Turbulence, the century problem of mechanics

It is natural for people to care about the movement of fluid, because the earth is surrounded by the atmosphere and two-thirds of the earth's surface is covered with water. Turbulence, as a scientific problem, was established after 1883 Reynolds made an experiment to distinguish laminar flow from turbulence. Since the 20th century, due to the development of engineering technology, there is an urgent need to understand the laws of turbulence, which greatly promotes the study of turbulence. In this 100 year, the understanding of turbulence has made great progress, otherwise, engineering technologies such as aviation, aerospace, ships, power, water conservancy, chemical engineering, marine engineering, and natural sciences such as meteorology and marine science could not have made great progress. On the other hand, people's understanding of turbulence is not comprehensive, which restricts the further development of these engineering technologies and natural sciences, and may also restrict the formation of some emerging science and technology in 2 1 century. Therefore, at the beginning of 2 1 century, it is necessary to bring this century's difficult problem to scientists again.

The root of the complexity of turbulent motion is that it is a strongly nonlinear system. The Naville-Stokes (N-S) equation governing turbulent motion is nonlinear. In most cases, its solution is unstable, which leads to multiple bifurcations of the flow, forming a complex flow pattern, and the nonlinearity of the equation couples flows of different scales, which cannot be studied separately.

For a century, mathematicians have done a lot of research on N-S equation, but due to the difficulties caused by its nonlinearity, the positive results are far less than those of other mathematical and physical equations. It seems important to further study the mathematical properties of N-S equation, but it is probably unrealistic to rely on this way to solve the turbulence problems raised in engineering technology and natural mathematics.

Physicists, mechanics and some mathematicians try to solve turbulence in another way, that is, to directly establish a model that can reflect some important characteristics of turbulence. For example, in the 1920s, by comparing molecular collision and molecular free path in the theory of molecular motion, the theory of vortex viscosity and mixing length was put forward, which solved some urgent engineering and technical problems. Although this theory is incomplete, and its parameters are determined by experiments, it is far from being considered as a real solution to the turbulence problem, but it is still used in many occasions because of its simplicity.

However, at present, this theory is only effective for homogeneous isotropic turbulence, which requires Reynolds number to be large enough and there is an inertia zone. The research object is actually a closed system in terms of physical mechanism, while most turbulence related to engineering technology is an open system, which has non-uniform and isotropic interaction with the surrounding environment, and the Reynolds number is not large enough to ensure the existence of inertia zone. Therefore, since the 1940s, people have explored another road. For many engineering and technical problems, it is enough to know the average quantity related to flow, and Reynolds has long derived the equation satisfied by the average quantity (Reynolds equation). Unfortunately, both turbulence and pulsation have second-order correlations, and because there is no universal pulsation probability density function, these correlations cannot be obtained in advance, thus becoming new unknowns, making the number of unknowns more than the number of equations.

Since 1950s, people have noticed that there are not only small-scale random pulsations, but also large-scale coherent structures with certain regularity, also known as coherent structures. By the end of 1960s, this discovery was confirmed. Moreover, these large-scale structures are the "intermediary" of the external environment affecting turbulence. Taking the boundary layer as an example, the outflow produces strong shear near the wall, which will produce large-scale structure (I believe there is also some evidence that this is the result of unstable flow). Although these large-scale structures are still random, they are quite certain. Their energy accounts for most of the total turbulence, and they split into small-scale pulsations through some unclear mechanism, which plays an important role in the occurrence and maintenance of the whole turbulence. Physically, this shear turbulence is a nonlinear open system, because it interacts with the external environment (produced by shearing force or heat exchange, etc.). It seems to be a universal law that this system has a wide range of incomplete random motion. The existence of this large-scale motion may be the reason why universal parameters cannot be found in the above model theory. Because the small-scale pulsation may have certain universality, as shown in the inertial region of the large Reynolds number homogeneous isotropic turbulence energy spectrum mentioned above. But large-scale movements are bound to be related to external conditions, which are obviously varied. Therefore, it is obviously an important or indispensable part to explore the laws of these large-scale motions and their relationship with small-scale pulsations. Especially, the Reynolds number of most engineering and technical problems is not large enough, perhaps the inertia zone does not exist or its span is too small, and the universal law found for the inertia zone can not provide direct help for solving practical engineering and technical problems.

