On the mechanism of mode interactions

Alfred M.Mukhamedov

Kazan State Technical University

Tatarstan, 9-15, Kazan, 420021, RUSSIA

e-mail: Rashidma@mail.ru

            Given paper is the development of the previous paper by the same author “Turbulence as a strange attractor of multimode evolution”. The main result is a construction of mode interaction mechanism and corresponding evolution equation

Mechanism of mode interactions consists of two levels. First level presents by the reactions of fast-relaxed modes on excited modes. These reactions take into account by introducing of connection  into phase space. For defining of it, we need for: 1) constraint equations that determine fast-relaxed modes, and 2) equipment of excited mode bundle that gives subspace created by reactions of fast-relaxed modes.

Second level presents interactions of excited modes. These interactions take into account by introducing of vector field  into phase space. For defining of it, we use phenomenological theory nonequilibrium thermodynamic. Let for considered regime with n freedom degrees we have n hydromechanical variables of extensive type that form full and independent set. For them, we postulate following balance equations

where  - flow of G,  - creation of G from external surround, and mode expressions of these variables

Substitution of marked expressions to balance equations lead to defining . The example, represented evolution equations for regime with three cinematic and one additional mode, is considered.

For the development of suggested mechanism, we produce the principle that defines the average flow. According to this principle, the steady drive of pulsation’s must vanish in the framework moving with average flow. Hence, in general case we deduce that the average flow must satisfy to classical Navier-Stokes equations. Nevertheless, for special regimes corresponded to special ensembles of turbulent trajectories there exist a possibility of modification of these equations. Example of such modification of average flow equations is given for the case of regime with four modes in which there exists impulse flow from additional mode to cinematic modes.

This principle becomes of great importance for flows with boundary restrictions. Indeed, boundary limitations choose some sort of turbulent trajectories in the neighbor of such boundaries. Therefore, they form special ensemble and corresponding average flow. In this case, equations of average flow generalize of equations for boundary layer.