The material contained in this chapter closely parallels the presentation in Chapter 2. In Section 3.1, we review the phenomenological description of turbulent mixing that is often employed in engineering models to relate the scalar mixing time to the turbulence time scales. In Section 3.2, the statistical description of homogeneous turbulent mixing is developed based on the one-point and two-point probability density function of the scalar field. In Section 3.3, the transport equations for one-point statistics used in engineering models of inhomogeneous scalar mixing are derived and simplified for high-Reynolds-number turbulent flows. Both inert and reacting scalars are considered. Finally, in Section 3.4, we consider the turbulent mixing of two inert scalars with different molecular diffusion coefficients. The latter is often referred to as differential diffusion, and is known to affect pollutant formation in gas-phase turbulent reacting flows (Bilger 1982; Bilger and Dibble 1982; Kerstein et al. 1995; Kronenburg and Bilger 1997; Nilsen and Kosály 1997; Nilsen and Kosály 1998).

Phenomenology of turbulent mixing

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum Eφ(k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998).