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This book provides a comprehensive overview of statistical descriptions of turbulent flows. Its main objectives are to point out why ordinary perturbative treatments of the Navier–Stokes equation have been rather futile, and to present recent advances in non-perturbative treatments, e.g., the instanton method and a stochastic interpretation of turbulent energy transfer. After a brief introduction to the basic equations of turbulent fluid motion, the book outlines a probabilistic treatment of the Navier–Stokes equation and chiefly focuses on the emergence of a multi-point hierarchy and the notion of the closure problem of turbulence. Furthermore, empirically observed multiscaling features and their impact on possible closure methods are discussed, and each is put into the context of its original field of use, e.g., the renormalization group method is addressed in relation to the theory of critical phenomena. The intended readership consists of physicists and engineers who want to get acquainted with the prevalent concepts and methods in this research area.

1. Introduction

Abstract
The physics of turbulence is a fascinating but at the same time inherently complex branch of classical physics. Although the underlying equations, i.e., the Navier-Stokes equations are known for nearly two centuries, we have yet to identify probabilistic methods that would allow for an accurate description of the spatio-temporal complexity exhibited by its velocity field fluctuations  [1].
Jan Friedrich

2. Basic Properties of Hydrodynamic Turbulence

Abstract
This introductory chapter will give an overview of basic hydrodynamic equations and the difficulties they present in the realm of fully developed turbulence. Furthermore, important quantities such as the Reynolds number or the local energy dissipation rate will be introduced.
Jan Friedrich

3. Statistical Formulation of the Problem of Turbulence

Abstract
Chapter 2 already highlighted the fact that a turbulent flow is a mechanical nonlinear system with a very large number of degrees of freedom, which can be estimated as a function of the Reynolds number according to $$\text {Re}^{9/4}$$. Therefore, in contrast to laminar fluid motion where only a few degrees of freedom are excited [1], it is nearly impossible to describe individual time variations of all generalized coordinates (velocity $$\mathbf{u}$$, pressure p, or temperature T) in a fully developed turbulent flow accurately. It is thus suggested to consider only ensembles of these generalized coordinates. Accordingly, successful approaches to hydrodynamic turbulence have to involve a treatment of the deterministic equations of fluid mechanics via statistical and stochastic methods.
Jan Friedrich

4. Overview of Closure Methods for the Closure Problem of Turbulence

Abstract
In the preceding chapter, we introduced a statistical formulation of hydrodynamic turbulence. An inherent difficulty in both the moment formulation in Sect. 3.​3 and the formulation via kinetic equations in Sect. 3.​4 is the hierarchical character of the corresponding system of equations. The latter can be considered as a signature of the spatio-temporal complexity that is inherent in turbulent systems. Despite the fact that moment and kinetic approach differ in the way the hierarchy of equations arises, they share the property that equations of statistical quantities of order n involve unclosed terms of order $$n+1$$. Hence, in both approaches, we are faced with the amply defined closure problem of turbulence.
Jan Friedrich

5. Non-Perturbative Methods

Abstract
The previous chapter highlighted the limitations of perturbative treatments of the Navier-Stokes equation such as the quasi-normal approximation and the renormalization or renormalization group method. It was found that nonlinearities are too strong to be grasped in a perturbative sense as perturbation expansions are set up in terms of powers of the Reynolds number.
Jan Friedrich

6. Outlook

Abstract
The longstanding problem of turbulence still awaits a thorough and complete understanding of the equations that govern turbulent fluid motion. As it has been stressed throughout this monograph, the overwhelmingly complex spatio-temporal organization of turbulent flows requires a comprehensive statistical treatment of the Navier-Stokes equation.
Jan Friedrich