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 Reynoldsaveraged 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.

In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

We are interested in the modelling of multi-dimensional turbulent hydraulic jumps in convergent radial flow. To describe the formation of intensive eddies (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 dissipative source terms. The structure of equations is reminiscent of generic Reynolds-averaged Euler equations for barotropic compressible turbulent flows. Two types of dissipative term are studied. The first one corresponds to a 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).