Think You're Smart Don'tcha: Figure This Out And Make A Million Bucks

In last week's post "Fluid Dynamics (and the filth on your phone)" I made the assertion "This is one of those fields of study that are so mind-bogglingly complex that....", without supplying any supporting statements or facts.

(in these situations the reader can assume I am relying on the Charlie Munger all-purpose turnaround: "Think about it a little more and you will agree with me because you're smart and I'm right.")

 

But for folks who require a bit of backup, here is Ars Technica, followed by the Clay Mathematics Institute, along with a cameo by Feynmann for added "Appeal to Authority":

 

Turbulence, the oldest unsolved problem in physics

The flow of water through a pipe is still in many ways an unsolved problem.

Werner Heisenberg won the 1932 Nobel Prize for helping to found the field of quantum mechanics and developing foundational ideas like the Copenhagen interpretation and the uncertainty principle. The story goes that he once said that, if he were allowed to ask God two questions, they would be, “Why quantum mechanics? And why turbulence?” Supposedly, he was pretty sure God would be able to answer the first question.

The quote may be apocryphal, and there are different versions floating around. Nevertheless, it is true that Heisenberg banged his head against the turbulence problem for several years.

His thesis advisor, Arnold Sommerfeld, assigned the turbulence problem to Heisenberg simply because he thought none of his other students were up to the challenge—and this list of students included future luminaries like Wolfgang Pauli and Hans Bethe. But Heisenberg’s formidable math skills, which allowed him to make bold strides in quantum mechanics, only afforded him a partial and limited success with turbulence.

Some nearly 90 years later, the effort to understand and predict turbulence remains of immense practical importance. Turbulence factors into the design of much of our technology, from airplanes to pipelines, and it factors into predicting important natural phenomena such as the weather. But because our understanding of turbulence over time has stayed largely ad-hoc and limited, the development of technology that interacts significantly with fluid flows has long been forced to be conservative and incremental. If only we became masters of this ubiquitous phenomenon of nature, these technologies might be free to evolve in more imaginative directions.

An undefined definition

Here is the point at which you might expect us to explain turbulence, ostensibly the subject of the article. Unfortunately, physicists still don’t agree on how to define it. It’s not quite as bad as “I know it when I see it,” but it’s not the best defined idea in physics, either.

So for now, we’ll make do with a general notion and try to make it a bit more precise later on. The general idea is that turbulence involves the complex, chaotic motion of a fluid. A “fluid” in physics talk is anything that flows, including liquids, gases, and sometimes even granular materials like sand.

Turbulence is all around us, yet it's usually invisible. Simply wave your hand in front of your face, and you have created incalculably complex motions in the air, even if you can’t see it. Motions of fluids are usually hidden to the senses except at the interface between fluids that have different optical properties. For example, you can see the swirls and eddies on the surface of a flowing creek but not the patterns of motion beneath the surface. The history of progress in fluid dynamics is closely tied to the history of experimental techniques for visualizing flows. But long before the advent of the modern technologies of flow sensors and high-speed video, there were those who were fascinated by the variety and richness of complex flow patterns.

For turbulence to be considered a solved problem in physics, we would need to be able to demonstrate that we can start with the basic equation describing fluid motion and then solve it to predict, in detail, how a fluid will move under any particular set of conditions. That we cannot do this in general is the central reason that many physicists consider turbulence to be an unsolved problem.....

....MUCH MORE

 

At the Max-Planck-Gesellschaft:

April 02, 2013

The American Nobel Prize Laureate for Physics Richard Feynman once described turbulence as “the most important unsolved problem of classical physics”, because a description of the phenomenon from first principles does not exist. This is still regarded as one of the six most important problems in mathematics today....  

And The Clay Mathematics Institute. In May 2000 the Institute offered a million dollar prize per problem for solving each of seven problems in math: 

Millennium Problems

 

Yang–Mills and Mass Gap

Experiment and computer simulations suggest the existence of a "mass gap" in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.

Riemann Hypothesis

The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

P vs NP Problem

If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.

Navier–Stokes Equation

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

Hodge Conjecture

The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.

Poincaré Conjecture

In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston's geometrization conjecture. Perelman's proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

Birch and Swinnerton-Dyer Conjecture

Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.

One of the problems, The Poincaré Conjecture, was solved by Russian mathematician  Grigori Perelman. He turned down the award and the million dollars. He has also turned down The Fields Medal, the highest award in mathematics.

The other six problems are still open, with the Navier–Stokes Equation being the object of our affection.