Turbulence, the oldest unsolved problem in physics

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 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.

I say “many” because some think it should be considered solved, at least in principle. Their argument is that calculating turbulent flows is just an application of Newton’s laws of motion, albeit a very complicated one; we already know Newton’s laws, so everything else is just detail. Naturally, I hold the opposite view: the proof is in the pudding, and this particular pudding has not yet come out right.

The lack of a complete and satisfying theory of turbulence based on classical physics has even led to suggestions that a full account requires some quantum mechanical ingredients: that’s a minority view, but one that can’t be discounted.

An example of why turbulence is said to be an unsolved problem is that we can’t generally predict the speed at which an orderly, non-turbulent (“laminar”) flow will make the transition to a turbulent flow. We can do pretty well in some special cases—this was one of the problems that Heisenberg had some success with—but, in general, our rules of thumb for predicting the transition speeds are summaries of experiments and engineering experience.

This figure at right is a nice illustration of this transition phenomenon. It shows the hot air rising from a candle flame, using a 19th century visualization technique that makes gases of different densities look different. Here, the air heated by the candle is less dense than the surrounding atmosphere.

For another turbulent transition phenomenon familiar to anyone who frequents the beach, consider gentle, rolling ocean waves that become complex and foamy as they approach the shore and “break.” In the open ocean, wind-driven waves can also break if the windspeed is high or if multiple waves combine to form a larger one.

For another visual aid, there is a centuries-old tradition in Japanese painting of depicting turbulent, breaking ocean waves. In these paintings, the waves are not merely part of the landscape but the main subjects. These artists seemed to be mainly concerned with conveying the beauty and terrible power of the phenomenon, rather than, as was Leonardo, being engaged in a systematic study of nature. One of the most famous Japanese artworks, and an iconic example of this genre, is Hokusai’s “Great Wave,” a woodblock print published in 1831.

For one last reason to consider turbulence an unsolved problem, turbulent flows exhibit a wide range of interesting behavior in time and space. Most of these have been discovered by measurement, not predicted, and there’s still no satisfying theoretical explanation for them.

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