This issue contains a selection of papers based on the plenary and invited lectures presented at the meeting ‘Theoretical fluid dynamics in the twenty-first century’, which was held at Imperial College London on 15–16 December 2006. The principal purpose of the meeting was to celebrate the distinguished career of Prof. J. Trevor Stuart, FRS, as well as his 40 years of active service at Imperial College. More than 80 participants, comprising three generations of fluid dynamicists together with numerous research students, attended this successful meeting.
Fluid mechanics, as a subject, has enjoyed huge success in the last century, with the spectacular advance in aeronautics and aerospace technology being a prime example. Theoretical fluid dynamics, being in the dominant position for most of this period, has played a crucial role in many technological breakthroughs. With the advent of powerful computers over the last two or three decades, the subject has arguably now come to crossroads. On the one hand, with computers able to tackle a variety of problems without making simplifications (as a mathematical theory usually does), it is easy to question the importance and relevance of theoretical fluid dynamics. On the other hand, long-standing fundamental problems remain unsolved while current technological developments are giving rise to many more of them. In this ever-changing landscape, the organizers decided that this occasion of celebration should also be an opportunity to reflect the role of theoretical fluid dynamics in the twenty-first century. The theme embodied in the title should be a fitting tribute to Prof. Stuart since his work continues to influence many aspects of current research and to inspire new generations of fluid dynamicists.
The underlying themes of the meeting were expertly surveyed by the panoply of invited speakers. These included many of Trevor Stuart's contemporaries, who are themselves prominent fluid dynamicists, as well as leading researchers from younger generations. A total of 16 lectures were presented, which reported on recent (and ongoing) theoretical, experimental and computational progress in fluid dynamics. Naturally, there was a great deal of emphasis on the topics closely related to, or influenced by, Prof. Stuart's own work and interests. In all, a broad spectrum of topics were covered, including hydrodynamic instability, laminar–turbulent transition, vortex dynamics, mixing, aeroacoustics and the mathematical properties of the Navier–Stokes equations. Owing to page limitations and the limited time frame for the production of this issue, only nine papers are featured here; more extended contributions will be, or have been, published elsewhere.
Each contributor has, in their own way, explored our underlying theme. The general tenor of this issue is aptly set in the paper by Prof. Hassan Aref (Aref 2008), which argues convincingly that theoretical fluid mechanics will remain a relevant and vibrant subject owing to the continued challenges presented by unresolved ‘classical’ problems as well as by new ones emerging from modern technology needs. One of the best-known unsolved classical problems is that of understanding the onset of turbulence in the flow through a circular pipe. This has puzzled and fascinated fluid dynamicists for more than a century. Willis, Peixinho, Kerswell and Mullin (Willis et al. 2008) report some recent progress in unlocking this persistent mystery. It is interesting to note that progress was made by combining mathematical concepts developed in dynamical systems theory with accurate numerical computations and laboratory experiments. Another problem in this ‘classical’ category is the instability of the flow over a plate oscillating in its own plane. Some linear instability results, completely unexpected until 5 or 6 years ago, are discussed in the paper by Blennerhassett & Bassom (2008).
Fundamental physical mechanisms and relatively simple exact solutions are an important component of theoretical fluid mechanics. They provide the framework for understanding complex flow phenomena as well as for interpreting experimental and computational data. The weakly nonlinear instability mechanisms uncovered by Prof. Stuart in 1950–1960s and the celebrated ‘Stuart vortex’ solution to the Euler equations have been playing an invaluable role for nearly half a century. Prof. J. T. C. Liu (Liu 2008) demonstrates recent applications of the former in predicting enhanced heat transfer caused by secondary instability of Görtler vortices, while Prof. Fraenkel (Fraenkel 2008) presents some interesting extensions of the Stuart vortex solution, extending the long sequence of contributions related to this problem (e.g. Meiron et al. 2000; Crowdy 2004). It should be added that what is known as the classical weakly nonlinear theory has also been developed into a ‘modern’ version in the context of high Reynolds number flows (see Goldstein 1995 and other related papers in vol. 352 of Phil. Trans. R. Soc. A).
Entering the twenty-first century, fluid dynamicists are destined to address new emerging problems which, increasingly often, possess multi-dimensional, multi-scale and highly nonlinear characteristics. More than ever before, theoretical fluid mechanics will be relied upon to identify key mechanisms and simplified rational models. This is exemplified in the paper by Kelly & Alves (2008) on the instability of a transverse jet, a complex problem closely related to high-speed combustion technology. But theoretical fluid mechanics must also take on the responsibility of providing direction and focus for the effective application of the extensive computational resources that are now available. The paper by Sandham & Salgado (2008), on subsonic jet noise, reflects this urgent need for closer interaction between theoretical and numerical approaches.
Fluid mechanics will continue to provide fascinating prototypical problems for applied mathematicians. The study of these often leads to elegant generic equations, which can offer insights into a broad range of nonlinear phenomena; a sample is provided in the paper by Akylas & Cho (2008) on three-dimensional solitons. The fundamental equations of fluid mechanics, the Navier–Stokes equations, continue to present grand challenges to mathematicians, both pure and applied. Among these are the issues of global (and local) existence and the regularity of solutions to these equations. A refreshing and highly promising approach to addressing these issues, centring on the Borel summability of divergent asymptotic series, is described by Tanveer and co-authors (Costin et al. 2008).
The present Theme Issue is meant to provide a snapshot of some current developments in fluid mechanics, but its scope is clearly limited. Fortunately, there exist complementary and expansive volumes devoted to the same subject, among which we mention two Theme Issues that have appeared in this journal: vol. 352 (1995, eds P. Hall & F. T. Smith) and vol. 363 (2005, eds J. S. B. Gajjar & F. T. Smith) as well as Perspectives in Fluid Mechanics (2002, eds G. K. Batchelor, H. K. Moffatt & M. G. Worster). The collection of the articles in this issue, along with those mentioned above, reflect and reaffirm the continued fascination, vitality and, most importantly, the relevance of theoretical fluid dynamics in the twenty-first century.
As organizers of the meeting, we would like to thank Prof. John Elgin, head of the Mathematics Department at Imperial College, for initiating this meeting and providing partial financial support. The generous financial support from the London Mathematical Society was also crucial to the success of this meeting. We are indebted to Prof. Philip Hall for his invaluable advice and suggestions, and for offering the Institute for Mathematical Sciences at Imperial College as the conference venue. Thanks are due to Prof. F. T. Smith for chairing the opening session and giving the memorable concluding speech. Finally, as editors of this issue, we thank Prof. J. T. C. Liu for his suggestion to publish the contributions in the present format, and Suzanne Abbott and Helen Ross of the Royal Society for their invaluable help and guidance throughout.
One contribution of 10 to a Theme Issue ‘Theoretical fluid dynamics in the twenty-first century’.
- © 2008 The Royal Society