Why does the vortex form in the von Karman Vortex Street?

Question

Why does the vortex form in the von Karman Vortex Street?

So far I have this for the reasoning: 'It occurs when a fluid flows around a cylindrical object. The pressure on a fluid particle rises to the stagnation pressure as it hits the leading edge of the object. As it trails along the boundary layer of the rounded surfaces (one on each side of the cylindrical object), the high pressure is not sufficient enough to force the flow about the back of the cylinder. Therefore the, near the widest section of the cylinder, the boundary layers separate from the surface of each side of the object and form two shear layers that flow into the wake. The slower of the shear layers, flows closest to the object. This is when the vortex start to occur. As this slower shear layer rolls into the wake before the faster layer, they fold on each other and form swirling vortices'.

The bolded part is specifically what I don't understand. High pressure isn't sufficient enough to force the flow about the back of the cylinder? What does this mean? I would assume that it IS the pressure that forces the flow down and into the start of a vortex. Any help would be appreciated.

Answer 1

So the vortex street appears as the stream goes faster and faster: the stream obeys Newton's laws, "objects in motion tend to stay in motion unless acted on by a net force."

The force that keeps the stream laminar around the cylinder is fluid pressure formed by the viscous forces of the fluid, which means there is a pressure gradient and a region of low pressure is being formed behind the cylinder. (This also doubles as one explanation of why, when you let go of the cylinder, it starts to flow downstream. Inverting that explanation, the fact that the rod has to be held in place means that these pressure gradients must exist.)

But as you increase the speed of the fluid and hence its momentum, if you do not alter other parameters to keep the Reynolds number constant, then the fluid flow lines must separate from the cylinder. You're increasing the inertial forces in the fluid but the viscous forces do not rise to compensate them and keep the flow laminar. This causes the fluid boundary to detach from the cylinder.

The fluid that's stuck inside the wake then receives shear forces from these two boundaries flowing around it on either side. For a certain parameter regime the cylinder just keeps two vortices in its wake, fed by the shear forces from the boundary layer.

The vortex street happens when these shear forces get to be so large that they actually push the vortices harder than the pressure gradients are pushing them back. If you could theoretically get the symmtery exactly right, both vortices would be shed together by symmetry: but it's like trying to balance a pencil on its point, the opposite rotation of the vortices forms effectively a force which repels them. Some asymmetry is inevitable, and they start to shed alternately. This creates the vortex street.

Does that help?

Answer 2

This is actually not a completely settled question, so don't feel too bad if you don't understand it in every detail! It's also possible that the cause is not the same in all situations, and be aware that my explanation is not universally accepted, but here's my interpretation:

There are actually two interacting phenomena going on: the formation of the attached eddies (vortices), and the wake instability. The formation of the eddies occurs because the fluid flowing past the cylinder, but at some distance away, is moving fast relative to the fluid directly behind the cylinder. The shear (or, if you like, the low pressure behind the cylinder) redirects the flow in towards the center of the wake and ultimately (some of it) back towards the cylinder, creating the two symmetrical eddies, one on each side. This is a steady-state solution to the Navier-Stokes equations given that the flow be zero at the boundary of the cylinder - but it is not necessarily a stable solution.

Meanwhile, in the wake instability, for certain flow parameters the center line of the cylinder wake begins to wobble. The reason for this still under study, and it is unclear exactly what role the attached eddies play at the beginning, if any, but it is a well-known instability. (This is the controversial part - many attempted explanations of the vortex shedding instability ignore the wake instability, or assume that the wake instability is a consequence of the instability of the attached eddies and therefore that it can't be a cause of the vortex shedding. I think both of those lines of reasoning are erroneous, though there is certainly a mutual interaction between the attached eddies and the wake instability.)

As you increase the flow rate, two things happen: the attached eddies get longer, and the amplitude of the wake instability increases. Eventually the eddies are long enough and the wake instability large enough that the peaks in the lateral wake flow can pinch off the eddies. Because the wake instability is wobbling back and forth, it pinches the eddies off alternately on one side and then the other, and you get the vortex street.

It is known that the interaction between the vortex shedding and the mean wake is nonlinear, in the sense that the most unstable mode of the initial steady wake initiates the instability, but then as the vortices grow they alter the mean flow, which changes the most unstable mode, etc., until a saturation point is reached. That saturation point is what is usually observed as the vortex street.