: Physical theory for near-bed turbulent particle-suspension capacity

The inability to capture the physics of solid-particle suspension in turbulent fluids in simple formulas is holding back the application of multiphase fluid dynamics techniques to many practical problems in nature and society involving particle suspension. We present a force balance approach to particle suspension in the region near no-slip frictional boundaries of turbulent flows. The force balance parameter Γ contains gravity and buoyancy acting on the sediment and vertical turbulent fluid forces; it includes universal turbulent flow scales and material properties of the fluid and particles only. Comparison to measurements shows that Γ = 1 gives the upper limit of observed suspended particle concentrations in a broad range of flume experiments and field settings. The condition of Γ > 1 coincides with the complete suppression of coherent turbulent structures near the boundary in direct numerical simulations of sediment-laden turbulent flow. Γ thus captures the maximum amount of sediment that can be contained in suspension at the base of turbulent flow, and it can be regarded as a suspension capacity parameter. It can be applied as a simple concentration boundary condition in modelling studies of the dispersion of particulates in environmental and man-made flows.

1.Much of the analysis is based upon hydraulically smooth data sets and their interpretation, a regime corresponding to u * d/ν 1.In this setting it is inevitable that the kinematic viscosity, ν, plays a vital role in the mechanics and associated dimensional reasoning.However, how relevant are hydraulically smooth conditions to natural settings?

2.
The key new empiricism is to write a quadratic function for the dimensionless acceleration (eqn 7) and hence a cubic equation for w 2 .This function is determined by fitting data from DeGraaff and Eaton (2000).Are there theoretical grounds for expecting a quadratic dependence?Or rather is the important insight to elucidate the dimensional dependence of the acceleration?Also given the comments in (i), how relevant is this fitted form for natural systems with non-vanishing hydraulic roughness.Could it be that roughness enters the expressions if the boundary is not smooth?
3. I wonder whether it is helpful to describe the dynamics in terms of an 'acceleration'.Instead, presumably, the mean pressure field is altered by non-vanishing gradients of w 2 in order to satisfy the 'vertical' momentum balance and it is through this pressure field that the particle-scale dynamics are affected.
4. Following (iii), I think more must be said to justify the particle motion and to explain why it is appropriate that the particles follow the motion of fluid elements.My approach would be to form the mean pressure field (see (iii)) and deduce the stresses on the particle due to it.It is presumably necessary to average over the particle size and therein it is necessary to assume that that the particle diameter d is much less than the flow lengthscale ν/u * , a condition that reduces to u * d/ν 1. Furthermore a more thorough analysis of the particle motion would naturally identify the Stokes number as the important measure of the effects of particle inertia.
5. Much is made of the realisation that Γ is independent of grain size (and/or settling velocity).For incipient motion in hydraulically smooth conditions, I suspect that this is already captured by the usual dependence of the critical Shields parameter θ c ≡ u 2 * /(Rgd) upon the particle Reynolds number, Re = u * d/ν.When Re 1, θ c = K/Re, where K is a constant.Thus incipient motion is determined by θ c Re = u 3 * /(Rgν) = K.This conclusion is identical to what is derived in the paper, but is surely in line with a 'competence'-approach to modelling sediment transport.
6.The need for a reference concentration or a flux boundary condition in gradient diffusion models of sediment transport: this is of course, a long standing issue in sediment transport research and one for which steady flows do not shed much insight.For steady flows, one might prescribe a reference concentration at a small elevation above the bed.Alternatively, one might prescribe an erosive flux, but then since the suspension is in a steady balance between erosion and settling, this also leads to a prescribed concentration at the base.It is only for unsteady dynamics that the two types of boundary conditions behave differently -and I wonder what this new formulation can say about the concentration, or its flux, in this scenario?
7. It is assumed that the concentrations are sufficiently dilute so that the particles do not affect the flow and thus the clear fluid correlations of eqn ( 7) are applicable.However the theory is used for volumetric concentrations in excess of 0.1 and so I wonder how secure this assumption is?It is also used qualitatively to describe the collapse of the turbulence, which might well be a phenomena associated with high concentration suspensions.
Other minor comments follow: