Kármán Vortex Streets and Hydraulic Jumps: Why Flowing Water Breaks Pattern

Steady flow goes in; a pattern comes out. Wind blowing past an island leaves a zig-zag of swirling clouds, and fast water sliding across a channel floor suddenly rears up into a tumbling wall. Both are one-way, energy-losing processes, so this post builds on our explainer Why Does Time Only Go Forward? The two effects are drawn by Vortessa, our vortex-street character, and Saltara, our hydraulic-jump character, both of whom you can find in our sticker collection.

What is a Kármán vortex street?

When fluid flows past a blunt object such as a cylinder, the flow peels away from both sides. Once the flow is fast enough (Roshko’s regular range starts at a Reynolds number of about 40, defined below), the two sides roll up into vortices that leave alternately from the top and bottom. The result is two staggered rows of whirlpools spinning in opposite directions: a vortex street.

Theodore von Kármán described it mathematically while working as a graduate assistant for Ludwig Prandtl, though “a French scientist was the first to photograph the feature” (NASA Earth Observatory^[1]). NASA dates his description to 1912; Anatol Roshko’s 1954 NACA report says “In 1911 Kármán gave his famous theory,” and credits Henri Bénard with pointing out the phenomenon in 1908 (Roshko 1954^[2]).

Why are the vortices staggered and not side by side?

Von Kármán asked which arrangement of two vortex rows could last. His answer, in his own later words: “I found that only the anti-symmetric arrangement could be stable, and only for a certain ratio of the distance between the rows and the distance between two consecutive vortices of each row” (NASA Earth Observatory^[1]). Side-by-side pairs quickly fall apart. Offset ones persist, which is why the street is always a zig-zag.

How fast do the vortices shed? The Strouhal number

Two dimensionless numbers do almost all the work:

St = fD/U and Re = UD/ν

Here f is the shedding frequency from one side of the cylinder (in hertz), D is the cylinder diameter (m), U is the speed of the oncoming flow (m/s) and ν is the fluid’s kinematic viscosity (m²/s). The Strouhal number St, named for Strouhal, whose first measurements date from 1878, says how many vortices leave per unit of flow passing by. The Reynolds number Re compares inertia with viscosity. Roshko uses S for St and R for Re (Roshko 1954^[2]).

Roshko measured the wakes of cylinders and fitted, for 50 < Re < 150:

St = 0.212 (1 − 21.2/Re)

with a slightly different fit, St = 0.212(1 − 12.7/Re), for 300 < Re < 2,000 (Roshko 1954^[2]). At Re = 100 the first fit gives St ≈ 0.167. So if a 1 cm cylinder sits in a 1 cm/s flow whose Reynolds number is 100, each side sheds f = St·U/D ≈ 0.167 × (0.01 m/s)/(0.01 m) ≈ 0.17 vortices per second, about one every six seconds from each side.

When does the street turn turbulent?

Roshko found three regimes. “At R=40 to 150, called the stable range, regular vortex streets are formed and no turbulent motion is developed.” Between 150 and 300 is a transition range. Above that, in the irregular range, the wake becomes fully turbulent within 40 to 50 diameters downstream (Roshko 1954^[2]).

Where can you see one?

From orbit. On 14 April 2025 NASA’s Aqua satellite (MODIS) imaged cloud vortex streets downwind of the Kuril Islands. The vortices “are sized in proportion to the diameter of the islands that created them”: 3 km-wide Chirinkotan Island made small whorls, while Ketoy Island, about 10 km across, left larger vortices in its wake (NASA Earth Observatory^[3]). Engineers watch for the same thing on land: “tall, cylindrical structures can also shed von Kármán vortices,” which can “exert damaging forces on structures,” and “devices such as dampers and fins can be used to counteract them” (NASA Earth Observatory^[3]).

What is a hydraulic jump?

Hubert Chanson’s definition: “A hydraulic jump is the turbulent transition from a high velocity into a slower flow.” Its moving cousin, the tidal bore that runs up some rivers, is a “hydraulic jump in translation” (Chanson 2012^[4]). You see a stationary jump below a dam spillway or sluice gate: shallow, fast water hits deeper, slow water and the surface leaps up in a churning roller.

