The Vortex Street

A Cylinder in a Stream

What the Solver Was Not Told

The viscosity is the only thing set by hand, through the relaxation time: ν = (τ − ½)/3 in lattice units, which fixes Re = UD/ν. Everything below the flow is then whatever the equations do.

Where the shedding starts. Below a critical Reynolds number the wake is steady and symmetric; above it, it oscillates. Nudging the wake once and watching whether the disturbance grows or dies:

So this flow starts shedding between Re = 44 and Re = 46. The published value for a circular cylinder is Re = 47. The panel was never given that number, and it landed within about five percent of it, low, which is the direction a channel of finite width pushes it.

How often it sheds. The Strouhal number St = fD/U is the shedding frequency made dimensionless. Measured off the probe at four Reynolds numbers:

Roshko’s form for this curve is St = A + B/Re. Fitting those four measured points gives A = 0.2139 and B = −3.94, or equivalently St = A(1 − 18.4/Re), with every residual under 1.2%. A is the asymptotic Strouhal number, the roughly 0.2 that a circular cylinder is famous for, and it comes out of a solver that was handed a viscosity and nothing else.

Where the third digit goes. Two convergence tests, which matter more than the agreement above because they say how much of it is physics. Holding everything else fixed and changing only the resolution, then only the speed:

Compressibility is not the error: doubling the Mach number moves the answer by 0.3%. Resolution is. Over a factor of two in cell size the answer moves 2.8% and, worse, it is still climbing: the steps are +0.0024 and +0.0026, so the trend has not begun to flatten and the converged value is somewhere above 0.186. The third digit of St is not earned, and this page does not claim it.

Why the Slider Stops at 180

A real cylinder wake stops being two-dimensional at about Re = 180, where a spanwise instability sets in and the vortices develop structure along the cylinder that no flat calculation can carry. Past that, this panel would go on showing a tidy alternating street, and the tidiness would be exactly the lie: the real flow has left the plane the solver is working in. So the range ends where the physics does, on the same principle as the 17.1 mT cap on the ferrofluid panel.

Why This Is an Attractlet

This is the one that nearly isn’t. Every other panel here has an obvious external hand: a field, a shaker, friction, a solver, a person doing arithmetic. The vortex street genuinely does something the others do not. Each vortex that peels off changes the pressure on the other side of the cylinder and starts the next one. The triggering is internal. That is a real feedback loop, and it is why the shedding has a frequency of its own rather than one imposed on it.

So the question has to be asked properly, and asking it properly is what the framework is for. The street produces the timing. It does not produce the energy. Every vortex in that train is made of momentum that arrived from upstream, and the structure that organizes it contributes precisely none of it. Press Cut the flow: the vortices do not reorganize, hold, or find another supply. They spin down where they are, and the street is gone.

Producing your own timing is not the same as producing your own continuation. That distinction is what the card tag means by supplied, and this panel is the sharpest place in the set to feel the difference, because here the first half is genuinely true.

The Shape of the Basin

Press Disturb the wake: a slug of dead fluid is dropped into the train a few diameters downstream, big enough to wipe out several vortices. The wake does not need to be restarted. The shedding at the cylinder never stopped, the hole washes downstream and out, and the train closes up behind it at the same frequency it had before.

That is a real basin, and it is a narrow one in an interesting way: what returns is the frequency and the spacing, not the phase. The street that reforms is not the one you disturbed; it is a street with the same measurements. Returning to a set, again, rather than to a state.

What This Model Is Not

Two-dimensional, as above, which is why the range ends at 180.

Weakly compressible. Lattice Boltzmann is not exactly incompressible; the error goes as the square of the Mach number. Measured above at 0.3% across a doubling, so this one is under control and it is said here only because it would otherwise be an unstated assumption.

Confined. The cylinder blocks about a tenth of the channel. Published Strouhal correlations are for an unbounded stream, and walls that close raise the shedding frequency. This is the most likely reason the measured onset sits a little below 47 and the fitted curve a little above Roshko’s.

Under-resolved. Twelve cells across the cylinder, which the convergence test above shows is not enough for three digits. It is enough for the shape of the answer and for the onset.

The staircase cylinder. A circle drawn on a square lattice is a staircase, and simple bounce-back puts the wall halfway between nodes. Finer boundary treatments exist; this panel does not use one, and part of the resolution drift above is that.

Jiang, H. & Cheng, L. (2017). Strouhal-Reynolds number relationship for flow past a circular cylinder. Journal of Fluid Mechanics, 832, 170–188. Accepted manuscript. Source of the onset at Re = 47 (after Henderson, 1997), of the form St = A + B/Re being Roshko’s, and of the three-dimensional transition near Re = 180.
Roshko, A. (1954). On the development of turbulent wakes from vortex streets. NACA Report 1191.
von Kármán, T. (1911). Über den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen.
Chen, S. & Doolen, G. D. (1998). Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30, 329–364.