In this book we have studied various effects produced by quenched disorder on thermodynamical properties of statistical systems. Considering different types of model, the emphasis has been made on the demonstration of the basic theoretical approaches and ideas. Although the considered systems and the corresponding problems involved (such as spin glasses, critical phenomena, directed polymers etc.) may at first look quite different, the aim of the book was to demonstrate that basically all these problems are deeply interconnected. I was trying to convince the reader that to work successfully on any particular problem in this field one needs to be familar with all the methods and ideas of statistical field theory. The physics of both the spin-glass state and critical phenomena in weakly disordered systems involve the ideas of the scaling theory of phase transitions, and the basic concepts of the replica theory of spin glasses. The aim of this book was to take the reader, starting from fundamentals and demonstrating well-established solutions of various problems, to the frontier of modern research. Here we are facing quite a few fundamental problems, both long-standing and new ones, still waiting for their solutions.

The most appealing problem in the scope of spin glasses had remained a question for almost two decades: whether or not the mean-field RSB physical picture (described in Chapters 2–6) is valid, at least at the qualitative level, for realistic spin glasses with finite-range interactions.

The nature of turbulent flows

There are many opportunities to observe turbulent flows in our everyday surroundings, whether it be smoke from a chimney, water in a river or waterfall, or the buffeting of a strong wind. In observing a waterfall, we immediately see that the flow is unsteady, irregular, seemingly random and chaotic, and surely the motion of every eddy or droplet is unpredictable. In the plume formed by a solid rocket motor (see Fig. 1.1), turbulent motions of many scales can be observed, from eddies and bulges comparable in size to the width of the plume, to the smallest scales the camera can resolve. The features mentioned in these two examples are common to all turbulent flows.

More detailed and careful observations can be made in laboratory experiments. Figure 1.2 shows planar images of a turbulent jet at two different Reynolds numbers. Again, the concentration fields are irregular, and a large range of length scales can be observed.

As implied by the above discussion, an essential feature of turbulent flows is that the fluid velocity field varies significantly and irregularly in both position and time. The velocity field (which is properly introduced in Section 2.1) is denoted by U(x, t), where x is the position and t is time.

Figure 1.3 shows the time history U1(t) of the axial component of velocity measured on the centerline of a turbulent jet (similar to that shown in Fig. 1.2).

From vector calculus we are familiar with scalars and vectors. A scalar has a single value, which is the same in any coordinate system. A vector has a magnitude and a direction, and (in any given coordinate system) it has three components. With Cartesian tensors, we can represent not only scalar and vectors, but also quantities with more directions associated with them. Specifically, an Nth-order tensor (N ≥ 0) has N directions associated with it, and (in a given Cartesian coordinate system) it has 3N components. A zeroth-order tensor is a scalar, and a first-order tensor is a vector. Before defining higher-order tensors, we briefly review the representation of vectors in Cartesian coordinates.

Cartesian coordinates and vectors

Fluid flows (and other phenomena in classical mechanics) take place in the three-dimensional, Euclidean, physical space. As sketched in Fig. A.1, let E denote a Cartesian coordinate system in physical space. This is defined by the position of the origin O, and by the directions of the three mutually perpendicular axes. The unit vectors in the three coordinate directions are denoted by e1, e2, and e3. We write ei to refer to any one of these, with the understanding that the suffix i (or any other suffix) takes the value 1, 2, or 3.

The basic properties of the unit vectors ei are succinctly expressed in terms of the Kronecker delta δij.

The most commonly studied turbulent free shear flows are jets, wakes, and mixing layers. As the name ‘free’ implies, these flows are remote from walls, and the turbulent flow arises because of mean-velocity differences.

We begin by examining the round jet. By combining experimental observations (Section 5.1) with the Reynolds equations (Section 5.2), a good deal can be learned, not only about the round jet, but also about the behavior of turbulent flows in general. In Section 5.3, we study the turbulent kinetic energy in the round jet, and the important processes of production and dissipation of energy. Other self-similar free shear flows are briefly described in Section 5.4; and further observations about the behavior of free shear flows are made in Section 5.5.

The round jet: experimental observations

A description of the flow

We have already encountered the round jet in Chapter 1, for example, Figs. 1.1–1.4. The ideal experimental configuration and the coordinate system employed are shown in Fig. 5.1. A Newtonian fluid steadily flows through a nozzle of diameter d, which produces (approximately) a flat-topped velocity profile, with velocity UJ. The jet from the nozzle flows into an ambient of the same fluid, which is at rest at infinity. The flow is statistically stationary and axisymmetric. Hence statistics depend on the axial and radial coordinates (x and r), but are independent of time and of the circumferential coordinate, θ.

The mean velocity 〈U(x, t)〉 and the Reynolds stresses 〈uiuj〉 are the first and second moments of the Eulerian PDF of velocity f(V; x, t) (Eq. (3.153)). In PDF methods, a model transport equation is solved for a PDF such as f(V; x, t).

The exact transport equation for f(V; x, t) is derived from the Navier–Stokes equations in Appendix H, and discussed in Section 12.1. In this equation, all convective transport is in closed form – in contrast to the term ∂〈uiuj〉/∂xi in the mean-momentum equation, and ∂〈uiuj〉/∂xi in the Reynolds-stress equation. A closed model equation for the PDF – based on the generalized Langevin model (GLM) – is given in Section 12.2, and it is shown how this is closely related to models for the pressure–rate-of-strain tensor, ℛij.

Central to PDF methods are stochastic Lagrangian models, which involve new concepts and require additional mathematical tools. The necessary background on diffusion processes and stochastic differential equations is given in Appendix J. The simplest stochastic Lagrangian model is the Langevin equation, which provides a model for the velocity following a fluid particle. This model is introduced and examined in Section 12.3.

A closure cannot be based on the PDF of velocity alone, because this PDF contains no information on the turbulence timescale. One way to obtain closure is to supplement the PDF equation with the model dissipation equation. A superior way, described in Section 12.5, is to consider the joint PDF of velocity and a turbulence frequency.