My point of view

In fact, Reynolds approach still the most widely used up to now. This approach is based on two explicit assumptions. The first one is the existence of some scale which separates macro characteristics (averaged velocity field) from the micro structure. The second assumption is that it is possible to build correct closure equations, i.e. the equations which express Reynolds stresses through the averaged fields. It seems that hundreds or even thousands of different closure equations have been suggested and used. But all of them have narrow range of applicability and, more important, they are based on some arbitrary assumptions without strong physical background.

Several more recent attempts to use some modern characteristics, such as fractal dimension for example, also did not lead to creation of a complete, closed set of equations which may be suitable for a description of turbulent flows.

So the research of different characteristic applications to the turbulent flows and the attempts to find out some universal relationships between such characteristics and the averaged ones is one of actual and required way toward the real theory of turbulence.

The present work is concerned with such investigation of multifractal characteristics with their application to several turbulent flows obtained from different sources.

An experimental investigation of turbulent flows with special experimental setup and equipment, which allow to obtain reliable data with good precision, is very difficult and expensive process. The measurements of natural phenomenon, such as atmospheric turbulence for example, require special equipment and research expeditions, which is even more expensive. The precision of experimental and natural measurements is rather poor because of many additional reasons which interfere and interact with a real phenomena, the subject of research.

This is why the direct numerical simulations of different turbulent flows become now more and more significant source of data sets, which may be used for further investigation by the same way as the data sets obtained from experimental and natural measurements. All three sources of data have their own advantages and disadvantages. For example it is well-known that the atmospheric turbulence is extremely unsteady process, and it is very difficult to separate a long-time trends from the middle and short-time pulsations. From another hand it is impossible for experimental setup to realize such high values of Reynolds numbers as for atmospheric flows. The numerical simulations have their own difficulties and limitations, such as the wide-known artificial viscosity and many others.

The natural assumption for the theory of turbulence is that there exist some (unknown yet) general and universal properties for wide range of different turbulent flows. A search of such properties is one of the main objectives of modern research at this area. The idea of present investigation is that it is naturally to searching for such universality based on the data sets obtained from different sources. It allows to avoid the spurious features, which may arise because of technology of observation, and to extract more clearly the general properties of turbulence.

Technically, many time series obtained for the natural atmospheric turbulence and for the direct numerical simulations of two different flows have been processed used double trace moment technique, with the same processing technology. The surprisingly good agreement of Levy index values have been found for all used data sets.

At the same time, we found close enough value of Levy index for the testing time series which have been generated as a sum of two and three sine waves with rationally independent frequencies. Clearly, such time series are far from representation of multifractal cascade process, which is the background for double trace moments technique. But from another side such time series are in the framework of the famous Landau hypothesis. The latter one state that a turbulent flow may be represented as a finite sum of sine waves with different (rationally independent) frequencies. This observation allows to suppose that the double trace moment technique has much more wide area of application then it may be assumed from its theoretical background. But the numerical aspects of this technique (such as an influence of data pre-processing, a finite number of points in time series, a finite step of discretization) are not well studied, and it is possible to expect also that these limitations lead to some spurious convergence of results, which is not the property of underlying processes.

It is clear that as wider some technique may be applied as more powerful it is. The assumption that double trace moments method may be applied not only to the processes with multifractal nature, but for many others leads to the conclusion that it is important and interesting to study the range of applicability and the numerical aspects of this method. Another very interesting direction of research is the clarification of common points between the processes of Landau type and the multifractal cascades.