The Strouhal Number — The Rhythm of Vortex Shedding

“Why do power lines sing in the wind? Why did the Tacoma Narrows Bridge oscillate itself to destruction? Why do fish and dolphins move their tails at a very specific frequency? A single dimensionless number — the Strouhal number, St ≈ 0.2 — governs all of these phenomena.”

1. Introduction

When a uniform flow encounters a bluff body — a cylinder, a bridge deck, a smokestack, a wire — the downstream wake does not remain steady. Instead, the boundary layer separates alternately from each side of the body, producing a staggered double row of counter-rotating vortices known as the Kármán vortex street. This periodic vortex shedding exerts an oscillating side force on the body at the shedding frequency $f$.

The Strouhal number (St) is the dimensionless ratio that collapses this frequency into a universal constant, independent of fluid properties and (to a good approximation) of flow speed:

2. Historical Background

Vincenc Strouhal (1850–1922) was a Czech physicist who, in 1878, systematically measured the frequencies of “aeolian tones” — the musical sounds produced by wires and strings exposed to wind. He found empirically that $f propto v/D$ with a proportionality constant close to 0.185. His paper, published in Annalen der Physik, established the dimensionless group that now bears his name.

Theodore von Kármán (1881–1963) provided the theoretical foundation in 1911, showing that the alternating vortex street is the only stable configuration downstream of a bluff body. He derived that the transverse spacing-to-longitudinal spacing ratio of the vortices must be $h/a pprox 0.281$ for stability.

3. The Kármán Vortex Street

Above Re ≈ 40, the symmetric pair of steady recirculating eddies behind a cylinder becomes unstable. Vortices are shed alternately from the top and bottom of the body, creating an asymmetric wake: a double row of vortices with opposite circulations, staggered with one vortex on each side per half-cycle.

Each shedding event creates a transverse (lift) force impulse on the body. This oscillating lift force has frequency $f_{shed}$ and can drive resonance in flexible structures.

4. Aeolian Tones — Wind Makes Music

Aeolian tones (from Aeolus, Greek god of wind) are the musical notes produced when wind flows over cylindrical obstacles such as strings, wires, and pipes. The tone frequency equals the vortex shedding frequency:

$$f_{aeolian} = rac{St cdot v}{D} pprox rac{0.20, v}{D}$$

5. The Tacoma Narrows Bridge Disaster (1940)

The Tacoma Narrows Bridge in Washington State, USA, collapsed on 7 November 1940, just four months after opening. Filmed footage shows the bridge oscillating in torsion with increasing amplitude before catastrophic failure in a wind of only 67 km/h (42 mph).

The bridge had a solid plate-girder deck — unlike the open-truss decks used on older suspension bridges. Wind flowing past this solid rectangular cross-section shed vortices at:

$$f_{shed} = rac{St cdot v}{D} pprox rac{0.20 imes 18.6}{12} pprox 0.31 ext{ Hz}$$

(where $D = 12$ m is the deck depth). The bridge’s natural torsional frequency was approximately $f_n pprox 0.20$ Hz. Near resonance, a self-excited mechanism (galloping aeroelasticity, more precisely than pure VIV) built oscillations to 8.5 m amplitude before the deck tore apart.

6. Vortex-Induced Vibration (VIV) and Lock-in

When a flexible structure vibrates at frequency $f_s$ close to the vortex shedding frequency $f_{shed}$, the shedding “locks in” to $f_s$ over a range of wind speeds. During lock-in, the vortices are shed at the structural frequency rather than the Strouhal frequency — even as wind speed varies within the lock-in window.

$$ ext{Lock-in range:} quad rac{f_n}{f_{Strouhal}} pprox 0.7 ext{ to } 1.4$$

Lock-in dramatically amplifies vibration amplitude because the fluid force is now always in phase with structural velocity. Structural fatigue and catastrophic failure can result.

7. Strouhal Number in Fish Locomotion

Taylor, Nudds and Thomas (2003) made a remarkable discovery: virtually all swimming fish and flying birds operate at Strouhal numbers in the range:

$$0.2 leq St leq 0.4$$

For swimming and flying animals, St is defined using the tail/wing stroke amplitude $A$ and flapping frequency $f$:

$$St = rac{fA}{v}$$

This range corresponds to the peak of propulsive efficiency in oscillating-foil theory. Evolution has therefore “discovered” and tuned to the same optimal St ≈ 0.25–0.30 that fluid mechanics predicts — from tiny bumblebees to blue whales.

