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

Chloé Mimeau,Iraj Mortazavi

doi:10.3390/fluids6020068

Chloé Mimeau, Iraj Mortazavi

Open Access

https://doi.org/10.3390/fluids6020068

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Abstract

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.

Highlights

IntroductionIn an Eulerian approach, the computational domain is described as a mesh and the solution vector Q is approximated on each node according to a numerical scheme involving the solution at the neighboring mesh nodes
Vortex methods (VM) belong to Lagrangian methods, called particle methods, used to solve continuous systems of the form: d dt Q(x, t) dx = ΩF(x, t, Q, ∇Q, ...) dx (1)where Q denotes the unsteady solution vector evolving in space (x) and time (t), composed of the physical quantities Qi, and where F is the source term depending on space and time and on the solution Q and its partial derivatives
The intensive development of the vortex methods between 1970 and the mid 1990s was motivated by the will to design robust and accurate numerical schemes able to overcome the main weaknesses of traditional grid methods, especially in the case of dominant advection effects in the flows

Summary

Introduction

In an Eulerian approach, the computational domain is described as a mesh and the solution vector Q is approximated on each node according to a numerical scheme involving the solution at the neighboring mesh nodes. In the Lagrangian approach, the solution Q is discretized on numerical particles and, given a location x in the domain, each quantity Q among the vector solution Q is approximated at this location x by: Q(x) = Q(y)δ(x − y) dy (2). To write this expression in a regularized and discrete form, the Lagrangian methods define a volume vp for the particles as well as a smoothing function W, of compact support and radial symmetry of radius ε, which tends to the Dirac distribution when ε tends to 0. The discrete approximation of the quantity Q at location x can be written as: Q(x) ∑ QpWε(x − xp)

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