The second question to the God: how to solve the mystery of turbulence?

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The second question to the God: how to solve the mystery of turbulence?
The second question to the God: how to solve the mystery of turbulence?
23 November 2020, 11:39Science
In the publishing house Corpus within the publishing program of the Polytechnic Museum and in the series "Books of the Polytechnic", James Glick's book "The Creation of a New Science" was published.

This is a world bestseller, translated into more than two dozen languages, which describes the history of the emergence of the science of chaos.

Starting with the accidental discovery of meteorologist Edward Lorenz, who tried to create a model for a long-term weather forecasting, Glick consistently reconstructs the entire chain of sudden insights and unusual experiments that led scientists to the realization that there are still unknown universal laws of nature. Glick not only tells the story of the birth of a new science, but also reflects on how scientific progress takes place and what is the role of crazy geniuses in it, who are looking for non-standard solutions in spite of existing knowledge.

The book was published as part of the publishing program of the Polytechnic Museum, whose mission is to popularize science. Novye Izvestia publishes a chapter from this book, Strange Attractors, which tells about an incredibly interesting and complex phenomenon - turbulence, to the study of which Soviet physicists have made a huge contribution:

“The problem of turbulence has a rich history. All great physicists have thought about it in one way or another. The smooth flow is broken up into shafts and eddy currents; erratic bends destroy the boundaries between liquid and solid surface; the energy of large-scale motion quickly flows into small eddies. Why?

Perhaps the most reasonable ideas came from mathematicians; most physicists were simply afraid to waste time studying turbulence, which seemed almost incomprehensible. Proof of this is the story of Werner Heisenberg, a famous scientist in quantum physics. On his deathbed, he admitted that he would like to ask the Lord God two questions: about the basics of relativity and about the cause of turbulence. “But I think the Lord can only answer the first of them,” Heisenberg remarked. Theoretical physics and the phenomenon of turbulence ended the game in a draw: science seemed to have approached an enchanted line and froze near it. Near this magical border, where fluids behave in an orderly manner, there is work to be done. Fortunately, a smoothly flowing liquid does not behave at all as if each of the countless molecules moved independently: droplets of liquid matter, which were initially nearby, usually remain close to each other, like horses in a team.

Hydraulic engineers have quite reliable equations describing the behavior of such a laminar flow: they use knowledge accumulated in the 19th century, when the movement of liquids and gases was one of the primary problems of physical science. By our time, this problem has already gone into the shadows, and even the deepest minds believed that there were no secrets left in the dynamics of liquids, except for one, unknown to heaven.

Гейзенберг

From the practical point of view, everything looked so clear that with a light heart it could be left at the mercy of specialist technicians. According to physicists, fluid dynamics has turned from a scientific problem into an engineering one. The young luminaries of physics found better things to do, and researchers of fluid dynamics came across only in engineering departments of universities. Among practitioners, interest in turbulence has always been in the foreground, but at the same time it remained somewhat one-sided and boiled down to the question of how to eliminate this phenomenon. Sometimes turbulence is even desirable (as, for example, in a jet engine, where efficient combustion depends on the rapid formation of a mixture), but in most cases it is tantamount to disaster. Turbulent air flow, acting on the wing of the aircraft, makes it difficult to take off. Turbulent flow inside the pipeline will retard fluid movement. Governments and corporations are investing heavily in the design of aircraft, turbine engines, propellers, submarines, and other similar devices that move in a liquid or gaseous medium. Researchers are interested in the blood flow in the vessels and through the heart valves, they are concerned about eddy currents and eddies, flames and shock waves from various types of explosions.

It is believed that nuclear physicists were involved in the project of the atomic bomb during World War II, but in reality all issues related to nuclear physics were resolved even before the start of work, and scientists at Los Alamos were engaged in gas and hydrodynamic aspects.

What exactly is turbulence? Complete disorder on all scales, tiny whirlwinds inside huge whirlpools. Turbulence is unstable and highly dissipative, that is, it has the ability to slow down movement by depleting energy. She is a disorderly movement. But how does the flow of a fluid change from smooth to turbulent? Imagine a flawlessly smooth hollow tube, an extremely stable source of water supply, and the entire structure is reliably protected from vibration. Now ask the question: how can something messy appear in the stream flowing inside the pipe? All the rules seem to be failing here. When the flow is smooth, or laminar, small disturbances disappear, but immediately after the appearance of turbulence, their number increases sharply, asking science a new riddle.

The stream bed at the foot of the cliff turns into a whirlpool, which grows, splits and whirls as the water moves downstream, and the plume of cigarette smoke that quietly winds over the ashtray suddenly accelerates and, having reached a critical speed, breaks up into violent vortices. The turbulence threshold can be observed with your own eyes and measured in laboratory experiments; it can be tested for each wing of an airplane or propeller in a wind tunnel test.

