3R40. Introduction to Interactive Boundary Layer Theory. - IJ Sobey (Comput Lab, Oxford Univ, UK). Oxford UP, Oxford, UK. 2000. 332 pp. ISBN 0-19-850675-9. \$80.00.

Reviewed by JH Lienhard V (Dept of Mech Eng, MIT, Rm 3-162, Cambridge MA 02139-4307).

Laminar boundary layer theory, as taught in most introductory fluid mechanics courses, involves asymptotic expansion of the Navier-Stokes equations to leading order in a small parameter equal to the minus one-half power of the Reynolds number. An inner viscous region (the boundary layer) and an outer inviscid region are matched to one another to fix the unknown terms in the expansions. Boundary layer theory at this order works very well for many engineering applications; it is, however, inadequate for the prediction of some very important phenomena, particularly in relation to flow separation.

If the asymptotic expansions are carried to higher order, difficulties arise in the form of singularities and an inability to match the higher-order terms. These problems were explored in detail by Goldstein and others, beginning in the 1930s. Only in 1969 was a resolution found, in the form of interactive boundary layer theory. To what interaction does this name refer? Specifically, it is an interaction between the pressure field and the streamline displacement, which is accommodated by a three layer, or triple deck, analysis of the boundary layer.

Sobey’s new monograph, Introduction to Interactive Boundary Layer Theory, is an effort to collect and summarize the fundamental developments in this very abstruse area of fluid dynamics. His approach is initially historical. After a short chapter recapitulating the equations of fluid dynamics, Sobey begins serious work with a 50-page summary of theoretical efforts to take boundary layer theory to higher order. These efforts focused on the simple situations of semi-infinite and finite length flat plates. Sobey thoroughly illustrates problems encountered by Goldstein and others as they attempted to complete a second-order theory, and he shows how a severe singularity in transverse velocity arises at the trailing edge of a plate.

The third chapter gives an account of the triple-deck theory, through which Stewartson and Messiter independently resolved the trailing edge singularity in about 1969. Their work introduced a three-layered structure near the trailing edge of a plate, each layer of which scales with a different power of the Reynolds number. The analysis is executed through multiple matched asymptotic expansions. The numerical solution of the resulting equations is described in Sobey’s next chapter. The triple-deck model successfully resolves the trailing-edge singularity found in the classical two-layer theory, and, in the author’s estimation, is “one of the outstanding achievements in theoretical fluid mechanics.”

In Part II of the text, Sobey turns to the problem of separated flow. After a brief introductory chapter, he devotes Chapter 6 to a historical survey of efforts to predict separation in crossflow over a cylinder. Free streamline theories, boundary layer theories for adverse pressure gradients, and combinations thereof are summarized through the year 1969. Chapter 7 describes efforts to use triple-deck ideas for this problem and outlines some of the continuing difficulties with the theory.

The final part of the book, in three chapters, applies interactive boundary layer theory to two-dimensional channel flows in which the walls are perturbed by indentations. Upstream influence and the Coanda effect are each considered at length. An Appendix provides problems and numerical exercises.

This book is not for the faint of heart. A working knowledge of asymptotics is essential, as is a thorough acquaintance with classical boundary layers and potential flow theory. For those having such a background, the mathematics will still demand careful reading. Of course, one of the rewards of having learned the classical theory is that a book like this one is both accessible and rewarding. On the other hand, despite the hopeful statement in the preface that readers with little background in fluid mechanics should be able to absorb the text, nonspecialists will find this material challenging.

The major flaw of the book is a lack of careful copy editing, as in the following passage from page 30:

“The factors of $2$ which appear in (2.25) are somewhat arbitrary, different authors have used slightly differing notation, in this case we are following Van Dyke (1964).”

Such phrasing permeates the text and poses an ongoing distraction. Sobey often moves to an imperative voice when introducing equations or procedures (“Now expand…,” “Define parabolic coordinates by…”), which creates the sensation of notes copied onto a chalkboard. Finally, although the book is typeset in LaTEX, the latter has not been used to its full potential. The delimiters in many equations are not sized properly, and the frequently-used symbol for “much less than” is set as < <, rather than via the glyph $≪$ (obtained with the LaTEX command \⅂⅂).

These concerns are, of course, minor; Introduction to Interactive Boundary Layer Theory will stand as a unique and valuable contribution to the literature. Students of theoretical fluid mechanics have much to gain from this book.