5R52. Stability and Transition in Shear Flows. Applied Mathematical Sciences, Vol 142. - PJ Schmid (Appl Math Dept, Univ of Washington, Seattle WA 98195-0001) and DS Henningson (Dept of Mech, Royal Inst of Tech, Stockholm, S-100 44, Sweden). Springer-Verlag, New York. 2001. 556 pp. ISBN 0-387-98985-4. \$79.95.

Reviewed by DF Jankowski (Dept of Mech and Aerospace Eng, Arizona State Univ, PO Box 875506, Tempe AZ 85287-5506).

Hydrodynamic stability is concerned with the fate of disturbances imposed on a basic flow. Since, in some flow situations, the initial growth of disturbances can lead to a transition to turbulence, the subject is particularly attractive to certain members of the fluid-dynamics community. The audience for this particular treatment of stability and transition is loosely defined in the preface where it is stated that the “book is foremost intended for researchers and graduate students with a basic knowledge of fluid dynamics.” The necessary level of this knowledge can only be inferred, and there is no mention of the considerable specialized mathematical knowledge that is also needed to fully understand the wide range of sophisticated mathematical tools that are used routinely in its nine chapters of nearly 500 pages, well over 1000 numbered equations and over 200 figures. In comparison to the emphasis on mathematical analysis and its associated modeling, the attention devoted to physical insights or experimental verification is limited, with the exception of the final chapter.

The tone of the book’s material on stability is immediately apparent in the brief first chapter, Introduction and General Results, wherein the general nonlinear disturbance equations and the kinetic-energy evolution equation are derived, and four definitions of temporal stability and four critical Reynolds numbers are introduced. A useful caution about the difference between the temporal and spatial evolution of disturbances is also provided. Three of the temporal definitions are related to the behavior of disturbance kinetic-energy in the asymptotic limit of large time. In later chapters, considerable attention is paid to short-time disturbance behavior. In the caption of Figure 1.2, “above $ReL:$ Possible instability” appears, contradicting a statement on page 6. This chapter is followed by the four chapters of Part I, Temporal Stability of Parallel Shear Flows, which presents the authors’ view of the “fundamental topics underlying stability theory,” and the four chapters of the longer Part II, Stability of Complex Flows and Transition, which “covers more advanced topics.”

An involved theoretical picture continues to develop in the second chapter, Linear Inviscid Analysis, which considers the linear stability of planar parallel flows with the effects of viscosity ignored. This modeling might have been motivated by noting that the effect of viscosity is, at first glance, expected to be stabilizing; a counter example is later mentioned in Section 3.2. The chapter begins with an examination of the pertinent governing equations for infinitesimal wave-like disturbances, the development of some related general results, and several idealized example solutions. It closes with a treatment of the inviscid initial-value problem for the time evolution of infinitesimal disturbances. The differences between approaches based on eigenvalue (modal) problems and initial-value (non-modal) problems are emphasized throughout the book. The third chapter, Eigensolutions to the Viscous Problem, roughly follows the pattern of the first portion of Chapter 2 although the inclusion of viscosity leads to a broader, more complicated, and often more subtle presentation. The primary governing equation for infinitesimal wavelike disturbances (the Orr-Sommerfeld (OS) equation) imposed on plane parallel flows is derived, and some related general results are presented. The corresponding eigenvalue problem is also derived for fully-developed flow in a pipe. Specific numerical results are presented and discussed for this flow, and for plane Couette flow, plane Poiseuille flow, and Blasius boundary-layer flow, a nearly-parallel flow. For the later two examples, a certain finite critical Reynolds number ($ReL$ from chapter 1) can be obtained. Experiments designed to verify these results are not mentioned. Some suitable numerical techniques for the OS equation and tables of eigenvalues for the four classical examples are provided in Appendix A. The chapter closes with topics related to various mathematical aspects of the OS equation, its adjoint equation, solutions, and ways of determining information about them. The next chapter, Viscous Initial Value Problem, introduces its subject by studying a simple model problem. Algebraic growth is found to be possible for small time and is attributed to the mathematical structure of the model system. The rest of the chapter pursues this possibility and related complications and extensions at considerable length for the four prototype flows mentioned earlier. It seems that $ReG$ of Section 4.6.2 should be $ReE$ of Section 5.6. Chapter 5 extends the discussion into the complicated realm of Nonlinear Stability. An analysis of a nonlinear model problem is followed by the derivation of the governing equations for finite-amplitude disturbances imposed on a parallel shear flow and selective (“mathematically advantageous”) treatments of the corresponding nonlinear initial-value problem. This is not easy going. The last topic of the chapter is energy stability theory about which a small comment can be made: $ReE$ is determined by solving the eigenvalue problem of Section 5.6.2, while a lower bound to $ReE$ follows from the relaxed eigenvalue problem associated with Eq. (5.153).

