Search

A long-wave approximation that describes travelling solitary waves within the framework of the model of a weakly stratified two-layer fluid is considered. It is demonstrated that wave regimes occur near the boundary of the parametric domain of shear instability in the stratified flow. This property offers an explanation for the mechanism of intense mixing in deep bottom layers.

The mathematical model of shear shallow water flows of constant density is studied. This is a 2D hyperbolic non-conservative system of equations that is mathematically equivalent to the 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 2D analytical solutions describing the flow with linear with respect to the space variables velocity, and on the solutions describing 1D roll waves. The capacity of the model to describe the full transition scenario as commonly seen in the formation of roll waves: from uniform flow to 1D roll waves, and, finally, to 2D transverse ‘fingering’ of the wave profiles, is shown.

International audience; We are interested in modelling of multidimensional turbulent hydraulic jumps in convergent radial flow. To describe the formation of intensive eddies near the free surface (rollers) at the front of the hydraulic jump, a new model of shear shallow water flows is used. The governing equations form a non–conservative hyperbolic system with dissipation source terms. The structure of equations is reminiscent of generic Reynolds-averaged Euler equations for barotropic compressible turbulent flows. Two types of dissipative terms are studied. The first one corresponds to Chézy-like dissipation rate, and the second one to a standard energy dissipation rate commonly used in compressible turbulence. Both of them guarantee the positive definiteness of the Reynolds stress tensor. The equations are rewritten in polar coordinates and numerically solved by using an original splitting procedure. Numerical results for both types of dissipation are presented and qualitatively compared with the experimental works. The results show both experimentally observed phenomena (cusp formation at the front of the hydraulic jump) as well as new flow patterns (the shape of the hydraulic jump becomes a rotating square).

The formation of a periodic roll-wave train in a long channel is studied for two sets of experimental parameters corresponding to Brock’s experiments [3], [4] who measured permanent wave profiles by introducing periodic perturbations at the channel inlet. In both cases, a formed free surface profile is found in good agreement with the experimental results. Qualitative properties of solutions to the model are studied in the case where the perturbation frequency is lower than the experimental one, so longer waves are generated at the channel inlet. It is observed that the corresponding roll-wave train is strongly modulated. In the case where the waves of two different lengths are generated at the channel inlet, the coarsening is observed, i.e. the process where shorter waves disappear progressively by transferring their energy to longer waves forming later a regular roll-wave train. The coarsening phenomenon is always accompanied by a strong modulation. A comparison with the Saint-Venant equations is also performed. The formation of a single wave composing a roll-wave train is also studied in a domain with periodic boundary conditions (called “periodic box”) for the same sets of experimental parameters. The free surface profile is found also in 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 stability of roll-waves. In particular, we study the stability of a single steady wave by increasing its length. It is shown that the wave becomes morphologically unstable after some critical wave length: it bifurcates into a system of two waves. In the framework of a simplified multi-D model of shear shallow water flows it is also proved that a single steady wave is stable under multi-dimensional perturbations.