Vortices

Vortices are regions of rotating flow where fluid elements follow curved trajectories around a common axis; they are characterized by high vorticity, defined as the curl of the velocity field, ω = ∇ × v, and by circulation, Γ = ∮ v · dl, around a closed loop that threads the vortex.

In idealized theory, two-dimensional point vortices or thick vortex tubes are used to model coherent rotational structures; real fluids exhibit three-dimensional vortices that can stretch and tilt, sustain as vortex lines and tubes, and can form vortex sheets. Vorticity can also arise in quantum fluids as quantized vortices.

Formation and dynamics: Vorticity is generated by shear, boundary layers, obstacles and impulsive motions. In inviscid, barotropic flows, Kelvin's circulation theorem states that the circulation around a material contour is conserved. Viscosity diffuses vorticity, causing decay and diffusion of vortical structures; in bluff-body wakes, vortex shedding forms alternating street patterns known as von Kármán vortex streets.

Examples and mechanisms: Atmospheric cyclones and tornadoes are large-scale vortices; whirlpools and ocean eddies are fluid vortices; smoke rings are coherent toroidal vortices. In engineering, wing tip and propeller vortices influence lift and propulsion and can affect downstream structures.

Measurement and applications: Vorticity is inferred from velocity fields via experimental techniques such as particle image velocimetry. Vortices play key roles in mixing, momentum transport, and energy dissipation in fluids; understanding their behavior informs aerodynamics, meteorology, oceanography, and industrial fluid mechanics.