The different aspects of vortex methods

This chapters discusses the different aspects of vortex methods enumerated in the following. The 1st part presents the fundamental equations and concepts of vortex methods. The 2nd part defines the discretization and initialization problem: the information carried by the particle and the way the vortex elements are discretized and initialized from a given vorticity field. The 3rd part presents the Viscous-Splitting algorithm/assumption which is used by some vortex methods to solve the convection and diffusion steps separately. The 4th part discusses the convection and stretching of vortex elements. The Lagrangian markers are convected by resolution of the trajectory equation using different numerical schemes. The stretching is applied by computation of the gradient of the velocity field. Different schemes exist. The 5th part presents grid-free and grid-based methods and provides references to coupled-Lagrangian–Eulerian solvers). The 6th part presents viscous diffusion and provides the general solution of the diffusion equation and different numerical implementations of viscous diffusion in vortex methods. The 7th part mentions bodies, boundaries and boundary conditions. Viscous and inviscid solid boundaries are discussed. Bodies are then presented as a particular case. The 8th part presents the regularization (also called kernel smoothing or mollification) required for the convergence of the (grid-free) method. The 9th part presents the topic of spatial adaptation (also called, or linked to, redistribution, rezoning or reinitialization) which is required due to the Lagrangian stretching of the computational domain which may results in “holes”. The 10th part briefly mentions the possibility to model turbulence via subgrid-scale models in vortex methods. The 11th part discusses the accuracy of vortex methods and provides guidelines and diagnostics to maintain or monitor accuracy within simulations.