Two Faces of a Whirl — Free Vortex, Forced Vortex, and Helmholtz

Pull the plug from a bathtub and the water spins. We call that scene "the water rotates." A fluid dynamicist, looking at most of that same region, will tell you the opposite: "nothing there is rotating (irrotational)." The contradiction is fake, and inside it lives the real definition of a vortex. This post separates the two whirls that share the same picture — the free vortex with $1/r$ velocity and the forced vortex with $\omega r$ velocity — explains what vorticity (twice the spin rate of a fluid particle) actually measures, and follows Helmholtz's theorem all the way to the claim that vortices are immortal.

Is a bathtub whirl really 'rotation'?#

Two meanings hide inside one word. The first is revolution — moving in a circle around some point. The second is spin — the particle turning on its own axis. In a bathtub whirl, water particles revolve. But spin? Far from the drain, almost none. A small leaf floating on top traces circular paths but never turns its body. The same face always points toward the drain.

In fluid mechanics, "rotation" means spin. So a free vortex is classified as non-rotating everywhere except at the singular center.

Forced vs. free: same picture, different soul#

Stir a bucket long enough and the entire liquid spins as a rigid body. This is the forced vortex. Each particle revolves and spins together.

u_\theta(r) = \omega \, r

Here $u_\theta$ is the tangential speed, $\omega$ the rigid-body angular velocity, $r$ the distance from the axis. Speed is proportional to radius.

The whirl that arises spontaneously in an ideal (inviscid) fluid is the free vortex, also called the potential vortex.

u_\theta(r) = \frac{\Gamma}{2\pi r}

$\Gamma$ is the circulation, the line integral of velocity around a closed loop. Speed falls off as $1/r$ and blows up at the center — viscosity tames that singularity in real flows.

	Forced	Free
Speed	$\omega r$	$\Gamma / 2\pi r$
Spin	yes	no
Energy input	continuous	once at birth
Everyday example	stirred cup	bathtub, tornado

Same circular streamlines, opposite spin behavior. One scalar quantifies the difference.

What vorticity measures#

Vorticity $\boldsymbol{\omega}$ is the curl of the velocity field.

\boldsymbol{\omega} = \nabla \times \mathbf{u}

In two dimensions only the $z$ -component survives:

\omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}

where $u, v$ are velocity components in $x, y$ . The value equals twice the angular velocity of a fluid particle's spin.

Plug in the forced vortex: $\omega_z = 2\omega$ — uniform across the whole region. Plug in the free vortex: $\omega_z = 0$ everywhere except the center. Identical round streamlines, opposite spin.

Rankine combined vortex — the tornado blueprint#

Real whirls follow neither formula in isolation. Near the center, viscosity is strong and the fluid spins like a solid; far away, viscosity is negligible and the free-vortex law takes over. The model that splices the two is the Rankine combined vortex.

u_\theta(r) = \begin{cases} \dfrac{U_{\max}}{R}\, r, & r \le R \\[6pt] \dfrac{U_{\max}\, R}{r}, & r > R \end{cases}

$R$ is the core radius and $U_{\max}$ the peak tangential speed at the core boundary. The two pieces match smoothly at $r = R$ . Tornado wind profiles, wing-tip vortices trailing aircraft, and the suction whirl above a drain all follow this shape. Inside the core: rotational. Outside: irrotational.

Helmholtz's theorem — the immortality of vortices#

In an inviscid fluid, the only force on a particle is pressure, and pressure points toward the particle's center, exerting no torque. Therefore a particle that is not spinning never starts, and a particle that is spinning never stops. That is Helmholtz's vortex theorem in one line.

A sharper statement is Kelvin's theorem: the circulation $\Gamma$ measured along a closed material curve does not change in time.

\frac{D\Gamma}{Dt} = 0

$D/Dt$ is the material derivative (following a fluid parcel). Vortex tubes cannot be cut or terminated; stretching one makes it spin faster — the same physics as a figure skater pulling her arms in. Real fluids have viscosity, so vortices eventually diffuse, but in low-viscosity media (water, air) they survive for days. That is why an aircraft contrail vortex stays visible long after the plane is gone.

