Vortex Strength and Circulation Calculator

Circulation Gamma is the line integral of the fluid velocity around a closed loop, Gamma = the closed loop integral of V dot dl. It measures the net tendency of the flow to rotate around that loop. For a circular path of radius r with uniform tangential velocity V_theta, this simplifies to Gamma = V_theta times 2*pi*r, since the velocity is everywhere parallel to the path.

For a free (potential, irrotational) vortex, tangential velocity varies as V_theta = Gamma / (2*pi*r), so rearranging gives Gamma = 2*pi*r*V_theta. This means circulation stays constant no matter which radius the loop is drawn at, as long as it encloses the vortex core, which is the defining property of a free vortex.

For a forced (solid-body) vortex rotating like a rigid disk at angular velocity omega, tangential velocity is V_theta = omega*r, and circulation around a circular loop of area A = pi*r squared is Gamma = 2*omega*A. Unlike a free vortex, circulation here grows with the enclosed area, not just the boundary radius.

In a free (potential) vortex, fluid particles move in circles but do not spin about their own axis, so vorticity is zero everywhere except at a singular core, and V_theta decreases as 1/r moving outward. In a forced (solid-body) vortex, every fluid particle rotates together like part of a rigid disk, vorticity is uniform (zeta = 2*omega) throughout, and V_theta increases linearly with r.

By Stokes theorem, circulation around a closed loop equals the vorticity integrated over the enclosed area. A solid-body rotation has uniform vorticity zeta = 2*omega everywhere, so Gamma = zeta*A = 2*omega*A. This can also be verified directly: Gamma = V_theta * 2*pi*r = (omega*r)(2*pi*r) = 2*pi*omega*r squared = 2*omega*(pi*r squared) = 2*omega*A.

Circulation Gamma has SI units of square metres per second (m squared / s), the same dimensional units as kinematic viscosity. This comes directly from velocity (m/s) multiplied by a length (m) in the line integral, or equivalently vorticity (1/s) multiplied by an area (m squared).

Yes, away from the core. A true free (potential) vortex is irrotational everywhere except at the central core point, where vorticity is technically infinite (a singularity) but concentrated in zero area. This is the key mathematical distinction from a forced vortex, where vorticity is finite and uniform (zeta = 2*omega) across the entire rotating region.

The Kutta-Joukowski theorem states that lift per unit span equals rho times V times Gamma, where rho is fluid density, V is the freestream velocity, and Gamma is the circulation generated around the airfoil section. This is the same circulation quantity computed here, applied to the bound vortex that forms around a lifting wing.

The outer region of a bathtub drain vortex, a tornado away from its core, or the flow field far from an aircraft wingtip vortex all approximate a free (potential) vortex, where tangential velocity falls off as 1/r and circulation stays constant with radius.

Fluid in a rotating container that has reached solid-body rotation, the eye of a tornado or hurricane close to its center, and a stirred cup of tea near the very center of rotation all approximate a forced (solid-body) vortex, where every particle rotates together at the same angular velocity omega.

A Rankine combined vortex models the core as a forced vortex and the outer region as a free vortex, joined at a core radius where both formulas give matching tangential velocity. Use the forced vortex mode for loops inside the core and the free vortex mode for loops outside it, keeping in mind circulation is continuous across the boundary in the idealized model.