation converges rapidly and efficiently during the shedding sub‐iteration process. This LL‐VP method shows good accuracy, even with moderate numbers of sections. Compared to its panel or vortex filaments counterparts, more frequently encountered in the literature, the present implementation inherently accounts for the diffusion term of the Navier‐Stokes equations, possibly with a turbulent viscosity model. Additionally, the present implementation can also account for more complex onset flows: upstream ambient turbulence and upstream turbine wakes. After validation on an analytical elliptic wing configuration, the model is tested on the Mexnext‐III wind turbine application, for three reduced velocities. Accurate results are obtained both on the analytical elliptic wing and on the New MEXICO rotor cases in comparison with other similar numerical models. A focus is made on the Mexnext‐III wake analysis. The numerical wake obtained with the present LL‐VP is close to other numerical and experimental results. Finally, a last configuration with three tidal turbines in interaction is considered based on an experimental campaign carried out at the IFREMER wave and current flume tank. Enhanced turbine‐wake interactions are highlighted, with favourable comparisons with the experiment. Hence, such turbine interactions in a farm are accessible with this LL‐VP implementation, be it wind or tidal energy field.

Effect of Surface Roughness on Aerodynamic Loads of Bluff Body in Vicinity of Smoothed Moving Wall

This paper contributes to a new Lagrangian vortex method for the statistical control of turbulence in two-dimensional flow configurations around a rough circular cylinder in ground effect when considering higher subcritical Reynolds numbers, namely 3 × 104 ≤ Re ≤ 2 × 105. A smoothed moving wall (active control technique) is used to include the blockage effect in association with the variation in cylinder surface roughness (passive control technique), characterizing a hybrid approach. In contrast with the previous approaches of our research group, the rough cylinder surface is here geometrically constructed, and a new momentum source term is introduced and calculated for the investigated problem. The methodology is structured by coupling the random Discrete Vortex Method, the Lagrangian Dynamic Roughness Model, and the Large Eddy Simulation with turbulence closure using the truncated Second-Order Velocity Structure Function model. This methodological option has the advantage of dispensing with the use of both a refined near-wall mesh and wall functions. The disadvantage of costly processing is readily solved with Open Multi-Processing. The results reveal that intermediate and high roughness values are most efficient for Reynolds numbers on the orders of 105 and 104, respectively. In employing a moving wall, the transition from the large-gap to the intermediate-gap regime is satisfactorily characterized. For the conditions studied with the hybrid technique, it was concluded that the effect of roughness is preponderant and acts to anticipate the characteristics of a lower gap-to-diameter ratio regime, especially with regard to intermittency.

Objective Lagrangian Vortex Cores and their Visual Representations.

The numerical extraction of vortex cores from time-dependent fluid flow attracted much attention over the past decades. A commonly agreed upon vortex definition remained elusive since a proper vortex core needs to satisfy two hard constraints: it must be objective and Lagrangian. Recent methods on objectivization met the first but not the second constraint, since there was no formal guarantee that the resulting vortex coreline is indeed a pathline of the fluid flow. In this paper, we propose the first vortex core definition that is both objective and Lagrangian. Our approach restricts observer motions to follow along pathlines, which reduces the degrees of freedoms: we only need to optimize for an observer rotation that makes the observed flow as steady as possible. This optimization succeeds along Lagrangian vortex corelines and will result in a non-zero time-partial everywhere else. By performing this optimization at each point of a spatial grid, we obtain a residual scalar field, which we call vortex deviation error. The local minima on the grid serve as seed points for a gradient descent optimization that delivers sub-voxel accurate corelines. The visualization of both 2D and 3D vortex cores is based on the separation of the movement of the vortex core and the swirling flow behavior around it. While the vortex core is represented by a pathline, the swirling motion around it is visualized by streamlines in the correct frame. We demonstrate the utility of the approach on several 2D and 3D time-dependent vector fields'.

Coupling of OpenFOAM with a Lagrangian vortex particle method for external aerodynamic simulations

In the field of computational aerodynamics, it is vital to develop tools that can accurately, but also efficiently, simulate the flow around bluff objects and calculate the aerodynamic forces acting on them. When strong body–vortex interactions take place, the simulations become more demanding, since complex phenomena appear. To address this issue, hybrid Eulerian–Lagrangian solvers have been developed and are increasingly used in the field. In this paper, a Vortex Particle Method (VPM) is coupled with the OpenFOAM software. The Eulerian solver (OpenFOAM) resolves the regions close to the solid boundaries, while the vortex particles evolve the wake downstream, significantly reducing artificial diffusion. The coupling strategy and the validation results of a hybrid code based on the domain decomposition technique are presented. This work is the first to couple OpenFOAM with a Lagrangian solver in the framework of a hybrid solver. Our objective is twofold: to verify the capability of OpenFOAM to run with a VPM and to validate the hybrid solver using benchmark cases. We demonstrate the validation of the solver on the Lamb–Oseen vortex case, the dipole case in the unbounded domain, and the flow around a cylinder at Re = 550. Our results show that coupling OpenFOAM with a VPM can be achieved without complications and efficiently reproduces the results of pure Eulerian simulations.

