Stroual instability

Principle of Stroual instability

When in a flow, an obstacle is introduced like a cylinder wich is the most commun example, we find just behind the obstacle a discontinuity of velocity called a slip-stream.

Flow behind a cylinder

In fact this slip-stream is as a double shearing layer with superposition of two vortex sheet but one in the positive direction, the other in the negative direction due to the direction of the vorticity.  This flow  can give instability which is a kind of double Kelvin-Helmholtz instability in different directions. The result is a Von Karman alley with alternated vortex detaching.

Von Karman alley behind a cylinder

This phenomenon of Stroual instability depends on Reynolds number. The instability begins to appear for a critical theoretical Reynolds number of 47 to approximatively Reynolds number of 200. After this limit we have turbulence.

Frequency of vortex detaching is given by Stroual number which is an adimensional number representing the ratio between frequency of vortices detaching f and an other frequency characteristic to movement with the velocity U. This number is constant for a flow behind a cylinder with diameter d and experimentally it is equal to 0.2.

We can see that frequency of vortices is directly linked to velocity in the fluid.

As this instability is like Kelvin-Helmholtz instability, we are not going to speak about mathematical theory (see Theory of Kelvin-Helmholtz instability) but we will prefer to show the manisfestations of such instability.

Manifestations of Stroual instability

Like for Kelvin-Helmholtz instabillity, manifestations of Stroual instability are often seen in industry or Nature. It can occur since there is an obstacle in a flow.
The consequence of such instability can be dangerous or not following the case.

For example, on the photography below you can see Von Karman alley which develops behind a wreck of a boat under sea water. The water flow is deviated by the wreck and inside the slip-stream a perturbation is amplified to give Stroual instability.

Stroual instability behind a wreck of a boat

In this case, Stroual instability has not irreversible consequence over the wreck or the sea water. But in some cases where frequency of vortex detaching is the same than resonance frequency of the obstacle this phenomenon of instability is very dangerous.
We are going to illustrate this with the example of Tacoma Bridge.

Tacoma bridge was a suspended bridge by cylindric ropes able to endure big stress. When wind blows Stroual instability can develop behind each ropes.
During few days, wind blew causing vortex detaching. The problem was that the frequency of vortex detaching was unfortunately identical to resonance frequency of the brigde so brigde enters in resonance. It began to move and like wind didn't stop blowing it broke down after some hours.

 Tacoma bridge entering in resonance Other view of the bridge in resonance Breaking down of the Tacoma bridge

This phenomenon can occur also in water behind pillars bridge. On offshore petrol platforms,
pillars are consolidated to avoid such inconveniences.

In aerodynamics also people try to avoid such instabilities because it reduces lifting capacity.

The obstacle can be a cylinder, like we have seen before, but the shape of the obstacle can be different.

Stroual instability behind different obstacles