Nevertheless, it is difficult to grasp its nature. As a rule, the data obtained lacks versatility: a trial and error study of the Boeing 707 wing does nothing for the design of an F-16 fighter wing. Even supercomputers are almost helpless in the face of the chaotic movement of matter.

Let's imagine that something shakes the liquid, causing waves inside it. The liquid has a viscosity, and for this reason, the energy imparted to it by shaking from it leaves. If you stop shaking the liquid, it will naturally come to rest. When you shake a liquid, you give it energy at a low frequency, or, in other words, you cause large vibrations, but the first thing you will notice after that is that the large vibrations seem to break up into small ones. Eddy currents are formed, and inside them there are smaller eddy currents, each of which dissipates the energy of the flow and does it in a characteristic rhythm.

Back in the 1930s, Andrei Kolmogorov proposed some mathematical description that helped to represent the dynamics of these eddy currents. The scientist looked at them on a smaller and smaller scale - until he reached the limit at which the vortices became so tiny that the viscosity of the substance no longer affected them. For simplicity of description, Kolmogorov imagined that the entire fluid consists of small vortex flows and in this sense is the same everywhere. But such an assumption about homogeneity is incorrect, as Poincaré guessed forty years ago, having observed in a turbulent river how water eddies alternate with sections of calm current. Thus, the flow instability is local and the energy is actually dissipated only in a part of the space. If you carefully examine a turbulent flow of any scale, you will notice that more and more areas of calm flow are being discovered.

Thus, the homogeneity hypothesis gives way to the intermittency hypothesis. This somewhat idealized description looks eminently fractal, with alternating rough and flowing zones that are noticeable at any scale, from large to small. But even this picture, to a certain extent, is not a complete reflection of reality. The question of how turbulence begins is very close to the formulated above, but at the same time independent is the question. How does fluid flow cross the line between smooth and turbulent? What intermediate stages does the turbulence go through before it makes itself felt in full? To answer these questions, there was a theory that looked somewhat more convincing.

Колмогоров

This generally accepted paradigm owes its appearance to Lev Landau, the great Russian scientist whose developments in the field of hydrodynamics set the standard in physical science. Landau's model describes a jumble of competing rhythms. He suggested that when more energy enters the system, one by one new frequencies appear, each of which is incompatible with the previous one, as if the violin string responds to the strengthening of the bow movement by sounding a second, dissonant tone, and then a third, fourth, and so on. , until the sounds merge into an incomprehensible cacophony. Any liquid or gaseous substance is a collection of single particles-molecules, the number of which is so great that it may seem infinite. If each particle moved by itself, there would be infinitely many variants of fluid motion (in scientific terms, infinitely many “degrees of freedom”) and the equations describing the motion would include an infinite number of variables.

However, nothing of the kind happens: the movement of each molecule largely depends on the movement of its neighbors and there can be only a few degrees of freedom (at least in a calm flow). Potentially complex motions remain connected, nearby particles do not diverge at all, or diverge smoothly and linearly, forming neat lines in photographs taken in a wind tunnel. Particles in a plume of cigarette smoke also rise for some time as a whole. Then indignation appears, a variety of mysterious stormy impulses. Sometimes such movements even received names: "oscillator", "cross rollers", "knot", "zigzag", "swollen veins" (which are the case with varicose veins).

According to Landau, the emerging unstable motions simply accumulated, overlapping one another and thus creating loops with overlapping velocities and sizes. Contemplatively, such a generally accepted model of turbulence seemed to fit real facts, and its uselessness from the point of view of mathematics was turned a blind eye. Landau's model made it possible to maintain dignity, although this was a complete fiasco. Imagine that water, with a faint whistle, flows slowly through a tube or flows inside a cylinder. We will mentally increase the pressure, thereby causing the appearance of rhythmic vibrations back and forth. The liquid hits the tube wall slowly. Turn the imaginary handle again, increasing the pressure. It is unknown where the second frequency will appear, not synchronized with the first. Disharmonious rhythms, as if competing, are superimposed on each other, and now a rather tangled movement has appeared: the waves hit the walls of the tube, mixing one with the other so that it is impossible to catch their rhythm. As the pressure rises, the third, then the fourth, fifth, sixth frequencies arise, and they all do not correspond to each other, so that the flow becomes unusually complex.

Perhaps this is turbulence. Physicists accepted this explanation, but none of them could predict when exactly the increase in energy would lead to the emergence of a new frequency or what it would be. No one discerned these mysteriously appearing frequencies during the experiment, because Landau's theory of the threshold of turbulence has not actually been tested yet..."

Ландау

Translated from English by Mikhail Nakhmanson, Yekaterina Barashkova.

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