Physics is full of equations that govern the laws of how everything works. Of all the equations, one set of equations has been an insurmountable mountain since it was first proposed 200 years ago (1822), the Navier–Stokes equation (NS equation).
Navier–Stokes equation. (Figure/Principle)
The mathematical difficulty of understanding the NS equation even exceeds that of Einstein's field equation, because behind it is one of the most difficult parts of the physical world to understand, but it is also a phenomenon that can be seen everywhere in our daily lives, turbulence.
What is turbulence?
What exactly is turbulence? The answer to this question may surprise you. In fact, in the scientific community, there is no universally agreed definition of turbulence, but this is again something that "you know when you see it."
For example, if you pour a little milk into a cup of black coffee, you can see the white liquid bloom like a cloud, spread in the liquid, appear complex swirls, and eventually merge into a perfect cup of coffee. Or, what was originally a gentle river flowing slowly, but when the river meets the piers, a large-scale turbulence event may occur.
Simulation of turbulence in a teacup. (Photo/sunysb.edu)
In fact, even if you casually wave your hand in the air, it will produce invisible turbulence, and they are more complex than imagined. In turbulent flows, there are a large number of vortices in the fluid, that is, many tiny vortices and the countercurrents they produce, which constantly change size, speed and direction, interact with each other, affect each other, and bring great challenges in calculation and simulation.
An example of a non-turbulent flow is a smooth river in which each part of the river moves at the same rate in the same direction. Turbulence is the rupture of the river, which allows different parts of the river to move in different directions at different speeds. (Photo: Joseasorrentino, Wiki)
Yet, despite its ubiquity, turbulence is a mathematical headache that even the giant of science, Werner Heisenberg, is trapped by. According to legend, In his later years, Heisenberg said that if he could ask God questions, he would ask two questions, "Why is there a theory of relativity?" Why turbulence? Then he went on to say, "I believe God will have the answer to the first question."
The story may not be as credible, but it is enough to see the "psychological shadow" that turbulence has left on scientists.
The NS Equation and the Millennium Award Puzzle
In the first half of the 19th century, the French engineer and physicist Claude-Louis Navier and the Irish mathematician and physicist George Gabriel Stokes gradually developed the NS equation that describes the motion of fluids.
Navier–Stokes equations and their implications. (Figure/Principle)
This is a set of nonlinear partial differential equations that describe how viscous, incompressible fluids move under a given viscosity, collective velocity, and external pressure. It can be understood so simply that it is like a fluid version of Newton's second law. Newton's second law connects acceleration with force, while the NS equation relates the rate of change in the velocity of a fluid to the force acting on the fluid.
For 200 years, the NS equation has helped countless physicists and engineers solve a large number of fluid problems, and this equation almost always predicts the motion of fluids well enough, and this consistency of prediction and experimentation may be enough for experimental physics and engineering.
Mathematicians, however, also have to solve more fundamental problems, the so-called NS equations of existence and smoothness. This problem was so difficult that it was listed by the Clay Institute for Mathematics as one of seven "Millennium Prize Conundrums."
This puzzle can be divided into two parts: the first is about the existence of solutions to equations, and the second is about whether these solutions have bounds (finite values).
The first part says that for a mathematical model, no matter how complex it is, to represent the physical world, it must first have a solution. At first glance, you might be wondering, if we can't be sure if these equations have a solution, why are we still using them? In practice, these equations provide many good predictions for the motion of the fluid, but these solutions are approximations of the complete solution of the NS equation. While we are very confident that these approximate solutions are correct, we lack a mathematical proof that can formally show that these equation solutions do exist.
The second part explores whether the solutions to these equations will have singularities (or infinities). The history of fluid mechanics is filled with simplified versions of the solutions of the NS equations that produce singular solutions. In this case, singular solutions often imply physical phenomena that were not previously considered in simplified models. Identifying this new physical phenomenon prompted the researchers to further refine their mathematical models, thereby improving the consistency between the model and reality. If NS does have singular solutions, then perhaps the next Millennium Award will be awarded to someone who discovers what new physics is needed to eliminate the singularity.
Turbulent mystery
When you look at a slowly flowing river and begin to analyze its turbulence, it can ruin the poetry of that moment. But if this gentle-flowing river becomes a destructive torrent, understanding the behavior of complex turbulence becomes crucial.
But because of the complexity of turbulence, even using the fastest supercomputers today, turbulence may only be able to simulate turbulence of a few centimeters around the wings of a civil aircraft. As a result, many scientists are trying to develop algorithms and models that "squeeze" out as much information as possible with existing techniques.
Turbulence also affects all aspects of energy consumption, and a more comprehensive understanding of turbulence can bring far-reaching economic benefits. It could also lead to huge advances in engineering, medical device design, vehicles, weather forecasting, and climate change research.
#创作团队:
Written by: Gaviota
Typography/Design: Wenwen
#参考来源:
https://www.sydney.edu.au/news-opinion/news/2022/04/05/wind-turbines--helicopters--weather-and-why-turbulence-matters.html
https://www.iflscience.com/physics/the-magic-and-mystery-of-turbulence/
https://www.quantamagazine.org/what-makes-the-hardest-equations-in-physics-so-difficult-20180116/
https://theconversation.com/millennium-prize-the-navier-stokes-existence-and-uniqueness-problem-4244
#图片来源:
Cover image: Giannandrea Inchingolo/Wikimedia Commons
First image: Gene Wilburn/Flickr