Opening remarks

Taylor's reputation

The obituary of G.I. Taylor (7 March 1886–27 June 1975), written by Sir Brian Pippard1 in 1975, begins thus: “Sir Geoffrey Ingram Taylor, who died at the age of 89, was one of the great scientists of our time and perhaps the last representative of that school of thought that includes Kelvin, Maxwell and Rayleigh, who were physicists, applied mathematicians and engineers – the distinction is irrelevant because their skill knew no such boundaries. Between 1909 and 1973 he published voluminously, and in a lifetime devoted to research left his mark on every subject he touched and on every one of his colleagues … his outgoing manner and complete lack of pomposity conveyed, as no formal exposition could have done, the enthusiasm and intuitive understanding that informed all his work.” These words, taken together with Pippard's closing sentence, “To his many friends he was an inspiration, at once a profound thinker and, it seemed, a truly happy man”, summarize the essential G.I. Taylor. Goldstein (1969) had this to say: “By the end of the first half-century there was a stronger and more widespread element of physics in thought and research on fluid mechanics than in the first twenty or thirty years, and this is much more so now. Several factors and several research workers contributed to this, but the greatest influence has been the example of G.I. Taylor.”

Introduction

Satish Dhawan was born on 25 September 1920 in Srinagar, Kashmir, the home town of his mother Lakshmi. His father, Devidayal, was from the North Western Frontier Province; both parents came from professional families, full of doctors, lawyers and academics – Devidayal retired as a respected judge of the High Court in Lahore, now in Pakistan. Satish's education began under private tutors at home, as his father kept getting transferred in his early career from one town to another in the North West (Kipling country to Indo-British readers). He completed his Indian education at the University of Punjab in Lahore with an unusual combination of degrees: BA in physics and mathematics (1938), MA in English literature (1941) and BE (Hons.) in mechanical engineering (1945). In 1946 he sailed to the USA on a government scholarship, and obtained an MS in aeronautical engineering from the University of Minnesota the following year. (The summer of 1947 saw much turmoil in the subcontinent preceding its imminent partition, and Satish's parents reluctantly left Lahore for India – never to return – a week before the formal end of colonial rule.) In the USA Satish moved to the California Institute of Technology where, with Hans W. Liepmann as his adviser, he obtained the degree of Aeronautical Engineer in 1949 and a PhD in aeronautics and mathematics in 1951. Dhawan made a strong impression, scientifically and otherwise, on everybody he came in contact with at Caltech.

Introduction

Theodore von Kármán, distinguished scientist and engineer with many interests, was born in Budapest on 11 May 1881. His father, Maurice von Kármán, a prominent educator and philosopher at the University of Budapest, had a significant influence over his early intellectual development. After graduating from the Royal Technical University of Budapest in 1902 with a degree in mechanical engineering, von Kármán published in 1906 the first of a long string of papers concerning solid mechanics problems outside the domain of linear elasticity theory, in this case on the compression and buckling of columns. In that same year, apparently at the urging of his father, von Kármán left Hungary for graduate studies at Göttingen. For his 1908 PhD, supervised by Ludwig Prandtl, he developed the concepts of reduced-modulus theory and their application to column behavior such as buckling. Later, with H.-S. Tsien and others, he developed a nonlinear theory for the buckling of curved sheets. His final work in solid mechanics was on the propagation of waves of plastic deformation published as a classified report in 1942 and in the open literature in 1950. In von Kármán's words:

Early years

On 3 April 1920, a few years after G.I. Taylor's far-reaching observations of turbulent diffusion aboard the SS Scotia (Taylor, 1921), and at the time Lewis Fry Richardson was imagining vast weather simulations of atmospheric flow by human ‘computers’ (Richardson, 1922), across the Atlantic in the city of Philadelphia, Stanley Corrsin was born. His parents, Anna Corrsin (née Schorr) and Herman Corrsin had both emigrated to the United States only 13 years before. They came from Romania, where many Russian Jews had settled after leaving Russia in the late 19th and early 20th century. Following further hostilities in Romania, many emigrated again, this time to America. Anna and Herman Corrsin arrived separately at Ellis Island in 1907, Anna in July, and Herman in October. After brief stays in the New York and New Jersey area, where they met and married in 1912, they settled in the city of Philadelphia, in a mixed middle-class neighborhood, not far from the University of Philadelphia. They went into business in the clothing industry and raised their children. Their first son Eugene died young and their second, Lester, was born in 1918. Stan was their third and youngest son.

