Vortex Shedding (Kármán Street)

How vortices peel off a bluff body

A streamlined airfoil sheds the boundary layer cleanly off its sharp trailing edge, producing a thin steady wake. A bluff body — a cylinder, a chimney, a square section — has no such edge. The boundary layer cannot follow the surface around to the back without separating, and the location of separation varies smoothly between the two shoulders.

What happens is that one shoulder forms a vortex first, growing and eventually pinching off into the wake. Pinching off shifts the local circulation balance and helps trigger formation of a vortex on the opposite shoulder. That second vortex then pinches off, and the process repeats. The resulting wake contains a staggered double row of counter-rotating vortices.

Each vortex shed off one shoulder produces a momentary suction on that side of the body. Because the shedding alternates, the body experiences an oscillating side force at the shedding frequency, plus an oscillating drag force at twice that frequency. For a stationary rigid body these forces are mostly a nuisance. For a flexible structure with a natural frequency near the shedding frequency, they're potentially destructive.

The Kármán street, side view

                ┌──┐                                   ↻       ↻
                │  │   →   shed vortices               wake at
   freestream → │  │   →   alternate sides:                ↺      ↺
   v ────────→  │  │   →
                │  │   →                              ↻       ↻
                └──┘   →   spacing ratio b/a ≈ 0.281
                                                        ↺      ↺
   stagnation pt    flow separation at ±80°
                    from front (sub-critical Re)

The vortices drift downstream at about 80% of the freestream velocity. The cross-stream spacing b and along-stream spacing a settle to a stable ratio b/a ≈ 0.281 — a result von Kármán proved by linear stability analysis of the inviscid double-row.

Strouhal numbers across geometries

Body shape	Strouhal St = fD/v	Re range	Notes
Circular cylinder (sub-critical)	0.21	300 to 2×10⁵	Most common reference
Circular cylinder (super-critical)	0.27–0.30	>3.5×10⁶	After drag crisis transition
Square cross-section, face-on	0.13	10⁴ to 10⁶	Sharp corners fix separation
Square, 45° orientation (diamond)	0.16	10⁴ to 10⁶	Effective width changes
Sphere	0.18	3×10⁴ to 2×10⁵	Wake is 3D, less coherent
Streamlined airfoil (small AoA)	~0.05	any	Trailing-edge wake only
Hexagonal prism, flat-on	0.14	10⁴ to 10⁵	Used in bridge piers
Triangular prism, point upstream	0.22	10⁴ to 10⁵	Sensitive to trip

The remarkable thing is how flat St is across many decades of Reynolds number for any given geometry. Once you know the Strouhal number for your shape, you can predict shedding frequency in any flow without further computation.

Worked example: chimney shedding frequency

Take a 100 m steel chimney, 4 m in diameter, in a 12 m/s gale. Strouhal St = 0.21 for a smooth circular cylinder. Shedding frequency:

f = St · v / D
  = 0.21 · 12 / 4
  ≈ 0.63 Hz

Now check the structural side. The first bending mode of a 100 m steel chimney typically falls in the 0.4–0.8 Hz band. The shedding frequency is squarely inside that band — and lock-in will probably engage at this wind speed. Without strakes or a tuned damper, fatigue cracks at the base will appear within months.

Compare a smaller case: a 10 cm cylindrical antenna at 2 m/s. f = 0.21·2/0.1 = 4.2 Hz. The antenna's first bending mode is much higher (typical 30 Hz), so lock-in does not engage and shedding is just a faint hum.

Lock-in: the dangerous resonance

Lock-in is the phenomenon where the structure's vibration captures the shedding rhythm. When the wind speed approaches v_crit = f_n·D/St (the speed at which natural shedding frequency equals the structural natural frequency f_n), three things happen at once:

1. The structure begins to vibrate transverse to the flow.
2. The vibrating motion synchronises vortex shedding spanwise — a much larger fraction of the chimney height sheds in phase, increasing the coherent forcing.
3. The shedding frequency stays locked to f_n over a wind-speed range of roughly ±20%, instead of tracking St·v/D.

Inside the lock-in band, vibration amplitudes can reach 1–2 diameters peak-to-peak. Damage is dose-dependent: total cycles × stress amplitude × stress concentration. Even a brief excursion through lock-in adds millions of fatigue cycles in days.

