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Recently, the authors presented two numerical studies for capturing the flow structure beneath water waves (Nachbin and Ribeiro-Junior 2014 , 3135-3153 (doi:10.3934/dcds.2014.34.3135); Ribeiro-Junior 2017 , 792-814 (doi:10.1017/jfm.2016.820)). Closed orbits for irrotational waves with an opposing current and stagnation points for rotational waves were some of the issues addressed. This paper summarizes the numerical strategies adopted for capturing the flow beneath irrotational and rotational water waves. It also presents new preliminary results for particle trajectories, due to irrotational waves, in the presence of a bottom topography.This article is part of the theme issue 'Nonlinear water waves'.