The Karman vortex street is one of the most striking images in fluid dynamics.
As a fluid flows past an obstacle, swirling vortices are shed periodically.
I'm Jeff Chasnov and I’m a professor of mathematics.
In this video, we’re going to have a close look at the flow around a circular cylinder.
Laboratory experiments have been performed And so have numerical computations
Flows past obstacles also occur in nature. The Karman vortex street has been observed
in atmospheric flows, when the wind blows over oceanic islands.
The phenomena of vortex shedding can also cause weird behavior on your local highway.
The ancient Aeolian harp sings its melancholic
tune because of vortex shedding and the Karman street.
Here is a particularly large and beautiful harp near the famous Exploratorium in San Francisco.
The Karman vortex street is named after the pioneering aerospace engineer
Theodore von Karman. Von Karman was born in Budapest in 1881
and immigrated to the United States before the second World War.
He was among a prominent group of Hungarian scientists who contributed to the American
war effort. The joke was that this group spoke a strange foreign language
and came from a little known country. With super human intelligence, perhaps they were
actually descendants of extra-terrestrials, and thus they became known as the Martians.
In 1963, then President John F. Kennedy presented von Karman with the National Medal of Science:
“Gentleman, Dr. von Karman. It is a great pleasure for me to select you as the first
recipient of the National Medal of Science. This nation and indeed the entire free world,
holds you in the highest esteem and respect for your devoted service,
for your scientific achievements and for your warmly human gifts as a teacher and councilman.
Your assistance to the United States air force and the NATO advisory group for aeronautical research
and development have been outstanding. We are also deeply indebted to you for your continuing
efforts in the promotion of international cooperation in science and engineering.
It is hard to visualize what the world would be like without aircraft and jet propulsion,
without the vision we have, just entering the realm of reality, of exploring space.
I’m especially glad to present this first national medal of science to one of the pioneers,
who has helped make all of this new and exciting age possible.”
Von Karman may have pioneered our understanding of aeronautics and astronautics, and we have become
quite good at building airplanes and spacecraft, yet fluid dynamics remains a difficult subject.
The governing equations are nonlinear partial differential equations called the Navier-Stokes
equations. Analytical solutions of these equations exist only for very simple flows,
so we need to solve these equations numerically, often using supercomputers.
In their simplest form, these equations contain one dimensionless parameter,
called the Reynolds number, Re,
named after Osborn Reynolds, who in the late 1800’s established the condition in which
fluid in pipes transitioned from a laminar smooth flow to a turbulent chaotic low.
We know that low Reynolds number flows are laminar And high Reynolds number flows are turbulent
In an often repeated story, we are told that the German physicist
Werner Heisenberg, one of the founders of quantum mechanics,
Said on his death bed, "When I meet God, I am going to ask him two questions:
Why relativity? And why turbulence?
I really believe he’ll have an answer for the first."
The Karman vortex street at low Reynolds numbers is a periodic laminar flow,
yet there is no simple analytical solution. Even the zero Reynolds number flow
past a cylinder, the polar opposite of turbulence, is a very difficult problem to solve.
George Stokes tried to find a solution in 1851 and his results are now known as the Stokes’ paradox.
You can’t solve this problem by simply setting the Reynolds number equal to
zero because the resulting equations together with their boundary conditions doesn't have a solution.
Oseen modified Stokes’ equations in 1910 and was able to provide a good approximate solution.
The analysis of the Karman street came soon after. Von Karman writes:
“In 1911, I was a graduate assistant in Gottingen, [working with the physicist
Ludwig Prandtl. At that time, Prandtl’s main interest was in the theory of
the boundary layer, the flow of a fluid very close to the surface of a body.
Prandtl had a doctoral candidate, Karl Hiemenz, to whom he gave the task of
constructing a water channel in which he could observe the separation of the flow
behind a cylinder. The object was to check
experimentally the separation point calculated by means of the boundary-layer theory.
For this purpose, it was first necessary to know the pressure distribution around the cylinder
in a steady flow. Much to his surprise,
Hiemenz found that the flow in his channel oscillated violently.
When he reported this to Prandtl, his mentor told him: “Obviously, your cylinder is not circular”
However, after very careful machining of the cylinder, the flow continued to oscillate.
Then Hiemenz was told that possibly the channel was not symmetric, and he started to adjust it.
I was not concerned with this problem, but every morning when I came in the laboratory I asked him,
“Herr Hiemenz, is the flow steady now?” He answered very sadly, “It always oscillates.” "
So von Karman went home for the weekend and worked out the stability of an arrangement of vortices
and determined that the vortex street was the stable configuration.
Von Karman never claimed to have discovered the street that was named after him.
He mentions a fresco from a church in Bologna, Italy where St. Christopher is shown carrying
the child Jesus across a flowing stream. Behind the saint’s foot the painter drew
alternating vortices that resembles a Karman street.
The vortices were also observed earlier by a French professor, Henri Benard,
now well-known for Rayleigh-Benard convection. von Karman claims that Benard was somewhat jealous
about the naming of the vortices. So von Karman told him,
I agree that what in Berlin and London is called Karman street
in Paris shall be called Avenue de Henri Benard. And von Karman claims that after that wisecrack,
they made peace and became good friends. Back in 1982, when I was still a student,
Milton van Dyke published his book, An Album of Fluid Motion. Here's a copy from our library.
