**Turbulent Flow**

The turbulent flow is characterized by the chaotic pulsation of the flow parameters, intermittence and complex eddy motion. The surface, which separates the turbulent flow and the outer flow, looks like the rough surface. The turbulent boundary layer is an example of turbulent flow.

The turbulent boundary layer is a typical self-organized flow formed around any rigid body moving in the viscose fluid at high Reynolds number. To illustrate the common problems of the boundary layer theory let us consider the structure of the flat plate boundary layer in adverse pressure gradient - Figure 1. This flow includes the laminar boundary layer (*1*), the transition flow (*2*), the turbulent boundary layer (*3*) and the separated turbulent flow (*4*).

The laminar boundary layer is the predictable and well-investigated flow. But this flow is not a stable at high Reynolds number, because it can be like the amplifier for the waves of small amplitude.

The transition layer has a complex structure included seven sub-regions:

- The laminar flow region in which the small disturbances are generated. This part of flow is considered often as a starting point of transition layer. The Reynolds number of initial point of transition layer is a very sensitive to the boundary conditions on the wall and in the outer flow. The estimated Reynolds number of transition zone is between 4*10
^{5}and 4*10^{6}; - The quasi-laminar flow region in which the amplitude of linear waves (called the Tollmien-Schlichting waves) grows up to the critical value 10
^{-2}. The typical scale of this region is about*L*=100_{x}*H*, where*H*is the local thickness of the boundary layer; - The nonlinear critical layer where the interaction between waves and main flow leads to the new unstable state. The typical scale of this region can be estimated as
*L*=10_{x}*H*; - 3D waves region with scale
*L*. In this region initial two-dimensional waves are transformed into three-dimensional waves;_{x}=H - The region of the secondary instability in which the short length waves are generated. The typical scale of this zone is about
*L*=0.1_{x }*H*; - The Emmons sports region with typical scale
*L*=_{x}*H*. In this part of flow the non-equilibrium process leads to the turbulent spectrum of velocity fluctuations; - The initial region of the turbulent flow in which the amplitude of velocity pulsation is about 3*10
^{-2}*U*.

Figure 1: A) The boundary layer in adverse pressure gradient: *1* - laminar boundary layer; *2* - transition layer; *3* - turbulent boundary layer; *4* - turbulent separated flow; B) the thickness of the laminar boundary layer in the air flow for the free stream velocity *U*=31.47* m/s*; C) the mean height of the separating boundary layer.

The transition from the laminar flow to the turbulent flow is a very attractive phenomenon from the mathematical point of view. Indeed the initial laminar flow, which is not consist of any chaotic waves, then suddenly transforms to the state with a chaotic behaviour.Many authors have investigated this problem of transformation called "dynamical chaos". See for instance

http://hypertextbook.com/Chaos/

The theory of the "dynamical chaos" is based mostly on the simplifier dynamical systems (Lorenz-like chaos) which can't be used directly for the boundary layer problem.

The fractal geometry theory developed by Benoit Mandelbrot also has been utilized to explain the chaos and intermittence in the hydrodynamic turbulence.