Slow-zone model for longitudinal dispersion in two-dimensional shear flows

A two-zone model is proposed for the longitudinal dispersion of contaminants in two-dimensional turbulent flow in open channels – a fast zone in the upper region of the flow, and a slow zone nearer to the bottom. The usual one-dimensional dispersion approach (Elder 1959) is used in each zone, but with different flow speeds U₁ and U₂ and dispersion coefficients D₁ and D₂ in the fast and slow zones respectively. However, turbulent vertical mixing is allowed at the interface between the two zones, with a small vertical diffusivity ε. This leads to a pair of coupled, linear, one-dimensional dispersion equations, which are solved by Fourier transformation. The Fourier-inversion integrals are analysed using two different methods.

In the first method asymptotically valid expressions are found using the saddle-point method. The resulting cross-sectional average concentration consists of a leading Gaussian distribution followed by a trailing Gaussian distribution. The trailing Gaussian cloud disperses (longitudinally) faster than the leading one, and this gives the long tail observed in most dispersion experiments. Significantly the peak value of the average concentration is found to decay exponentially with time at a rate which is close to the rate observed by Sullivan (1971) in the early stage of the dispersion process. The solution is useful for fairly small times, and both the calculated value of D₁ and the predicted bulk concentration distribution are in meaningful agreement with the experimental and simulation data of Sullivan (1971).

In the second method an exact solution is found in the form of a convolution integral for the case D₁ = D₂ = D₀. Explicit expressions which are valid for small times and for large times from the release of contaminant are found. For small times this exact solution confirms the basic results obtained by the saddle-point method. For large times the exact solution gives a contaminant concentration which approaches a Gaussian distribution travelling with the bulk speed as predicted by the Taylor model. The overall longitudinal dispersion coefficient at large times, D(∞), consists of the diffusivity D₀ plus a contribution D_{[ell ]}(∞) which depends entirely on the vertical mixing. D(∞) is in good agreement with Chatwin's (1971) interpretation of Fischer's (1966) experimental data.