Monograph. 2nd revised and augmented edition. Frick P.G. – Moscow – Izhevsk: Institute of Computer Science, 2010. P. 342.

Monograph. 2nd revised and augmented edition. Frick P.G. – Moscow – Izhevsk: Institute of Computer Science, 2010. P. 342.

Preface to the Second Edition:

Five years have passed since the first edition of this book was published. Turbulence has been and still remains one of the most important yet not fully understood topics in the modern theory of fluid mechanics. A century of its investigation is marked by numerous approaches related to the most promising research directions in physics and mathematics over the corresponding periods of time. Statistical physics and probability theory, dimension theory, Fourier’s analysis and direct numerical methods, theory of dynamic systems, fractal theory and wavelet analysis fall far from being a complete list of fields of science that give impetus to research concerning turbulence. Scientists are still a long way from completing the theory of turbulence. New approaches to its study continue to appear, and the number of models conveying essential information about its specific features ever increases.

The purpose of this book is to provide a deeper insight into the main research ideas that current approaches are based upon, to demonstrate the capabilities of these approaches, to exhibit the problems that remain to be solved by these approaches, and to present models that are as yet unknown.

The book is based on a lecture course delivered by the author to the students of the Faculty of Physics at Perm State National Research University and to the students of specialty “Mathematical modeling of systems and processes” at Perm National Research Polytechnic University. The lecture course has been designed for students that are going to work at research institutions and departments and whose interests are focused on the study of fluid mechanics problems. It also reviews general approaches to modeling complex dynamic systems, which can be useful for those who are involved in modeling all sorts (not only mechanical) of systems and phenomena.

Chapter 1 provides an overview of the basic ideas underlying incompressible fluid dynamics, including the derivation of equations of motion for ideal and viscous fluids, free convection equations and magnetohydrodynamic equations. It also gives the fundamentals of the stability theory which is of paramount importance in understanding the laminar-turbulent transition. Two problems are thoroughly discussed: the problem of the stability of plane Poiseuille flow and the Rayleigh problem of onset of convective stabilities in a horizontal layer of incompressible fluid heated from below. Special attention is given to issues regarding a dimensionless form of equations of motion, similarity laws and dimensionless parameters and their role in the description of the processes of transition to chaotic behavior.

Over the last 20 years there has been significant progress in understanding the nature and properties of turbulence related to the advances in the theory of dynamic systems that allow us to realize how the chaotic behavior occurs in deterministic systems. These results are presented in Chapter 2, where one can find the description of the key concepts of the theory of dynamic systems, some applications of this theory, different methods of investigation of the transition to chaos, and the characteristics of dynamic systems under periodic and chaotic behavior conditions. It also includes the description and analysis of some scenarios (Landau, Ruelle-Takens, and subharmonic cascade) for the transition from order to chaos. The examples of simple hydrodynamic systems exhibiting chaotic behavior are given at the end of this chapter.

Chapter 3 provides an introduction to the methods of describing fully developed turbulence, namely, the Reynolds approach which is a pioneer and best known approach for dealing with turbulent flows. In addition, some semi-empirical turbulence models stemmed from this method are described. The chapter begins with the determination of the statistical moments of random fields characterizing turbulent flows. Then it offers the derivation of the mean-field Reynolds equation and a brief discussion of general approaches to the development of semi-empirical models. Semi-empirical models have received relatively little attention in this book. The reason for this neglect lies in the following. First, this approach has been almost completely surveyed in the literature, and, secondly, the intent of this book is to familiarize readers with the methods of investigation of the properties of small-scale turbulence, which is just the one that keeps beyond the control of semi-empirical models.

Chapter 5 describes the role of conservation laws in generating cascade processes. The chapter also provides a detailed analysis of the distinguishing features of the behavior of two-dimensional turbulence, where a supplementary conservation law leads to a new, qualitatively different, behavior of small-scale turbulence.

Chapter 6 considers the processes of stirring passive impurities in turbulent flows and the specific character of cascade processes in turbulent flows where the force fields generating turbulence experience by themselves its effect. Temperature fields in convective turbulence and magnetic fields in conducting fluid turbulence are examples of such active impurities.

Chapter 7 presents models in which a special functional basis is used to represent the structure of turbulence flows. Such bases, known as hierarchical, fall into the category of wavelet bases (to use contemporary terminology). The wavelet analysis (appeared in much more recent time than the hierarchical models) has become a well-developed area of mathematical physics, and its importance for studying stochastic hydrodynamic systems and turbulence is not restricted to the use of wavelet bases in numerical modeling of flows.

Chapter 8 is devoted to shell models of turbulence – the simplest models of fully developed turbulence that prove to be effective in modeling the properties of turbulence within the inertial intervals at very high Reynolds numbers. These models are the dynamic systems of relatively high order (several dozens of equations), and therefore they are able to describe cascade processes in an extended range of scales.

Chapter 9 presents examples of constructing the grid-shell and combined models of complex turbulent flows. In the chapter, the model of convective turbulence is described, the model of MHD turbulence is presented, and the results of investigation of cascade and dynamo-processes in the fully developed turbulence of a conducting fluid are given. It also includes the description of a combined dynamo-model that involves two equations for magnetic field large-scale modes and a grid-shell model for small-scale turbulence. The chapter ends with a description of a grid-shell method recently proposed for turbulent flows at high Reynolds numbers. The approach combines grid-shell methods of solving mean-field equations and a description of subgrid turbulence in terms of grid-shell models.

Chapter 10 provides basic facts about the spectral, correlation and wavelet analysis of random fields. The advantages of the wavelet analysis to explore time series and spatial fields are shown through illustrative examples. The specific problems of the analysis of spectral distribution of turbulent fields and the correlation of different characteristics of turbulent flows are discussed.


(Yekaterinburg: Ural branch of RAS 2009, p – 337)


This book presents the results of theoretical and experimental investigations of electrovortex flows in the flat channels of engineering devices, conditions for generation of these flows, free surface instability of a plane layer and pumping effect in a flat MHD-channel with electrovortex flows. Different models of engineering MHD-devices having a flat channel are described. The operating principle of these models is based on the interaction between the current and its own magnetic field. Theoretical and experimental data supporting the effectiveness of these devices are provided.


Monograph. Frick P.G. – Moscow – Izhevsk: Institute of Computer Science, 2010. P. 292.

Monograph. Frick P.G. – Moscow – Izhevsk: Institute of Computer Science, 2010. P. 292.


Monograph. / Zimin V.D., Frick P. G. Moscow: Nauka, 1988. P. 173.


The book describes the results obtained by the authors in the study of turbulent convection and laminar-to-turbulent convection transition region. Much emphasis is placed on testing large-scale coherent structures examined from the threshold of onset of stochastic fluctuations to the establishment of fully developed turbulence. Special attention is given to optical techniques that have been applied to the study of convection. An approach to exploring turbulence which rests on the functional basis describing the hierarchy of vortices and thermals of decreasing size is presented in detail.