Multilayer methods for geophysical flows: modelling and numerical approximation.

Esta tesis se enmarca en el ámbito de la Matemática Aplicada y la Mecánica de Fluidos Computacional. Concretamente, aborda el modelado matemático y la simulación numérica de flujos geofísicos mediante modelos multicapa. Las contribuciones principales se encuentran en los Capítulos 2, 3 y 4. En el Capítulo 1 se revisa brevemente la aproximación multicapa para las ecuaciones de Navier-Stokes con viscosidad constante, así como el procedimiento para obtener un modelo multicapa. Las avalanchas granulares se han estudiado principalmente mediante modelos integrados. Sin embargo, esos modelos no reproducen variaciones en tiempo de los per les de velocidad. En el Capítulo 2 se presenta un modelo multicapa para avalanchas granulares secas considerando una viscosidad variable de nida por la ley constitutiva (I). En este modelo no se prescribe el per l normal de velocidad horizontal, lo que permite reproducir fuertes cambios en tiempo de estos per les. En el Capítulo 3 se extiende el modelo multicapa anterior al caso de una masa granular con nada en un canal rectangular, para lo que se añade un nuevo término de fricción en las paredes laterales. Se presenta también un esquema numérico bien equilibrado para este modelo, con un tratamiento espec co de los términos correspondientes a la fricción y la reologa. Se muestra que el término de fricción lateral modi ca signi cativamente la evolución de la avalancha. En particular, altera completamente el per l vertical de velocidad, dando lugar a zonas donde el material queda estático bajo una capa superior que se mueve. As mismo, se prueba que incluir el término de fricción lateral en modelos integrados de una capa puede dar lugar a soluciones carentes de sentido desde el punto de vista físico. En el Capítulo 4 se presenta una discretización semi-implícita en tiempo para modelos multicapa, para los que se obtiene una condición CFL menos restrictiva en el caso de un flujo subcrítico, lo que permite reducir notablemente el coste computacional. La descripción multicapa propuesta es novedosa, ya que el número de capas verticales puede cambiar a lo largo del dominio computacional, sin una pérdida de precisión relevante. Estas técnicas se aplican a problemas de flujos oceánicos y de transporte de sedimento

Flexible and efficient discretizations of multilayer models with variable density

We show that the semi-implicit time discretization approaches previously introduced for multilayer shallow water models for the barotropic case can be also applied to the variable density case with Boussinesq approximation. Furthermore, also for the variable density equations, a variable number of layers can be used, so as to achieve greater flexibility and efficiency of the resulting multilayer approach. An analysis of the linearized system, which allows to derive linear stability parameters in simple configurations, and the resulting spatially semi-discretized equations are presented. A number of numerical experiments demonstrate the effectiveness of the proposed approach

Layer-averaged approximation of Navier-Stokes system with complex rheologies

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In this work, we present a family of layer-averaged models for the Navier–Stokes equations. For its derivation, we consider a layerwise linear vertical profile for the horizontal velocity component. As a particular case, we also obtain layer-averaged models with the common layerwise constant approximation of the horizontal velocity. The approximation of the derivatives of the velocity components is set by following the theory of distributions to account for the discontinuities at the internal interfaces. Several models has been proposed, depending on the order of approximation of an asymptotic analysis respect to the shallowness parameter. Then, we obtain a hydrostatic model with vertical viscous effects, a hydrostatic model where the pressure depends on the stress tensor, and fully non-hydrostatic models, with a complex rheology. It is remarkable that the proposed models generalize plenty of previous models in the literature. Furthermore, all of them satisfy an exact dissipative energy balance. We also propose a model that is second-order accurate in the vertical direction thanks to a correction of the shear stress approximation. Finally, we show how effective the layerwise linear approach is to notably improve, with respect to the layerwise constant method, the approximation of the velocity profile for some geophysical flows. Namely, a Newtonian fluid and some complex viscoplastic (dry granular and Herschel–Bulkley) materials are considered

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

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

A weakly non-hydrostatic shallow model for dry granular flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope

