Vortex dynamics

Definitions

Vorticity ω

Curl of velocity field: ∇ × u. Units: 1/s. Measures rotation.

Circulation Γ

Line integral of velocity around a closed curve: Γ = ∮ u · dl. Units: m²/s. Conserved in inviscid flow.

Rankine vortex

Simple model: solid body rotation (ω = const) inside core radius r_c; irrotational (ω = 0) outside. Discontinuous vorticity at r_c.

Lamb–Oseen vortex

Viscous vortex: core grows with time. Vorticity profile: ω(r,t) = (Γ/πνt) exp(−r²/4νt), where ν is kinematic viscosity.

Kármán vortex street

Alternating vortices shed by a cylinder at Re ≈ 20–200. Strouhal number St ≈ 0.2. Named after Theodore von Kármán.

Physics

A vortex is a region of rotating fluid. The Rankine vortex is the simplest model: inside a core radius r_c, fluid rotates as a solid body with constant vorticity ω = Γ/(πr_c²). Outside, the flow is irrotational and resembles a sink or source—velocity falls as 1/r.

In reality, viscosity smooths the transition. The Lamb–Oseen vortex solution to the Navier–Stokes equation shows how a vortex core diffuses outward over time: the vorticity distribution widens and flattens, while circulation is conserved. This is visible in smoke rings and tornado dissipation.

When a fluid flows past an obstacle, vortices are shed periodically. In the Kármán vortex street, two rows of opposite-sign vortices form behind the object—a wake. The frequency is given by the Strouhal number: St = f·d/U ≈ 0.2, where f is shedding frequency, d is obstacle width, and U is flow speed. This explains why power lines hum and tall buildings sway in wind.

Sources

• Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2016). Fluid Mechanics, 6th ed. Academic Press.
• Lamb, H. (1932). Hydrodynamics, 6th ed. Dover Publications. (Mathematical theory of vortex dynamics.)
• Kármán, T. v., & Burgers, J. M. (1935). General aerodynamic theory. In Aerodynamic Theory, vol. II. Springer.