Elements of Turbulent Flow

Abstract

The unidirectional flow becomes unstable as soon as the Reynolds number increases above a certain threshold; beyond which flow enters in a turbulent regime. Turbulence is characterized by a complex three-dimensional motion that, despite it is governed by deterministic laws (Navier–Stokes equation), displays apparently random properties; the unpredictability of deterministic laws is explored through a parallel with simple chaotic systems. The randomness of individual solution suggests to seek a description of turbulent flow in terms of average statistical properties. The classic averaging operator is introduced and applied to the Navier–Stokes equation to obtain an equation for the average velocity, the Reynolds equation. However, the simplification introduced by the well-behaving average variables is paid by the appearance of novel terms in the averaged equation that is non-closed and requires additional assumptions to be solved. The Reynolds equation is used to obtain the general solution for a fluid moving in a turbulent regime over a wall, the result is then used to estimate the friction associated with turbulent flowing inside a duct. Finally, the concept of phase-averaging for time-periodic flows is introduced.