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.