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).

The field of static aeroelasticity is the study of flight-vehicle phenomena associated with the interaction of aerodynamic loading induced by steady flow and the resulting elastic deformation of the lifting-surface structure. These phenomena are characterized as being insensitive to the rates and accelerations of the structural deflections. There are two classes of design problems that are encountered in this area. The first and most common to all flight vehicles is the effects of elastic deformation on the airloads, as well as effects of airloads on the elastic deformation, associated with normal operating conditions. These effects can have a profound influence on performance, handling qualities, flight stability, structural-load distribution, and control effectiveness. The second class of problems involves the potential for static instability of the lifting-surface structure to result in a catastrophic failure. This instability is often termed “divergence” and it can impose a limit on the flight envelope.

“Aeroelasticity” is the term used to denote the field of study concerned with the interaction between the deformation of an elastic structure in an airstream and the resulting aerodynamic force. The interdisciplinary nature of the field is best illustrated by Fig. 1.1, which originated with Professor A. R. Collar in the 1940s. This triangle depicts interactions among the three disciplines of aerodynamics, dynamics, and elasticity. Classical aerodynamic theories provide a prediction of the forces acting on a body of a given shape. Elasticity provides a prediction of the shape of an elastic body under a given load. Dynamics introduces the effects of inertial forces. With the knowledge of elementary aerodynamics, dynamics, and elasticity, students are in a position to look at problems in which two or more of these phenomena interact. The field of flight mechanics involves the interaction between aerodynamics and dynamics, which most undergraduate students in an aeronautics/aeronautical engineering curriculum have studied in a separate course by their senior year. This text considers the three remaining areas of interaction, as follows:

• between elasticity and dynamics (i.e., structural dynamics)

• between aerodynamics and elasticity (i.e., static aeroelasticity)

• among all three (i.e., dynamic aeroelasticity)

Because of their importance to aerospace system design, these areas are also appropriate for study in an undergraduate aeronautics/aeronautical engineering curriculum. In aeroelasticity, one finds that the loads depend on the deformation (i.e., aerodynamics) and that the deformation depends on the loads (i.e., structural mechanics/dynamics); thus, one has a coupled problem.

The purpose of this chapter is to convey to students a small introductory portion of the theory of structural dynamics. Much of the theory to which the students will be exposed in this treatment was developed by mathematicians during the time between Newton and Rayleigh. The grasp of this mathematical foundation is therefore a goal that is worthwhile in its own right. Moreover, as implied by the da Vinci quotation, a proper use of this foundation enables the advance of technology.

Structural dynamics is a broad subject, encompassing determination of natural frequencies and mode shapes (i.e., the so-called free-vibration problem), response due to initial conditions, forced response in the time domain, and frequency response. In the following discussion, we deal with all except the last category. For response problems, if the loading is at least in part of aerodynamic origin, then the response is said to be aeroelastic. In general, the aerodynamic loading then will depend on the structural deformation, and the deformation will depend on the aerodynamic loading. Linear aeroelastic problems are considered in subsequent chapters, and linear structured dynamics problems are considered in the present chapter. Other important phenomena, such as limit-cycle oscillations of lifting surfaces, must be treated with sophisticated nonlinear-analysis methodology; however, they are beyond the scope of this text.

From First Edition

A senior-level undergraduate course entitled “Vibration and Flutter” was taught for many years at Georgia Tech under the quarter system. This course dealt with elementary topics involving the static and/or dynamic behavior of structural elements, both without and with the influence of a flowing fluid. The course did not discuss the static behavior of structures in the absence of fluid flow because this is typically considered in courses in structural mechanics. Thus, the course essentially dealt with the fields of structural dynamics (when fluid flow is not considered) and aeroelasticity (when it is).

As the name suggests, structural dynamics is concerned with the vibration and dynamic response of structural elements. It can be regarded as a subset of aeroelasticity, the field of study concerned with interaction between the deformation of an elastic structure in an airstream and the resulting aerodynamic force. Aeroelastic phenomena can be observed on a daily basis in nature (e.g., the swaying of trees in the wind and the humming sound that Venetian blinds make in the wind). The most general aeroelastic phenomena include dynamics, but static aeroelastic phenomena are also important. The course was expanded to cover a full semester, and the course title was appropriately changed to “Introduction to Structural Dynamics and Aeroelasticity.”

Aeroelastic and structural-dynamic phenomena can result in dangerous static and dynamic deformations and instabilities and, thus, have important practical consequences in many areas of technology.