The ability of modern computers to store and rapidly manipulate large quantities of data has allowed the development of flow simulations that complement, and sometimes even replace, traditional laboratory experiments. In industry, the considerable expense of constructing and exploiting physical models motivates the search for reliable simulation techniques to reduce the cost of the design and development phases. Full, detailed coverage of numerical methods for turbulent flows goes well beyond the scope of this book and would require a volume to itself. This chapter is therefore limited to a relatively brief overview of the three main types of numerical techniques, represented by the following three sections, in which the reader desiring further details of numerical methods will find references to more specialized works.

Direct Numerical Simulation (DNS)

The first, and most obvious, approach to the numerical simulation of turbulent flows treats them like any others, solving the unsteady, three-dimensional Navier-Stokes equations numerically without further ado. Such methods are referred to as direct simulations, in contrast with those based on the statistically averaged equations considered in Section 8.3, which involve closure approximations, but are much less computationally costly than DNS at high Reynolds numbers. Direct simulation allows numerical experiments to be carried out that are often difficult or impossible to realize in the laboratory, and yields detailed information concerning the flow field in individual realizations and its statistical properties that again often surpasses what can be measured in real flows. Furthermore, unlike other simulation techniques for turbulent flows, DNS does not involve approximations, other than those due to discretization, which are inherent in any numerical solution of differential equations, and so provides benchmarks against which less costly simulation methods may be evaluated and parameterized, just as they can by using the results of physical experiments. However, the continuum of different spatial (and temporal) scales intrinsic to turbulence, a continuum whose width increases with the turbulent Reynolds number, ReL, leads to computer time and storage requirements that increase rapidly with ReL. In consequence, DNS is limited to relatively low Reynolds numbers in practice and it is currently very expensive to conduct a DNS calculation at even moderately high Reynolds number.

Besides its apparent randomness, one of the important characteristics of turbulent flows at high Reynolds number (which we presume throughout this chapter) is the wide range of different space and time scales which they contain. As illustrated in Figure 3.1a, the velocity measured at a given point within a turbulent flow fluctuates randomly and does not yield a smooth function of time, but rather one with finescale “fur.” If part of such a graph is examined at successively higher magnifications, instead of approaching a straight line, as would a smooth function, the fur is revealed as being itself “furry,” and so on, through a succession of smaller and smaller time scales, until eventually the smallest of all is reached. At this scale, the velocity finally appears as a smooth function of time, with no further structure revealed by additional magnification. On the other hand, the largest time scales of turbulence are characterized by the time interval required for statistical decorrelation of the turbulent velocity fluctuations. At high Reynolds number, the separation of scales between the largest and smallest is wide, and gets wider as the Reynolds number increases.

The continuum of time scales described above is also reflected in the spatial structure of the flow. The order of size of the largest spatial scales of turbulence is measured by the velocity correlation length L, which characterizes the spatial separation required for significant decorrelation of the velocity fluctuations. Successively higher magnification of the spatial structure of the velocity field shows smaller and smaller scales down to the smallest, known as the Kolmogorov scale, at which it appears smooth. Thus, turbulence contains a continuum of scales, ranging from the large ones, whose size is of order the correlation length, to those of order the Kolmogorov scale. Both the Kolmogorov scale, r\, and correlation scale, L, are best considered as orders of magnitude: one talks of scales comparable with, much smaller than, or much larger than these quantities. The spatial and temporal “furriness” of the velocity field are related; indeed, one can think of the small time scales as resulting from convection of fine spatial structures past the measurement probe, as we will see later.

If one attempts to take the derivative of the velocity field, either with respect to time or one of the spatial coordinates, its furry nature becomes markedly more apparent.

The theory of turbulence has seen considerable progress in the last fifty years, although these advances are, at first sight, less marked than might have been hoped at the outset. Prior to that period, Reynolds, Taylor, Prandtl, Von Karman, and Kolmogorov, to name just some of the eminent workers, had laid the foundations of the subject and a number of excellent books and surveys appeared in the 1950s. Batchelor's book, The Theory of Homogeneous Turbulence, was first published in 1953 and mainly concerned the development of spectral methods, quickly gaining a following among existing and aspiring specialists in turbulence. Shortly afterwards (1956), another classic appeared in the shape of Townsend's The Structure of Turbulent Shear Flow, whose approach was based more on physical reasoning and placed more emphasis on the more realistic, but less theoretically tractable case of inhomogeneous turbulent flows.

Many other excellent works have appeared since then, among them the books given in the General References list following this preface. In each of them, the authors view turbulence from their own perspective, different from that of other authors, for, owing to the nature of the subject, an author's personal viewpoint colors the presentation perhaps more than in most areas of science. The book by Tennekes and Lumley is somewhat different from most of the others because it gives considerable weight to physical ideas, order-of-magnitude discussions, and other means for gaining intuition about turbulent flows, while the authors remind us that “several dozen introductory texts in general fluid dynamics exist, but the gap between monographs and advanced texts in turbulence is wide.” The book gives the reader excellent food for thought before tackling more specialized ones, such as Monin and Yaglom's two-volume work, which remains the undisputed (if slightly dated) bible for the subject.