As a child, Stan Corrsin attended school in Philadelphia and, showing early signs of a highly gifted analytical mind, went on to skip two grades. He enjoyed following the ups and downs of his favorite baseball team, the Philadelphia Athletics.

Introduction

Robert Harry Kraichnan (1928–2008) was one of the leaders in the theory of turbulence for a span of about forty years (mid-1950s to mid-1990s). Among his many contributions, he is perhaps best known for his work on the inverse energy cascade (i.e. from small to large scales) for forced two-dimensional turbulence. This discovery was made in 1967 at a time when two-dimensional flow was becoming increasingly important for the study of large-scale phenomena in the Earth's atmosphere and oceans. The impact of the discovery was amplified by the development of new experimental and numerical techniques that allowed full validation of the conjecture.

How did Kraichnan become interested in turbulence? His earliest scientific interest was in general relativity, which he began to study on his own at age 13. At age 18 he wrote at MIT a prescient undergraduate thesis, Quantum Theory of the Linear Gravitational Field; he received a PhD in physics from MIT in 1949 for his thesis, Relativistic Scattering of Pseudoscalar Mesons by Nucleons, supervised by Herman Feshbach. His interest in turbulence arose in 1950 while assisting Albert Einstein in search for highly nonlinear, particlelike solutions to unified field equations.

Introduction

The nature of turbulent flow has presented a challenge to scientists over many decades. Although the fundamental equations describing turbulent flows (the Navier–Stokes equations) are well established, it is fair to say that we do not yet have a comprehensive theory of turbulence. The difficulties are associated with the strong nonlinearity of these equations and the non-equilibrium properties characterizing the statistical behaviour of turbulent flow. Recently, as predicted by von Neumann 60 years ago, computer simulations of turbulent flows with high accuracy have become possible, leading to a new kind of experimentation that significantly increases our understanding of the problem. The largest numerical simulations nowadays use a discretized version of the Navier–Stokes equations with several billion variables producing many terabytes of information that may be analyzed by sophisticated statistical tools and computer visualization. None of these tools were available in the 1920s when some of the most fundamental concepts in turbulence theory were introduced through the work of Lewis Fry Richardson (1881–1953). Although his name is not as well-known as other contemporary eminent scientists (e.g. Einstein, Bohr, Fermi) and although his life was spent outside the mainstream of academia, his discoveries (e.g. the concept of fractal dimension) are now universally known and essential in understanding the physics of complex systems.

Introduction

George Batchelor (1920–2000), whose portrait (1984) by the artist Rupert Shephard is shown in Figure 8.1, was undoubtedly one of the great figures of fluid dynamics of the twentieth century. His contributions to two major areas of the subject, turbulence and low-Reynolds-number microhydrodynamics, were of seminal quality and have had a lasting impact. At the same time, he exerted great influence in his multiple roles as founder Editor of the Journal of Fluid Mechanics, co-Founder and first Chairman of EUROMECH, and Head of the Department of Applied Mathematics and Theoretical Physics (DAMTP) in Cambridge from its foundation in 1959 until his retirement in 1983.

I focus in this chapter on his contributions to the theory of turbulence, in which he was intensively involved over the period 1945 to 1960. His research monograph The Theory of Homogeneous Turbulence, published in 1953, appeared at a time when he was still optimistic that a complete solution to ‘the problem of turbulence’ might be found. During this period, he attracted an outstanding group of research students and post-docs, many from his native Australia, and Senior Visitors from all over the world, to work with him in Cambridge on turbulence. By 1960, however, it had become apparent to him that insurmountable mathematical difficulties in dealing adequately with the closure problem lay ahead.

Introduction

Philip G. Saffman was a leading theoretical fluid dynamicist of the second half of the twentieth century. He worked in many different sub-fields of fluid dynamics and, while his impact in other areas perhaps exceeded that in turbulence research, which is the topic of this article, his contributions to the theory of turbulence were significant and remain relevant today. He was also an incisive and, some might conclude, a somewhat harsh critic of progress or what he perceived as the lack thereof, in solving ‘the turbulence problem’. This extended to his own work; he stated in a preface to lectures on homogeneous turbulence (Saffman, 1968) that

In this article, we will try to survey Saffman's thinking and contribution to turbulence research from the mid 1950s, when he began to mature as a scholar, until the late 1970s when he moved away from the study of turbulence to concentrate on the related but separate area, of the dynamics of isolated and interacting vortices. Although, for the most part, the evolution of his ideas and their application to turbulence in this period developed both thematically and chronologically together, where there are departures we will tend to focus on the former.

