A Co‐Spectral Budget Model Links Turbulent Eddies to Suspended Sediment Concentration in Channel Flows

Abstract

The vertical distribution of suspended sediment concentration remains a subject of active research given its relevance to a plethora of problems in hydraulics, hydrology, ecology, and water quality control. Many of the classical theories developed over the course of 90 years represent the effects of turbulence on suspended sediments (SS) using an effective mixing length or eddy diffusivity without explicitly accounting for the energetics of turbulent eddies across scales. To address this gap, the turbulent flux of sediments is derived using a co‐spectral budget (CSB) model that can be imminently used in SS and other fine particle transport models. The CSB closes the pressure‐redistribution effect using a spectral linear Rotta scheme modified to include isotropization of production and interactions between turbulent eddies and sediment grains through a modified scale‐dependent de‐correlation time. The result is a formulation similar in complexity to the widely used Rouse's equation but with all characteristic scales, Reynolds number, and Schmidt number effects derived from well‐established spectral shapes of the vertical velocity and accepted constants from turbulence models. Finally, the proposed CSB model can recover Prandtl's and Rouse's equations under restricted conditions.

Reduced Sediment Settling in Turbulent Flows Due To Basset History and Virtual Mass Effects

“…Thus, $${u}_{\ast }^{2}$$ = g o S o H and the bulk velocity U b = Q /( BH ). The bulk turbulent kinetic energy dissipation rate can now be determined from bulk variables as $${{\epsilon}}_{b}={U}_{b}\left({g}_{o}{S}_{o}\right)=\left({U}_{b}/H\right){u}_{\ast }^{2}$$ enabling an estimate of a bulk Kolmogorov micro‐scale eddy size $${\eta }_{b}={\left({\nu }^{3}/{{\epsilon}}_{b}\right)}^{1/4}$$ to be compared with reported grain diameter d p without any z dependency for convenience (S. Li et al., 2022). In all model calculations, the local ϵ ( z ) is used in the estimation of $$\eta (z)={\left[{\nu }^{3}/{\epsilon}(z)\right]}^{1/4}$$.…”