The wind tunnel: one number decides the flow

A moving fluid has two tendencies. Its momentum carries it straight on; its viscosity, the internal friction that makes honey different from water, smooths out differences in speed. The Reynolds number compares them:

with U the speed of the stream, D the size of the object and $\nu$ the viscosity. At low Re, friction wins and the flow is smooth and orderly. At high Re, momentum wins, and the flow separates from the object and breaks into eddies.

Osborne Reynolds introduced it in 1883 after watching dye in water flowing through glass pipes. Its power is that the flow depends on U, D and $\nu$ only through Re: a model aircraft in a wind tunnel and the real aircraft behave alike if their Reynolds numbers match. That is why the slider here sets Re rather than a speed or a size.

• Re below about 5: the flow closes up neatly behind the cylinder, almost the same front and back.
• Re from about 5 to 47: the flow separates from the back of the cylinder, and two still eddies sit side by side behind it. They grow longer as Re rises.
• Above Re ≈ 47: the pair cannot stay balanced. The wake starts to swing, and eddies break away from alternate sides, forming two staggered rows of vortices spinning in opposite directions. Theodore von Kármán explained in 1911 why that staggered arrangement is the stable one, and it bears his name: the Kármán vortex street.

Vortex streets are everywhere once you look: behind islands in satellite images of clouds, behind bridge piers, behind the cables that sing in a strong wind.

Each eddy that breaks away gives the cylinder a sideways push, alternately up and down. The chart shows that push as the lift coefficient swinging back and forth, and the lab counts the swings. Their frequency f, made dimensionless, is the Strouhal number:

Experiments put it near 0.16 at Re = 100, rising towards 0.2 by a few hundred (Williamson, 1996). Here it comes out a little higher, for a known reason: the tunnel is narrow, and the object takes up about an eighth of its height (the blockage in Controls). Squeezed between the walls, the flow past the object speeds up, and so does the shedding. Real wind tunnels correct their measurements for exactly this.

The shedding frequency is also why wires hum in the wind. A telephone wire a few millimetres across in a moderate breeze sheds eddies a few hundred times a second, which is in the range you can hear. The sound is called an Aeolian tone, after Aeolus, the Greek keeper of the winds, and it is what makes an Aeolian harp play.

Tilt an aerofoil slightly and it turns the flow downwards behind it. Pushing air down pushes the wing up: that is lift. The Wing at 6° preset shows the flow staying attached to both surfaces and leaving the trailing edge smoothly, which is the condition, named after Wilhelm Kutta, that fixes how much lift a wing makes.

Tilt it further and the flow over the upper surface has to climb an ever steeper rise in pressure towards the back. Eventually it cannot, and it tears away from the surface in big eddies, shed one after another: the wing has stalled. A full-size wing, at Reynolds numbers in the millions, stalls at around 15° and its lift drops sharply. At the few hundred this tunnel can reach, stall is gentle. Here the lift stops growing between about 15° and 20° while the drag nearly doubles. Beyond that the tilted wing shoves air down like a flat board, gaining lift again but at a heavy cost in drag. Sweep the angle up from 0 and watch both coefficients in the readout.

Rather than solving the Navier–Stokes equations for the flow directly, the tunnel uses the lattice Boltzmann method. Each cell of the grid keeps nine numbers: how much fluid is at rest there, and how much is moving towards each of its eight neighbours. Every step, each of those moves one cell along its direction, then the nine at each cell relax a little towards the balanced mix for their local density and velocity. Averaged over many cells, this simple shuffle obeys the Navier–Stokes equations, with a viscosity set by how quickly the relaxation happens.

The method suits graphics cards, because every cell does the same small job at once, and this page runs it on yours: about 170,000 cells, at a few thousand steps a second on a typical laptop. Fluid that runs into the object bounces back where it came from, which makes the surface non-slip. The force on the object is added up from all those bounces. Where there is no WebGL2, the page runs the same method on the processor at a smaller size.

A few details keep it honest. The stream moves at about a sixth of the speed of sound in the simulation, slow enough that the air behaves as if incompressible. The collision step uses two relaxation rates (the TRT scheme of Irina Ginzburg), which keeps it stable at much lower viscosity than the simplest version. Bands of extra viscosity at the inlet and outlet absorb pressure waves and eddies so they do not bounce back. And Re is capped at the point where the grid becomes too coarse to follow the flow: about 20 times the object’s size in cells.

This tunnel is two-dimensional: the object is really an infinitely long cylinder or wing, and nothing can move in the third direction. Real wakes are different. Behind a real cylinder the vortex street begins to twist in three dimensions at Re ≈ 190, and well before Re reaches the thousands the wake is turbulent. In two dimensions eddies tend to merge into bigger ones, where in three they break down into smaller ones. So read the high-Re end of the slider as irregular flow, not as turbulence.

• Start at Re = 20 and raise it in steps of 5, giving the flow time to settle each time. Find the Re where the wake starts to swing.
• At Re = 150, release a streak of dye just upstream of the cylinder and follow it into the street.
• Compare drag for the cylinder, the square and the flat plate at the same Re. Which is worst?
• Draw a second cylinder in the first one’s wake. Racing cyclists ride close behind each other for the same reason.
• Switch the walls to solid and watch boundary layers grow along the top and bottom.

• O. Reynolds (1883), An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, Phil. Trans. R. Soc..
• C. H. K. Williamson (1996), Vortex Dynamics in the Cylinder Wake, Annu. Rev. Fluid Mech..
• T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva and E. M. Viggen, The Lattice Boltzmann Method: Principles and Practice, Springer (2017).
• I. Ginzburg (2008), Two-relaxation-time lattice Boltzmann scheme: about parametrization, velocity, pressure and mixed boundary conditions, Commun. Comput. Phys. 3, 427.
• D. V. Schroeder, Fluid Dynamics Simulation, an earlier lattice Boltzmann tunnel in plain JavaScript.