The present book does not seek to replace these existing works, but to complement them, giving the reader a solid grounding in the theory of turbulence and developing both physical insight and the mathematical framework needed to express the theory. Turbulence is treated using one-point statistics in the early chapters of the book, while multipoint statistics, analyzed by spectral methods, are introduced later, the two approaches being complementary and mutually illuminating.

Thomas Paine was a unique political thinker. Writing at the end of the eighteenth Century while a revolutionary tide swept over North America and Europe, he had an unrivalled reputation not just as a democratic polemicist but also as an activist politician. He lived in England, the new United States, and France, and was hailed by all those eager for an anti-monarchical political order in the Atlantic world. Hated and feared by conservatives in England and America, Paine was nonetheless suspect to the most radical French revolutionaries. Extraordinary because he came not from the cultured classes but from the common folk, he expressed their political feelings and ushered in a new era of political rhetoric and public commitment.

Little is known of the details of Paine's life, especially of the first thirty-seven years he spent in England. He was born at Thetford in Norfolk, in 1737, and was early on subject to the unconventional religious influence of his Quaker father. For a time he followed his father's occupation as a maker of corsets, but also went to sea as a common sailor for two years, ran a small tobacco shop, and had a minor job with the English government collecting customs duties, an “excise officer.”

Despite the absence of information, Paine's biographers have rightly speculated on what shaped his life in England in the hope of understanding his later career. For after what appeared to be an entirely ordinary and nondescript life in England, filled with the failures and woes of the lower ranks of that time, he left for the American colonies in 1774, where he began to write in favor of their Separation from the Mother Country. Within a short time he became known in both the Old and New Worlds as a political theorist of great significance. What had gone on in England to prepare him for his novel role?

Paine had obtained enough education to master reading, writing, and arithmetic in Norfolk, a tribute to his family's zeal for education and a sign of the growing aspirations of the class of craftspeople in England to which his family belonged. As an adult, in addition to his other employments, Paine had even briefly been a school teacher. In his early manhood he supplemented his rudimentary knowledge by attending populär lectures on the physics of Isaac Newton that had transformed eighteeenth-century intellectual life.

It has been my intention for several years past to publish my thoughts upon Religion. I am well aware of the difficulties that attend the subject; and from that consideration had reserved it to a more advanced period of life. I intended it to be the last offering I should make to my fellow-citizens of all nations, and that at a time when the purity of the motive that induced me to it could not admit of a question, even by those who might disapprove the work.

The circumstance that has now taken place in France, of the total abolition of the whole national order of priesthood and of everything appertaining to compulsive Systems of religion and compulsive articles of faith, has not only precipitated my intention, but rendered a work of this kind exceedingly necessary; lest, in the general wreck of superstition, of false Systems of government, and false theology, we lose sight of morality, of humanity, and of the theology that is true. As several of my colleagues, and others of my fellow-citizens of France, have given me the example of making their voluntary and individual profession of faith, I also will make mine; and I do this with all that sincerity and frankness with which the mind of man communicates with itself.

I believe in one God, and no more; and I hope for happiness beyond this life.

I believe in the equality of man, and I believe that religious duties consist in doing justice, loving mercy, and endeavoring to make our fellow-creatures happy.

But lest it should be supposed that I believe many other things in addition to diese, I shall, in the progress of this work, declare the things I do not believe and my reasons for not believing them.

I do not believe in the creed professed by the Jewish church, by the Roman church, by the Greek church, by the Turkish church, by the Protestant church, nor by any church that I know of. My own mind is my own church.

All national institutions of churches - whether Jewish, Christian, or Turkish - appear to me no other than human inventions set up to terrify and enslave mankind and monopolize power and profit.

From the part Mr. Burke took in the American Revolution it was natural that I should consider him a friend to mankind; and as our acquaintance commenced on that ground, it would have been more agreeable to me to have had cause to continue in that opinion than to change it.

At the time Mr. Burke made his violent speech last winter in the English Parliament against the French Revolution and the National Assembly, I was in Paris, and had written to him but a short time before to inform him how prosperously matters were going on. Soon after this I saw his advertisement of the pamphlet he intended to publish: As the attack was to be made in a language but little studied and less understood in France, and as everything suffers by translation, I promised some of the friends of the Revolution in that country that whenever Mr. Burke's pamphlet came forth, I would answer it. This appeared to me the more necessary to be done when I saw the flagrant misrepresentations which Mr. Burke's pamphlet contains; and that while it is an outrageous abuse on the French Revolution and the principles of Liberty, it is an imposition on the rest of the world.

I am the more astonished and disappointed at this conduct in Mr. Burke, as (from the circumstances I am going to mention) I had formed other expectations.

I had seen enough of the miseries of war to wish it might never more have existence in the world, and that some other mode might be found out to settle the differences that should occasionally arise in the neighborhood of nations. This certainly might be done if courts were disposed to set honestly about it, or if countries were enlightened enough not to be made the dupes of courts. The people of America had been bred up in the same prejudices against France which at that time characterized the people of England; but experience and an acquaintance with the French nation have most effectually shown to the Americans the falsehood of those prejudices; and I do not believe that a more cordial and confidential intercourse exists between any two countries than between America and France.

