J. Fluid Mech. (2015), vol. 775, pp. 328–348. c  Cambridge University Press 2015 doi:10.1017/jfm.2015.308 328 A simple model of wave–current interaction Nicoletta Tambroni 1 , Paolo Blondeaux 1, † and Giovanna Vittori 1 1 Department of Civil, Chemical and Environmental Engineering, University of Genoa, Via Montallegro 1, 16145 Genova, Italy (Received 4 September 2014; revised 24 February 2015; accepted 27 May 2015; first published online 23 June 2015) The interaction between a steady current and propagating surface waves is investigated by means of a perturbation approach, which assumes small values of the wave steepness and considers current velocities of the same order of magnitude as the amplitude of the velocity oscillations induced by wave propagation. The problems, which are obtained at the different orders of approximation, are characterized by a further parameter which is the ratio between the thickness of the bottom boundary layer and the length of the waves and turns out to be even smaller than the wave steepness. However, the solution is determined from the bottom up to the free surface, without the need to split the fluid domain into a core region and viscous boundary layers. Moreover, the procedure, which is employed to solve the problems at the different orders of approximation, reduces them to one-dimensional problems. Therefore, the solution for arbitrary angles between the direction of the steady current and that of wave propagation can be easily obtained. The theoretical results are compared with experimental measurements; the fair agreement found between the model results and the laboratory measurements supports the model findings. Key words: coastal engineering, surface gravity waves, waves/free-surface flows 1. Introduction In many shallow coastal environments, sediment transport is often due to the combined action of surface waves and steady currents. The oscillatory flow induced by wave propagation contributes substantially to the pick-up of sediment from the bottom. Then, even though a net flux of sediment can also be induced by the waves because of the steady streaming effects and the possible asymmetry of the velocity oscillations, the largest contribution to the sediment transport is possibly induced by the currents, which steadily drag the sediment in their direction. In the modelling of the sediment transport a major role is played by accurate evaluation of the bottom shear stress and flow in the bottom boundary layer (Blondeaux & Vittori 1999). Hence, a large number of studies have been devoted to the study of flow in the boundary layer generated close to the bottom by the interaction of a wave train and a steady current. Attention has been focused on the turbulent regime, and turbulence models of different complexity have been used. Some of them are described in the review paper by Soulsby et al. (1993), who † Email address for correspondence: blx@dicat.unige.it https:/www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2015.308 Downloaded from https:/www.cambridge.org/core. Open University Library, on 05 Feb 2017 at 07:33:41, subject to the Cambridge Core terms of use, available at