Hydrodynamic Instabilities and the Transition to Turbulence
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The problem of explaining the origin of turbulent flows has been recognized for more than a century. Early discussions of hydrodynamic stability included an article on the instability of fluid jets by Rayleigh [1.1] in 1879 and a paper by Reynolds [1.2] in 1883 on "direct" and "sinuous" flow in pipes. Much is now known about a variety of hydrodynamic instabilities [1.3-5]. In addition, strongly turbulent flows have been rather well characterized using statistical methods I-1.6]. However, a convincing and quantitative explanation of the origin of chaotic fluid motion remains elusive. It is natural to ask why this problem has proven to be so difficult, and what recent progress justifies the writing of this book. A number of the experimental difficulties that have hindered progress toward understanding the transition problem are described in Sect. 1.1. On the theoretical side, the basic difficulty is the intractability of the nonlinear hydrodynamic equations ; although they are believed to be correct even for strongly turbulent flows, the mathematical challenge of obtaining explicit solutions is formidable, and solutions are not generally unique, except at low Reynolds number. In the past decade both theory and experiment have undergone profound development. This progress, which provided the principal motivation for writing this book, is described briefly in the following sections and in detail in the chapters of this book.
1.1 Experimental Difficulties and Advances An operational definition of turbulence has been lacking, so that the concept of the onset of turbulence has been ill defined. Until recently, the practical definition has been the appearance of apparent randomness in photographs of flows containing materials which permit visualization of streamlines or other features. However, this approach omits the possibility of complex flow patterns that are nevertheless highly ordered. The qualitative appearance of a flow is not necessarily a reliable indicator of its fundamental behavior. A second experimental difficulty has been the absence of sensitive and quantitative experimental techniques capable of measuring the time dependence of the fundamental dynamical variables : the velocity field, temperature field, vorticity, etc. The prediction and measurement of time-averaged
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L. Swinney and J.P. Gollub
fields (e.g., velocity profiles) is of limited use in achieving a fundamental understanding of the onset of turbulence. A third difficulty has been an inadequate degree of control over boundary conditions and other experimental parameters. Flows which are near a hydrodynamic instability are extremely sensitive to external perturbations. Moreover, precision control is essential to successful discrimination between random flow produced by internal dynamics and random flow produced by external influences. Finally, experimentation has been hindered by the lack of an adequate conceptual framework. Since much of the theoretical work on turbulence has concentrated on the strongly turbulent regime rather than the trans
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