How high does the water jump? The Bélanger equation

Conservation of mass and momentum across the jump gives, for a horizontal rectangular channel without friction, the classical Bélanger equation (Bélanger 1841) (Chanson 2012^[4]; Ponce^[5]):

d₂/d₁ = ½(√(1 + 8Fr₁²) − 1) and Fr₁ = V₁/√(g d₁)

Here d₁ is the water depth before the jump and d₂ the depth after (m), called the conjugate or sequent depths; V₁ is the incoming speed (m/s); g ≈ 9.81 m/s² is gravitational acceleration; and Fr₁ is the upstream Froude number, the flow speed compared with the speed of small surface waves. A jump needs Fr₁ > 1, called supercritical flow. At Fr₁ = 1 the formula gives d₂ = d₁ and there is no jump.

A worked number

Take water 0.10 m deep moving at 5.0 m/s. Then Fr₁ = 5.0/√(9.81 × 0.10) ≈ 5.05, and the Bélanger equation gives d₂/d₁ ≈ 6.66: the water leaps from 10 cm to about 67 cm deep and slows to about 0.75 m/s. Using the same mass and momentum balance, the energy per unit weight carried by the flow (its “specific energy,” d + V²/2g) drops from about 1.37 m to about 0.69 m. Roughly half the incoming mechanical energy is turned into turbulence and finally heat. You cannot run a jump backward, which is why it belongs with the story of entropy.

Is the ring in your kitchen sink a hydraulic jump?

Open point. When a tap jet hits a flat sink, it spreads as a thin fast film and then thickens abruptly in a ring. For channels, gravity sets the jump, as above. For this thin-film ring the cause is debated. In 2018 Bhagat, Jha, Linden and Wilson argued that “surface tension and viscous forces balance the momentum in the liquid film and gravity plays no significant role,” backed by experiments that showed no dependence on the orientation of the surface (Bhagat et al. 2018^[6]). In 2019 Duchesne, Andersen and Bohr replied: “We show that this approach is incorrect,” arguing that surface tension’s effect is “fully contained in the Laplace pressure” and, for thin films, “much smaller than claimed” (Duchesne et al. 2019^[7]). We have not seen a settled verdict, so we leave it open.

FAQ

What causes a Kármán vortex street?

Flow separating from both sides of a blunt body rolls up into vortices that leave alternately. Von Kármán showed only the staggered arrangement can be stable.

What is a typical Strouhal number for a cylinder?

Roshko’s fits give about 0.17 at Re = 100, rising to about 0.21 at Reynolds numbers near 2,000.

What does a hydraulic jump do?

It turns a fast, shallow flow into a slow, deep one in a short turbulent roller, and it dissipates a large share of the flow’s energy.

Is a tidal bore a hydraulic jump?

Yes, a moving one. Chanson calls tidal bores and positive surges a “hydraulic jump in translation,” and the same mass and momentum balance applies.

References

1. NASA Earth Observatory, “The Stability of von Kármán’s Vortices”. https://science.nasa.gov/earth/earth-observatory/the-stability-of-von-karmans-vortices-147743/
2. A. Roshko (1954), “On the Development of Turbulent Wakes from Vortex Streets,” NACA Report 1191 (NASA NTRS). https://ntrs.nasa.gov/api/citations/19930092207/downloads/19930092207.pdf
3. NASA Earth Observatory, “Vortex Variety Hour” (Kuril Islands, MODIS Aqua, 14 April 2025). https://science.nasa.gov/earth/earth-observatory/vortex-variety-hour-154293/
4. H. Chanson (2012), “Momentum Considerations in Hydraulic Jumps and Bores,” J. Irrig. Drain. Eng. 138, 382–385 (DOI 10.1061/(ASCE)IR.1943-4774.0000409). https://www.uq.edu.au/~e2hchans/reprints/jide_2012_4.pdf
5. V. M. Ponce, San Diego State University, “Calculation of the sequent depth of a hydraulic jump” (online calculator). https://ponce.sdsu.edu/onlinechannel11.php
6. R. K. Bhagat, N. K. Jha, P. F. Linden & D. I. Wilson (2018), “On the origin of the circular hydraulic jump in a thin liquid film,” J. Fluid Mech. 851, R5 (DOI 10.1017/jfm.2018.558). https://doi.org/10.1017/jfm.2018.558
7. A. Duchesne, A. Andersen & T. Bohr (2019), “Surface tension and the origin of the circular hydraulic jump in a thin liquid film,” Phys. Rev. Fluids 4, 084001 (arXiv:1903.11495). https://arxiv.org/abs/1903.11495

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.