**1. Introduction**

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The problem of turbulence arose more than hundred years ago to explain the nature of chaotic motion of the nonlinear continuous medium and to find ways for its description; so far it remains one of the most attractive and challenging problems of classical physics. Researchers of this problem have met with exclusive difficulties and there was an understanding of that the problem of turbulence always considered difficult, is actually extremely difficult. This problem is named by Clay Mathematics Institute as one of seven millennium mathematical problems [1] and it is also in the list of 18 most significant mathematical problems of XXI century formulated by S.Smale [2].

The nature of turbulence - the disordered chaotic motion of a nonlinear continuous medium, the causes and mechanisms of chaos generation remain the main issue in the turbulence problem. Several models trying to explain the mechanisms of turbulence generation in nonlinear solid media were suggested at different time. Among such models the most known are Landau-Hopf and Ruelle-Takens models, explaining generation of turbulence by the infinite cascade of Andronov-Hopf bifurcations and, accordingly, by destruction of three-dimensional torus with generation of strange attractor. However, these models have not been justified by experiments with hydrodynamic turbulence.

The universal unified mechanism of transition to dynamical chaos in all nonlinear dissipative systems of differential equations including autonomous and nonautonomous systems of ordinary and partial differential equations and differential equations with delay argument was theoretically and experimentally proven in number of recent papers by the authors [3–12]. The mechanism is developing by FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario through subharmonic and homoclinic bifurcation cascades of stable cycles or stable two dimensional or many-dimensional tori.

In this chapter we are presenting a consistent numerical solution method for 3D evolutionary Navier-Stokes equations with an arbitrary initial-boundary value problem posed. Then we

©2012 Magnitskii and Evstigneev, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

consider two well studied problems for incompressible Navier-Stokes equations, namely flow over a backward facing step and Rayleigh-Benard convection in cubic cavity. Numerical solutions of these problems for transitional regimes indicated existence of complicated scenarios formed by theory FSM. Thus, it seems reasonable, that there is no unified laminar-turbulent transition scenario, it can be a cascade of stable limit cycles or stable two dimensional or many dimensional tori, but all these scenarios lay in the frameworks of the FSM-theory.
