Karman Vortex Simulator — Strouhal Number and Lock-In
Evaluate the Karman vortex shedding frequency fs = St V / D from a circular cylinder in real time. From the flow speed, diameter, Strouhal number and the structural natural frequency the tool reports the Reynolds number, the fs / fn ratio and the lock-in regime, and visualizes the wake vortex street and the V-fs chart to explain flow-induced vibration physics.
Parameters
Flow speed V
m/s
Cylinder diameter D
mm
Strouhal number St
—
Natural frequency fn
Hz
With the defaults (V = 5.0 m/s, D = 50 mm, St = 0.200, fn = 25 Hz, air with nu = 1.5e-5 m^2/s) fs = 20.0 Hz, Re about 1.67e4, fs / fn = 0.800, lock-in is no. Increasing V to about 6.25 m/s gives fs / fn = 1.00 and the system enters the lock-in regime.
Results
—
Shedding frequency fs
—
Reynolds number
—
fs / fn
—
Lock-in
Cylinder wake vortex street
The grey circle at the centre is the cylinder. Flow (white arrows from left to right) impinges on it and forms alternating vortices in the wake: red on the upper side (counterclockwise) and blue on the lower side (clockwise). The spacing corresponds to the wavelength lambda = V / fs. Below Re = 300 no vortex street forms, and above Re = 2e5 the wake becomes broadband turbulent.
V-fs chart (Strouhal line)
Horizontal axis: flow speed V (m/s, 0.1 to 30). Vertical axis: shedding frequency fs (Hz). Blue solid line: fs = St V / D. Green horizontal line: structural natural frequency fn. Yellow marker: current operating point (V, fs). When the yellow marker meets the green line the system enters lock-in (0.85 < fs / fn < 1.15). Design practice detunes fs and fn or disrupts shedding by damping and strakes.
Theory & Key Formulas
Shedding frequency: the Karman vortex street from a circular cylinder is released alternately at a frequency set by the Strouhal number, flow speed and diameter.
$$f_s = \mathrm{St}\cdot\frac{V}{D}$$
Reynolds number (air at 20 deg C, $\nu = 1.5\times10^{-5}$ m^2 / s):
$$\mathrm{Re} = \frac{V\,D}{\nu}$$
Lock-in criterion (vortex shedding synchronizes with the structural vibration; amplitude grows sharply):
$$0.85 < \frac{f_s}{f_n} < 1.15$$
$V$ is the flow speed (m/s), $D$ is the cylinder diameter (m), $\mathrm{St}$ is the Strouhal number (about 0.20 for a circular cylinder) and $f_n$ is the structural natural frequency (Hz). The Karman vortex street is steady in the range Re = 300 to 2e5.
What is the Karman Vortex Simulator?
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The Karman vortex street is the row of swirls behind a bridge pier in a river, right? Why are the vortices so neatly arranged in alternating top and bottom?
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Good question. When flow impinges on a bluff body like a circular cylinder or a pier, the Reynolds number Re = V D / nu controls the wake. Once Re passes about 300, vortices start to shed alternately from the top and bottom of the wake at the frequency fs = St V / D, where St is the Strouhal number (about 0.20 for a cylinder over a wide Re range). With the tool defaults (V = 5 m/s, D = 50 mm, St = 0.200) the result is fs = 20.0 Hz and Re about 1.67e4. The alternation arises from a feedback mechanism: once a vortex sheds from one side, the resulting pressure field favors growth on the opposite side, and the process repeats periodically.
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I have heard of lock-in but never quite understood what is dangerous about it. Why does fs / fn = 0.8 show "no"?
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Important question. Normally you would expect resonance only when fs equals fn exactly. For Karman shedding, however, once fs falls within about plus or minus 15 percent of fn the vortices get captured and synchronize with the structure, so fs = fn over a finite band. The lock-in condition is roughly 0.85 < fs / fn < 1.15. With the defaults fs / fn = 20 / 25 = 0.800, just outside the band, so the indicator shows "no". Raise V to 6.25 m/s and you get fs = 25 Hz and fs / fn = 1.000, deep inside lock-in, and the amplitude explodes. The 1940 Tacoma Narrows bridge failed through this kind of coupled VIV and flutter.