8. Vortex Flowmeters

The vortex flowmeter is one of the most widely used industrial flow measurement instruments. It exploits St ≈ 0.20 directly: a bluff strut (shedder bar) of known width $D$ is inserted into the pipe; vortices are shed at frequency $f = St cdot v/D$, detected by a piezoelectric or capacitive sensor. The pipe-average velocity and volumetric flow rate follow:

$$v = rac{f cdot D}{St}, qquad Q = v cdot A_{pipe}$$

Advantages: no moving parts, wide turndown ratio (typically 10:1 to 30:1), accuracy ±0.5–1%, suitable for steam, gas, and liquid.

9. Strouhal Number in Aeronautics

In aircraft aerodynamics, St appears in several unsteady flow contexts:

• Wing flutter: Aeroelastic instability occurs when the flutter frequency matches the Strouhal frequency of wake vortices from upstream components. Structural dynamic coupling must be checked against VIV.
• Wind turbines: Blade wake vortices shed at St ≈ 0.2 based on blade chord; these must not coincide with tower bending frequency.
• Car aerodynamics: Antenna rods, wing mirrors, and roof pillars all shed vortices. The resulting buffeting noise and vibration are mitigated by aerodynamic shaping (oval cross-sections increase St and shift the tone frequency out of the objectionable range).

10. St for Different Cross-Section Shapes

Cross-section	St (approximate)	Re range
Circular cylinder	0.20–0.21	300–2×10⁵
Square cylinder	~0.12	Large
Rectangular (W/H=2)	~0.14	Large
Flat plate (normal to flow)	~0.12	Large
D-shaped section	~0.17	Large
Ellipse (2:1 aspect)	~0.15	Large
Triangle (apex upstream)	~0.18	Large

11. Strouhal Number in Cardiovascular Flows

Blood flow in large arteries is pulsatile — driven by the periodic beating of the heart at frequency $f_{heart} pprox 1$–1.5 Hz. A Strouhal number (sometimes called the Womersley number in this context) characterises the ratio of oscillatory inertia to viscous forces:

$$St_{cardio} = rac{f_{heart} D_{vessel}}{v_{mean}}$$

In the aorta: $f pprox 1.2$ Hz, $D pprox 0.025$ m, $v_{mean} pprox 0.2$ m/s, giving $St pprox 0.15$. This parameter influences the development of flow separation, secondary flows, and shear stress distributions that are linked to atherosclerosis risk.

12. IYPT 2027 Connections

13. Summary of Key Applications

Application	Typical St	Characteristic length	Key issue
Power line wind song	0.20	Wire diameter	Aeolian tone frequency
Bridge deck (VIV)	0.10–0.20	Deck depth	Resonance, flutter
Industrial chimney	0.20	Chimney diameter	Lock-in wind speed
Offshore platform leg	0.20	Leg diameter	Fatigue from VIV
Fish (cruising)	0.25–0.35	Tail amplitude	Max propulsive efficiency
Bird flight	0.20–0.40	Wing stroke amplitude	Optimal flapping gait
Vortex flowmeter	0.20	Shedder bar width	Flow measurement
Heat exchanger tubes	0.20	Tube diameter	Tube fatigue, acoustic resonance
Car antenna	~0.20	Antenna diameter	Buffeting noise

14. Key Equations Summary

15. Dimensional Analysis Perspective

By the Buckingham Pi theorem, for periodic shedding from a bluff body, the relevant variables are $f$, $v$, $D$, $
ho$, and $mu$. This gives two dimensionless groups: St and Re. The empirical result that St ≈ 0.20 is nearly independent of Re (for $Re$ in the turbulent shedding regime) means that a single universal constant collapses all cylinder-wake oscillation data across many decades of Re. This is a profound and practically powerful result — the natural frequency of a wind-excited structure can be predicted from its diameter and the wind speed alone, without knowing the fluid viscosity.