“Will no-one rid me of this turbulent priest?” So, according to tradition, cried Henry II, King of England, in the year 1170, even then conveying a hint of present frustration and future trouble. The noun form ‘la turbulenza’ appeared in the Italian writings of that great genius Leonardo da Vinci early in the 16th century, but did not appear in the English language till somewhat later, one of its earliest appearances being in the quotation above from Shakespeare. In his “Memorials of a Tour in Scotland, 1803”, William Wordsworth wrote metaphorically of the turmoil of battles of long ago: “Yon foaming flood seems motionless as ice; its dizzy turbulence eludes the eye, frozen by distance …”. Perhaps we might speak in similar terms of long-past intellectual battles concerning the phenomenon of turbulence in the scientific context.

Turbulence in fluids, or at least its scientific observation, continued to elude the eye until Osborne Reynolds in 1883 conducted his brilliant ‘flow visualisation’ experimental study “of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels”.

Introduction

Scope

Articles on Osborne Reynolds' academic life and published works have appeared in a number of publications beginning with a remarkably perceptive anonymous obituary notice published in Nature within eight days of his death (on 21 February 1912) and a more extensive account written by Horace Lamb, FRS, and published by the Royal Society (Lamb, 1913) about a year later. More recent reviews have been provided by Gibson (1946), a student of Reynolds and later an academic colleague, by Allen (1970), who provided the opening article in a volume marking the passage of 100 years from Reynolds taking up his chair appointment at Manchester in 1868, and by Jackson (1995), in an issue of Proc. Roy. Soc. celebrating the centenary of the publication of Reynolds' 1895 paper on what we now call the Reynolds decomposition of the Navier–Stokes equations, about which more will be said later in the present chapter. A significant portion of the present account is therefore devoted to Reynolds' family and background and to hitherto unreported aspects of his character to enable his contributions as a scientist and engineer to be viewed in the context of his life as a whole. While inevitably some of what is presented here on his academic work will be known to those who have read the articles cited above, archive material held by the University of Manchester and The Royal Society and other material brought to light in the writers' personal enquiries provide new perspectives on parts of his career.

Early years

Albert Alan Townsend was born on the 22nd of January 1917 in Melbourne Australia son of Albert Rinder Townsend and Daisy Townsend née Gay. At the time of his birth his father was a clerk in the accounts branch of the Department of Trade and Customs – he also served as secretary of the Commonwealth Film Censorship Board. His father went on to have a very successful career in the Commonwealth public service. As his career evolved he moved the family to Canberra in the ACT (Australian Capital Territory) which is the seat of the government in Australia. Albert and Daisy had three children: Alan, Elisabeth and Neil. In 1933 Albert Rinder Townsend was awarded the OBE.

Alan obtained his Leaving Certificate in 1933 from the Telopea Park High School with an outstanding pass, including first-class honours in mathematics, and the Canberra University College Council awarded him a scholarship of £120 a year to pursue a science course at Melbourne University. He completed his Bachelor of Science in 1936, graduating with first-class honours, and started his Master of Science. Just before his 20th birthday (1937) he graduated Master of Science, with honours in natural philosophy and pure mathematics. He was awarded the Dixson Research Scholarship and the Professor Kernot Research Scholarship.

The towering figure of Kolmogorov and his very productive school is what was perceived in the twentieth century as the Russian school of turbulence. However, important Russian contributions neither start nor end with that school.

