It was a pleasure to read this important book. To understand and predict the development of turbulent flows represents both a continuing scientific challenge and also a serious practical problem in many different fields. Professor Pope has based his book on graduate level lecture courses on turbulence that he has presented at MIT and at Cornell University. It is intended for students in engineering, applied mathematics, oceanography and atmospheric sciences, as well as researchers and practising engineers. The emphasis is on turbulent flows, rather than on the theory of homogeneous turbulence, and only constant-density, nonreacting flows are considered. The author states that his aim is to explain concepts and develop the necessary mathematical tools, rather than to provide a practical guide to turbulence modelling. The text is divided into two parts. Part I provides an introduction to turbulent flows and the fundamental physical processes involved. Topics discussed in separate chapters include: the equations of fluid motion, the statistical description of turbulent flows, the mean flow equations, free shear flows, scales of turbulent motion and wall flows. The chapter on statistical methods for turbulence is particularly good; together with the related appendices it represents a valuable and accessible introduction to the subject. Several approaches for modelling or simulating turbulent flows are then described in Part II, in which the various chapters describe direct numerical simulation (DNS), turbulent viscosity models such as the k-ε model, Reynolds stress and related second-moment models, probability density function (pdf) models and large-eddy simulation (LES) techniques. Finally, some necessary mathematical tools are summarized in ten Appendices dealing with a wide range of relevant topics including tensors, Dirac delta functions, Fourier transforms, random processes, derivation of pdf equations, characteristic functions and stochastic descriptions of diffusion processes. I particularly enjoyed the chapters on modelling and simulation of turbulent flows. A unified treatment of the different types of model has enabled the author to make valuable connections between them and to reach clear and logical conclusions about the strengths and weaknesses of each approach. Over many years Professor Pope has made very important contributions to the development and application of pdf methods for the prediction of nonreactive and also reactive turbulent flows. The chapter on this subject is an exceptionally clear description of these powerful and often under-utilized simulation methods. The final chapter, on LES, provides an excellent introduction, with valuable and novel insights into both the promise and the problems of this relatively new and rapidly developing approach. It should be compulsory reading for many LES practitioners. Each section of the book contains a number of exercises, which typically invite the reader either to derive an expression that is quoted in the text or to generalize an analysis set out in the text. These exercises are often divided into several stages, and contain appropriate instructions, so that a diligent student will find his or her way through them. An advantage of the extensive use of appendices and student exercises is that analytical clarity and physical understanding are not obscured by unnecessary algebraic detail or by the need to review basic mathematical tools. The result is an exceptionally clear presentation, together with an often penetrating critique of both classical methods and recent developments in the theory and modelling of turbulent flows. There is excellent cross-referencing between one section and another, and an extensive and up-to-date bibliography. I strongly recommend this book to advanced students of fluid mechanics, to their teachers and to all researchers, engineers and others with a professional interest in turbulent flows. K N C Bray

Turbulent Flows: FUNDAMENTALS

Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112095000462

On the identification of a vortex

The role of coherent structures in the production and dissipation of turbulence in a boundary layer is characterized, summarizing the results of recent investigations. Coherent motion is defined as a three-dimensional region of flow where at least one fundamental variable exhibits significant correlation with itself or with another variable over a space or time range significantly larger than the smallest local scales of the flow. Sections are then devoted to flow-visualization experiments, statistical analyses, numerical simulation techniques, the history of coherent-structure studies, vortices and vortical structures, conceptual models, and predictive models. Diagrams and graphs are provided.

