The aim of this work is the application of a 2D depth-averaged URANS simulation model to study the resonant coupling between standing gravity waves and vortex shedding in shallow water cavity flows. The URANS turbulence modelling approach is adopted for this purpose thanks to the particularities of these turbulent shallow flows: they are characterized by large scale horizontal eddies (2D), which coexist with small-scale 3D turbulence. The proposed model uses a high-resolution WENO-ADER numerical scheme, which allows to accurately resolve the large 2D eddies with a low numerical diffusion and dispersion. The small-scale 3D turbulence cannot be resolved by the 2D depth-averaged URANS, hence it is modelled using a closure relation. Experimental data recorded in straight channels with lateral cavities is considered as benchmark. The ability of the model to predict the resonant gravity waves observed in the experimental setup is assessed. Furthermore, the numerical results are not only analyzed in the physical domain (i.e. space and time) but also in the frequency domain. The study of the spectral characteristics of the time evolution of the water surface and vorticity allows to obtain 2D maps of the seiche amplitude and coupling intensity.

Turbulent shallow flows are characterized by the presence of horizontal large-scale vortices, caused by local variations of the velocity field. Apart from these 2D large vortices, small scale 3D turbulence, mainly produced by the interaction of the flowing water with the solid boundaries, is also present. The energy spectrum of turbulent shallow flows shows the presence of a 2D energy cascade at low wave numbers and a 3D energy cascade at high wave numbers, with a well-defined separation region between them. Horizontal flow movements (e.g. 2D large-scale vortical structures) at low wave numbers mostly determine the hydrodynamic behavior of these flows. Moreover, the generation of standing waves often occurs closely associated to the interaction of 2D horizontal flows with lateral boundaries, this is the case of seiches. To adequately reproduce these phenomena, a mathematical and numerical model able to resolve 2D turbulence is required. We herein show that depth-averaged (DA) unsteady Reynolds averaged Navier Stokes (URANS) models based on the Shallow Water Equations (SWE) are a suitable choice for the resolution of turbulent shallow flows with sufficient accuracy in an affordable computational time. The 3D small-scale vortices are modeled by means of diffusion terms, whereas the 2D large-scales are resolved. A high order numerical scheme is required for the resolution of 2D large eddies. In this work, we design a DA-URANS model based on a high order augmented WENO-ADER scheme. The mathematical model and numerical scheme are validated against observation of complex experiments in an open channel with lateral cavities that involve the presence of resonant phenomena (seiching). The numerical results evidence that the model accurately reproduces both longitudinal and transversal resonant waves and provides an accurate description of the flow field. The high order WENO-ADER scheme combined with a SWE model allows to obtain a powerful, reliable and efficient URANS simulation tool.