## Section: New Results

### Free surface geophysical flows

Participants : Emmanuel Audusse [ Université Paris 13, Institut Galilée ] , Marie-Odile Bristeau, Marica Pelanti, Benoît Perthame, Jacques Sainte-Marie [ Saint-Venant Laboratory-CETMEF and MACS project-team ] .

We are involved in research concerning the numerical simulation of free surface geophysical flows such as rivers, lakes, coastal areas and also overland flows. Many applications related to environmental problems are concerned : floodings, dam breaks, swell, transport and diffusion of pollutants, water quality, upwellings, sustainability of aquatic ecosystems, ...

The basic model for these problems is the 3D free surface Navier-Stokes system leading to a 3D solver [42] with a moving mesh. However for efficiency reasons, vertically averaged models such as the Saint-Venant system [49] are often used.

We have developed extensions of the Saint-Venant system where the basic Saint-Venant solver [38] is still used and, in that way, the robustness, the efficiency and the easiness to treat the free surface are preserved while the domain of validity is larger. These extensions are derived from the free surface Navier-Stokes equations:

- 1D section-averaged Saint-Venant model,

- Multilayer Saint-Venant model with mass exchanges,

- Multilayer Saint-Venant system with varying density,

- Vertically averaged models for the free surface Euler system.

The Multilayer Saint-Venant model with varying density is compared with the Navier-Stokes solver “Ophélie” developed at EDF/LNHE and generalized by M. Pelanti.

One of the applications of the Saint-Venant solvers concerns overland flows.

#### 1D section-averaged Saint-Venant model

Even if efficient 2D Saint-Venant solvers are available, in many studies of natural rivers hydraulics on large domains, the 1D Saint-Venant equations are used. However for these rivers, it is important to take into account the width variations. So the section-averaged Saint-Venant system is derived asymptotically from the 3D incompressible Navier-Stokes system for free surface flows and some additional source terms are induced by the cross section shape.

We have proposed a kinetic interpretation
of this section-averaged Saint-Venant system and derived an associated numerical scheme.
The numerical treatment of the source terms related to the cross section has to preserve
the equilibrium of the “lake at rest”, it is
based on an extension of the hydrostatic reconstruction technique [37]
developed for the topography source term. So the numerical scheme -formally 2^{nd} order
in space and time- is stable and well-balanced. We also have studied the computation
of the friction term either by a centered implicit scheme or using the apparent topography
approach. The accuracy of the proposed method has been
proved by comparison with analytic solutions and also with experimental measures.

This work [22] has been done in collaboration with N. Goutal (EDF/LNHE).

#### Multilayer Saint-Venant system with mass exchanges

Considering flows with large friction coefficients, with significant water depth or with important wind effects, the horizontal velocity can hardly be approximated – as in the Saint-Venant system – by a vertically constant velocity.

To drop this limitation, a first multilayer Saint-Venant model has been introduced in [39] where the interfaces are advected by the flow and so there is no mass exchange between the layers. A new multilayer approximation has been proposed that allows the fluid to circulate from one layer to the connected ones. The total water height is divided at each time step in a given distribution, then there is only one continuity equation for the total height and a momentum equation for each layer.

We have studied the derivation
of this model, its main properties (energy equality, kinetic interpretation,...) and proved its validity through numerical simulations [4] .
The basic tool remains the
Saint-Venant solver [38] with a kinetic
scheme and an hydrostatic reconstruction [37] to take into account
the bottom topography. A formally 2^{nd} order extension has been done for the 1D code
and has to be done in the 2D code.

These multilayer models give a precise description of the vertical profile of the horizontal velocity while preserving the computational efficiency of the classical Saint-Venant system. The interest of the second approach is that it allows to simulate recirculating area as, for instance, the effect of the wind on a lake. We are now validating this application by comparison with the Navier-Stokes solver “Mistral” developed at Cermics.

This multilayer solver is also the basic tool for the system with varying density presented in the following section.

#### Multilayer Saint-Venant system with varying density. Comparison with a Navier-Stokes solver.

We are now considering a free surface flow with a varying density (related to salinity or temperature) in order to simulate stratifications and upwelling phenomena.

We start with the free surface hydrostatic Euler equations and a transport equation for a tracer T , typically salinity or temperature in the applications considered. The density is given by = (T) (this means that the flow remains incompressible).

The vertical
discretization of the horizontal velocity u and tracer T consists
in a Galerkin approximation (P_{0} type) in Lagrangian
formulation. After dividing the flow domain in a given number of layers,
the vertically discretized system is obtained by integrating the continuous equations
on each layer. Then, using the Leibnitz rule, we introduce the averaged values of the
variables and the exchange terms.

A kinetic interpretation of this system is also proposed leading to a numerical scheme. Due to the density coupling, non linear systems have to be solved on each water column. This model is well suited to simulate stratified flows and upwellings (see Fig.4 ), it is an important milestone in view of the coupling with bio-dynamics.

Besides this work on the multilayer system, we are working on the generalization of the code “Ophélie” initially developed at EDF/LNHE. This code is based on the hydrostatic Navier-Stokes solver with the Boussinesq approximation for the density variations. For the free surface, the rigid lid assumption is done.

This study is done in collaboration with M.J. Salençon (EDF/LNHE) and is the object of a postdoctoral grant.

#### Vertically averaged models for the free surface Euler system

To deal with small amplitude waves (swell, waves induced by a rapid opening or closing of a gate,...), the Saint-Venant system is not sufficient, actually these equations rely on the assumption that the vertical velocities are negligible and the resulting pressure is hydrostatic.

To improve the model, from the free surface Navier-Stokes system, a first non-hydrostatic Saint-Venant system (pressure depends of the vertical acceleration) including friction and viscosity has been derived [41] . The interest of this model has been proved by comparison of numerical results with experiments.

Then we have considered the full Euler system with free surface (keeping all the terms related to vertical velocity). In a first step, we use the shallow water assumption and the associated small parameter that is the ratio between two typical characteristic lengths (horizontal and vertical) of the fluid domain. We obtain a single layer averaged system approximating the Euler system up to terms. Secondly, we drop the assumption and we decompose the water height H into N layers with N possibly large. This means we have shallow layers instead of a global shallow water assumption. Using this multilayer approach, we derive a system approximating the Euler system up to terms. For the proposed model, we also give a kinetic type interpretation based on a Boltzmann transport coupled with a reaction term.

From this complex kinetic interpretation, a numerical scheme should be deduced leading to a free surface Euler solver for deep flows without a moving mesh to deal with.

#### Overland flows

Overland flows on agricultural soils induce problems of environmental resources preservation (decrease of soil thickness by erosion, nutrients losses, decrease in water quality). To improve watershed management, a good prediction of the surface flow network is needed.

For agricultural areas, empirical works showed that the interaction between furrows and topography strongly controls the geometry of the flow network: at low flux, overland flow follows the furrow direction, while, at high flux, overland flow follows the topographic slope too. We intend to model this type of flow in order to better understand and predict the effect of surface morphology on overland flow.

As a first step, we have compared numerical simulations with different friction terms (Darcy-Weisbach, Manning) to laboratory measurements. This example which has been developed at Inra (Orléans) concerns the overland flow due to some rain on a sloping channel with different roughnesses.

This work is a participation in the ANR project “METHODE” (url http://methode.netcipia.net ).