Turbulent attractors in chaotic states of evolution of continuous medium

Alfred M. Mukhamedov

Kazan State Technical University of A.N.Tupolev's name

Tatarstan 9-15, Kazan 420021, RUSSIA

e-mail: Alfared@Yandex.ru

 

Present paper is an attempt to produce the rational foundation for the problem of description of turbulent regimes in continuum media as a self-organization processes that distinguish turbulent and thermodynamic chaos.

The main result that stimulates the beginning of such program of investigations consists in the formula (7) which gives the explicit expression for turbulent pulsations of thermodynamic entropy. It leads to the new definition of turbulence based on entropy pulsations as a criterion for differentiating of structural turbulence and unstructured chaos. Structural turbulence (or turbulent attractor) is a medium motion in which contribution into entropy that comes from the turbulent pulsations is negative. Otherwise the beginning of turbulent pulsations lead to decreasing of entropy from it average value.

Definition of turbulent attractors becomes the first step for investigations of chaos from the point of view of the unified theory of irreversible processes and self-organization. For the development it is introduced the notion on local attractor's structure defined by the set of linear operators each of them is a linearization of non-linear turbulent equations at the proper stationary points (definition 5). The main unit for qualitative estimation of turbulent portrait becomes an ensemble of turbulent motions. In particular, it is the set of meta-stable states of motion. These states and their attractive characteristics consist of local features of structural turbulence.

Turbulent attractor's definition introduces new thermodynamic criterion for classification of turbulent motions. For finite dimensional dynamic systems, this criterion agrees with one of the most essential feature of strange attractors, namely, the compression of phase volume accompanied by the stretch of this volume in a certain directions. In contrast to attractors of finite dimensional systems, structural condition (9) defines the constraint of spatial coordinates. Hence, it gives the foundation for searching of spatial heterogeneity of turbulent chaos. Known types of strange attractor turn out particular examples, which one can consider as a special case of turbulent attractor.

In order to appreciate the adequateness of turbulent dynamic equation (2) we discuss the method of increasing of phase space's dimension. Complication of turbulent regimes demands not only the increasing but also proper decreasing of number of independent dynamic coordinates. Condition (14) help for introducing new coordinates. If (14) does not realize for some coordinates then the compensatory values become a new coordinates. It gives some arguments for revealing the turbulent stage of medium evolution and for spatial and temporal characteristics of this stage.