Scientific papers on how to represent in mathematical form the types of fluid motion we call turbulent flow have been appearing for over a century while, for the last sixty years or so, a sufficient body of knowledge has been accumulated to tempt a succession of authors to collect, systematize and distil a proportion of that knowledge into textbooks. From the start a bewildering variety of approaches has been advocated: thus, even in the 1970s, the algebraic mixing-length models presented in the book by Cebeci and Smith jostled on the book-shelves with Leslie's manful attempt to make comprehensible to a less specialized readership the direct-interaction approach developed by Kraichnan and colleagues. As the progressive advance in computing power made it possible to apply the emerging strategy of computational fluid dynamics to an ever-widening array of industrially important flows, however, eddy-viscosity models (EVMs) based on the solution of two transport equations for scalar properties of turbulence (essentially, length and time scales of the energy-containing eddies) emerged as the modelling strategy of choice and, correspondingly, have been the principal focus in several textbooks on the modelling of turbulent flows (for example, Launder and Spalding, Wilcox and Piquet).

Today, two-equation EVMs remain the work-horse of industrial CFD and are applied through commercially marketed software to flows of a quite bewildering complexity, though often with uncertain accuracy. However, there has been a major shift among the modelling research community to abandon approaches based on the Reynolds-averaged Navier–Stokes (RANS) equations in favour of large-eddy simulation (LES) where the numerical solution for any flow adopts a three-dimensional, time-dependent discretization of the Navier–Stokes equations using a model to account simply for the effects of turbulent motions too fine in scale to be resolved with the mesh adopted – that is, a sub-grid-scale (or sgs) model. While acknowledging that LES offers the prospects of tackling turbulence problems beyond the scope of RANS, a further major driver for this changeover has been the manifold inadequacies of the stress-strain hypothesis adopted by linear eddy-viscosity models. While such a simple linkage between mean strain rate and turbulent stress seemed adequate for a large proportion of two-dimensional, nearly parallel flows, its weaknesses became abundantly clear as attention shifted to recirculating, impinging and three-dimensional shear flows. Although an LES approach will, most probably, also adopt an sgs model of eddy-viscosity type, the consequences are less serious for two reasons. First, the majority of the transport caused by the turbulent motion will be directly resolved by the large eddies and secondly, the finer scale eddies that must still be resolved by the sub-grid-scale model of turbulence will arguably be a good deal closer to isotropy. Thus, adopting an isotropic eddy viscosity as the sgs model may not significantly impair the accuracy of the solution.

Introduction

In the early decades of the 20th century Göttingen was the center for mathematics. The foundations were laid by Carl Friedrich Gauss (1777–1855) who from 1808 was head of the observatory and professor for astronomy at the Georg August University (founded in 1737). At the turn of the 20th century, the well-known mathematician Felix Klein (1849–1925), who joined the University in 1886, established a research center and brought leading scientists to Göttingen. In 1895 David Hilbert (1862–1943) became Chair of Mathematics and in 1902 Hermann Minkowski (1864–1909) joined the mathematics department. At that time, pure and applied mathematics pursued diverging paths, and mathematicians at Technical Universities were met with distrust from their engineering colleagues with regard to their ability to satisfy their practical needs (Hensel, 1989). Klein was particularly eager to demonstrate the power of mathematics in applied fields (Prandtl, 1926b; Manegold, 1970). In 1905 he established an Institute for Applied Mathematics and Mechanics in Göttingen by bringing the young Ludwig Prandtl (1875–1953) and the more senior Carl Runge (1856–1927), both from the nearby Hanover. A picture of Prandtl at his water tunnel around 1935 is shown in Figure 2.1.

Prandtl had studied mechanical engineering at the Technische Hochschule (TH, Technical University) in Munich in the late 1890s. In his studies he was deeply influenced by August Föppl (1854–1924), whose textbooks on technical mechanics became legendary.

To supplement the foregoing chapters, we offer below a table listing some key developments in turbulence research over the period covered by this book, i.e. roughly up to mid-1970s. Later developments involving massive computations, low-dimensional dynamics, the renormalization group, turbulence control, modern instrumentation, and so on, are not included; nor do we include such closely related areas as turbulent thermal convection, combustion, wave turbulence, or the vast field of applications in geophysics, astrophysics and plasma physics. Moreover, the table is ‘internal’ to the subject, in that we make no attempt to relate the events to developments in other scientific fields or to the wider historical context. Despite these limitations, it is our hope that the table, necessarily subjective to some extent, will provide a useful point of reference for the reader. We thank the authors of this book for their comments on the table, especially Professor R. Narasimha for the inspiration he provided.

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.