Most existing analytical and mathematical models for sediment-laden flows are based on theverning equations for single-phase flows, and are valid only for low sediment concentration situations. This paper presents an analysis, on the basis of the fundamental equations for fluid-solid two-phase flows, of the velocity and sediment concentration profiles in open-channel flows. A new diffusion equation is established for suspended sediment concentration from a rigorous derivation of the water-sediment mixture's normal velocity in the sense of mass flux conservation. The differences between this new equation and Schmidt's aswell as Hunt's are shown to be attributable to the different approximations of sediment velocity. Previous formulations for velocity and sediment concentration distributions are special cases of the present model for low sediment concentration flows. The developed model is extensively tested against available measurements, and satisfactory or fairly good agreement is obtained.

One of the basic impediments to a clear understanding of a variety of fundamental problems in the context of sediment transport has been the lack of a well-grounded formulation of bed sediment entrainment. This is dealt with herein, physically based on the mechanism that bed sediment particles are actually entrained by the bursting process inherent in wall turbulent flows. A simple theoretical model for sediment entrainment from flat, loose bed is established using the averaged bursting period scaled on inner variables and the spatialscales of turbulent bursts. Sediment entrainment is shown to depend strongly on bed-shear velocity. The theoretical entrainment flux is compared to available laboratory data sets covering both hydraulically smooth andtransitional bed situations. Generally good agreement is obtained, representing the best performance of the present model in relation to existing entrainment functions. It appears to characterize a rather encouraging aspect for a new approach to sediment transport, given the enhanced understanding of turbulent bursting.

A new approach is presented for calculating the equilibrium near-bed concentration of suspended sediment in an alluvial channel flow. It is formulated from the balance between bed sediment entrainment and suspended sediment deposition across the near-bed boundary. The entrainment flux is determined making use of a turbulent bursting outer-scale-based function and the flux of deposition by the product of near-bed concentration and hindered settling velocity of sediment. A number of flume data records in the literature are analyzed to calibrate and verify the present approach. The observed near-bed concentrations for the data records are obtained by first isolating the suspended load transport rate from the observed total load transport rate using Engelund and Fredsoe's bed-load formula and then equating the suspended load transport rate to the shape integration of Dyer and Soulsby. The present approach is shown to perform satisfactorily compared to the results of data analysis. It is found that the near-bed concentration is evidently dependent on sediment particle size in addition to the Shields parameter due to skin friction. This finding seems to challenge previous relationships that simply represent the near-bed concentration as empirical functions of the purely skin-friction-related Shields parameter.

Existing numerical river models are mostly built upon asynchronous solution of simplified governing equations. The strong coupling between water flow, sediment transport, and morphological evolution is thus ignored to a certain extent. An earlier study led to the development of a fully coupled model and identified the impacts of simplifications in the water-sediment mixture and global bed material continuity equations as well as of the asynchronous solution procedure for aggradation processes. This paper presents the results of an extended study along this line, highlighting the impacts on both aggradation and degradation processes. Simplifications in the continuity equations for the water-sediment mixture and bed material are found to have negligible effects on degradation. This is, however, in contrast to aggradation processes, in which the errors purely due to simplified continuity equations can be significant transiently. The asynchronous solution procedure is found to entail appreciable inaccuracy for both aggradation and degradation processes. Further, the asynchronous solution procedure can render the physical problem mathematically ill posed by invoking an extra upstream boundary condition in the supercritical flow regime. Finally, the impacts of simplified continuity equations and an asynchronous solution procedure are shown to be comparable with those of largely tuned friction factors, indicating their significance in calibrating numerical river models. It is concluded that the coupled system of complete governing equations needs to be synchronously solved for refined modeling of alluvial rivers.

The last half a century has seen more and more developments and applications of mathematical models for fluvial flow, sediment transport and morphological evolution. However, the quality of this modelling practice has emerged as a crucial issue for concern, which is widely viewed as the key that could unlock the full potential of computational fluvial hydraulics. The major factors affecting the modelling quality comprise: (a) poor assumptions in model formulations; (b) simplified numerical solution procedure; (c) the implementation of sediment relationships of questionable validity; and (d) the problematic use of model calibration and verification as assertions of model veracity. An overview of mathematical models for alluvial rivers is provided in this and the companion paper 'Part I: General review'. This paper is the second part, dealing with three special issues of mathematical river models. First, turbulence closure models are highlighted, particularly with respect to the role of sediment in modulating turbulence and its implications for adapting turbulence closure models for fluvial sediment-laden flows. Second, the bottom boundary conditions are discussed in detail as one of the main sources of model uncertainty. And third, the commonly used calibration and verification/validation methodology in mathematical river modelling is addressed. It is argued that model calibration can be subjective, verification is impossible because models are not closed systems, and validation does not necessarily establish model truth. Confirmation of observations by models only supports model probability, rather than demonstrating model veracity. It is vital for model developers and end-users to keep aware of what mathematical river models can realistically reflect, and therefore avoid misleading decisionmaking. Additionally, some strategies are proposed which can improve the practice of mathematical river modelling.