Shedding some light on b-factors

The use of a Fickian (infinitesimal–mixing–length) framework for the case of turbulent mixing can necessitate the use of ad hoc modifications (e.g. β–factors) in order to reconcile experimental data with theoretical expectations. This is because in many cases turbulent mixing occurs on scales which cannot be considered infinitesimal. In response to this problem a Finite–Mixing– Length (FML) model for turbulent mixing was derived by Nielsen and Teakle. This paper considers the application of this model to the scenario of suspended sediment in steady, uniform channel flows. It is shown that, unlike the Fickian framework, the FML model is capable of explaining why β– factors are required to be an increasing function of ws/u*. The FML model does not on its own explain observations of β < 1, seen in some flat–bed experiments. However, some potential reasons for β < 1 are considered.

Differential diffusion: Often a finite-mising length effect

Many instances of differential diffusion, i e, different species having different turbulent diffusion coefficients in the same flow, can be explained as a finite mixing length effect. That is, in a simple mixing length scenario, the turbulent diffusion coefficient has the form 1 ( m )2 m m c l K w l OL   =  +    where, wm is the mixing velocity, lm the mixing length and Lc the overall distribution scale for a particular species. The first term represents the familiar gradient diffusion while the second term becomes important when lm/Lc is finite. This second term shows that different species will have different diffusion coefficients if they have different overall distribution scales. Such different Lcs may come about due to different boundary conditions and different intrinsic properties (molecular diffusivity, settling velocity etc) for different species. For momentum transfer in turbulent oscillatory boundary layers the second term is imaginary and explains observed phase leads of shear stresses ahead of velocity gradients.