Introduction

This chapter introduces the fundamental aspects of the book and the theoretical treatment of fully developed turbulence in fluids governed by the Navier-Stokes equations and lists the basic assumptions for turbulent flows of incompressible, Newtonian fluids in various compact and non-compact, i.e., open domains \(\mathcal {D}\) : the fluid is a continuum, and Knudsen number limits are given; the Navier-Stokes pdes are valid; the IBVPs for the Navier-Stokes pdes are always well posed. The latter condition reflects the fundamental gap in the theory of the Navier-Stokes pdes. Properties of turbulent flows of Newtonian fluids observed in experiments and direct numerical simulations are systematically collected and condensed to a working definition of turbulence in later Chap. 3 . Possible methods of analysis of turbulent flows are listed and their relative merits are discussed: DNS approach moment approach, Pdf/Cdf approach, characteristic function/functional approach, and mapping approach. The turbulence problem is defined and notational conventions are listed. The notation of functional derivatives is defined and explained with an example.