Physicist and pilot

What seems to be the first major Russian contribution to the turbulence theory was made by Alexander Alexandrovich Friedman, famous for his work on non-stationary relativistic cosmology, which has revolutionized our view of the Universe. Friedman's biography reads like an adventure novel. Alexander Friedman was born in 1888 to a well-known St. Petersburg artistic family (Frenkel, 1988). His father, a ballet dancer and a composer, descended from a baptized Jew who had been given full civil rights after serving 25 years in the army (a so-called cantonist). His mother, also a conservatory graduate, was a daughter of the conductor of the Royal Mariinsky Theater. His parents divorced in 1897, their son staying with the father and becoming reconciled with his mother only after the 1917 revolution. While attending St. Petersburg's second gymnasium (the oldest in the city) Friedman befriended a fellow student Yakov Tamarkin, who later became a famous American mathematician and with whom he wrote their first scientific works (on number theory, received positively by David Hilbert).

Introduction

When a fluid is locally perturbed by an impulse (for example, by an impact) or a periodic excitation (the vibration of a membrane, string, or mechanical blade), the perturbation may propagate from the source in the form of a wave. Examples include acoustic waves, surface waves, and internal waves in a stratified fluid (Lighthill, 1978). The solution of the linearized equations for small amplitude perturbations leads to a major result: the wave number k and frequency ω (or the wave speed c = ω/k) are not independent, they are related by a dispersion relation. Since this relation is obtained from linearized equations, another major result is that dispersion does not depend on the amplitude of the perturbation. However, if the amplitude exceeds some level, new effects arise that the dispersion relation of linear theory obviously does not describe. To explain these new effects it is necessary to include nonlinear terms neglected in the linear study, i.e., to develop a theory of nonlinear waves. Such waves are also referred to as finite-amplitude waves, in contrast to the waves of infinitesimal amplitude considered in a linear analysis.

The objective of the present chapter is to give an elementary account of the theory of nonlinear waves. We will show (i) how nonlinear waves can be constructed by a perturbation method (essentially the multiple-scale method presented in the preceding chapter), and (ii) how the linear stability of these waves can be studied.

Introduction

In this chapter we present an introduction to dense granular flows and their stability by discussing two classes of phenomena: avalanches on an inclined plane, and particle transport on an erodible bed sheared by a fluid flow. These granular flows lead to the appearance of surface waves, called ripples or dunes depending on whether their wavelength is of a few centimeters or a few meters (the relevance of this common distinction will be discussed later on). Owing to the difficulty – both experimental and theoretical – of studying granular media, the mechanisms responsible for these waves remain poorly understood, and so the results presented in this chapter are definitely less well established than those in the preceding chapters.

Avalanches, ripples, and dunes present serious problems for human activities. Among natural phenomena, snow and mud avalanches are well known for their destructive nature; the displacement of a sand dune by the wind – the so-called aeolian dunes – while less dramatic, can cut communication links and threaten habitation and industrial installations. Subaqueous dunes perturb navigation in rivers and shallow seas such as the North Sea, while on river bottoms such dunes increase friction and raise the water level, thereby contributing to flooding. Granular flows are also omnipresent in industry: flow and transport of coal, construction materials (cement, sand, gravel), agricultural foodstuffs, pharmaceutical materials, and sand from oilfields are some examples. Instabilities occur in the conduits used to transport these materials, giving rise to dunes which perturb the flow and may form obstructions, causing serious damage to operating equipment.

Hydrodynamic instabilities occupy a special position in fluid mechanics. Since the time of Osborne Reynolds and G. I. Taylor, it has been known that the transition from laminar flow to turbulence is due to the instability of the laminar state to certain classes of perturbations, both infinitesimal and of finite amplitude. This paradigm was first displayed in a masterful way in the studies of G. I. Taylor on the instability of Couette flow generated by the differential rotation of two coaxial cylinders. From then on, the theory of hydrodynamical instability has formed a part of the arsenal of techniques available to the researcher in fluid mechanics for studying transitions in a wide variety of flows in mechanical engineering, chemical engineering, aerodynamics, and in natural phenomena (climatology, meteorology, and geophysics).

The literature on this subject is so vast that very few researchers have attempted to write a pedagogical text which describes the major developments in the field. Owing to the enormity of the task, there is a temptation to cover a large number of physical situations at the risk of repetition and of wearying the reader with just a series of methodological approaches. François Charru has managed to avoid this hazard and has risen to the challenge. With this book he fills the gap between the classical texts of Chandrasekhar and Drazin and Reid, and the more recent book of Schmid and Henningson.

Classical instability theory essentially deals with quasi-parallel or parallel shear flows such as mixing layers, jets, wakes, Poiseuille flow in a channel, boundary-layer flow, and so on.