Multilayer models for hydrostatic Herschel-Bulkley viscoplastic flows

This is an open access article under the CC BY-NC-ND licenseStarting from Navier-Stokes’ equation we derive two shallow water multilayer models for yield stress fluids, depending on the asymptotic analysis. One of them takes into account the normal stress contributions, making possible to recover a pseudoplug layer instead of a purely plug zone. A specific numerical scheme is designed to solve this model thanks to a finite volume discretization. It involves well-balancing techniques to be able to compute accurately the transitions between yielded and unyielded (or pseudoplug) zones, an important feature of the original partial differential equations’ model. We perform numerical simulations on various test cases relevant to these physics: analytical solution of a uniform flow, steady solutions for arrested state, and a viscoplastic dam break. Simulations agree well when we perform comparisons with physical experiments of the group of Christophe Ancey (EPFL) and we make a comparative study including shallow water models and lubrication models that they present in Ancey et al. (2012) [3]. Thanks to the multilayer structure of our model, we can go further on the description of the vertical structure associated to the (bottom) sheared layer and the top (pseudo-)plug layer

Shallow Water Moment models for bedload transport problems

In this work a simple but accurate shallow model for bedload sediment transport is proposed. The model is based on applying the moment approach to the Shallow Water Exner model, making it possible to recover the vertical structure of the flow. This approach allows us to obtain a better approximation of the fluid velocity close to the bottom, which is the relevant velocity for the sediment transport. A general Shallow Water Exner moment model allowing for polynomial velocity profiles of arbitrary order is obtained. A regularization ensures hyperbolicity and easy computation of the eigenvalues. The system is solved by means of an adapted IFCP scheme proposed here. The improvement of this IFCP type scheme is based on the approximation of the eigenvalue associated to the sediment transport. Numerical tests are presented which deal with large and short time scales. The proposed model allows to obtain the vertical structure of the fluid, which results in a better description on the bedload transport of the sediment layer

A general vertical decomposition of Euler equations: Multilayer-moment models

In this work, we present a general framework for vertical discretizations of Euler equations. It generalizes the usual moment and multilayer models and allows to obtain a family of multilayer-moment models. It considers a multilayer-type discretization where the layerwise velocity is a polynomial of arbitrary degree N on the vertical variable. The contribution of this work is twofold. First, we compare the multilayer and moment models in their usual formulation, pointing out some advantages/disadvantages of each approach. Second, a family of multilayer-moment models is proposed. As particular interesting case we shall consider a multilayer-moment model with layerwise linear horizontal velocity. Several numerical tests are presented, devoted to the comparison of multilayer and moment methods, and also showing that the proposed method with layerwise linear velocity allows us to obtain second order accuracy in the vertical direction. We show as well that the proposed approach allows to correctly represent the vertical structure of the solutions of the hydrostatic Euler equations. Moreover, the measured efficiency shows that in many situations, the proposed multilayer-moment model needs just a few layers to improve the results of the usual multilayer model with a high number of vertical layers

Discussion on different numerical treatments on the loss of hyperbolicity for the two-layer shallow water system

This is an open access article under the CC BY-NC licenseThis paper focus on the numerical approximation of two-layer shallow water system. First, a new approximation of the eigenvalues of the system is proposed, which satisfies some interesting properties. From this approximation, we give an accurate estimation of the non-hyperbolic region, which improves significantly the one computed with the classic eigenvalues approximation. In particular, we estimate both the lower and upper boundaries of the non-hyperbolic region. We also give a simple algorithm that allows us to compute bounds for the external eigenvalues, even when complex eigenvalues arise. We design an efficient FV solver depending on the hyperbolic nature of the system, that combines the PVM-2U-FL scheme and a new solver introduced in this work, named IFCP-FL, which degenerates to the Lax–Wendroff method in smooth areas. Different strategies for the numerical treatment for the loss of hyperbolicity are considered and discussed. Two of them are based on adding a friction term, depending on the classic or the new eigenvalues approximations, and the third one considers a simplified hyperbolic model in areas with complex eigenvalues. Some numerical tests are performed, including the case of two-layer fluids with different ratio of densities. The application to a two-layer model for submarine landslides is also considered. In the latter case, we show how the treatments based on adding friction are not appropriate for this kind of applications, whereas the treatment based on changing to a hyperbolic model produces much better results.Junta de AndalucíaMinisterio de Ciencia e InnovaciónUnión Europe