Evolutionary Airfoil Shape Optimization Coupling Parameterization Methods with Lagrangian Vortex Method

In the present paper, the evolutionary algorithm for single-objective optimization is developed using a genetic algorithm and employing polynomial-based (PARSEC) and radial basis function (RBF) functions for NACA 2412 airfoil parameterization. The determination of the objective functions, being aerodynamic coefficients, are performed using the Lagrangian vortex particle method. The results of the lift coefficient at the wide range of angle of attack using the vortex particle method shows a good agreement with experimental data listed in the literature. For the optimization results, the lift coefficient obtained from the PARSEC method is optimized to be larger for the whole range of angle of attack; while it still keeps the stall region at the upper surface of the airfoil to be the same as that of the original airfoil. In addition, the RBF method illustrates the lift coefficients larger at the range of angle of attack from -50 to 140 but stall occurs earlier than the original airfoil.

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.

Singularity resolving in solution of the boundary integral equation in two-dimensional vortex methods

The problem of 2D incompressible flow simulation around airfoils with sharp edges and corner point is considered. The solution of the boundary integral equation with respect to vortex sheet intensity arising in Lagrangian vortex method has weak singularity that cannot be resolved correctly in the framework of the existing Galerkin-type numerical schemes. It is shown that for piecewise-smooth bounded solutions the known schemes allow for solution reconstruction with high quality and provide the 2-nd order of accuracy, while for singular solution their order of accuracy goes down to the 1-st. A numerical scheme is suggested that allows for solution singularity resolving and provides the 2-nd order of accuracy. As a model problem, the added mass tensor components computation is considered, since its exact value is known for the Joukowsky wing airfoil with sharp edge (cusp point).

Lagrangian Coherent Structures in the Mediterranean Sea: Seasonality and basin regimes

The dynamics of fluid flows give rise to robust, persistent circulation features that underpin the flow and exert strong control over the advection of water masses, either enhancing it or suppressing it, collectively known as lagrangian coherent structures. Lagrangian approaches and metrics have been shown to be better suited than eulerian ones at locating and delineating such structures and capturing the effect they have on the formation and dispersion of water masses, particularly at the smaller scales. In this paper, we use the framework of lagrangian coherent structures to analyse the ocean velocity fields over a climatological year obtained from a high-resolution eddy-resolving model in order to investigate the lagrangian regimes that affect the motion, separation and mixing of water masses in the Mediterranean Sea. The lagrangian regimes that develop in each sub-basin over the course of the year are characterised and regions of persistent lagrangian activity and coherent structure formation and presence are identified. A quantitative picture of the seasonal variability of the lagrangian coherent structure-induced horizontal mixing and vortex formation is obtained.

The VM2D Open Source Code for Two-Dimensional Incompressible Flow Simulation by Using Fully Lagrangian Vortex Particle Methods

This article describes the open-source C++ code VM2D for the simulation of two-dimensional viscous incompressible flows and solving fluid-structure interaction problems. The code is based on the Viscous Vortex Domains (VVD) method developed by Prof. G. Ya. Dynnikova, where the viscosity influence is taken into account by introducing the diffusive velocity. The original VVD method was supplemented by the author’s algorithms for boundary condition satisfaction, which made it possible to increase the accuracy of flow simulation near the airfoil’s surface line and reduce oscillations when calculating hydrodynamic loads. This paper is aimed primarily at assessing the efficiency of the parallelization of the algorithm. OpenMP, MPI, and Nvidia CUDA parallel programming technologies are used in VM2D, which allow performing simulations on computer systems of various architectures, including those equipped with graphics accelerators. Since the VVD method belongs to the particle methods, the efficiency of parallelization with the usage of graphics accelerators turns out to be quite high. It is shown that in a real simulation, one graphics card can replace about 80 nodes, each of which is equipped with 28 CPU cores. The source code of VM2D is available on GitHub under GNU GPL license.

Neural vortex method: From finite Lagrangian particles to infinite dimensional Eulerian dynamics

In fluid analysis, there has been a long-standing problem: lacking a rigorous mathematical tool to map from a continuous flow field to finite discrete particles, hurdling the Lagrangian particles from inheriting the high resolution of a large-scale Eulerian solver. To tackle this challenge, we propose a novel learning-based framework, the neural vortex method (NVM). NVM builds a neural-network description of the Lagrangian vortex structures and their interaction dynamics to reconstruct the high-resolution Eulerian flow field in a physically-precise manner. The key components of our infrastructure consist of two networks: a vortex detection network to identify the Lagrangian vortices from a grid-based velocity field and a vortex dynamics network to learn the underlying governing interactions of these finite structures. By embedding these two networks with a vorticity-to-velocity Poisson solver and training its parameters using the fluid data obtained from grid-based numerical simulation, we can predict the accurate fluid dynamics on a precision level that was infeasible for all the previous conventional vortex methods. We demonstrate the efficacy of our method in generating highly accurate prediction results with low computational cost by predicting the evolution of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Navier–Stokes (NS) equations with different external forces. We compare the prediction results made by NVM and the Lagrangian vortex method (LVM) for solving the NS equation in the periodic box and find that the relative error of the predicted velocity using NVM is more than 10 times lower than that of the LVM. Moreover, our method only requires data collected from a very short training window, more than 100 times smaller than the prediction period, which potentially facilitates data acquisition in real systems.

