9R8. Statistical Mechanics of Turbulent Flows. - Edited by S Heinz (Dept of Fluid Mech, Tech Univ of Munich, Boltzmannstrasse 15, Garching 85747 Germany). Springer-Verlag, Berlin. 2003. 214 pp. ISBN 3-540-40303-2. $59.95.

Reviewed by AC Buckingham (Center for Adv Fluid Dyn Appl, LLNL, Mail Code L-023, PO Box 808, Livermore CA 94551).

This book is a descriptive monograph with a focus on stochastic probability density function (pdf) procedures and their implementation for turbulent flow. Special emphasis is placed on exposition and adaptation of Lagrangian single-point pdf methods with emphasis on application to chemically reactive turbulent flows. Compressibility considerations enter as a consequence of the temperature and closely associated component species and global mass density changes (and their statistical fluctuations) generated by the component chemical reactions. The stochastic field is treated as a Markov process. The reviewer recommends Pope’s (SB Pope (1994)) exposition and review of these methods, their development, potential applications, and physical significance as an informative complement to the author’s more concise description, limited, perhaps, by his desire to provide students with a broader survey and comparison of other turbulence closure procedures.

The book is based on the author’s upper division and perhaps beginning graduate student level university lectures with stated emphasis given to the instruction of students in environmental sciences and process (chemical) engineering.

Unfortunately the title, statistical mechanics of turbulent flows, perhaps inadvertently, exaggerates the book’s content. The term statistical mechanics labels the discipline devoted to study and prediction of average behavior/state of a collection (ensemble) of similarly composed systems. It necessarily includes development of asymptotic predictive formulas for the average behavior where the state of evolution of the individual system members in space-time is too complex for precise determination. The book certainly describes one such promising entry for predicting the evolution of average turbulence features, and surveys several other existing approaches when viewed relative to the author’s perspective and experience with stochastic pdf methods. However, theoretical understanding, experimental definition, and specifically applicable and successful turbulence modeling procedures continue to emerge as more aspects of the turbulence problem become evident. Describing the presently available collection of approaches and procedures as a statistical mechanics of turbulence entity appears premature if not optimistic.

The book includes a usefully extensive reference list, author and subject index. However, for background introduction to statistical turbulence this reviewer feels obliged to add the following references to seminal investigations. The origin of the treatment of turbulence as a statistical phenomenom is usually attributed to Taylor in the early part of the 20th Century (GI Taylor (1921)). Out of a host of important contributions to the growing set of statistical mechanics turbulence contributions by Kraichnan the reviewer is compelled to cite the early work on incompressible hydromagnetic turbulence (RH Kraichnan (1958)), structure of isotropic turbulence at high Reynolds number (RH Kraichnan (1959)), Kolmogorov’s hypothesis and eulerian turbulence theory (RH Kraichnan (1964)), and Lagrangian history closure approximation (RH Kraichnan (1965)). In addition, the reader’s attention should be drawn to Leslies useful and clear summary of contributions to turbulence theory and statistical mechanics in the very productive years up to the early 1960s (DC Leslie (1973)).

The first three chapters provide a useful, if compact, review of stochastic processes and the mathematical framework used in their analysis. A special flavor in this review is provided with regard to the author’s obviously extensive experience with generation of conditional pdf’s in Markov processes exploiting the Fokker-Planck equation. The 4th chapter introduces development of the contiuum flow and thermodynamic state equations, multi-component reaction equations, Reynolds averaged Navier-Stokes (RAN) single point closure, and RAN moment closure hierarchies commencing from the underlying molecular statistical collisional field governed by the Boltzmann equation. Turbulence concepts are introduced and identified such as: inherent macro and microscales, energy spectrum, correlations (and their defining pdf’s), and inertial range separation of scales. A discussion of direct numerical simulations and their limitations provides a framework for describing the utility, albeit added complexity of combining Lagrangian pdf projection methods with stochastic Lagrangian model differential equations. The Langevin equation is introduced as the prototypical stochastic model differential equation for generation of the large scale directly computable motions and scalar fluctuations. A hierarchy of generalizations to the Langevin model permits simulation of more complex, multidimensional, multi-component flows for chemically reactive flow.

The next to last chapter introduces the combination of specified pdf scalar models with large eddy simulations LES where the large scale flow may be conventionally computed but the convolution filtering includes scalar pdf approximations for the small scale reactive components, and advances to consideration of stochastic subgrid scale models for scalar component reaction distributions with stochastic convolution filter functions and ordered terms in the large (computable) range. Note is made of advances in computing both the large scale range and the filtered sub=grid pdf scalar with stochastic models. The last chapter outlines the practical limitations to utilization of the complex stochastic flow and scalar distribution models for reactive flow in industrial applications. Attention is given to appropriate scaling and parameterization of the appropriate length and time scale coefficients in single point RAN closure methods for practical economical computation. This sought for unification of model approaches for industrial usage is left as a valuable but open, question for discussion.

Among the many topics the author was not able to cover but that must be considered in conjunction with filling out the statistical mechanics of turbulent flows include: multiphase flows, critical phenomena, tracking and appropriate modeling of phase discontinuities, shock wave interaction with turbulence, shock wave enhanced mixing of components and shock-induced critical phase changes, unsteady shockwave turbulence induced motions and resulting heating, turbulent boundary layer separation and reattachment dynamics and the general problems associated with rapidly rotating and swirling combustion flows. The book, within the constraints of size does provide a valuable survey of stochastic turbulent flow procedures supplementary to and in comparison with more conventional closure procedures. Considering the book’s relatively modest price it would be a useful addition for personal as well as institutional acquisition. 1 

Added References
RH Kraichnan (1958). Phys. Rev. 109: 1047. RH Kraichnan (1959). Journ. Fluid Mech. 5: 497. RH Kraichnan (1964). Phys. Fluids 7: 1723. RH Kraichnan (1965). Phys. Fluids 8: 575. DC Leslie (1973). Developments in the Theory of Turbulence. Clarendon Press, Oxford. SB Pope (1994). Annu. Rev. Fluid Mech. 26: 23. GI Taylor (1921). Proc. London Math. Soc. 20: 196.