von Kármán Vortex Street - An Amazing Phenomenon!
What is von Kármán vortex street?
“It is a repeating pattern of swirling vortices, caused by vortex shedding”
“The Kármán vortex street is an exquisite flow pattern that can be seen in a variety of flow situations involving bluff or blunt bodies”
We’ll begin our stroll down the street with an elaborative explanation to what exactly it is? where is it actually seen? Last, why is it formed?
This phenomenon is ubiquitous but can be latent to our eyes. This can be seen in a broad spectrum of examples, for instance this flow pattern exists around an Aeolian harp string which is a few millimeters wide to a large island which is a couple hundred kilometers wide. This pervasiveness combined with its beauty and simplicity has resulted in it being one of the most visualized and familiar (or maybe not) of all flow phenomena.
Surely one of those images don’t fit in!
The top two satellite images are taken from the NASA Earth Observatory, the panoramic image on the bottom is the infamous Tacoma Narrows Bridge. From the satellite images, the repeating pattern of the Kármán vortex street around the archipelago can be easily spotted. But nothing of that sort is seen on the bottom image. This bridge collapsed in 1940 due to an aeroelastic flutter. Check out the story on Tacoma Narrows Bridge failure by SimScale for an intricate engineering explanation. In gist, the cause is forced mechanical resonance. The frequency of vortex shedding and the natural frequency (frequency at which a system tends to oscillate in the absence of any driving or damping force) of the bridge were matched and this led to cataclysmal. This phenomenon has happened to be favorable in recent years. One of its major applications is an energy harvester for sensors which have power consumption in ~mW.
The earliest observation of the Kármán vortex street was recorded in a painting in 14th century, even before the Renaissance period!
Let’s begin with the why…
It’s imperative to understand how these flow patterns are formed and this is going to be by far the most interesting part. I know the quoted definitions are not intuitive, so I’ll try to explain every word in the best possible manner. I’ll give a brief understanding on three concepts before I explain the reason behind the formation of Kármán vortex street.
Boundary layer
Consider a stack of similar books kept on top of each other and perfectly aligned, placed on the surface of a table. Since every surface, to a certain degree, has roughness and due to this there exists friction between the books as well as the books and the table. When the stack of books are pushed across the table, the book which is in contact with the table travels slower than the books which or on the top. The primary reason for this is the frictional force. The above macroscopic illustration is just an inflated depiction of boundary layer concept. It is applied in molecular level. Now replace the stack of books with stack of fluid layers and replace the frictional forces with viscous forces, then read the quoted definition below.
“a boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant”
The above illustration is similar to the one before with the books, except that the books are replaced with fluid. The change in gradient of red color shows the variation in velocity and how it increases from the wall. Here, the wall is just an arbitrary surface and it could be anything. The blue color shows the velocity of the fluid before it comes in contact with the wall and this velocity is called the free stream velocity. The region between the wall and the free stream velocity is known as boundary layer thickness. The difference between the left and right diagrams is its viscosity. Fluids with different viscosities will have different velocity profiles. (I hope this analogy was intuitive enough to get an understanding of boundary layer theory)
Adverse pressure gradient
A gradient points in the direction of the steepest ascend. It’s a vector function which associates itself with the position at a given time.
The pressure gradient describes in which direction and at what rate the pressure increases the most rapidly around a particular location.
Generally, the static pressure decreases in the direction of flow and this enables the fluid to accelerate, this is known as favorable pressure gradient. But why? The fluid accelerates because, the static pressure which can be considered as the fluid energy reduces and this correlates to decrease in potential energy. The consequence of reduction in potential energy is increase in kinetic energy (from law of conservation of energy) , hence the fluid accelerates.
What could possibly be the adverse version of this?
Consider a fluid flow in which the static pressure increases instead of decreasing. This will lead to deceleration of the fluid. The effect of this is an abrupt increase in pressure gradient in the opposite direction. This is known as the adverse pressure gradient.
The above figure illustrates a boundary layer on in which the flow has decelerated and completely reversed the direction of flow due to adverse pressure gradient.
Flow separation:
Flow separation is also known as boundary layer separation and as the name suggests it is the detachment of the boundary layer from a surface. The detachment of boundary layers is mainly caused due to adverse pressure gradient. In adverse pressure gradient the fluid in continuously decelerated and at one point it becomes zero and reverses its direction. This reversal leads to flow separation. The reversed flows causes recirculation in the low pressure region. The point where the velocity of the fluid layer becomes zero is the point of separation, this is where the adverse pressure gradient and the boundary layer detachment.
Lastly, the Kármán vortex
Now consider a sphere as a blunt or a bluff body which is placed in the fluid flow. Combining the three concepts which we reviewed earlier, we can see that there is boundary layer detachment i.e., flow separation due to adverse pressure gradient. The decrease in the momentum of the flow is the primary cause of adverse pressure gradient. Since the bluff body is symmetric in nature, the flow separation is almost identical on top half of the sphere and bottom half of the sphere. The adverse pressure gradient is observed in the downstream or the wake region of the sphere. The pressure variations along boundary layer is generated predominately due the curved surface if the sphere. Vortices are also generated alongside recirculation in the region of reversed flow. These vortices start to swirl in repeating manner. These low-pressure alternating vortices tend to repeat at certain frequency, this is called frequency of vortex shedding. Thus, the name vortex street?!
Hope you liked it :)