Coherent Motions in the Turbulent Boundary Layer

The evolution of a single hairpin vortex-like structure in the mean turbulent field of a low-Reynolds-number channel flow is studied by direct numerical simulation. The structure of the initial three-dimensional vortex is extracted from the two-point spatial correlation of the velocity field by linear stochastic estimation given a second-quadrant ejection event vector. Initial vortices having vorticity that is weak relative to the mean vorticity evolve gradually into omega-shaped vortices that persist for long times and decay slowly. As reported in Zhou, Adrian & Balachandar (1996), initial vortices that exceed a threshold strength relative to the mean flow generate new hairpin vortices upstream of the primary vortex. The detailed mechanisms for this upstream process are determined, and they are generally similar to the mechanisms proposed by Smith et al. (1991), with some notable differences in the details. It has also been found that new hairpins generate downstream of the primary hairpin, thereby forming, together with the upstream hairpins, a coherent packet of hairpins that propagate coherently. This is consistent with the experimental observations of Meinhart & Adrian (1995). The possibility of autogeneration above a critical threshold implies that hairpin vortices in fully turbulent fields may occur singly, but they more often occur in packets. The hairpins also generate quasi-streamwise vortices to the side of the primary hairpin legs. This mechanism bears many similarities to the mechanisms found by Brooke & Hanratty (1993) and Bernard, Thomas & Handler (1993). It provides a means by which new quasi-streamwise vortices, and, subsequently, new hairpin vortices can populate the near-wall layer.

Mechanisms for generating coherent packets of hairpin vortices in channel flow

The turbulent boundary layer on a flat plate, with zero pressure gradient, is simulated numerically at four stations between R sub theta = 225 and R sub theta = 1410. The three-dimensional time-dependent Navier-Stokes equations are solved using a spectra method with up to about 10 to the 7th power grid points. Periodic spanwise and stream-wise conditions are applied, and a multiple-scale procedure is applied to approximate the slow streamwise growth of the boundary layer. The flow is studied, primarily, from a statistical point of view. The solutions are compared with experimental results. The scaling of the mean and turbulent quantities with Reynolds number is examined and compared with accepted laws, and the significant deviations are documented. The turbulence at the highest Reynolds number is studied in detail. The spectra are compared with various theoretical models. Reynolds-stress budget data are provided for turbulence-model testing.

/pdf/direct-simulation-of-a-turbulent-boundary-layer-up-to-r-sub-4zua6gh54r.pdf

Direct simulation of a turbulent boundary layer up to R sub theta = 1410

This paper describes an experimental and numerical investigation of concentrated vortex flow past a sphere in a constant-diameter pipe. As the swirl was increased at a fixed sphere Reynolds number of approximately 1100, the length of the mean downstream separation bubble decreased. For a small range of swirl intensity, an almost stagnant separation bubble formed on the upstream hemisphere. A further increase in swirl caused the bubble to become unstable and develop into an unsteady spiral disturbance. At very high swirl the downstream separation bubble was eliminated and an unsteady separation zone extended far upstream. Calculations of the vorticity field from surface fits to azimuthal and axial velocity data suggest that upstream separation is caused by the distortion of vortex filaments in the diverging flow approaching the sphere. Numerical solutions of steady inviscid axisymmetric flow past a sphere exhibit a fold in the vicinity of upstream separation. It is suggested that this accounts for the extreme sensitivity encountered in the experiments.

/pdf/vortical-flow-part-2-flow-past-a-sphere-in-a-constant-p2rny69029.pdf

Vortical flow. Part 2. Flow past a sphere in a constant-diameter pipe

Study of turbulence structure using the invariants of the velocity gradient tensor

Experimental investigation of turbulent boundary layer developing along a streamwise external corner (chine)

Experimental results are presented for high Reynolds number turbulent boundary layers. A flat plate zero pressure gradient layer has been studied in a new high Reynolds number boundary layer tunnel. Measurements were made of the mean flow in the Reynolds number range of 3.6 × 103 < R θ < 6.0 × 104 based on momentum thickness. The data supports the existence of a logarithmic law of the wall in the overlap region and constants κ= 0.41 and A = 5.0 are found to best fit the data.

Scaling of the turbulent boundary layer at high Reynolds numbers

RSM (Reynolds Stress Model) simulations of turbulent flow on
a streamwise external corner were carried out using the CFD
code FLUENT to obtain skin friction. Both wall function and
near-wall treatment approaches were used. Experimental data
from an earlier experimental study [J. Fluid Mech. 511 (2004)1], measured at the streamwise station x = 0.54 m was used as the inlet condition for the numerical simulation. The objective of the numerical investigation is to determine the accuracy of RSM prediction of skin friction coefficients (Cf) for a complex three-dimensional flow. Calculated values were compared with the experimental data and only wall function model data is found to be in reasonable agreement.

/pdf/skin-friction-cfd-calculation-for-complex-flow-turbulent-3z9qzh7na5.pdf

Skin friction CFD calculation for complex flow: turbulent flow along an external corner

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