Author(s): Joe C. Willis

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Abstract: Derivations of distributions of diffusivity, flow velocity, and sediment concentration over the flow depth were made for a generalized normalization of the relative distance from the flow boundary. Boundary condition restraints suggest a normalization function with a finite lower limit and an infinite upper limit. In analytical considerations, the distribution functions were discussed relative to an unspecified mean turbulent diffusivity. An analysis of the distributions was made in terms of their limits at the channel boundary and the free surface (or top of the boundary layer). The analytical developments were then applied to three models that have been used previously. Besides the parabolic and error function distributions of diffusivity, an expression for the diffusivity distribution was derived from the power function model for the velocity distribution. A numerical solution for the concentration distribution was made based on this diffusivity distribution. Two new normalization functions were investigated in some detail. These were the Rayleigh function and the gamma function. Of the models investigated herein, the gamma function offers the greatest promise as a generally applicable model for turbulent diffusivity. If appropriate parameters are selected, it satisfies both boundary conditions on the velocity and is asymptotic to the error function, which also satisfies the velocity-slope boundary condition at the water surface. Any normalized probability function was shown to be expressible as a potential model for the turbulent diffusivity. A consideration of all available functions is beyond the scope of this paper; however, the analysis procedure used herein in terms of the functional form and boundary conditions should be generally applicable and should serve as a basis for comparison of potential models. Possibly these concepts will stimulate the development and experimental verification of better diffusivity models for describing fluid flow phenomena.

DOI: https://doi.org/10.1080/00221687909499586

Year: 1979