Chapter 6, Temporal Stability of Complex Flows, the first chapter of Part II, moves away from the simple incompressible shear flows of Part I to a series of examples that involve new physical features and varying levels of mathematical complexity: similarity boundary layers with adverse and favorable pressure gradients; three-dimensional boundary layers with cross-flow; channel flows with span-wise rotation and slight streamline curvature; free-surface flow down an inclined plane; channel flow with an oscillatory pressure gradient; general time-dependent parallel basic flow; and the inclusion of compressibility in boundary layers. It is not possible to discuss all of the related issues to these examples in the space alloted. The next chapter, Growth of Disturbances in Space, is the longest in the book and contains 400 equations. Its issues are the particular complications associated with disturbances that grow spatially. Differences and connections between the temporal and spatial cases are effectively displayed with model problems and with some specific results for channel and boundary-layer flows. This is followed by a treatment of absolute and convective instability, and approximate applications to the plane wake behind a circular cylinder and the classical rotating-disk flow. The spatial initial-value problem, a multi-faceted look at non-parallel effects, and a brief look at receptivity theory close the chapter. The later topic brings the discussion somewhat closer to the transition process since it involves the study of how ever-present disturbances are drawn into a boundary layer and how it responds.

The subject of the short Chapter 8 is Secondary Instability, which considers the fact that the growth of a disturbance can, in certain circumstances, lead to an altered basic flow that has its own set of stability conditions. Some of the ideas presented are of direct relevance to several of the special examples of Chapter 6. The tone of the final chapter, Transition to Turbulence, is more descriptive and tutorial in nature than the previous eight chapters; there are only seven equations in a chapter that is about 75 pages long. These remarks should not be interpreted as implying that the chapter is somehow less challenging than its predecessors. The goal of this chapter is to connect concepts and results associated with instability to the transition to turbulence. Three possible transition scenarios in a temporal setting are presented in its introductory first section. It is followed by a sequence of sections, which considers the “complete transition process in spatially evolving flows.” Results from a number of physical and numerical experiments are routine parts of these sections. The chapter ends with a largely empirical section on transition modeling and a warning about the lack of a general transition model.

Four appendices and a bibliography of over 300 items, including some from 1999 and 2000, close the book. Appendix A, Numerical Issues and Computer Programs, discusses the numerical solution of the classical OS eigenvalue problem for parallel and nearly parallel flows, and provides several MATLAB codes for this purpose. It is followed by two brief appendices that treat some mathematical details. The final appendix contains 14 student problems related to the material of Part I.

Stability and Transition in Shear Flows is an ambitious and personal book, really a monograph because of its in-depth treatment of the two major topics announced in its title. It contains much to learn and think about. With its primary focus on their mathematical side, it has found an appropriate home as a volume in the long established Springer-Verlag series on applied mathematical sciences. Every serious technical library should have a copy as should a limited number of individuals, mostly specialists and those who want to become one. It is not obvious how to use the book in a graduate course since considerable specific background material must be available in order to appreciate the single chapter on transition. Working through the book is not for the light-hearted. An author index and a list of symbols would have helped. In general, the number of ideas, concepts, and special conditions that must be kept track of are close to overwhelming. More motivation and guidance as to the sense in which topics and results are important and how they ultimately relate to transition would have smoothed the way. Unanswered research issues might have been given more direct attention, but astute and careful readers should still be able to uncover some interesting possibilities. In spite of these comments, there is no need for the authors to apologize for the fact that learning about a complicated subject is, in turn, complicated.