Gladys Narbona-Reina

Extension of WAF Type Methods to Non-Homogeneous Shallow Water Equations with Pollutant

This paper deals with the extension of the WAF method to discretize Shallow Water Equations with pollutants. We consider two different versions of the WAF method, by approximating the intermediate waves using the flux of HLL or the direct approach of HLLC solver. It is seen that both versions can be written under the same form with different definitions for the approximation of the velocity waves. We also propose an extension of the method to non-homogeneous systems. In the case of homogeneous systems it is seen that we can rewrite the third component of the numerical flux in terms of an intermediate wave speed approximation. We conclude that—in order to have the same relation for non-homogeneous systems—the approximation of the intermediate wave speed must be modified. The proposed extension of the WAF method preserves all stationary solutions, up to second order accuracy, and water at rest in an exact way, even with arbitrary pollutant concentration. Finally, we perform several numerical tests, by comparing it with HLLC solver, reference solutions and analytical solutions.

A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model

We are interested in the derivation of an integrated Herschel-Bulkley model for shallow flows, as well as in the design of a numerical algorithm to solve the resulting equations. The goal is to simulate the evolution of thin sheet of viscoplastic materials on inclined planes and, in particular, to be able to compute the evolution from dynamic to stationary states. The model involves a variational inequality and it is valid from null to moderate slopes. The proposed numerical scheme is well balanced and involves a coupling between a duality technique (to treat plasticity), a fixed point method (to handle the power law) and a finite volume discretization. Several numerical tests are done, including a comparison with an analytical solution, to confirm the well balanced property and the ability to cope with the various rheological regimes associated with the Herschel-Bulkley constitutive law.

A second order PVM flux limiter method. Application to magnetohydrodynamics and shallow stratified flows

In this work we propose a second order flux limiter finite volume method, named PVM-2U-FL, that only uses information of the two external waves of the hyperbolic system. This method could be seen as a natural extension of the well known WAF method introduced by E.F. Toro in [23]. We prove that independently of the number of unknowns of the 1D system, it recovers the second order accuracy at regular zones, while in presence of discontinuities, the scheme degenerates to PVM-2U method, which can be seen as an improvement of the HLL method (see [6,10]). Another interesting property of the method is that it does not need any spectral decomposition of the Jacobian or Roe matrix associated to the flux function. Therefore, it can be easily applied to systems with a large number of unknowns or in situations where no analytical expression of the eigenvalues or eigenvectors are known. In this work, we apply the proposed method to magnetohydrodynamics and to stratified multilayer flows. Comparison with the two-waves WAF and HLL-MUSCL methods are also presented. The numerical results show that PVM-2U-FL is the most efficient and accurate among them.

On a Shallow Water Model for Non-Newtonian Fluids

The aim of this work is to modelize the evolution of a viscoelastic fluid through a Shallow Water system. The fluid hydrodynamic in this situation comes from the Navier-Stokes equations but the difficulty lies in the definition of the stress tensor for this non-Newtonian fluid. In order to get an expression for it we focus on the microscopic properties of the fluid by considering a diluted solution of polymer liquids. A kinetic theory for this type of solutions gives us “constitutive equations” that relate the stress tensor to the velocity. They are known as the Fokker–Planck equations. Once the stress tensor is defined we shall derive the model by developing the asymptotic analysis of the joined system of equations to obtain a Shallow Water type model following [6]. Finally we show a numerical test to check the influence of the polymers in the behavior of the flow.

Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy

In this work we present a deduction of the Saint-Venant–Exner model through an asymptotic analysis of the Navier–Stokes equations. A multi-scale analysis is performed in order to take into account that the velocity of the sediment layer is smaller than the one of the fluid layer. This leads us to consider a shallow water type system for the fluid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classic Saint-Venant–Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classic models do not have an associated energy and how they can be modified in order to recover a model with this property. (ii) The model incorporates naturally a necessary modification that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that in general this modification is different from the ones considered classically. Nevertheless, it coincides with a classic one in the case of constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, which allows to ensure sediment mass conservation. Moreover, we include a simplified version of the model for the case of quasi-stationary regimes. Some of these simplified models correspond to a generalization of classic ones such as Meyer-Peter and Muller and Ashida–Michiue models. Three numerical tests are presented to study the evolution of a dune for several definition of the repose angle, to see the influence of the proposed definition of the effective shear stress in comparison with the classic one, and by comparing with experimental data.

