Depth-averaged unsteady RANS simulation of resonant shallow flows in lateral cavities using augmented WENO-ADER schemes

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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.