A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation

The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighborhing layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.Comment: Submitted to M2A

An energy-consistent depth-averaged Euler system: derivation and properties

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by aminimal energy constraint instead of an asymptotic expansion. The model slightly differs from thewell-known Green-Naghdi model and is confronted with stationary andanalytical solutions of the Euler system corresponding to rotationalflows. At the end of the paper, we givetime-dependent analytical solutions for the Euler system that are alsoanalytical solutions for the proposed model but that are not solutionsof the Green-Naghdi model. We also give and compare analytical solutions of thetwo non-hydrostatic shallow water models

Layer-averaged Euler and Navier-Stokes equations

In this paper we propose a strategy to approximate incompressible hydrostatic free surface Euler and Navier-Stokes models. The main advantage of the proposed models is that the water depth is a dynamical variable of the system and hence the model is formulated over a fixed domain.The proposed strategy extends previous works approximating the Euler and Navier-Stokes systems using a multilayer description. Here, the needed closure relations are obtained using an energy-based optimality criterion instead of an asymptotic expansion. Moreover, the layer-averaged description is successfully applied to the Navier-Stokes system with a general form of the Cauchy stress tensor

A 2D model for hydrodynamics and biology coupling applied to algae growth simulations

Cultivating oleaginous microalgae in specific culturing devices such as raceways is seen as a future way to produce biofuel. The complexity of this process coupling non linear biological activity to hydrodynamics makes the optimization problem very delicate. The large amount of parameters to be taken into account paves the way for a useful mathematical modeling. Due to the heterogeneity of raceways along the depth dimension regarding temperature, light intensity or nutrients availability, we adopt a multilayer approach for hydrodynamics and biology. For free surface hydrodynamics, we use a multilayer Saint-Venant model that allows mass exchanges, forced by a simplified representation of the paddlewheel. Then, starting from an improved Droop model that includes light effect on algae growth, we derive a similar multilayer system for the biological part. A kinetic interpretation of the whole system results in an efficient numerical scheme. We show through numerical simulations in two dimensions that our approach is capable of discriminating between situations of mixed water or calm and heterogeneous pond. Moreover, we exhibit that a posteriori treatment of our velocity fields can provide lagrangian trajectories which are of great interest to assess the actual light pattern perceived by the algal cells and therefore understand its impact on the photosynthesis process.Comment: 27 pages, 11 figure

Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems

From the free surface Navier-Stokes system, we derive the non-hydrostatic Saint-Venant system for the shallow waters including friction and viscosity. The derivation leads to two formulations of growing complexity depending on the level of approximation chosen for the fluid pressure. The obtained models are compared with the Boussinesq models

Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system

International audienceA lot of well-balanced schemes have been proposed for discretizing the classical Saint-Venant system for shallow water flows with non-flat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example Godunov, Roe, kinetic,...). If this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semi-discrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We prove also that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term

A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows

We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

Boundary Conditions for the Shallow Water Equations solved by Kinetic Schemes

Projet M3NWe consider the Saint-Venant system for Shallow Water which is an usual model to describe the flows in rivers, coastal areas or floodings. The hyperbolic system of conservation laws is solved on unstructured meshes using a finite volume method together with a kinetic solver.We add to this system a friction term, the role of which is important when small water depths are considered. In this paper we address the treatment of the boundary conditions, the difficulty is due to the fact that in some cases (fluvial flows) the given boundary conditions are not sufficient to apply directly the scheme, we discuss here how to treat these boundary conditions using a Riemann invariant.Some numerical results illustrate the ability of the method to treat complex problems like the filling up or the draining off of a river bed