You can see the von Karman street on the cover and there are many visualizations of the street
in the book. This ensured the popularity of the street in the fluid dynamics community.
Nowadays, with fast computers and commercial software,
students are able to simulate the von Karman Street in their engineering courses.
Some adventurous students may even try to write their own code.
A two-dimensional simulation may start with the Navier-Stokes equations for the stream function
and the vorticity in log-polar coordinates; the obstacle, and a finite difference grid.
The equations are discretized And a numerical scheme is chosen.
Sounds easy? Well …. the numerical method needs to be stable and efficient, and the code needs to be
bug-free and fast. Still not an easy problem, so most students use packaged software.
Like Open FOAM Or ANSYS Fluent
I wrote my own code and can show you some of my solutions.
It’s easy to visualize the stream function and the vorticity of the fluid using color
and contour lines of constant values. In a steady flow, the contour lines of the stream function
show us the direction of the fluid motion. You can literally see how the fluid flows by
simply following the contour lines. The vorticity shows us the local spinning motion of the fluid.
It's a bit harder to interpret, but we'll see that for an unsteady flow, the vorticity will give us
the best visualization of the Karman street. Let’s start with the zero Reynolds number flow.
Here’s a plot of the stream function. And here’s a plot of the vorticity.
The fluid moves from left to right, and yet the flow fields are symmetrical
upstream and downstream of the cylinder. This is a special feature of the zero Reynolds number,
and its not a realistic flow. Let’s first look at the stream function as the
Reynolds number increases from zero to six. The stream lines shift to the right
as the fluid particles get displaced a larger distance behind the cylinder than in front of it.
And if we look at the vorticity, we see it’s being swept behind the cylinder as the Reynolds number
increases. The fluid is spinning more behind the cylinder than in front of it.
Something new and interesting happens when the Reynolds number becomes larger than around six.
Here’s what the stream function looks like for a Reynolds number of
ten. We now see two attached eddies in the flow behind the cylinder.
If we draw arrows on the streamlines to denote the direction of the fluid flow, we can see that
the fluid in the upper eddy must be rotating clockwise,
and the fluid in the lower eddy must be rotating counterclockwise.
The fluid is trapped in these attached eddies, recirculate, and never moves off downstream.
We can see how these recirculation regions grow as we increase the Reynolds number.
Here’s a video of the stream function as the Reynolds number increases from six to fifty.
And here’s a video of the vorticity. The most prominent feature here is the vorticity that
comes off the front of the cylinder and extends into the flow behind the cylinder.
At these Reynolds numbers, the flow is still steady.
The fluid is moving, but its direction and its speed is always the same.
These were the steady flows that Prandtl and his post-doc were trying to find.
The steady flows at higher Reynolds numbers become unstable. Unstable flows
are not seen in nature because …. they’re unstable. They’re replaced by stable flows.
Let’s have a look at how a steady flow at a Reynolds number of 45 becomes unstable
as we increase the Reynolds number to 60. First, we develop the steady flow at the
lower Reynolds number of 45. This is what we would see in an experiment as
we turn on the water flow in a channel. We then raise the Reynolds number to 60.
In an experiment, we would increase the flow rate and the velocity of the fluid.
We see that an oscillation begins to develop. And the vorticity eventually breaks up.
The resulting flow is a rather damped Karman street. The flow’s no longer steady,
but we see that it’s periodic. Now let’s lower the Reynolds
number back down to 45 and see what happens. The oscillations slowly die down and we see
that the flow returns back to its steady state. For those of you who know some nonlinear dynamics,
this sudden transition from a steady flow to an oscillating flow is called a supercritical
Hopf bifurcation. If you’re interested, you can look up the mathematics on the web.
Experiments have estimated that this transition occurs at a Reynolds number of around 46.
Let’s look again at the Karman street when the Reynolds number is sixty.
We can zoom into the vorticity field near the cylinder.
We can see the attached eddies oscillating with the vortex street,
but they don’t seem to interact with the vortex shedding.
This dynamics changes slightly at higher Reynolds numbers.
Let’s first compute the flow field at a Reynolds number of a hundred and five.
Again, see the flow field develop, become unstable, and form a Karman street.
Now if we zoom into the attached eddies, we can see them interacting with the street, something
that wasn’t obvious at the lower Reynolds numbers. What happens at even higher Reynolds numbers?
Turbulence, of course Turbulence is a three-dimensional instability,
and I don’t have the software or the computer fire power to simulate three-dimensional turbulence.
So let’s have a look at some of the videos that I could find on YouTube.
Here’s a developing turbulent flow at a Reynolds number of 3900.
And here’s the flow field right next to the cylinder.
Here’s another beautiful visualization of a turbulent flow
at a Reynolds number of 2200, but around a square cylinder.
Let me conclude with a beautiful clip of an airplane coming into land,
and shedding two absolutely gorgeous vortices. I’m Jeff Chasnov! Thanks for watching!