We investigate swash on an erodible beach using the one-dimensional shallow-water equations fully coupled to a bed-evolution (Exner) equation. In particular, the dam-break/bore-collapse initial condition of Shen & Meyer (J. Fluid Mech., vol. 16, 1963, pp. 113–125) and Peregrine & Williams (J. Fluid Mech., vol. 440, 2001, pp. 391–399) is investigated using a numerical model based on the method of characteristics. A sediment-transport formula (cubic in velocity u: Au3) is used here; this belongs to a family of sediment-transport formulae for which Pritchard & Hogg (Coastal Engng, vol. 52, 2005, pp. 1–23) showed that net sediment transport under the Shen & Meyer (1963) bore collapse is offshore throughout the swash zone when a non-erodible bed is considered. It is found that full coupling with the beach, although still resulting in the net offshore transport of sediment throughout the swash zone, leads to a large reduction in the net offshore transport of sediment from the beach face. This is particularly true for the upper third of the swash zone. Moreover, in contradistinction to swash flows over non-erodible beds, flows over erodible beaches are unique to the bed mobility and porosity under consideration; this has very important implications for run-up predictions. The conclusion is that it is essential to consider full coupling of water and bed motions (i.e. full morphodynamics) in order to understand and predict sediment transport in the swash, regardless of other physical effects (e.g. turbulence, infiltration, pre-suspended sediment, etc.).

SUMMARY A well‐balanced total variation diminishing–McCormack scheme is used to simulate the fast evolving flow on a mobile coarse sediments bed. The scheme is chosen because of its shock capturing capabilities and its relative simplicity, which allow different sediment transport formulae to be slotted in easily. A typical example of the kind of flows treated here is bore‐driven wave run‐up. The analogy with a dam‐break on a mobile bed is used here to analyze the performance of the model. The model solves the nonlinear shallow water equations coupled with the Exner sediment balance equation for the mobile bed. Quasi‐analytical solutions to this problem for different expressions for instantaneous sediment discharge formulae are used to test the performance of the scheme. Together with the existing solution for the Grass formula, a further solution is obtained for a different formula. Numerical tests were also carried out for a further formula that is an industry standard. The agreement of the results with the solutions is very good and consistent results were obtained for all the formulae tested. Copyright © 2011 John Wiley & Sons, Ltd.

This paper details a novel numerical approach for solution of the Navier-Stokes equations for free surface flows involving two-way fluid-solid interaction in arbitrary domains. The approach, which is hybrid Eulerian Lagrangian in nature, is based on the full particle particle-in-cell (PIC) method applied to incompressible flows. An extension of the distributed Lagrange multiplier (DLM) technique proposed by Patankar et al. [Int. J. Multiphase Flow, 26 (2000), pp. 1509-1524] is employed for the two-way fluid-solid coupling. The resulting code is called PICIN. Solid bodies can be mobile, either having prescribed motion or moving as a consequence of interaction with the fluid. Numerical results for three distinct example applications of the model in two spatial dimensions are given. A comparison of PICIN predictions with state-of-the-art numerical results of other researchers is made for each of the test cases presented.

In a recent paper Kelly et al. (2015) [PICIN: A Particle-In-Cell solver for incompressible free surface flows with two-way fluid-solid coupling. SIAM Journal on Scientific Computing 37 (3), B403-B424.] detailed the PICIN full particle Particle-In-Cell (PIC) solver for incompressible free-surface flows. The model described in that paper employed a tailored version of the Distributed Lagrange Multiplier (DLM) method for the strong coupling of fluid-solid interaction. In this paper we propose an alternative strong fluid-solid coupling algorithm based on a modification to the cut cell methodology that is informed by the variational approach. The solid velocity flux/integral on the boundary is expressed purely in terms of pressure leading to a revised pressure Poisson equation that is discretised in a finite volume sense. This approach allows the PICIN model to simulate the motion of floating bodies of arbitrary configuration. 2D test cases involving floating bodies with one or more degrees of freedom (DoF) are used to validate the modified PICIN model. The results presented show that the modified PICIN model is able to both efficiently and robustly predict the motions of surface-piercing floating structures under either regular or extreme wave action.