《湍流流动》：流体力学关键教材

流体力学

turbulence

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List of tables

5.1 Spreading rate parameters of turbulent round jets 101

5.2 Timescales in turbulent round jets 131

5.3 Spreading parameters of turbulent axisymmetric wakes 153

5.4 Statistics in homogeneous turbulent shear ﬂow 157

6.1 Characteristic scales of the dissipation spectrum 238

6.2 Characteristic scales of the energy spectrum 240

6.3 Tail contributions to velocity-derivative moments 259

7.1 Wall regions and layers and their deﬁning properties 275

7.2 Statistics in turbulent channel ﬂow 283

8.1 Computational diﬃculty of diﬀerent turbulent ﬂows 338

9.1 Numerical parameters for DNS of isotropic turbulence 349

9.2 Numerical parameters for DNS of channel ﬂow 355

9.3 Numerical parameter for DNS of the ﬂow over a backward-facing step 356

10.1 The turbulent Reynolds number of self-similar free shear ﬂows 366

10.2 Deﬁnition of variables in two-equation models 384

11.1 Special states of the Reynolds-stress tensor 394

11.2 Mean velocity gradients for simple deformations 415

11.3 Tensors used in pressure–rate-of-strain models 426

11.4 Coeﬃcients in pressure–rate-of-strain models 427

11.5 Coeﬃcients in algebraic stress models 452

11.6 Integrity basis for turbulent viscosity models 453

11.7 Attributes of diﬀerent RANS turbulence models 457

12.1 Comparison between ﬂuid and particle systems 518

12.2 Diﬀerent levels of PDF models 555

13.1 Resolution in DNS and in some variants of LES 560

13.2 Filter functions and transfer functions 563

13.3 Estimates of ﬁltered and residual quantities in the inertial subrange 589

13.4 Deﬁnition of the diﬀerent types of triad interactions 608

B.1 Operations between ﬁrst- and second-order tensors 662

D.1 Fourier-transform pairs 679

E.1 Spectral properties of random processes 689

G.1 Power-law spectra and structure functions 700

I.1 Relationships between characteristic functions and PDFs 710

xviii Preface

approaches. The present selection of topics and approaches has evolved over

the 20 years I have been teaching graduate courses on turbulence at MIT

and Cornell. The emphasis on turbulent ﬂows – rather than on the theory

of homogeneous turbulence – is appropriate to applications in engineering,

atmospheric sciences, and elsewhere. The emphasis on quantitative theories

and models is consistent with the scientiﬁc objective – of developing a

tractable, quantitatively accurate theory of the phenomenon – and is ideal for

providing a solid understanding of computational approaches to turbulent

ﬂows, e.g., turbulence models and LES.

With the exceptions of LES and direct numerical simulation (DNS), the

theories and models presented stem from the statistical approach, pioneered

by Osborne Reynolds, G. I. Taylor, Prandtl, von K

arm

an, and Kolmogorov.

A sizable fraction of the academic research work in the last 25 years has

emphasized a more deterministic viewpoint: for example experiments on

coherent structures, and models based on low-dimensional dynamical systems

(e.g., Holmes, Lumley, and Berkooz (1996)). At this stage, this alternative

approach has not led to a generally applicable quantitative model, neither

– for better or for worse – has it had a major impact on the statistical

approaches. Consequently, the deterministic viewpoint is neither emphasized

nor systematically presented.

The book consists of two parts followed by a number of appendices. Part

I provides a general introduction to turbulent ﬂows, including the Navier–

Stokes equations, the statistical representation of turbulent ﬁelds, mean-ﬂow

equations, the behavior of simple free-shear and wall-bounded ﬂows, the

energy cascade, turbulence spectra, and the Kolmogorov hypotheses. In the

ﬁrst ﬁve chapters, the focus is ﬁrst on the mean velocity ﬁelds, and how they

are aﬀected by the Reynolds stresses. The concept of ‘turbulent viscosity’ is

introduced with a thorough discussion of its deﬁciencies. The focus then shifts

to the turbulence itself, in particular to the production and dissipation of

turbulent kinetic energy. This sets the stage for a description (in Chapter 6) of

the energy cascade and the Kolmogorov hypotheses. The spectral description

of homogeneous turbulence in terms of Fourier modes in wavenumber space

is developed in some detail. This provides an alternative perspective on the

energy cascade; and it is also used in subsequent chapters in the descriptions

of DNS, LES, and rapid distortion theory (RDT).

Simple wall-bounded ﬂows are described in Chapter 7, starting with the

mean velocity ﬁelds and proceeding to the Reynolds stresses. The exact

transport equations for the Reynolds stresses are introduced, and their

balances in turbulent boundary layers are examined.

The simulation and modelling approaches described in Part II are: DNS,

Preface xix

turbulent viscosity models (e.g., the k–ε model), Reynolds-stress models, PDF

methods, and LES. It is natural to consider DNS ﬁrst (in Chapter 9) since it

is conceptually the most straightforward approach. However, its restriction

to simple, low-Reynolds-number ﬂows motivates the consideration of other

approaches. The most widely used turbulence models are the turbulent-

viscosity models described in Chapter 10. Reynolds-stress models (Chapter

11) provide a more satisfactory connection to the physics of turbulence. The

Reynolds-stress balance equations can be obtained from the Navier–Stokes

equations, and the various contributions to this balance have been measured

in experiments and simulations. Rapid-distortion theory is introduced to

shed light on the eﬀects that mean velocity gradients have on the Reynolds

stresses. In developing and presenting modelled Reynolds-stress equations,

the emphasis is on the fundamental concepts and principles, rather than on

the detailed forms of particular models.

Chapter 12 deals with PDF methods. The primary object of study is the

(one-point, one-time, Eulerian) joint probability density function (PDF) of

velocity. The ﬁrst moments of this PDF are the mean velocities; the second

moments are the Reynolds stresses. For several reasons it is both natural

and advantageous to proceed from the Reynolds stress to the PDF level of

description: in the PDF equation, convection (by both mean and ﬂuctuating

velocity) appears in closed form, and hence does not have to be modelled;

the eﬀect of rapid distortions on turbulence can (in a limited sense) be treated

exactly; and PDF methods are becoming widely used for turbulent reactive

ﬂows (e.g., turbulent combustion) because they are able to treat reaction

exactly – without modelling assumptions.

Essential ingredients in PDF methods are stochastic Lagrangian models,

such as the Langevin model for the velocity following a ﬂuid particle. These

models are also described in the context of turbulent dispersion (where they

originated with G. I. Taylor’s 1921 classic paper).

The ﬁnal chapter describes LES, in which the large-scale turbulent motions

are directly represented, while the eﬀects of the smaller, subgrid-scale motions

are modelled. Many of the concepts and techniques developed in Chapters

9–12 ﬁnd application in the modelling of the subgrid-scale processes.

I use this book in a one-semester course, taught to students who previously

have taken one or more graduate courses in ﬂuid mechanics and applied

mathematics. For most students, there is a good deal of new material, but I

ﬁnd that they can successfully master it, provided that it is clearly and fully

explained. Accordingly there are many appendixes that provide the necessary

development and explanation of mathematical techniques and results used

in the text. In my experience, it is best not to rely upon the students’ prior

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