Numerical ability of hyperbolic flux solvers to compute 2D shear layers in turbulent shallow flows

Abstract

Current computational power allows the modelling of complex time dependent flows. In particular, shallow water flows are frequently simulated to achieve solutions for different problems in the field of civil and environmental engineering. The mesh resolution and related computational cost are primarily associated to the scale of the flow structures to be investigated. Nonetheless, turbulence also plays an important role on computational cost since it participates in the generation, shedding and support of such flow structures. Different mathematical models can be considered to numerically simulate the turbulence of unsteady shallow flows depending on the degree of turbulent scale resolution required. Depending on the adopted approach, the level of accuracy required (i.e. the range of scales that must be resolved with a low diffusive and dispersive error) is different. Such accuracy, namely the dispersive and dissipative characteristic, is directly related with the numerical scheme used to discretize the equations. In finite volume schemes, the range of scales of turbulent motion that a numerical model can accurately resolve strongly depends on the Riemann solver used (via its intrinsic numerical diffusion), apart from the order of accuracy and degrees of freedom of the method. In this work, we aim at the analysis of two well-known Riemann solvers in the framework of the classical shallow water equations (i.e. considering the full convective terms and neglecting dissipation): the ARoe and HLLS solvers. An important difference between the ARoe and HLLS solvers is the numerical diffusion inherent to each of them. This artificial diffusion combined with the mesh resolution determine the cut-off scale resolved by each numerical technique. For this purpose, we assess the suitability of each solver by means of the analysis of the kinetic energy cascade of the numerical solution using a double shear layer configuration. This analysis is combined with the examination of the analytical expression of he approximate solution for a shear wave, provided by the aforementioned solvers. The study herein presented allows to assess whether or not all the relevant turbulent flow structures are resolved and if the phenomenon of interest is thus accurately modeled. The numerical results evidence that a diffusive profile appears at the shear line during the first steps of the simulation, determining the duration of the linear regime prior to the turbulent motion. The strength of this profile, shown to be higher for the HLLS solver, is associated to the numerical diffusion of the solver. The analysis of the energy cascade also agrees with this observation.