Turbulent diffusion of momentum and suspended particles: A finite-mixing-length theory

A finite-mixing-length theory is presented for turbulent mixing. This theory contains Fickian diffusion as the limiting case for $lm/L→0,$ where $lm$ is the mixing length and L is the scale of the distribution under consideration. The new model is of similar generality to that of Taylor (1921), “Diffusion by continuous movements.” However, while Taylor’s model, being strictly Lagrangian, is difficult to apply to inhomogeneous scenarios, the new model is Eulerian and easily applicable to bottom boundary layers and other inhomogeneous flows. When applied to steady suspended sediment concentrations $c(z),$ the theory predicts the observed trend of apparent Fickian diffusivities $εFick=−wsc/(dc/dz)$ being larger for particles with larger settling velocity $ws,$ in a given flow. The corresponding nature of the ratios between apparent Fickian sediment diffusivities and eddy viscosities, $β=εFick/vt,$ for different particles in the same flow, is also revealed. That is, β is an increasing function of the particle settling velocity and may be greater or smaller than unity depending on the relative magnitude of the vertical scales for concentrations $Lc$ and for horizontal velocity $Lu.$ Applied to the case of wave motion over ripples, even the observed trend of fine sand indicating decreasing $εFick(z)$ while coarse sand shows increasing $εFick(z)$ can be modelled with realistic turbulence parameters. Different Fickian diffusivities displayed by different species like density, heat, and salinity in a given turbulent flow may also be qualitatively explainable in terms of finite-mixing-length effects.