After more than 30 years of research, people have made great progress in the study of coherent structure. It has been proved that coherent structure in free shear turbulence is caused by flow instability, which can be calculated by flow stability theory. Because this stability is the so-called inertial instability, it is not affected by viscosity and small-scale pulsation. The coherent structure in wall turbulence, such as boundary layer turbulence, is considered to be caused by instability in principle, but this instability is greatly influenced by viscosity and small-scale turbulence, so it is difficult to completely separate its research from small-scale turbulence. In recent years, with the efforts of Chinese scholars, some progress has been made in explaining the coherent structure of wall turbulence with the theory of flow stability.

From the results of experiments and direct numerical simulation of turbulence, coherent structures play an important role in turbulent transport. Therefore, it is necessary to study it not only in theory but also in practice. Moreover, as a kind of "self-organization" phenomenon in chaotic behavior produced by nonlinear open systems, the research on it may have an impact beyond the scope of mechanics both in methodology and in actual results.

From the current understanding, because the turbulence in most engineering and technical problems is shear turbulence, and Reynolds number is not very high, it seems that the study of coherent structure, including its relationship with small-scale turbulence, is the key to turbulence research.

The development of engineering technology and science constantly puts forward new requirements for turbulence research. For example, the development of supersonic civil aircraft and aerospace aircraft urgently needs a deeper understanding of turbulence in supersonic boundary layer, including the transition from laminar flow to turbulence and the full development of turbulence properties. We still know very little about it. It is not easy to generalize the law obtained at low speed to high speed, because there are many "small shock waves" in the supersonic boundary layer, which essentially increases its complexity. In order to establish a more perfect weather system model, in addition to considering the turbulence in the atmosphere, the interaction between the atmosphere and the underlying surface (ocean and land) is also a link that cannot be ignored, and it is precisely because of the unclear grasp of the turbulence law that the establishment of a more accurate model is hindered. Limited by space, it is impossible to list more specific examples one by one.

The development of large computers and computational science provides a powerful means for turbulence research. Now some simple turbulence with small overall scale and Reynolds number can be numerically simulated by directly solving the N-S equation. Many flow details that can't be obtained by experimental methods can be obtained, which provides a basis for establishing a more reasonable turbulence model theory. The process of putting forward the scaling law cascade model in the uniform isotropic turbulent inertia region is a good example.

Some people think that maybe one day the turbulence problem can be solved by solving the N-S equation numerically, so the importance of in-depth mechanical research on it will be reduced. This view is incorrect. If this method is used to calculate the complete flow field of aircraft and ships, including the turbulence in their boundary layer, the speed and storage capacity of the computer are at least 107~ 108 times higher than that of the current supercomputer. Dealing with turbulence in the ocean and atmosphere is even more unimaginable. At present, the numerical grid of weather forecast is still vertical 100 m, horizontal 10 km, and the small-scale turbulence is in millimeters, which does not include the complexity brought by two-phase flow. Therefore, in 2 1 century, in order to promote the development of science and engineering technology, it is still essential to study the mechanism of turbulence. The development of computing technology has indeed brought new possibilities for the development of many engineering technologies, and it is possible to optimize their design. However, as long as these engineering technologies involve turbulence, if there is no correct turbulence model, the optimization result is not real optimization. Incorrect turbulence model may even bring wrong results. Therefore, the development of computing technology not only does not reduce the urgent need for turbulence research, but also increases the need for further understanding of turbulence, otherwise the benefits brought by the development of computing technology will not be fully exerted.

Because the calculation results of the model theory are not very satisfactory, it is not practical to solve the engineering technical problems directly by solving the N-S equation, and people also put forward a "compromise" method, that is, the large eddy simulation method. Its essence is to try to use an "average" N-S equation to calculate the pulsation whose spatial scale is larger than a predetermined value, and add the influence of pulsation whose scale is smaller than this given value to this equation through some modeling method. This method not only has the advantages of direct numerical simulation and model theory, for example, a lot of details of pulsation can be obtained, but the calculation workload is still quite large, so it can not be used in practical engineering and technical problems. On the other hand, the modeling of small-scale pulsation is not perfect. However, in the near future, it seems that this method may replace modal theory and become a more reliable calculation method in some problems.

In a word, turbulence will still be an important scientific problem that must be taken seriously in the development of many engineering and technical problems and some natural sciences in the 2 1 century. Because it is a chaotic motion generated by a strongly nonlinear system, it has very complex properties. However, the mathematical properties of its governing equations are difficult to understand. Therefore, with the cooperation of experiment and direct numerical simulation (when the control equation and calculation method are reliable, it is equivalent to a numerical experiment), it will be a more realistic way to propose a reasonable turbulence model to solve the problem.