Most flows encountered in nature and in engineering practice are turbulent. It is therefore important to understand the fundamental mechanisms at work in such flows. Turbulent flows are unsteady and contain fluctuations that are random in space and time. An important characteristic is the richness of scales of eddy motion present in such flows: In a fully developed turbulent flow all scales appear to be fully occupied or saturated in a sense, from the largest ones that can fit within the size of the flow region down to the smallest scale allowed by dissipative processes. Turbulent flows are also highly vortical, a consequence of vortex stretching and tilting by larger random vorticity fields.

The reason turbulence is so prevalent in fluids of low viscosity is that steady laminar flows tend to become unstable at high Reynolds or Rayleigh numbers and therefore cannot be maintained indefinitely as steady laminar flows. Instability to small disturbances is an initial step in the process whereby a laminar flow goes through transition to turbulence. In investigations of instability of flows that are homogeneous in one or more spatial dimensions, one usually formulates a linear problem of the evolution of an infinite train of small-amplitude waves so as to find whether such waves will grow or decay with time. More general disturbances may be analyzed by Fourier superposition.

A complete description of the transition process requires one to consider the development of disturbances of finite amplitudes. This is generally a difficult theoretical task since it leads to nonlinear problems. A few simplified model problems, giving some insight into the nature of the transition process, are tractable, however, such as the evolution of a finiteamplitude wave train in a parallel shear flow, finite-amplitude density interface waves, weak nonlinear interaction of several wave trains, and the influence of distortion by large-scale motion on smaller-scale wave trains. The treatment of non-wave-like amplitude disturbances lacking spatial and temporal periodicity is more difficult.

Because of the random nature of fully developed turbulent flow fields, statistical methods are usually employed for their description. However, in the statistical averages much of the information that may be relevant to the understanding of the turbulent mechanisms may be lost, especially phase relationships. This may not seem too serious for flows in which the motion appears to be completely disorganized, such as in nearly isotropic or homogeneous turbulent flows.

In this chapter we introduce the concept of an ensemble average, which allows one to form averages for time-dependent processes. One such ensemble average statistical measure is the autocovariance (or autocorrelation) function. It gives information about the average time dependence of a process. The Fourier transform of the autocovariance, in turn, describes the frequency contents of the process. For two random functions of time one can define cross covariance between values of the two functions at different times. The Fourier transform of the cross covariance with respect to delay time gives the cross-spectral density. When these measures are independent of the choice of time origin, the processes are stationary. We shall look at examples of how one can derive a propagation speed from cross covariances or cross spectra and also see how one can find decay times and other properties of a random process. A useful application of spectra and covariance functions is to the relationship between input and output statistical measures for a linear system. From observations of the excitation and the response one is able to draw conclusions about the dynamics of a system. If one knows the system dynamics and some of the statistical properties of the input, one can find the statistical properties of the output, and vice versa.

Many fluid flows can be approximated by linear systems of equations. This means that, in turn, some flows may react to excitation in ways that we can analyze, especially if the excitation is weak.

An example of a linear response of a flow field to turbulence is the emission of acoustic waves from a turbulent jet, as first analyzed by Lighthill (1952) and discussed in Chapter 10. Flows that respond to excitation by divergent oscillation are unstable; such flows are discussed in Chapter 7.

Correlations and spectra depend upon the second moments of a joint probability density. In order to relate correlations and probability distributions, this chapter also outlines some of the elements of probability theory, including the central limit theorem and the normal distribution.

As an illustration of a non-normal distribution the log-normal distribution is presented.

Ensemble averages

A random function is a function that cannot be predicted from its past. An example of a random function of space and time is the velocity field in a turbulent jet.

Fluid flow turbulence is a phenomenon of great importance in many fields of engineering and science. It presents some of the most difficult problems both in the fundamental understanding of its physics and in applications, many of which are still unresolved. Turbulence and related areas have therefore continued to be subjects of intensive research over a period that has lasted for more than a century, and the interest in this field shows no signs of abatement.

In recognition of the need for helping graduate students prepare for their own research in this and related areas of fluid dynamics, a course with the cover title was started by one of us (E. M.-C.) some 20 years ago. Our joint efforts in producing a set of notes for this course has resulted in the present book. The course and its subject matter has evolved over this time period of teaching a mixed group of students from all fields of engineering and from many areas of science, including astrophysics, physics, chemistry, applied mathematics, meteorology, oceanography, and occasionally biology and physiology. With students of such widely different backgrounds we could not assume much commonality in preparation beyond the basics. Hence we found it necessary to start each topic at a fundamental level, and very few concepts could be borrowed from common professional experiences. Many of the students in the course were looking for a thesis topic or needed more insight into turbulence in support of their ongoing research. Discussions with students have resulted in the start of successful research subjects in many instances.

The main aim of the book is to give the students the background enabling them to follow the literature and understand current research results. The book stresses fundamental concepts and basic methods and approaches, although attempting to introduce some recent ideas that we think will prove important in future work on turbulence and related fields. The flavor of a course based on this book will be strongly dependent on the instructor and on the emphasis and the examples of research results chosen for presentation, since the book in itself is not a complete course. Reading of the literature and monographs are also needed.

In addition to correcting misprints and errors in the text, the equations, and the figures of the first edition, we also have further clarified points that have proved difficult for students. We also have benefited from reviews of the book and made other additions and changes as needed.

We added in Chapter 8 a short description of a simplified model for the temporal and spatial evolution of three-dimensional disturbances in a strong mean shear, which we thought might give some theoretical framework for the study of bursting in the near-wall region of a turbulent boundary layer. We also have added a short chapter (Chapter 12) on numerical modeling of turbulence, the lack of which many reviewers pointed to as a shortcoming of the first edition.

A few reviewers have questioned the need to include stability and wave motions in an introductory book on turbulence. In our view, research on hydrodynamic instability has contributed significantly to our understanding of how turbulence is created and maintained. The work in the new field of nonlinear dynamical systems and their chaotic behavior has added further insights showing, for example, that nonlinear waves may show chaotic behavior.

Additions notwithstanding, we have tried strenuously to retain the compactness of the book. It is intended to be a graduate-level introduction and overview of the subject suitable for a one-term course.