A comprehensive review is presented of the computational methods based upon Helmholtz’s powerful concepts of vortex dynamics, making use of Lagrangian or mixed Lagrangian-Eulerian schemes, the Biot-Savart law or the Vortex-in-Cell methods. The ingenious approximations and smoothing schemes developed in search of predictive models, qualitative solutions, new insights, or just some inspiration in the simulation of often two-dimensional, occasionally three-dimensional, and almost always incompressible fluids are described in detail. One is forewarned at the onset that chaos awaits at the end of the road. The challenge is to produce results in the face of ever accumulating errors within a time scale appropriate for the investigation. The review is organized around two major sections: Theoretical foundations and practical applications of vortex methods. The first covers topics such as vorticity and laws of transportation, evolution equations for a vortex sheet, real vortices and instabilities, Biot-Savart law, smoothing techniques (cutoff schemes, amalgamation of vortices, subvortex methods), cloud-in-cell or vortex-in-cell methods, body representation (Routh’s rule, surface singularity distributions), operator splitting and the random walk method (description and convergence), and asymmetry introduction. The next section covers contra flowing streams, vortical flows in aerodynamics (vortex sheet roll-up; slender-body, two-vortex, multi-discrete vortex, and segment or panel methods; three-dimensional flow models, and vortex-lattice methods), separated flow about cylindrical bodies (circular cylinder, sharp-edged bodies, arbitrarily-shaped bodies), general three-dimensional flows (vortex rings, turbulent spots, temporally and spatially-growing shear layers, and other applications (vortex-blade interactions, combustion phenomena, acoustics, contour dynamics, interaction of line vortices, chaos, and turbulence). The review is concluded with a brief comparison of these methods with others used in computational fluid dynamics and a personal view of their future prospects.

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