Kármán Vortex Street

Equivalent forms

St = \frac{f D}{v}

St \approx 0.198\left(1 - \frac{19.7}{Re}\right)

f = \frac{0.2\,v}{D}\ (\text{typical})

Turbulent chaos organizing itself into a metronome — a tidy alternating street of spinning vortices.

Symbol	Name	SI unit	Dimension	Range
$St$	Strouhal Number Dimensionless shedding frequency	dimensionless	1	0.1 – 0.3
$v$	Flow Speed Free-stream flow speed	m/s	L·T⁻¹	0 – 60
$D$	Body Diameter Characteristic cross-stream size of the body	m	L	0.001 – 2
$f$	Shedding Frequencyoutput Frequency of alternate vortex shedding	Hz	T⁻¹	0 – 5000

Unit systems

Symbol	SI	CGS	Imperial
$St$	—	—	—
$v$	m/s	cm/s	ft/s
$D$	m	cm	ft
$f$	Hz	Hz	Hz

Where it holds

Periodic laminar/transitional shedding for bluff bodies over roughly 47 < Re <

2\times 10^{5}

, where

St \approx 0.2

is nearly constant. Outside this range St drifts and the wake becomes irregular or fully turbulent.

Dimensional analysis

$[St v / D] = (1)(L\cdot T^{-1})/(L) = T^{-1} = Hz \checkmark$ (frequency)

Discovery

Theodore von Kármán · 1911

Von Kármán analysed the staggered double row of vortices behind a cylinder and showed only a specific spacing ratio is stable. Vincenc Strouhal had measured the related 'aeolian tone' of wires in 1878; the shedding street now governs everything from chimney design to the Tacoma Narrows collapse.

Try this

Wind humming through wires and bridges swaying to a steady beat — both are vortices peeling off in rhythm.

Air at 10 m/s flows past a 0.005 m wire. With Strouhal number St ≈ 0.2, what frequency of alternating vortices (and audible tone) does it shed?

Research status: stable

Real-world applications

Vortex flowmeters
Chimney, cable and bridge wind-loading design (strakes, spoilers)
Aeolian tones in wires and antennas
Energy-harvesting from vortex-induced vibration

Common misconceptions

Vortices shed simultaneously from both sides — they alternate, which is what produces the sideways forcing
The street is fully turbulent chaos — over a wide range it is strikingly periodic
Tacoma Narrows fell from simple resonance with St shedding — it was aeroelastic flutter, though vortex forcing is the cousin phenomenon

Experimental verification

Hot-wire and flow-visualization studies confirm the constant St and the staggered street; the audible aeolian tone of wires matches

f = 0.2v/D

, and the 1940 Tacoma Narrows failure is the textbook lock-in example.

Derivation

Dimensional analysis of bluff-body shedding gives a single dimensionless group, the Strouhal number

St = fD/v

, which experiments show is nearly constant

(\sim 0.2)

over a broad Reynolds range.

Solving for frequency yields

f = St\cdot v/D

.

Von Kármán's stability analysis fixes the staggered vortex spacing ratio h/ℓ

\approx 0.281

.

Limiting cases

Re < 47⟶ No sheddingBelow the critical Reynolds number the wake is a steady pair of attached vortices — no alternating street.

47 < Re < 10^5⟶ St \approx 0.2 (steady street)Over a wide Reynolds-number range the Strouhal number is nearly constant, giving a clean periodic tone.

f \to f_{natural}⟶ Lock-in resonanceWhen shedding frequency matches a structure's natural frequency, oscillations amplify dangerously.

What if…

What if the wind speed doubles?

Shedding frequency doubles $(f \propto v)$ , so the wire's hum jumps an octave — a direct audible readout of wind speed.

What if f matches a chimney's natural frequency?

Lock-in resonance amplifies the sway; engineers add helical strakes to break up the coherent shedding.

1

Tone shed by a wire

Given ·

St:: 0.2
v:: 10
D:: 0.005

Find · f

Steps

Apply Strouhal relation: $f = St\cdot v/D$
$f = 0.2 \times 10 / 0.005$
$f = 400\,\mathrm{Hz}$ — the aeolian 'singing' of a wire in the wind

Result ·

f = St\cdot v/D = 0.2\times 10/0.005 = 400\,\mathrm{Hz}

— an audible hum near musical G4.

Reynolds Number→Drag Force (Quadratic)→Froude Number→