Review of Non-Hydrostatic Free Surface Flows by Oscar Castro-Orgaz and Willi H. Hager

Non-Hydrostatic Free Surface Flows by Oscar Castro-Orgaz and Willi H. Hager is an exceptional book that masterfully takes the reader on a brave voyage into a universe of Free-Surface Flows that do not presume a hydrostatic pressure distribution in the vertical direction. For almost two centuries, the shallow-water theory proposed by Barré de Saint-Venant has been the dominant approach of computing free-surface flow in natural and built channels. Although de St. Venant’s theory is an excellent approximation of real channel flow, it fails to capture a plethora of phenomena that are commonly observed in nature. This deficiency was resolved by Boussinesq’s higher-order approximation, which allowed the pressure distribution in the vertical to deviate from hydrostatic. Unfortunately, Boussinesq’s brilliant theory is more demanding, both mathematically and computationally, thus its presentation has been generally avoided by the majority of open-channel flow textbooks. Non-Hydrostatic Free Surface Flows fills a century-long void in the literature, and elevates both the rigorousness and elegance of a subject that is of paramount importance in civil and environmental engineering.

The book consists of seven chapters that are characterized by detailed derivation of the governing equations, meticulous presentation of underlying assumptions, and a thorough discussion of the physical importance of each segment of the theory. Each chapter is skillfully complemented by technical drawings and stunning photographs that give the reader a true laboratory observer’s view of the phenomena described in the text. The presentation begins with a historical review of depth-averaged, open-channel flow theory that sheds light on the remarkable developments of 19th century hydraulics. Chapter 2 gives a detailed derivation of the vertically integrated non-hydrostatic, free-surface flow equations. The presentation should be a student’s delight, as no steps are omitted, and the level of explanations rivals that offered in the classroom environment. Chapter 3 is a tour de force of irrotational flow solutions to open-channel flow problems that exhibit appreciable curvature of the free surface. The presentation includes a comprehensive description of both analytical and numerical models, and an exhaustive literature review of past work on the topic. Chapter 4 describes saturated flow in a homogeneous aquifer with a phreatic surface, which is somewhat distracting from the main theme of the book. However, it adheres to the promise of addressing a problem with a free-surface albeit phreatic, and also does justice to another of Boussinesq’s brilliant approximations. Chapter 5 is arguably the most fascinating part of the book. The chapter is literally a thesaurus of models and solutions for viscous, depth-integrated flows in open channels, and succinctly depicts the reasons that de St. Venant’s theory is insufficient for describing problems with pronounced free-surface curvature. For students that have never seen these phenomena, the laboratory photographs verify the existence of undular bores and jumps that otherwise remain hidden and mathematically intractable by professional engineers. The chapter also introduces the class of Serre equations, thus bridging the gap between the various orders of approximation in the shallow-water theory. Finally, Chapter 6 addresses the topic of granular flows, extending the non-hydrostatic theory to a variety of environmental problems, from rock avalanches to saturated debris flow, thus opening new horizons for the interested reader.

Non-Hydrostatic Free Surface Flows differs from the classical books of Bakhmeteff (1932), Chow (1959), and Henderson (1966) in both content and style. Rather the text is an extension of Boussinesq’s own essay (1877), offering an alternative view of depth-integrated, free-surface flow. After more than 40 years of teaching senior or master’s level courses in open-channel flow, Castro-Orgaz’ and Hager’s book forces me to question why the education of civil engineers over the last one hundred years has followed the path of de Saint-Venant instead of that of Boussinesq. Is the content of Non-Hydrostatic Free Surface Flows too difficult for civil and environmental engineering students? Are the practical analysis and design considerations adequately described by gradually-varied flow conditions, thus the civil engineer can have a successful career limited to computing backwater profiles? Why have the majority of authors of modern open-channel flow texts veered away from the non-hydrostatic theory? Finally, are both the Boussinesq and de St. Venant equations destined to become obsolete, as computational fluid dynamics codes become widely available, meeting the same extinction with conformal mapping when the finite-element method was introduced to civil engineering education? I have had long discussions with colleagues regarding these questions, however, the opposing argument has always been that the use of Boussinesq-type equations in open-channel flows was limited to research papers, thus it was extremely difficult to attempt to teach a course in non-hydrostatic channel flow. Non-Hydrostatic Free Surface Flows provides the means to bring closure to all these arguments for the first time in one hundred and fifty years. For not only does the book cover all the developments since Boussinesq’s original essay, but does so in the form of a real textbook, written for the student and not only for the researcher. In that regard, the book is a sequel to Henderson’s open-channel flow, for it explains every detail rather than superficially expose the reader to a survey of methods and results.

After reading Non-Hydrostatic Free Surface Flows from cover to cover, I have concluded that there is nothing impossible for our senior and master’s students. These students have typically strong analytical and computational skills, and since all derivations in the text are based on fundamentals, nothing extraordinary is required for completing a course based on this book. Furthermore, the non-hydrostatic treatment of open-channel flow opens new horizons for practical analysis and design, from weirs to spillways to dam-break flood-wave mitigation that should expand the portfolio of every professional engineer. Regarding the approach taken by most other books on open-channel flow, I can venture to speculate that in the time the classic book by Henderson was written, it already required extraordinary courage to pursue applications of the de St. Venant equations since the method of characteristics was the only means of solution of these quasi-linear partial differential equations. Deriving the Boussinesq or Serre equations while their solution remained intractable was impertinent for an engineering textbook, thus Henderson’s objectives were realigned to truncation of the governing equations and pursuing solutions to non-inertial and kinematic waves. That was a brave undertaking in a book that also offered short-cut solutions using a slide rule. Although this does not justify the present trend in open-channel flow books, it brings into perspective the dilemma between having to solve a third-order partial differential equation or predicting that all waves will break in a horizontal channel. The resolution of this conundrum is actually the strength of Non-Hydrostatic Free Surface Flows, which manages to complement the derivation of the associated equations with practical solutions, and eventually offer a unified theory of first- and second-order shallow-water approximations. The result is not just elegant; it opens a window to an amazing mathematical model that can capture the gradual steepening of an undular bore until its breaking. Finally, I have to admit that computational fluid dynamics solutions represent the future of free-surface flow education because they do not require any approximation of the pressure distribution over the depth of flow. However, “natura valde simplex est,” thus a rigorous approximation should always be preferred over a superfluous elaborate solution. In that regard, Non-Hydrostatic Free Surface Flows is a timeless contribution that is destined to guide students and teachers in the years to come.

In closing, I have only one regret after reading Non-Hydrostatic Free Surface Flows by Castro-Orgaz and Hager: that I was not fortunate enough to have this textbook as a student of hydraulic engineering, and as a teacher who tried to cover this material by searching the library for old books and papers. The wealth of information amassed in this text is extraordinary. Both students and researchers will find that the quality of the presentation is outstanding, and the didactic value of the text is unparalleled.