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On the chart the blue line is fs = St V / D and the green line is fn. Is the intersection the most dangerous flow speed?
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Exactly. The crossing of the two lines defines the critical speed Vcr = fn D / St. With the defaults (fn = 25 Hz, D = 0.05 m, St = 0.20) Vcr = 25 times 0.05 over 0.20 equals 6.25 m/s. Move the slider close to 6.25 m/s and the yellow marker lands on the green line and the lock-in indicator flips to "yes". In practice you detune fn so that Vcr lies outside the operating speed range, or you fit helical strakes (the spiral protrusions you see on tall steel chimneys) to break the spanwise coherence of the shedding. Offshore riser pipes treat VIV suppression as the single most important design item.
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If the Strouhal number is essentially fixed at St = 0.20, why does the slider let me change it between 0.18 and 0.22?
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Good catch. St for a circular cylinder is indeed close to 0.20 over the subcritical range Re = 1e2 to 1e5, but it does drift with Re, surface roughness and free-stream turbulence and ends up in the band 0.18 to 0.22. Non-circular sections diverge much more, with St of about 0.13 for a square cylinder, 0.06 to 0.15 for bridge decks and so on. So even for the same V, D and fn the predicted fs and fs / fn ratio shift, and you can see the resulting uncertainty by moving the slider. In real design you fix St from wind-tunnel data and check the most conservative bound for chimneys and bridges.
Frequently Asked Questions
The Karman vortex street is the alternating row of vortices shed from a bluff body such as a circular cylinder or a bridge pier when the Reynolds number Re is roughly in the range 300 to 2e5. The shedding frequency is fs = St V / D, where St is the Strouhal number (about 0.20 for a circular cylinder), V is the flow speed and D is the cylinder diameter. With the default values of this tool (V = 5 m/s, D = 50 mm, St = 0.200) the result is fs = 20.0 Hz with Re about 1.67e4. The phenomenon is responsible for vibration and noise on bridges, stacks, cables and pipes, and has caused major failures such as the 1940 Tacoma Narrows bridge collapse.
The Strouhal number St = fs D / V is a dimensionless shedding frequency that captures the rhythm of vortex release. For a circular cylinder St remains close to 0.20 over the wide Re range 300 to 2e5, which is the most striking feature of the Karman vortex street. In the subcritical regime (Re = 1e2 to 1e5) St lies in 0.18 to 0.22, while in the supercritical regime (Re above 3.5e5) the turbulent boundary layer can push St up to 0.27 to 0.30 before it drops again. The tool lets you vary St in 0.18 to 0.22 to assess design uncertainty.
Lock-in is the phenomenon where the vortex shedding frequency fs is captured by the natural frequency fn of a flexible structure once they come close, so that fs synchronizes with fn and the vibration amplitude grows sharply. The condition is approximately 0.85 < fs / fn < 1.15. With the tool defaults (fs = 20 Hz, fn = 25 Hz, fs / fn = 0.800) lock-in is not active, but increasing V to 6.25 m/s gives fs / fn = 1.00 and the system enters the lock-in regime. The standard mitigation is to detune fs and fn or to disrupt shedding with damping and helical strakes.
Historical examples include: (1) the Tacoma Narrows bridge collapse (USA, 1940) at a 19 m/s wind through coupled VIV and flutter, (2) the simultaneous buckling of three Ferrybridge power-station stacks (UK, 1965) due to wind-driven Karman shedding, (3) fatigue failures of offshore marine risers due to current-induced shedding, (4) galloping of overhead power lines, and (5) vibration fatigue of heat-exchanger tubes. Standard countermeasures include helical strakes, damping plates, tuned mass dampers and streamlined fairings on bridge decks. The tool visualizes the dangerous fs / fn = 1 region.
