A Review of Vortex Methods and Their Applications: From Creation to Recent Advances

This review paper presents an overview of Vortex Methods for flow simulation and their different sub-approaches, from their creation to the present. Particle methods distinguish themselves by their intuitive and natural description of the fluid flow as well as their low numerical dissipation and their stability. Vortex methods belong to Lagrangian approaches and allow us to solve the incompressible Navier-Stokes equations in their velocity-vorticity formulation. In the last three decades, the wide range of research works performed on these methods allowed us to highlight their robustness and accuracy while providing efficient computational algorithms and a solid mathematical framework. On the other hand, many efforts have been devoted to overcoming their main intrinsic difficulties, mostly relying on the treatment of the boundary conditions and the distortion of particle distribution. The present review aims to describe the Vortex methods by following their chronological evolution and provides for each step of their development the mathematical framework, the strengths and limits as well as references to applications and numerical simulations. The paper ends with a presentation of some challenging and very recent works based on Vortex methods and successfully applied to problems such as hydrodynamics, turbulent wake dynamics, sediment or porous flows.

On the Scope of Lagrangian Vortex Methods for Two-Dimensional Flow Simulations and the POD Technique Application for Data Storing and Analyzing

The possibilities of applying the pure Lagrangian vortex methods of computational fluid dynamics to viscous incompressible flow simulations are considered in relation to various problem formulations. The modification of vortex methods—the Viscous Vortex Domain method—is used which is implemented in the VM2D code developed by the authors. Problems of flow simulation around airfoils with different shapes at various Reynolds numbers are considered: the Blasius problem, the flow around circular cylinders at different Reynolds numbers, the flow around a wing airfoil at the Reynolds numbers 104 and 105, the flow around two closely spaced circular cylinders and the flow around rectangular airfoils with a different chord to the thickness ratio. In addition, the problem of the internal flow modeling in the channel with a backward-facing step is considered. To store the results of the calculations, the POD technique is used, which, in addition, allows one to investigate the structure of the flow and obtain some additional information about the properties of flow regimes.

Fluid Dynamical 2D Simulations of Jupiter's South Polar Region Based On JunoCam Image Data

<p>During almost every perijove pass in more than three years of Juno's ~53-day polar orbits around Jupiter, its wide-angle visible-light camera, JunoCam [1], has imaged Jupiter's south polar region [2].</p><p>We sought to determine whether these images could be used for prognostic “weather forecasts” in Jupiter. One of the simplest fluid dynamical models suitable for forecasting dynamical behavior of essentially barotropic incompressible flows of very low viscosity is the 2D Euler fluid. Vortex methods [3] are particularly suitable for modeling the resulting turbulence.</p><p>Sequences of images taken with a cadence of several minutes reveal small motions of the cloud tops within the illuminated area of the pole. The south pole itself has been visible in the twilight.</p><p>Raw JunoCam image data are transformed into an equidistant south-polar azimuthal map, roughly illumination-adjusted, high-passed with local contrast-normalization, and registered.<br>A streamfunction describing the velocity field approximately is derived from a sequence of consecutive maps of a common perijove flyby. Running a Monte-Carlo approach for stereo correlation repeatedly with different pseudo-random number sets returns an ensemble of streamfunctions.<br>The Laplacian of a streamfunction returns the vorticity values for a randomized 2D vortex particle seed as initial conditions of a grid-free vortex method. Applying the Biot-Savart law [3, p.19ff] on a 2-spherical geometry to the vorticity field returns the velocity field. A single-step explicit Runge-Kutta method of order 4 or 5 and fixed time steps advects the 4th-degree Gaussmollified vortex particles. Measuring the area of their Voronoi cells (Dirichlet/Thiessen polygons) reassesses the radius of the vortex particles. The method allows for some divergence. An approximately inviscid and incompressible 2D-flow is simulated over 2 up to 54 real-time days or about one Juno orbital period. The randomized nature of the method induces simulation ensembles for a given streamfunction by repeated runs.</p><p>Reducing the streamfunction to a Morse-Smale complex returns idealized model vortex seeds.</p><p>JunoCam images of the south polar region taken during a perijove pass provide an ensemble of dynamical data. These initial conditions extend to ensembles of forecast runs of the 2-spherical dynamics of the visible cloud tops in Jupiter's south polar region. We find that JunoCam images of<br>Jupiter's south polar region allow for reasonably plausible forecasts of the dynamics of the observed area with grid-free 2D vortex methods over at least a few days.</p><p>[1] C.J. Hansen, M.A. Caplinger, A. Ingersoll, M.A. Ravine, E. Jensen, S. Bolton, G. Orton.<br><em>Junocam: Juno’s Outreach Camera.</em> Space Sci Rev 2013:475-506, <em>2017</em><br>[2] F.Tabataba-Vakili,J.H.Rogers,G.Eichstädt,G.S.Orton,C.J.Hansen,et al. <em>Long-term Tracking of</em><br><em>Circumpolar Cyclones on Jupiter From Polar Observations with JunoCam</em>. Icarus 335, 113405,<br><em>2020</em>.<br>[3] G.-H. Cottet, P. D. Koumoutsakos, <em>Vortex Methods: Theory and Practice</em>, Cambridge University<br>Press, <em>2000</em></p>