Some writers have so confounded society with government as to leave little or no distinction between them; whereas they are not only different, but have different origins. Society is produced by our wants and government by our wickedness; the former promotes our happiness positively by uniting our affections, the latter negatively by restraining our vices. The one encourages intercourse, the other creates distinctions. The first is a patron, the last a punisher.

Society in every State is a blessing, but government, even in its best State, is but a necessary evil; in its worst State an intolerable one; for when we suffer or are exposed to the same miseries by a government, which we might expect in a country without government, our calamity is heightened by reflecting that we furnish the means by which we suffer. Government, like dress, is the badge of lost innocence; the palaces of kings are built upon the ruins of the bowers of paradise. For were the impulses of conscience clear, uniform, and irresistibly obeyed, man would need no other lawgiver; but that not being the case, he finds it necessary to surrender up a part of his property to furnish means for the protection of the rest; and this he is induced to do by the same prudence which in every other case advises him out of two evils to choose the least. Wherefore, security being the true design and end of government, it unanswerably follows that whateveryora thereof appears most likely to ensure it to us, with the least expense and greatest benefit, is preferable to all others.

In order to gain a clear and just idea of the design and end of government, let us suppose a small number of persons settled in some sequestered part of the earth, unconnected with the rest; they will then represent the first peopling of any country, or of the world. In this State of natural liberty, society will be their first thought. A thousand motives will excite them thereto; the strength of one man is so unequal to his wants, and his mind so unfitted for perpetual solitude, that he is soon obliged to seek assistance and relief of another, who in his turn requires the same.

The following little piece was written in the winter of 1795 anc* ‘96; and, as I had not determined whether to publish it during the present war, or to wait till the commencement of a peace, it has lain by me, without alteration or addition, from the time it was written.

What has determined me to publish it now is a sermon preached by Watson, Bishop of Llandaff. Some of my readers will recollect, that this Bishop wrote a book entitled “An Apology for the Bible,” in answer to my second part of “The Age of Reason.” I procured a copy of his book, and he may depend upon hearing from me on that subject.

At the end of the Bishop's book is a list of the works he has written. Among which is the sermon alluded to; it is entitled: “The Wisdom and Goodness of God, in having made both Rieh and Poor; with an Appendix, containing Reflections on the Present State of England and France.”

The error contained in this sermon determined me to publish my “Agrarian Justice.” It is wrong to say God made rieh and poor, He made only male and female; and He gave them the earth for their inheritance …

Instead of preaching to encourage one part of mankind in insolence … It would be better that priests employed their time to render the general condition of man less miserable than it is. Practical religion consists in doing good: and the only way of serving God is that of endeavoring to make His creation happy. All preaching that has not this for its objeet is nonsense and hypoerisy.

THOMAS PAINE

Agrarian justice

To preserve the benefits of what is called civilized life, and to remedy at the same time the evil which it has produced, ought to be considered as one of the first objeets of reformed legislation.

Whether that State that is proudly, perhaps erroneously, called civilization, has most promoted or most injured the general happiness of man, is a question that may be strongly contested. On one side, the spectator is dazzled by splendid appearances; on the other, he is shocked by extremes of wretchedness; both of which it has erected. The most affluent and the most miserable of the human race are to be found in the countries that are called civilized.

These are the times that try men's souls. The summer soldier and the sunshine patriot will, in this crisis, shrink from the service of his country; but he that Stands it now, deserves the love and thanks of man and woman. Tyranny, like hell, is not easily conquered; yet we have this consolation with us, that the harder the conflict, the more glorious the triumph. What we obtain too cheap, we esteem too lightly: it is dearness only that gives everything its value. Heaven knows how to put a proper price upon its goods; and it would be stränge indeed if so celestial an article as FREEDOM should not be highly rated. Britain, with an army to enforce her tyranny, has declared that she has a right (not only to TAX) but “to BIND US in ALL CASES WHATSOEVER”; and if being bound in that manner is not slavery, then is there not such a thing as slavery upon earth. Even the expression is impious; for so unlimited a power can belong only to God.

Whether the independence of the continent was declared too soon, or delayed too long, I will not now enter into as an argument; my own simple opinion is, that had it been eight months earlier it would have been much better. We did not make a proper use of last winter; neither could we, while we were in a dependent State. However, the fault, if it were one, was all our own; we have none to blame but ourselves. But no great deal is lost yet. All that Howe has been doing for this month past is rather a ravage than a conquest, which the spirit of the Jerseys a year ago would have quickly repulsed, and which time and a little resolution will soon recover.

I have as little superstition in me as any man Irving; but my secret opinion has ever been, and still is, that God Almighty will not give up a people to military destruction, or leave them unsupportedly to perish, who have so earnestly and so repeatedly sought to avoid the calamities of war, by every decent method which wisdom could invent.