Real-world cases

• Tacoma Narrows Bridge, 1940. Crosswind at 19 m/s past the H-section deck of width 12 m gave a shedding frequency near 0.2 Hz — close to the deck's first torsional mode at 0.18 Hz. The deck began oscillating, but the catastrophic 6 m amplitude that killed it was sustained by self-excited torsional flutter, a different aeroelastic instability that vortex shedding had triggered.
• Submarine periscopes. A periscope drawn through water at 5–10 knots sheds vortices at 5–15 Hz, exactly in the band where the slim mast can resonate. Modern designs use helical strakes or active damping; the 1960s solution was just to accept the rattle.
• Power-line galloping. Iced transmission cables develop a non-circular cross-section that shifts St into a low-frequency, large-amplitude regime. Cables can swing through 5+ metres, contacting adjacent phases and tripping the line. Stockbridge dampers — small dumbbell weights on stranded cable — bleed energy at the resonant frequency.
• Heat-exchanger tube failure. Tube banks with cross-flow on the shell side shed vortices that excite the tubes; "fluid-elastic instability" beyond a critical velocity makes adjacent tubes whip into each other and crack. A standard mode of failure for first-of-class designs that didn't run a TEMA-AE check.
• Vortex flow meters. Industrial in-line flow meters deliberately use a bluff body in the pipe and count the shed vortices to measure flow. f = St·v/D rearranged: v = f·D/St gives the velocity, hence flow rate, with one moving part (the piezo crystal).
• Burj Khalifa. The Y-plan footprint of the world's tallest building is partly chosen to disrupt vortex shedding — the corners create a different shedding pattern at every height as the tower steps down, preventing any single mode from being excited coherently.

Variants

• Strouhal vs reduced-velocity formulation. Reduced velocity U_r = v/(f_n·D) is the structural-engineering form. Lock-in onset is U_r ≈ 5 for circular cylinders — equivalent to St ≈ 0.2. Two ways to write the same condition.
• Single-mode vs multi-mode shedding. Long structures (cables, chimneys) can have multiple bending modes inside the wind speed range; lock-in shifts mode-to-mode as wind picks up. Designs must pass a fatigue check across all relevant modes.
• Two-cylinder (tandem) shedding. A second cylinder placed in the wake of a first one suppresses the upstream shedding for spacing < 3.5D, then triggers irregular bistable shedding for 3.5–4D, then reverts to independent shedding for >4D. Cooling-tower clusters use this.
• Galloping (across-wind, 1-DOF flutter). Square or D-section bodies above a critical reduced velocity can extract energy from the flow on their own, separate from vortex shedding. Different criterion (Den Hartog), different remedies.
• Wake-galloping. A downstream cylinder driven by the wake of an upstream one — common in twin-cable bridges. Leads to oval orbital motion of the downstream cable.
• 3D streamwise vortices on bluff bodies. At high Re, primary spanwise Kármán vortices coexist with secondary streamwise pairs, complicating the wake structure beyond the textbook 2D picture.

Common failure modes

• Vortex-induced vibration in chimneys. The classic case. The 1965 Ferrybridge cooling-tower collapse killed three towers in a single windstorm; the eventual fix added strakes and tuned dampers to all stacks of similar slenderness.
• Cable-stayed bridge cables. Long, slender, lightly damped — almost a worst-case textbook example. Cross-tying, stockbridge dampers, MR-fluid active dampers, and helical fillets on the cable jacket are all standard mitigations now.
• Tall-building cross-wind acceleration. Even when stress isn't a problem, vortex-induced lateral acceleration above 15–20 milli-g makes a tower's top floors uninhabitable; dampers like the Taipei 101 tuned mass damper (660 t, 5.5 m diameter sphere) are sized for this serviceability limit, not strength.
• Heat-exchanger tube whipping. Cross-flow shell-side velocity above the fluid-elastic critical value, combined with vortex shedding lock-in, can crack tubes within weeks. Standard inspection finding on retrofitted feed-water heaters.
• Subsea risers and pipelines. Currents past long deepwater riser pipes generate 24/7 VIV that accumulates fatigue damage; strakes installed over 80–90% of riser length are the routine mitigation.
• Naval periscope/mast hum. Often a livability issue rather than structural — but a humming periscope at 50 m depth is also a sound signature that an enemy hydrophone can pick up.