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 e nergetic 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.