In this chapter, we review selected results from the statistical description of turbulence needed to develop CFD models for turbulent reacting flows. The principal goal is to gain insight into the dominant physical processes that control scalar mixing in turbulent flows. More details on the theory of turbulence and turbulent flows can be found in any of the following texts: Batchelor (1953), Tennekes and Lumley (1972), Hinze (1975), McComb (1990), Lesieur (1997), and Pope (2000). The notation employed in this chapter follows as closely as possible the notation used in Pope (2000). In particular, the random velocity field is denoted by U, while the fluctuating velocity field (i.e., with the mean velocity field subtracted out) is denoted by u. The corresponding sample space variables are denoted by V and v, respectively.

Homogeneous turbulence

At high Reynolds number, the velocity U(x, t) is a random field, i.e., for fixed time t = t* the function U(x, t*) varies randomly with respect to x. This behavior is illustrated in Fig. 2.1 for a homogeneous turbulent flow. Likewise, for fixed x = x*, U(x*, t) is a random process with respect to t. This behavior is illustrated in Fig. 2.2. The meaning of ‘random’ in the context of turbulent flows is simply that a variable may have a different value each time an experiment is repeated under the same set of flow conditions (Pope 2000). It does not imply, for example, that the velocity field evolves erratically in time and space in an unpredictable fashion. Indeed, due to the fact that it must satisfy the Navier–Stokes equation, (1.27), U(x, t) is differentiable in both time and space and thus is relatively ‘smooth.’