3. Mixing in Rivers: Turbulent Diffusion and Dispersion

In previous chapters we considered the processes of advection and molecular diffusion and have seen some example problems with so called “turbulent diffusion” coefficients, where we use the same governing equations, but with larger diffusion (mixing) coefficients. In natural rivers, a host of processes lead to a non-uniform velocity field, which allows mixing to occur much faster than by molecular diffusion alone. In this chapter, we formally derive the equations for nonuniform velocity fields to demonstrate their effects on mixing. First, we consider the effect of a random, turbulent velocity field. Second, we consider the combined effects of diffusion (molecular or turbulent) with a shear velocity profile to develop equations for dispersion. In each case, the resulting equations retain their previous form, but the mixing coefficients are orders of magnitude greater than the molecular diffusion coefficients. We start by giving a description of turbulence and its effects on the transport of contaminants. We then derive a new advective diffusion equation for turbulent flow and show why turbulence can be described by the regular advective diffusion equation derived previously, but using larger turbulent diffusion coefficients. We then look at the effect of a shear velocity profile on the transport of contaminants and derive one-dimensional equations for longitudinal dispersion. This chapter concludes with a common dye study application to compute the effective mixing coefficients in rivers.