Real-world Applications
Wind loading and aeroelastic design of long-span bridges: on suspension and cable-stayed bridges the Karman shedding from the deck excites vertical and torsional modes, accumulates fatigue over decades and in extreme cases reproduces a Tacoma-style collapse. Designers model the deck section (height D about 2 to 4 m) with St about 0.10 to 0.15, scan design wind speeds V = 20 to 40 m/s and obtain fs about 0.5 to 3 Hz, then verify that the deck bending and torsion modes fn are well separated from this band. Enter equivalent values in the tool to compute Vcr and check that the in-service wind speed range avoids the danger window. Common mitigations are streamlined fairings and tuned mass dampers.
Industrial stacks and transmission towers with helical strakes: steel chimneys of 100 to 200 m height with D about 5 m have St near 0.20 and yield fs about 0.8 Hz at a design wind speed V = 20 m/s, which easily coincides with the lowest bending mode fn near 0.5 to 1 Hz. Helical strakes (spiral protrusions on the upper third of the stack with pitch 5 D and width 0.1 D) destroy the spanwise coherence of shedding and remove the lock-in risk. The collapse of three uncladded chimneys at the Ferrybridge power station (UK, 1965) made strakes the de-facto industry standard from then on.
VIV suppression on offshore marine risers: deepwater risers (D about 0.3 m, St about 0.20) connecting a floating platform to a wellhead are exposed to currents of V = 0.5 to 2 m/s, generating fs about 0.3 to 1.3 Hz. The risers are long and slender with densely packed natural frequencies, so lock-in is almost always present somewhere along the riser. Counter-measures include helical strakes, fairings and internal-flow damping. With the V-sweep button you can watch the marker move along the blue line and successively cross the green natural-frequency line.
Vortex-induced vibration of heat-exchanger tubes: in shell-and-tube exchangers the shell-side cross-flow (V = 1 to 5 m/s) sheds vortices from each tube (D = 19 to 25 mm) with fs = 8 to 50 Hz. Adjacent tubes are fluid-elastically coupled, and once a critical reduced velocity is exceeded the coupled mode goes unstable through fluid-elastic instability, leading to tube-to-tube impact and rupture. Designers apply the Connors criterion and add baffle plates to detune the flow. Use the tool to sweep V and D over the typical operating envelope.
Common Misunderstandings and Cautions
The most common misconception is that St = 0.2 can be used as a fixed constant for all design. In reality St drifts with Re even for a smooth circular cylinder: it sits in 0.18 to 0.22 over the subcritical range Re = 1e2 to 1e5, becomes irregular in the transition range Re = 2e5 to 3.5e5 and then climbs to 0.27 to 0.30 in the supercritical range. Square cylinders give St about 0.13, bridge decks give 0.06 to 0.15 and tube banks give different values again. The slider in this tool is calibrated for a circular cylinder, but real designs should use wind-tunnel data appropriate to the section.
The next most common misconception is that moving slightly outside lock-in is always safe. In fact the boundaries 0.85 < fs / fn < 1.15 are soft: amplitudes start growing gradually before they reach the lock-in plateau, and very lightly damped systems (steel chimneys, marine risers, low-tension cables) can lock in across the much wider band 0.7 < fs / fn < 1.4. A quantitative analysis uses the Skop-Griffin number SG = m delta / (rho D^2) (m mass per length, delta logarithmic decrement, rho fluid density) to estimate the amplitude itself. The tool only flags whether the operating point is inside or outside the band.
The last misconception is that Karman shedding occurs only behind circular cylinders. Periodic shedding actually appears behind any bluff section at sufficiently high Re: square cylinders, ellipses, bridge decks, trailing edges of airfoils, chimneys and overhead lines all generate similar wakes. What changes is the value of St (about 0.13 for a square cylinder, 0.06 to 0.15 for bridge decks). When you analyse a new section, always pick St from the literature or from a wind-tunnel test rather than assuming the circular-cylinder value. The tool can give an order-of-magnitude estimate even with non-circular St if used carefully.