Turbulence

Abstract

This Chapter addresses modern theory of turbulence. Recall that chaotic dynamics, of which turbulence is the most extreme form, computationally started in 1963, when Ed Lorenz from MIT took the Navier–Stokes equations from viscous fluid dynamics and reduced them into three first–order coupled nonlinear ODEs (1.21), to demonstrate the idea of sensitive dependence upon initial conditions and associated chaotic behavior. Starting from the simple Lorenz system, in this Chapter we develop the comprehensive theory of turbulent flows. For start–off, recall from Introduction that D. Ruelle and F. Takens argued in a seminal paper [RT71] that, as a function of an external parameter, the route to chaos in a fluid flow is a transition sequence leading from stationary (S) to single periodic (P), double periodic (QP₂), triple periodic (QP₃) and, possibly, quadruply periodic (QP₄) motions, before the flow becomes chaotic (C).