Appl Math Optim 33:103-105 (1996)

"ell 9 1996 Springer-Verlag New York Inc.

Book Review The journal will start a book review section where, from time to time, books on topics in applied mathematics will be reviewed.

Analysis of the K-Epsilon Turbulence Model, by B. Mohammadi and O. Pironneau. Wiley, New York and Masson, Paris, 1994. 196 pp. This book is included in the series "Research in Applied Mathematics" published by Masson together with Wiley and Springer-Verlag. It fits this series excellently: it is definitely a book on applied mathematics and not on fluid mechanics or mechanical engineering as it is possible to suppose according to its title. Fluid mechanicians and mechanical engineers will apparently find that too much space in the book is devoted to mathematical refinements and not enough to discussions of fluid dynamical essence and the details of possible applications. Moreover, the notations used are taken from mathematical texts and, being unfamiliar to most of the researchers and engineers dealing with fluid dynamics, can sometimes hamper the understanding of the book by such readers. The contents of the book are centered on the classical k-e model, which is only one of the many suggested and used turbulence models; true, some other models are also briefly considered in the book, but their consideration is too short to be really useful. Therefore it is clear that the recent book by D. C. Wilcox, Turbulence Modeling for CFD, DCW Industries, 1993, 460 pp., is much more suitable for practical workers dealing with problems requiring the use of turbulence modeling and for students wishing to familiarize themselves with the modern state of the subject. However, pure and applied mathematicians who are interested in applications of their subjects to the theory of turbulence and turbulence modeling can find in the book by Mohammadi and Pironneau much interesting material and many references to appropriate mathematical texts with short descriptions of their contents. Note also that the book under review is much shorter than that by Wilcox and the contents of these two books do not have too much of an overlap; therefore the first of these two books can also be useful as supplementary reading for those readers of the second one who are interested in more rigorous mathematical approaches to the problem of turbulent modeling. The book consists of twelve chapters divided into three parts and the introductory Chapter 1. This chapter begins with the formulation of Navier-Stokes equations, their generalization to the case of a compressible fluid and the necessary boundary conditions;

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then the notions of the Reynolds number, boundary layer, and wall laws are introduced and mathematical problems related to the existence and uniqueness of solutions of the dynamical equations, regularity of these solutions and their long-time behavior are briefly discussed. Part I is called "Incompressible Turbulence" and consists of five chapters. Chapter 2 is devoted to homogeneous incompressible turbulence; here the Fourier decomposition of such turbulence, invariants of fluid flows, the Kolmogorov spectral law (for inertial range only), and the form of the spectral tensor of a homogeneous and isotropic turbulence are considered and the general idea of Direct Numerical Simulations (DNS) and Subgrid Scale Modeling (SSM, now more often called LES-Large Eddy Simulation) of turbulent flows is formulated. Chapter 3 is called "Reynolds Hypothesis"; it begins with the introduction of the notions of a linear filter and the Reynolds decomposition used for the derivation of the Reynolds equations leading to the definition of Reynolds stresses. Then the simplest closure assumption about the linear dependence of Reynolds stresses on velocity gradients is formulated, the consequences of the existing symmetries are listed, the simplest algebraic subgrid-scale models are considered and their usefulness is illustrated by results of some appropriate numerical computations. Chapter 4 contains the definition of the classical k-s turbulence model, i.e., of the approximate closed system of dynamical equations for a turbulent flow consisting of the equations for the mean velocities U/, mean kinetic energy of turbulent fluctuations k, and mean rate of the energy dissipation by viscosity s. (The supplementary unknowns entering these equations are represented here through Ui, k, and e with the aid of specific empirical hypotheses.) Some arguments justifying the model are also presented in Chapter 4 together with the determination of numerical coefficients in the model equations from comparison of some computational results with experimental data. Chapter 5 is devoted to additional numerical results and to more special mathematical questions related to the k-s model (namely, those about the existence, uniqueness and stability of the solutions for a k-e system, positivity of the obtained values for k and s, and the most appropriate numerical methods for computation of solutions). The final chapter of Part I consists of remarks about some other turbulence models; here the Reynolds stress models (which include the semiempirical equations for all Components of the Reynolds stress tensor), and also nonlinear modifications based on the renormalization group theory (RNG) of the classical k-s model, are briefly considered. Part II is devoted to k-s models for compressible turbulent flows and consists of four chapters: Chapter 7 "Numerical Simulation of Compressible Flows," Chapter 8 "Compressible Reynolds Equations," Chapter 9 "Numerical Tools for Compressible k-s," and Chapter 10 "Numerical Results and Extensions." The presentation is similar to that in Part I but all the model equations are now considerably more complicated and use a greater number of nonstrict hypotheses. To make the presentation shorter some of the details in the derivations are omitted in this part. Part III, consisting of two chapters on "Convection of Microstructures," is the most complicated in the book; it is interesting but is clearly intended for mathematically more sophisticated readers. Chapter 11 is devoted to the study of convection of a scalar field c(x, t) by a turbulent velocity field; however, instead of semiempirical closure of the equation for the mean value C of the field c more complicated approaches are considered here. These approaches use the tools from functional analysis and the general

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theory of nonlinear oscillations and such nonelementary techniques as homogenization and multiple scale asymptotic expansions. In the last chapter (Chapter 12) the approaches developed in Chapter 11 are applied to the derivation of the two equations of the turbulence model, which justifies some of the assumptions used in the introduction of the k - e model and show some limits for its applicability. The book ends with a reference list, index, and a short appendix about software for computation of turbulent flows. Since the presentation of all the material is very brief here and many results are given either without any proof or with a short sketch of a proof only and the recommended approaches are often only mentioned without any details and explanations, -the references are a very important part of the book. Unfortunately, the reference list and the remarks accompanying references in the text are prepared very negligently and contain a number of errors. For instance, in the very beginning, on page xiii of the Introduction, it is stated that "the idea of using two convection equations for turbulence modeling was around in the sixties (see for example von Karman [1948] and Rotta [1951])"; however the paper by von Karman has no relation to the turbulence modeling studied in this book and Rotta's paper has a relation but only an indirect one (it contains a one-equation model). In fact the first two-equation model was proposed by Kolmogorov in 1942; the reference to this paper is included in the reference list but I could not find it in the text of the book (in the text here there is a reference to Kolmogorov [1941] which is missing in the reference list). There are also some other references in the text which are erroneously missing in the reference list (e.g., to Prandtl [1945]). In some cases the year of the publication is indicated differently in the text and in the reference list; in cases where two papers of the same author published in the same year are mentioned in the book it is usually impossible to understand which paper is being cited. The reference to Orszag [1973] is related to the unpublished book; it must be replaced by the reference to the well-known long paper of the same title published in 1977. There are also many other misprints and small defects in the text and the reference list (for example, the reference to the important paper by Launder, Reece, and Rodi of 1975 on p. 51 does not contain the names of the authors). Of course, the remarks in this paragraph are not to be constructed as serious criticism, but the great number of minor defects diminishes considerably the usefulness of this valuable book. A. M. Yaglom 1 Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA and Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia

1 Professor Yaglom is the coauthor with A. S. Monin of Statistical Fluid Mechanics which was first published by Nauka in Moscow in 1965 and 1967 (the two years correspond to the two volumes). The English edition was published by the MIT Press in 1971 and 1975 and was reprinted four times (the last time in 1988). Professor Yaglom is now at work on a fully revised English edition.