Vortex Lens: Interactive Vortex Core Line Extraction using Observed Line Integral Convolution.

This paper describes a novel method for detecting and visualizing vortex structures in unsteady 2D fluid flows. The method is based on an interactive local reference frame estimation that minimizes the observed time derivative of the input flow field v(x, t). A locally optimal reference frame w(x, t) assists the user in the identification of physically observable vortex structures in Observed Line Integral Convolution (LIC) visualizations. The observed LIC visualizations are interactively computed and displayed in a user-steered vortex lens region, embedded in the context of a conventional LIC visualization outside the lens. The locally optimal reference frame is then used to detect observed critical points, where v=w, which are used to seed vortex core lines. Each vortex core line is computed as a solution of the ordinary differential equation (ODE) · w(t)=w(w(t), t), with an observed critical point as initial condition (w(t0), t0). During integration, we enforce a strict error bound on the difference between the extracted core line and the integration of a path line of the input vector field, i.e., a solution to the ODE · v(t)=v(v(t), t). We experimentally verify that this error depends on the step size of the core line integration. This ensures that our method extracts Lagrangian vortex core lines that are the simultaneous solution of both ODEs with a numerical error that is controllable by the integration step size. We show the usability of our method in the context of an interactive system using a lens metaphor, and evaluate the results in comparison to state-of-the-art vortex core line extraction methods.

Intensification of magnetic field in merging magnetic flux tubes driven by supergranular vortical flows

ABSTRACT The spatiotemporal dynamics of vorticity and magnetic field in the region of a photospheric vortex at a supergranular junction of the quiet Sun is studied, using Hinode’s continuum intensity images and longitudinal magnetograms. We show that in a 30-min interval during the vortex lifetime, the magnetic field is intensified at the centres of two merging magnetic flux tubes trapped inside the vortex boundary. Moreover, we show that the electric current density is intensified at the interface boundary layers of merging tubes, resulting from strong vortical downflows in a supergranular vertex. Evidence of Lagrangian chaos and vortex stretching in the photospheric plasma turbulence responsible for driving the intensification of magnetic fields is analysed. In particular, we report the first solar observation of the intensification of electromagnetic energy flux resulting from the merger of magnetic flux tubes.

A Lagrangian vortex method for smoke simulation with two-way fluid–solid coupling

We propose a pure Lagrangian vortex particle dynamics framework to simulate the various phenomena of turbulent smoke and two-way coupling between fluid and solid. In the framework, we model the fluid using vorticity and velocity fields that are discretized on the vortex particles. We extend a segment-based vortex stretching strategy with the subdivision of long segments to capture the fine details of smoke while maintaining good numerical stability. In addition, a vorticity field correction mechanism based on the merging and splitting of particles is proposed to avoid excessive particle counts and reduce the associated unnecessary computational overhead. We also present a fast and approximate two-way coupling method for simulating the interaction between solid and fluid. Our method can be used to solve various fluid phenomena, including Kármán vortex streets, jets, and some examples of two-way fluid–solid coupling.

A Ship Firefighting Training Simulator with Physics-Based Smoke

Under the current background of navigation education, students generally lack a comprehensive grasp of ship firefighting equipment’s operation. Therefore, we develop a novel ship firefighting training simulator with a multi-sensory human–computer interaction function for teaching and training marine students. In the simulator, we consider a container ship of 1.8w containers as the prototype ship, and the entire ship models are built using three-dimensional modeling technology. We design various interactive modes and realize a full-process operation simulation of several standard ship firefighting equipment. Furthermore, we propose a purely Lagrangian vortex dynamics framework to simulate smoke and flame in fire scenarios. In this framework, we model fluids using velocity and vorticity fields discretized on discrete vortex segments. The main components of the framework include a stable geometric stretching solution and particle strength exchange method for solving the diffusion term. The simulation results show that the simulator has good behavioral realism and scene immersion and can be applied to ship firefighting training. To the best of our knowledge, this is the first study on real-time smoke simulation using a physics-based method in a firefighting training simulator.

Research on smoke simulation with vortex shedding.

The Lagrangian vortex method has the advantage of producing highly detailed simulations of fluids such as turbulent smoke. However, this method has two problems: the construction of the velocity field from the vorticity field is inefficient, and handling the boundary condition is difficult. We present a pure Lagrangian vortex method, including a nested grid to accelerate the construction of the velocity field, and a novel boundary treatment method for the vorticity field. Based on a tree structure, the nested grid algorithm considerably improves the efficiency of the velocity computation while producing visual results that are comparable with the original flow. Based on the vortex-generating method, the least square method is used to compute the vorticity strength of the new vortex elements. Further, we consider the mutual influence between the generated vortex particles. We demonstrate our method’s benefits by using a vortex ring and various examples of interaction between the smoke and obstacles.