## Dunes

Dunes and antidunes are extremely widespread bedforms which are important for in a large number of environmental and engineering problems. They interact very much with river navigation and with fluvial infrastructure, such as bridges or tunnel crossing under the river bed (a remarkable example of infrastructure threatened by bedforms is the tunnel under the Rio Paranà, in Argentina). Bedforms are also very important paleo climatic proxies (for instance, paleo-antidunes can be the clue of a high discharge rate). Finally, the hyporeic fluxes which arise inside the bedforms affect a number of key bio-geochemical processes that occur in river corridor. Dunes and antidunes have been investigated thoroughly over the last decades, and well established mathematical models have been developed. The main results have been to determine which kind of bed form will develop under given flow condition and sediment characteristics (through a linear stability analysis) and the amplitude of such bedforms (through a non-linear analysis). Basically, the aim of any morphodynamic study is to evaluate the evolution of an erodible bottom given the flow characteristics. It is therefore a key point to correctly modeling the fluid phase and the sediment-fluid interaction, which, in turn, influence the bed evolution. By now, very refined model are used, in which the fluid phase is modelled by a 2D rotational model, and the sediemnt transport by the Meyer-Peter-Mueller formulation. Anyway, several questions are still open, and our research is concentrated in developing new tecniques for the solution of the complicated 2D rotational model as well as the use of more advanced models of sediment transport.

Figure 1 |

On the research branch of the solution of the 2D rotational model of the fluid phase we have recently derived a novel Orr–Sommerfeld-like equation for gravity-driven turbulent open-channel flows over a granular erodible bed, and developed the linear stability analysis. This allowed us to compute and analyze the whole spectrum of eigenvalues and eigenvectors of the complete generalized eigenvalue problem. A key feature is that the fourth-order eigenvalue problem presents singular non-polynomial coefficients with non-homogenous Robin-type boundary conditions that involve first and second derivatives. Furthermore, the Exner condition is imposed at an internal point. In order to face these problems we have developed a numerical discretization of spectral type based on a single-domain Galerkin scheme. In order to manage the presence of singular oefficients, some properties of Jacobi polynomials have been carefully blended with umerical integration of Gauss–Legendre type. The results (reported in figure 1, where the grow rates of different morphological and hydrodynamical instabilities are reported as a function of the wavenumber and the Froude's number) show a positive agreement with the classical experimental data. The eigenfunctions allow two types of boundary layers to be distinguished, scaling, respectively, with the roughness height and the saltation layer for the bedload sediment transport.

Figure 2 |

On the research branch devoted to improve the bed forms understanding with advanced sediment transport modelling, we are facing morphodynamic problems by coupling the fluid dynamics and the bottom evolution through mechanistic approaches. The key point of such approach is to write a sediment transport model starting from a dynamic equilibrium balance of all the forces acting on the sediment grains (see figure 2), removing as much empiricism as possible. This allow us to have a clear picture of the different forces acting on the different sediment grains, and thus allowing us to elucidate how the different phenomena contribute in the inception, growth and migration of the bedforms.

Selected papers

A spectral approach for the stability analysis of turbulent open-channel flows over granular beds, Camporeale, C., Canuto, C. and Ridolfi, L., Theoretical and Computational Fluid Dynamics 26(1), 51-80 (2012) DOI: 10.1007/s00162-011-0223-0

A shallow-water theory of river bedforms in supercritical conditions. Physics of Fluids **24**, 094104 (2012) https://doi.org/10.1063/1.4753943