2D granular flows with the μ(I) rheology and side walls friction: A well-balanced multilayer discretization

Abstract We present here numerical modelling of granular flows with the μ ( I ) rheology in confined channels. The contribution is twofold: (i) a model to approximate the Navier–Stokes equations with the μ ( I ) rheology through an asymptotic analysis; under the hypothesis of a one-dimensional flow, this model takes into account side walls friction; (ii) a multilayer discretization following Fernandez-Nieto et al. (2016) [20] . In this new numerical scheme, we propose an appropriate treatment of the rheological terms through a hydrostatic reconstruction which allows this scheme to be well-balanced and therefore to deal with dry areas. Based on academic tests, we first evaluate the influence of the width of the channel on the normal profiles of the downslope velocity thanks to the multilayer approach that is intrinsically able to describe changes from Bagnold to S-shaped (and vice versa) velocity profiles. We also check the well-balanced property of the proposed numerical scheme. We show that approximating side walls friction using single-layer models may lead to strong errors. Secondly, we compare the numerical results with experimental data on granular collapses. We show that the proposed scheme allows us to qualitatively reproduce the deposit in the case of a rigid bed (i.e. dry area) and that the error made by replacing the dry area by a small layer of material may be large if this layer is not thin enough. The proposed model is also able to reproduce the time evolution of the free surface and of the flow/no-flow interface. In addition, it reproduces the effect of erosion for granular flows over initially static material lying on the bed. This is possible when using a variable friction coefficient μ ( I ) but not with a constant friction coefficient.

Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization

Abstract We propose an extension of the discretization approaches for multilayer shallow water models, aimed at making them more flexible and efficient for realistic applications to coastal flows. A novel discretization approach is proposed, in which the number of vertical layers and their distribution are allowed to change in different regions of the computational domain. Furthermore, semi-implicit schemes are employed for the time discretization, leading to a significant efficiency improvement for subcritical regimes. We show that, in the typical regimes in which the application of multilayer shallow water models is justified, the resulting discretization does not introduce any major spurious feature and allows again to reduce substantially the computational cost in areas with complex bathymetry. As an example of the potential of the proposed technique, an application to a sediment transport problem is presented, showing a remarkable improvement with respect to standard discretization approaches.

Application of the WAF Method to Shallow Water Equations with Pollutant and Non-Constant Bottom

In this work we perform the extension of the WAF method [3] to discretize non-homogeneous Shallow Water Equations with pollutant.

The mur neutralisant as an active thermal system: Saint Gobain tests (1931) versus CFD simulation (2015)

Abstract: At the same time as the initial development of air conditioning systems for indoor climate control in buildings were occurring in USA, Le Corbusier and Lyon made truly innovative proposals for different projects he was working on in Europe. These served to generate homogenous thermal environments and focused on the combined effect of his mur neutralisant and respiration exacte. The clearest example of their shortcomings is the City of Refuge in Paris (1930-33). Given the technological and economic mistrust towards these proposals, as it was impossible to execute these according to the original plan these were not pursued. CFD simulations carried out by our research team confirm that the mur neutralisant and respiration exacte for the City of Refuge in Paris would have functioned together if they had been executed following the original plans. The main aim of this paper is to confirm the validity of the mur neutralisant as an active thermal system for buildings. Firstly, the results of the tests carried out by the engineers of Saint Gobain are compared to the results of the CFD simulations. Based on the comparison of the results from the physical models tested in Saint Gobain laboratories and CFD energy model simulations, a possible calibration is proposed for CFD which might prompt the establishment of other operation hypotheses. Keywords: Le Corbusier; mur neutralisant; The City of Refuge; Active Facade System; Computational Fluid Dynamics (CFD); Numerical Simulation. DOI: http://dx.doi.org/10.4995/LC2015.2015.899

Numerical Approximation of Convection-Diffusion Problems Through the PSI Method and Characteristics Method

In this work we present some numerical methods for solving evolutive convection-diffusion problems. In order to obtain a physically admissible solution we search for monotone and accurate methods that are also non-linear due to the Godunov’s theorem. We will center in Fluctuation Splitting methods, [8], in particular in PSI scheme, and characteristic type methods, where a new Lagrangian method is proposed. Finally, a numerical test is presented to assess the performance of the numerical methods described in the present work.