Kármán Vortex Street: Flow Instability and the Strouhal Number

Reynolds regimes and the birth of shedding

The Reynolds number Re = UD/ν governs the wake topology behind a cylinder of diameter D in flow of velocity U. Below Re ≈ 5, the wake is steady and symmetric; between Re ≈ 5 and 47, two steady recirculation vortices sit behind the cylinder. At Re ≈ 47 the symmetric wake undergoes a supercritical Hopf bifurcation — a complex eigenvalue pair crosses the imaginary axis and periodic alternating vortex shedding is born: the Kármán vortex street. Above Re ≈ 190 the wake develops spanwise 3D structure, and by Re ~ 10³–10⁵ the boundary layer itself is turbulent.

The Strouhal number and shedding frequency

The dimensionless shedding frequency is the Strouhal number St = f·D/U. For circular cylinders in the laminar shedding regime, Williamson & Brown (1998) give the empirical fit below; the wake settles near St ≈ 0.2 across the turbulent regime, relatively insensitive to Re.

St = 0.2663 − 1.019/√Re        (47 < Re < 200)
Re = 100 → St ≈ 0.167   Re = 200 → St ≈ 0.185
h/a = (1/π)·arcsinh(1) ≈ 0.2806   ← von Kármán stability ratio

Two opposite-sign vortices shed per cycle, one from each side. Von Kármán (1911) showed the lateral-to-longitudinal vortex spacing ratio must equal h/a ≈ 0.2806 for the double row of point vortices to remain stable; any other ratio disrupts the pattern quickly.

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Vortex-induced vibration and lock-in

Shedding produces an oscillating lift force at frequency f_s and a weaker drag ripple at 2f_s (C_D ≈ 1.35, C_L amplitude ≈ 0.33 at Re = 100). If a cylinder is elastically mounted, shedding frequency can "lock in" to the structure's natural frequency f_n over reduced velocities U_r = U/(f_n·D) ≈ 4–8, driving oscillation amplitudes of 1–2 diameters — far beyond the ≈0.01D seen outside lock-in. Modern bridges use aerodynamically refined box-girder cross-sections with Strouhal numbers chosen to dodge resonance across expected wind speeds; the same physics is exploited constructively in piezoelectric flow-energy harvesters and even in fish locomotion, where a trout "surfs" a Kármán street shed by an obstacle ahead of it.

Frequently asked questions

At what Reynolds number does vortex shedding begin?

Periodic alternating vortex shedding — the Kármán vortex street proper — sets in around Re ≈ 47, a supercritical Hopf bifurcation where the steady symmetric wake loses stability. Below Re ≈ 5 the flow is steady and attached; above roughly Re ≈ 190 the wake becomes three-dimensional.

What is the Strouhal number?

The Strouhal number St = f·D/U is the dimensionless vortex shedding frequency, where f is shedding frequency, D is cylinder diameter and U is free-stream velocity. In the laminar shedding regime it follows St ≈ 0.2663 − 1.019/√Re, settling near St ≈ 0.2 in the turbulent regime.

What caused the Tacoma Narrows Bridge collapse?

The 1940 Tacoma Narrows collapse is the most cited example of vortex-induced resonance, though modern analysis shows torsional flutter — a different aeroelastic instability — was the primary mechanism, with vortex shedding contributing to the initial oscillations. It remains why long-span bridges use aerodynamically refined cross-sections today.

Try it live

Everything above runs in your browser — open Kármán Vortex Street and watch a real-time Lattice-Boltzmann (D2Q9) simulation shed vortices as you sweep the Reynolds number. Nothing is installed, nothing is uploaded.

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