Dispersion of diffusive tracers in steady flows

When considering large spatial and temporal scales the transport of a concentration field by advection and molecular diffusion can be be approximately described by a diffusion equation with an effective diffusion coefficient. The main question then is to find an expression for the effective diffusivity as a function of the flow parameters and molecular diffusivity. A range of this type of problems are discussed in the review by Majda and Kramer (1999). Here we consider two simple examples of this problem in the case of steady two-dimensional flows with open and closed streamlines, respectively. 

Advection and diffusion in a uni-directional shear flow of the form v = [u(y), 0] is a classical problem studied by Taylor (1953), considered as a simple model for the dispersion of a contaminant in a river or in a flow through a pipe (Fig. 2.3). 

The concentration field satisfies the advection-diffusion equation 

In the following we use an overbar to denote the mean of a quantity averaged along the transversal y direction, A(x) = L 1 II dyA(x,y), where L is the width of the channel. The velocity field can be 

After sufficiently long time t L2/D, diffusion smoothes out the fluctuations in the y direction so that the dominant non-uniformity is along the longitudinal direction, i.e. C C. Assuming that in the longitudinal direction advection is much faster than molecular diffusion the dominant balance in (2.46) is 

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