The door velocity potential opens#

In an irrotational region ( $\boldsymbol{\omega} = 0$ ) a vector identity hands you a free gift:

\nabla \times \mathbf{u} = 0 \quad \Longrightarrow \quad \mathbf{u} = \nabla \phi

The scalar $\phi$ is the velocity potential. If the flow is also incompressible, $\nabla \cdot \mathbf{u} = 0$ , then

\nabla^2 \phi = 0

Laplace's equation. Inviscid, irrotational, incompressible flow reduces to a single scalar problem. That is how nineteenth-century fluid mechanics handled airfoils, sources, and sinks analytically. The free vortex itself has $\phi = (\Gamma / 2\pi)\, \theta$ — a potential proportional to the angular coordinate.

Two vortices in code#

Plotting Rankine speed and discrete vorticity makes the difference obvious.

import numpy as np
 
def rankine_speed(r, R_core, U_max):
    """Tangential speed of a Rankine combined vortex."""
    speed = np.where(r <= R_core,
                     U_max * r / R_core,
                     U_max * R_core / np.maximum(r, 1e-9))
    return speed
 
def vorticity_field(u, v, dx, dy):
    """Discrete curl in 2D: omega_z = dv/dx - du/dy."""
    dvdx = (v[1:-1, 2:] - v[1:-1, :-2]) / (2 * dx)
    dudy = (u[2:, 1:-1] - u[:-2, 1:-1]) / (2 * dy)
    return dvdx - dudy
 
# Sample on a 200x200 grid
N, L = 200, 2.0
x = np.linspace(-L, L, N)
y = np.linspace(-L, L, N)
X, Y = np.meshgrid(x, y)
R = np.hypot(X, Y)
THETA = np.arctan2(Y, X)
 
U_max, R_core = 1.0, 0.5
u_theta = rankine_speed(R, R_core, U_max)
u =  -u_theta * np.sin(THETA)
v =   u_theta * np.cos(THETA)
 
dx = dy = x[1] - x[0]
omega = vorticity_field(u, v, dx, dy)
 
# Mean vorticity inside and outside the core
mask_in  = R[1:-1, 1:-1] <  R_core
mask_out = R[1:-1, 1:-1] >  1.5 * R_core
print(f"core mean omega_z ~ {omega[mask_in].mean():.3f}")   # ~ 2 * U_max / R_core
print(f"outer mean omega_z ~ {omega[mask_out].mean():.3f}") # ~ 0

Inside the core, $\omega_z \approx 2 U_{\max}/R$ and stays nearly constant. Outside, it drops to zero. The same circular flow carries a sharp boundary at $r = R$ that splits spinners from non-spinners.

Play with it directly#

Try the simulation below. Yellow tracers trapped in the central disc form the forced-vortex core; blue tracers drifting outside live in the free-vortex region.

core radius R =70U_max =110highlight core (ω ≠ 0)

Yellow tracers sit inside the forced-vortex core and rotate as a solid body (ω = 2U/R). Blue tracers outside follow u_θ = ΓR/r — circular streamlines but zero local spin. The Rankine combined vortex is the simplest model for a bath-tub drain or tornado.

Shrinking $R$ narrows the spinning region — the model of a tornado eye getting tighter. Raising $U_{\max}$ scales the core vorticity $\omega = 2 U_{\max}/R$ . The inset plot in the upper right shows the linear core ( $\omega r$ ) joining the hyperbolic outside ( $1/r$ ) at $r = R$ .

Next time you see a whirl#

Separate the spinning region from the non-spinning region. That is the one sentence to take from this post. The next time a bathtub drain, a teacup eddy, or a satellite typhoon image catches your eye, recall three things.

Revolution and spin are different. Identical circular streamlines can carry zero vorticity.
The Rankine combined vortex is the simplest realistic model. Solid-body inside, $1/r$ outside.
Helmholtz answers why vortices are so persistent — the lower the viscosity, the longer they live.

Carry these three lines and the next time a turbulence lecture rolls out the "vorticity transport equation," it will not feel new. Vorticity itself is no longer conserved, but the spirit lives in the same place.