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Chapter 6 presents a discussion of instabilities in coordinate systems other than Cartesian. In this context, the Taylor problem, Görtler vortices, pipe flow, the rotating disk problem, the trailing vortex and the round jet are all presented. In each case the linearized disturbance equations are derived.
The instability of geophysical flows are covered in Chapter 7. From the class of geophysical flows, there are three classes that are distinct and that illustrate the salient properties when viewed from the basis of perturbations. These cases include the effects of density variations and rotation. The cases considered in this chapter are stratified flow, rotation (Rossby waves) and the Ekman layer.
Chapter 4 addresses the important topic of spatial instability for spatially evolving flows, such as shear layers, jets and wakes. The chapter starts out with a derivation of Gaster’s transformation that allows spatial growth rates to be computed from temporal growth rates. The chapter also presents a dicussion of absolute and convective instabilites, and of wavepackets. It concludes with a discussion of dicrete and continuous spectra.
Chapter 8 addresses the intial value problem, x, where the effect of initial conditions are sought within the linear disturbance regime. Laplace transforms, moving coordinates and numerical approaches are all discussed. Examples of the latter include channel flows and the Blasius boundary layer. The chapter concludes with an in-depth discussion of optimizing the initial conditions for subcritical Reynolds numbers to obtain the maximum energy as a function of time. The concept of algebraically instability is discussed within this context, such that when the normalized energy density is greater than one, the flow is said to be algebraically unstable.
Chapter 13 addresses issues associated with experimental techniques for investigating hydrodynamic instabilties. These issues include the experimental facility, model configuration and instrumentation, all of which impact our understanding of hydrodynamic instabilities.
Chapter 12 summarizes techniques of flow control and optimization. The reader is introduced into both passive and active flow control. Techniques such as flexible boundaries, wave induced forcing, feed-forward and feedback control and optimal control theory are all discussed in some detail.
Chapter 1 introduces the basic concepts of hydrodynamic stability theory. The chapter begins with a discussion of the classical experiments of Reynolds, and moves the reader quickly through other examples of instability found in nature. The basic equations of motion and their linearization are then introduced, which sets the up the foundation for the rest of the book.
Chapter 11 introduces the reader to the world of direct numerical simulations. Temporal and spatial formulations are covered along with boundary and initial conditions. Time-marching methods and spatial discretization methods are also discussed. A variety of applications are then presented.
Chapter 5 examines the role of compressibility on the instability of boundary layers and mixing layers. For the compressible mixing layer, a thorough discussion of the mean flow, compressible Rayleigh equation and neutral stability curves is presented. In the discussion of invisicd distrubances, the compressible vortex sheet is also discussed. For the compressible boundary layer, the viscous stability equations are derived, followed by the neutral stability boundaries.
Chapter 9 moves beyond linear theory by examining weakly nonlinear theory, secondary instability theory and resonant wave interactions. The chapter concludes with a discussion of the parabolized stability equation theory, which sets linear, secondary and nonlinear instabilities within a single framework.
Chapter 3 examines the stability of viscous flowsusing the Orr–Sommerfeld equation. In particular, the stability of channel flows, the Blasius boundary layer, and the Falkner–Skan family are examined. The chapter concludes with a discussion of the spectrum for unbounded flows.