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.

The random nature of turbulence

In a turbulent flow, the velocity field U(x, t) is random. What does this statement mean? Why is it so?

As a first step we need to understand the word ‘random.’ Consider a fluid-flow experiment that can be repeated many times under a specified set of conditions, C, and consider an event A, such as A ≡ {U < 10 m s−1}, where U is a specified component of velocity at a specified position and time (measured from the initiation of the experiment). If the event A inevitably occurs, then A is certain or sure. If the event A cannot occur, then it is impossible. The third possibility is that A may occur or it may but need not occur. In this case the event A is random. Then, in the example A ≡ {U < 10 m s−1}, U is a random variable.

A mistake that is sometimes made is to attribute incorrectly additional significance to the designation ‘random,’ and then to dispute the fact that turbulence is a random phenomenon. That the event A is random means only that it is neither certain nor impossible. That U is a random variable means only that it does not have a unique value – the same every time the experiment is repeated under the same set of conditions, C. Figure 3.1 illustrates the values U(n)(n = 1, 2, …, 40) taken by the random variable U on 40 repetitions of the experiment.

The challenge

In the study of turbulent flows – as in other fields of scientific inquiry – the ultimate objective is to obtain a tractable quantitative theory or model that can be used to calculate quantities of interest and practical relevance. A century of experience has shown the ‘turbulence problem’ to be notoriously difficult, and there are no prospects of a simple analytic theory. Instead, the hope is to use the ever-increasing power of digital computers to achieve the objective of calculating the relevant properties of turbulent flows. In the subsequent chapters, five of the leading computational approaches to turbulent flows are described and examined.

It is worthwhile at the outset to reflect on the particular properties of turbulent flows that make it difficult to develop an accurate tractable theory or model. The velocity field U(x, t) is three-dimensional, time-dependent, and random. The largest turbulent motions are almost as large as the characteristic width of the flow, and consequently are directly affected by the boundary geometry (and hence are not universal). There is a large range of timescales and lengthscales. Relative to the largest scales, the Kolmogorov timescale decreases as Re−1/2, and the Kolmogorov lengthscale as Re−3/4. In wall-bounded flows, the most energetic motions (that are responsible for the peak turbulence production) scale with the viscous lengthscale δv which is small compared with the outer scale δ, and which decreases (relative to δ) approximately as Re−0.8.

In contrast to the free shear flows considered in Chapter 5, most turbulent flows are bounded (at least in part) by one or more solid surfaces. Examples include internal flows such as the flow through pipes and ducts; external flows such as the flow around aircraft and ships' hulls; and flows in the environment such as the atmospheric boundary layer, and the flow of rivers.

We consider three of the simplest of these flows (sketched in Fig. 7.1), namely: fully developed channel flow; fully developed pipe flow; and the flat-plate boundary layer. In each of these flows the mean velocity vector is (or is nearly) parallel to the wall, and, as we shall see, the near-wall behaviors in each of these cases are very similar. These simple flows are of practical importance and played a prominent role in the historical development of the study of turbulent flows.

Central issues are the forms of the mean velocity profiles, and the friction laws, which describe the shear stress exerted by the fluid on the wall. In addition the mixing length is introduced in Section 7.1.7; the balance equations for the Reynolds stresses are derived and examined in Section 7.3.5; and the proper orthogonal decomposition (POD) is described in Section 7.4.

Channel flow

A description of the flow

As sketched in Fig. 7.1, we consider the flow through a rectangular duct of height h = 2δ. The duct is long (L/δ ≫ 1) and has a large aspect ratio (b/δ ≫ 1).