Statistical Theory of Turbulence in theory of turbulence

Generally, the concept of statistical average is used to study turbulence. The statistical result is the average of the fine structure of turbulence, which describes some general situations of fluid movement, which should be sensitive to the actual details of turbulence. Therefore, it can be considered that almost all theory of turbulence (including the theories mentioned in the last two sections) are statistical theories, but the statistical theory mentioned in general works actually refers to the statistical theory after introducing multi-point correlation. When Taylor studied turbulent diffusion in the early 1920s, he introduced the correlation of fluctuating velocity at the same point in the flow field at different times, thus initiating the study of turbulent statistical theory. This correlation is called Lagrangian correlation, which can describe the diffusion ability of flow. This ability is expressed by the diffusion coefficient εd, then

formula

It is called correlation coefficient. Knowing the Lagrangian correlation, the turbulent diffusion coefficient can be calculated. In 1935, Taylor introduced the correlation of velocity components at different points at the same time to describe the turbulent fluctuation field, which is called Euler correlation. Corresponding correlation coefficient

Taylor used this correlation to study an ideal turbulence-uniform isotropic turbulence. The simple idealized definition of this turbulence is that the average velocity and all average quantities keep the spatial coordinate translation unchanged, and the correlation function is the same in any direction. It is difficult to simulate this turbulence approximately even in the laboratory. However, in this kind of turbulence, there will be no interaction between average flow and pulsation, and there will be no turbulence energy diffusion effect caused by heterogeneity and turbulence energy redistribution effect caused by anisotropy. Therefore, this turbulence can be used to study the energy attenuation law of turbulence and the energy distribution and exchange law between vortices at all levels in the turbulence field. Because there is no energy generation and diffusion of turbulence, this turbulence will gradually decay once it is generated. The attenuation law of Taylor-induced turbulence energy is as follows:

Where λ is Taylor microscale of turbulence; U is the pulsating velocity. All the second-order velocity correlations of this turbulence can be expressed by a longitudinal correlation function.

Where l represents the direction of the connecting line between point p and point P'; R is the distance between two points; Ul(0) and u'l(r) are the pulse velocity components of point P and point P' in the L direction, respectively; It is the autocorrelation of pulse velocity in L direction, which is called longitudinal autocorrelation, and its 1.5 times is turbulent energy. Carmen and L howarth derived the dynamic equation of f(r).

Equation (7) is called Carmen-howarth equation, which describes the change of correlation with time. The attenuation law of the flow field can be obtained by solving f, and the equation is expanded according to the power of r, the first term of which is formula (6), and the subsequent terms are related to κ. κ is a third-order correlation coefficient and an unknown number, so the equation is not closed. The early homogeneous isotropic correlation theory is to study various closed methods and solutions of this equation.

Perform Fourier transform on uiu'j' to obtain a three-dimensional energy spectrum function:

Where k is the wave number. Let E(k, t)=2πk2Eij(k, t), which is also a three-dimensional energy spectrum function. The energy spectrum equation corresponding to the Carmen-howarth equation is:

Where f is related to the third-order velocity correlation function. So the energy spectrum equation is not closed, it contains two unknown quantities, e and f, and the turbulent energy can be obtained by integrating the energy spectrum functions e and k:

Therefore, E(k, t)dk is the energy of turbulent vortex with wave number between k and dk. As shown in the figure, in the energy spectrum curve (curve of E versus K), the wavelet number corresponds to a large turbulent vortex, and the large wave number corresponds to a small turbulent vortex. For mesoscale vortex, A.H. Kolmogorov shows that its energy spectrum changes with the -5/3 power of k, that is, in the inertia sub-region of the graph, the energy spectrum curve can be expressed as E=Aε2/3k-5/3, where ε is the turbulent energy dissipation rate. This form is called Andrey Kolmogorov spectral law. A large number of observation data support this deterministic result.

There is a cascade view of the relationship between turbulent eddies at all levels. Once the turbulence is formed, the general trend is that the large vortex gradually evolves into the middle vortex, and the middle vortex evolves into the small vortex. Reflected in the evolution of the energy spectrum curve, due to the weakening of the large vortex, the e value at the small k gradually decreases; On the one hand, the value of e in the middle k receives the energy of the smaller k-value region, on the other hand, it transmits the energy to the larger k-value region. Finally, due to the effect of fluid viscosity, energy is converted into heat on some micro-scale vortices and dissipated. The spectral theory of homogeneous isotropic turbulence deduces the specific form of energy spectrum curve and its attenuation law from the closed method of studying spectral equation (8).

In 194 1, Kolmogorov put forward the concept of local isotropy. He thinks that the actual flow is always affected by the boundary, so the motion of large-scale vortex which is greatly affected by the boundary cannot be isotropic, while the small-scale vortex which is less affected by the boundary may be isotropic. In order to eliminate the influence of large eddy, he studied the relative velocity wi=vi-v'i and the structure function derived from it, and thought that the average property determined by the pulsating field wi was isotropic, so this kind of turbulence was called local uniform isotropic turbulence. Zhou Peiyuan and others solved the Navier-Stokes equation from another way, and then statistically averaged the obtained basic vortices to study the homogeneous isotropic turbulence, and obtained the attenuation law of related quantities. In addition, the uniform shear turbulence is also studied. R.H. krishnan put forward the theory of direct interaction; Grossman introduced renormalization group theory into turbulence research; S. Zug, M. B. Lewis and B. B. Struminski have studied the aerodynamic theory of turbulence, but they have not made significant progress. Turbulence has been studied for 100 years, but only a few quantitative predictions have been obtained. 1. Some new discoveries about the structure of turbulence in recent twenty years, and about the mechanical system and mathematical system that lead to chaos due to instability and bifurcation may provide a new way to understand the occurrence of turbulence. Scientists and engineers began to think more about the mechanism of turbulence. However, this kind of thinking about the mechanism will not quickly make a thorough understanding of the fully developed turbulence, but only provide the conditions for building a statistical hypothesis that more accurately reflects the basic mechanism of the turbulence process. The establishment of theory of turbulence is a very arduous task. The task now is to improve the technology of controlling instability and enhance the prediction ability of turbulence statistical model, so as to promote the design of new industrial products and enhance the prediction ability of weather and ocean currents.