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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.
Flows of fluids of low viscosity may become unstable when large gradients of kinetic and/or potential energy are present. The flow field set up by the instability generally tends to smooth out the velocity and temperature differences causing it. The available kinetic or potential energy released by the instability may be so large that transition to a fully developed turbulent flow occurs.
Transition is influenced by many parameters. An important one is the level of preexisting disturbances in the fluid; a high level would generally cause early transition. Another cause for early transition in the case of wall-bounded shear flows is surface roughness. The manner in which transition occurs may also be very sensitive to the detailed flow properties.
For shear flows the basic nondimensional flow parameter measuring the tendency toward instability and transition is the Reynolds number; for high Re values, kinetic energy differences can be released faster into fluctuating motion than viscous diffusion will have time to smooth them out. For a heated fluid subject to gravity the Rayleigh number is the main stability parameter.
Of crucial importance for the tendency of a flow to become unstable and go through transition is the detailed distribution of mean velocity and/or temperature in the field. The analysis that follows is intended to illustrate this.
Although the flow processes involved in instability and transition might at a first glance appear to have only a slight resemblance to those observed in fully developed turbulence, they are nevertheless related to it in important ways. In a gross sense turbulence may be regarded as a manifestation of flow instability occurring randomly in space and time. The linear instability problem is the simplest flow model incorporating the interaction between unsteady fluctuations and a background shear or density distribution. With the aid of nonlinear instability theory one may also possibly be able to clarify some of the mechanisms whereby turbulence is maintained.
Instability to small disturbances
Because of the mathematical difficulties in the analysis of flow instability, only idealized cases for which the basic fluid flow properties vary with one spatial coordinate can be analyzed in a reasonably simple manner.
The novelist's novelist', wrote Henry James of Flaubert, declaring Madame Bovary his masterpiece. Novelist and masterpiece have been decisively influential and remain an inescapable - at times obsessive - fact of modern literary experience. Doubtless the most remarkable testimony to this is the massive study of Flaubert as author of Madame Bovary undertaken by the philosopher Jean-Paul Sartre and still unfinished at his death, after nearly 3,000 pages. Conceptions of the book have varied strongly and importantly: Zola saw it as providing ‘the code of the new art’ he was developing as naturalism; Nabokov regarded it as essentially ‘a prose poem'; Robbe-Grillet today considers it ‘a nouveau roman before its time', unsettling our assumptions of realism and initiating a whole modern ‘practice of writing'.
From the start, moreover, Flaubert's novel had an intense social reverberation. Brought to trial for offences against family and religion, it gained a notoriety that focused it at once as part of a questioning of marriage, sex, and the role of women. Its achievement was to transpose those given social elements into a new configuration that captured and articulated a fundamental experience of the post-romantic, commercialindustrial, democratic period. Indeed, the depiction of Emma Bovary was appropriated as a general representation and bovarysme entered French and other Western languages as the word for a typical attitude and its understanding. The disturbing aspect of the achievement, to which the trial was one response, involved quite directly Flaubert's artistic creed of impersonality, which was perceived as leaving his work dangerously indifferent, with no clear moral, no message; only what Nietzsche would call ‘the desire for nothing’ and D. H. Lawrence condemn as a withdrawal from life ‘as from a leprosy'. Nietzsche's attack bore too on Madame Bovary as exemplary of what he considered the damaging ‘feminisation’ of modern art and feeling, on the terms of its presentation of - its own implication in - bovarysme.
Then they spoke of the mediocrity of life in the provinces' (11,8)
The full title of Flaubert's novel is Madame Bovary Moeurs deprovince. The second half - ‘provincial manners’ - gives a certain code of the novel, announces a relation of this novel to the realism associated with Balzac and his influence. In the 1842 ‘Avant-Propos’ Balzac had defined his ambition in the Comedie Humaine as that of writing ‘the history forgotten by so many historians, that of manners [celle des moeurs]’ and the first and major part of the Comedie was to have the overall heading Etudes de moeurs, so many ‘studies in manners’ to provide ‘the representation of society in all its effects'. One of the sections into which the Etudes de moeurs were in turn to be divided was Scenes de la vie de province, comprising provincial subjects of the kind to be found in Eugenie Grandet or La Muse du departement, the latter characteristically seeking to bring to light, in its own words, ‘one of those long and monotonous marital tragedies which would remain forever unknown did not the avid scalpel of the nineteenth century … go foraging into the darkest recesses of the heart'. Writing here in 1843, Balzac already has that image of the scalpel that will become, via Sainte-Beuve, the dominant image for the contemporary understanding of Madame Bovary. the novelist opens up social existence, explores its reality through the hidden stories of people with all their passions and dramas. By the time of Madame Bovary, the provinces are a specifically novelistic subject with a specifically novelistic conception - constant monotony, obscure lives, silent intensity - to which Flaubert immediately alludes: Moeurs de province.
This Balzacian resonance is sustained in the novel itself, which does indeed stand as a powerful representation of provincial life in the Normandy Flaubert knew so well, a life that he is concerned to get right: ‘I really have to see an agricultural show’ (17-18 July 1852, B), and the next day he does, making faithful use of it for the equivalent scene in the novel (11,8).