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Section: Research Program
Geophysical flows – modelling and analysis
Hazardous flows are complex physical phenomena that can hardly be represented by shallow water type systems of partial differential equations (PDEs). In this domain, the research program is devoted to the derivation and analysis of reduced complexity models – compared to the Navier-Stokes equations – but relaxing the shallow water assumptions. The main purpose is then to obtain models adapted to the physical phenomena at stake and eventually to simulate them by means of robust and efficient numerical techniques.
Even if the resulting models do not strictly belong to the family of hyperbolic systems, they exhibit hyperbolic features: the analysis and discretization techniques we intend to develop have connections with those used for hyperbolic conservation laws. It is worth noticing that the need for robust and efficient numerical procedures is reinforced by the smallness of dissipative effects in geophysical models which therefore generate singular solutions and instabilities.
More precisely, the derivation of the Saint-Venant system from the Navier-Stokes equations is based on two main approximations – valid because of the shallow water assumption – namely
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the horizontal fluid velocity is well approximated by its mean along the vertical direction,
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the pressure is hydrostatic or equivalently the vertical acceleration of the fluid can be neglected compared to the gravitational effects.
As a consequence the objective is to get rid of these two assumptions, one after the other, in order to obtain models accurately approximating the incompressible Euler or Navier-Stokes equations.
Multilayer approach
As for the first assumption, multi-layer systems were proposed describing the flow as a superposition of Saint-Venant type systems [21] , [25] , [26] . Even if this approach has provided interesting results, it implies to consider each layer as isolated from its neighbours and this is a strong limitation. That is why we proposed a slightly different approach [22] , [23] based on Galerkin type decomposition along the vertical axis of all variables and leading, both for the model and its discretization, to more accurate results.
A kinetic representation of our multilayer model allows to derive robust numerical schemes endowed with properties such as: consistency, conservativity, positivity, preservation of equilibria,...It is one of the major achievements of the team but it needs to be analyzed and extended in several directions namely:
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The convergence of the multilayer system towards the hydrostatic Euler system as the number of layers goes to infinity is a critical point. It is not fully satisfactory to have only formal estimates of the convergence and sharp estimates would enable to guess the optimal number of layers.
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The introduction of several source terms due for instance to Coriolis forces or extra terms from changes of coordinates seems necessary. Their inclusion should lead to substantial modifications of the numerical scheme.
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Its hyperbolicity has not yet been proved and conversely the possible loss of hyperbolicity cannot be characterized. Similarly, the hyperbolic feature is essential in the propagation and generation of waves.
Non-hydrostatic models
The hydrostatic assumption (ii) consists in neglecting the vertical acceleration of the fluid. It is considered valid for a large class of geophysical flows but is restrictive in various situations where the dispersive effects (like wave propagation) cannot be neglected. For instance, when a wave reaches the coast, bathymetry variations give a vertical acceleration to the fluid that strongly modifies the wave characteristics and especially its height.
When processing an asymptotic expansion (w.r.t. the aspect ratio for shallow water flows) into the Navier-Stokes equations, we obtain at the leading order the Saint-Venant system. Going one step further leads to a vertically averaged version of the Euler/Navier-Stokes equations integrating the non-hydrostatic terms. This model has several advantages:
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it admits an energy balance law (that is not the case for most of the models available in the literature),
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it reduces to the Saint-Venant system when the non-hydrostatic pressure term vanishes,
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it consists in a set of conservation laws with source terms,
The main challenge in the study of this model is the derivation of a robust and efficient numerical scheme endowed with properties such as: positivity, wet/dry interfaces treatment, consistency.
It has to be noticed that even if the non-hydrostatic model looks like an extension of the Saint-Venant system, most of the known techniques used in the hydrostatic case are not efficient as we recover strong difficulties encountered in incompressible fluid mechanics due to the extra pressure term. These difficulties are reinforced by the absence of viscous/dissipative terms.
It is important to point out that the modelling and efficient simulations of non-hydrostatic models allow to answer important and various questions such as: