Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel

Abstract : The formation of a periodic roll-wave train in a long channel is studied for two sets of experimental parameters (noted as Case 1 and Case 2) corresponding to Brock's experiments [3], [4]. In both cases, a formed free surface profile was found in a very good agreement with the experimental results. Mathematical properties of the model were also tested in the case where the perturbation frequency was lower than the experimental one, so longer waves were generated at the channel inlet. It was observed that the amplitude and the enstrophy of the corresponding roll-waves train are strongly modulated. In the case where the waves of two different lengths were generated at the channel inlet, the coarsening was observed. The coarsening phenomenon is always accompanied by a strong modulation. A comparison with the Saint-Vennat equations is also done. The formation of a single wave composing a roll-wave train was also studied in a domain with periodic boundary conditions (called " periodic box ") for the same sets of experimental parameters. The free surface profile was found also in a very good agreement with the experimental results. This allows us to justify the use of the " periodic box " as a simple mathematical tool for a qualitative study of roll-waves stability. In particular, we studied the stability of a single steady wave by increasing its length. It was shown that the wave becomes morphologically unstable after some critical wave length : it transforms into a system containing several waves. It is also proved that a single steady wave corresponding to Case 1 is stable under multi-dimensional perturbations in the framework of a model which is a simplification of a general multi-D model of shear shallow water flows.
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Kseniya Ivanova, Sergey Gavrilyuk, Boniface Nkonga, Gael Richard. Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel. Computers and Fluids, Elsevier, 2017, 159, pp.189-203. ⟨hal-01527469⟩

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