Multi–dimensional shear shallow water flows : problems and solutionsReport as inadecuate

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1 AMU - Aix Marseille Université 2 IUSTI - Institut universitaire des systèmes thermiques industriels

Abstract : The mathematical model of shear shallow water flows of uniform density is studied. This is a 2D hyperbolic non-conservative system of equations which is reminiscent of a generic Reynolds-averaged model of barotropic turbulent flows. The model has three families of characteristics corresponding to the propagation of surface waves, shear waves and average flow contact characteristics. The system is non-conservative : for six unknowns the fluid depth, two components of the depth averaged horizontal velocity, and three independent components of the symmetric Reynolds stress tensor one has only five conservation laws conservation of mass, momentum, energy and mathematical -entropy-. A splitting procedure for solving such a system is proposed allowing us to define a weak solution. Each split subsystem contains only one family of waves either surface or shear waves and contact characteristics. The accuracy of such an approach is tested on exact 2D solutions describing the flows where the velocity is linear with respect to the space variables, and 1D solutions. The capacity of the model to describe the full transition observed in the formation of roll waves : from uniform flow to one-dimensional roll waves, and, finally, to 2D transverse -fingering- of roll wave profiles is shown.

Keywords : roll waves Godunov–type methods Non-conservative hyperbolic equations

Author: Sergey Gavrilyuk - Kseniya Ivanova - Nicolas Favrie -



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