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

Critical points in the skin friction field of wall-bounded ows were investigated using data from direct numerical simulations of channels at Reτ = 934 and Reτ = 1834. A method for their detection and characterisation is outlined, and a statistical description of their properties is reported. Their lifetime, average distance, velocity and area density were computed. Conditionally averaged fields were calculated in order to examine the average ow in the vicinity of a critical point. It was found that the critical points are connected to surprisingly large features in the channel. A mechanism for the generation of the critical points is postulated, based on the conditional averages and analysis of time sequences in the ow above them.

https://iopscience.iop.org/article/10.1088/1742-6596/506/1/012009/pdf

Skin-friction critical points in wall-bounded flows

A statistical description of flow regions with negative streamwise velocity is provided based on simulations of turbulent plane channels in the Reynolds number range -dependent dynamics further away from the wall.

The structure and dynamics of backflow in turbulent channels

Turbulence-driven secondary motion (known as secondary flow of Prandtl's second kind) is the result of the anisotropy of Reynolds stresses. It is associated with internal or external turbulent flow in the vicinity of streamwise corners. In the last six decades the flow along the streamwise internal corner has attracted the attention of many researchers, but it is only recently at the University of Melbourne that an experimental investigation on a flow over an external corner has been undertaken. Although the turbulence-driven secondary motions are generally much smaller than those resulting from skewing of the mean spanwise vorticity (known as secondary flow of Prandtl's first kind), they have a pronounced effect on peripheral wall shear stress distributions and heat transfer rates in the corner region. In the present study the details of turbulence-driven secondary motion over the external corner are explored experimentally and a counter-rotating helical streamwise vortex pair symmetrically placed around the corner bisector is found. As expected, all mean velocity and Reynolds stress profiles on both surfaces at the same spanwise distance from the corner agree quite well, having nominal deviation. The secondary flow is observed to be directed away from the corner bisector and then flow back towards the wall. It is also revealed that far from the corner, the flow develops as a nominal two-dimensional flat-plate boundary-layer flow.

Experimental investigation of turbulence-driven secondary motion over a streamwise external corner

This paper contains the details of an experimental study of the vortex formed in front of a piston as it moves through a cylinder. The mechanism for the formation of this vortex is the removal of the boundary layer forming on the cylinder wall in front of the advancing piston. The trajectory of the vortex core and the vorticity distribution on the developing vortex have been measured for a range of piston velocities. Velocity field measurements indicate that the vortex is essentially an inviscid structure at the Reynolds numbers considered, with viscous effects limited to the immediate corner region. Inviscid flow is defined in this paper as being a region of the flow where inertial forces are significantly larger than viscous forces. Flow visualization and vorticity measurements show that the vortex is composed mainly of material from the boundary layer forming over the cylinder wall. The characteristic dimension of the vortex appears to scale in a self-similar fashion, while it is small in relation to the apparatus length scale. This scaling rate of t0.85+0.7m, where the piston speed is described as a power law Atm, is somewhat faster than the t3/4 scaling predicted by Tabaczynski et al. (1970) and considerably faster than a viscous scaling rate of t1/2. The reason for the structure scaling more rapidly than predicted is the self-induced effect of the secondary vorticity that is generated on the piston face. The vorticity distribution shows a distinct spiral structure that is smoothed by the action of viscosity. The strength of the separated vortex also appears to scale in a self-similar fashion as t2m+1. This rate is the same as suggested from a simple model of the flow that approximates the vorticity being ejected from the corner as being equivalent to the flux of vorticity over a flat plate started from rest. However, the strength of the vorticity on the separated structure is 25% of that suggested by this model, sometimes referred to as the ‘slug’ model. Results show that significant secondary vorticity is generated on the piston face, forming in response to the separating primary vortex. This secondary vorticity grows at the same rate as the primary vorticity and is wrapped around the outside of the primary structure and causes it to advect away from the piston surface.

Vortex formation in front of a piston moving through a cylinder

Interpretation of flow visualization

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