A two-phase shallow debris flow model with energy balance

This paper proposes a thin layer depth-averaged two-phase model provided by a dissipative energy balance to describe avalanches of solid-fluid mixtures. This model is derived from a 3D two-phase model based on the equations proposed by Jackson [The Dynamics of Fluidized Particles. Cambridges Monographs on Mechanics (2000)] which takes into account the force of buoyancy and the forces of interaction between the solid and fluid phases. Jackson’s model is based on mass and momentum conservation within the two phases, i.e. two vector and two scalar equations. This system has five unknowns: the solid volume fraction, the solid and fluid pressures and the solid and fluid velocities, i.e. three scalars and two vectors. As a result, an additional equation is necessary to close the system. Surprisingly, this issue is inadequately accounted for in the models that have been developed on the basis of Jackson’s work. In particular, Pitman and Le [Philos. Trans. R. Soc. A 363 (2005) 799–819] replaced this closure simply by imposing an extra boundary condition. If the pressure is assumed to be hydrostatic, this condition can be considered as a closure condition. However, the corresponding model cannot account for a dissipative energy balance. We propose here a closure equation to complete Jackson’s model, imposing incompressibility of the solid phase. We prove that the resulting whole 3D model is compatible with a dissipative energy balance. From this model, we deduce a 2D depth-averaged model and we also prove that the energy balance associated with this model is dissipative. Finally, we propose a numerical scheme to approximate the depth-averaged model. We present several numerical tests for the 1D case that are compared to the results of the model proposed by Pitman and Le

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 uid layer. This leads us to consider a shallow water type system for the uid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classical Saint- Venant-Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classical models do not have an associated energy and how to modify them in order to recover a model with this property. (ii) The model incorporates naturally a necessary modi cation that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that this modi cation is di erent of the ones considered classically, and that it coincides with a classical one only if the solution has a constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, what allows to ensure sediment mass conservation. Moreover, we include a simpli ed version of the model for the case of quasi-stationary regimes. Some of these simpli ed models correspond to the generalization of classical ones such as Meyer- Peter&M uller and Ashida-Michiue models. Three numerical tests are presented to study the evolution of a dune for several de nition of the repose angle, to see the in uence of the proposed de nition of the e ective shear stress in comparison with the classical one, and by comparing with experimental data

IFCP Riemann solver: Application to tsunami modelling using GPUs

In this work, we present a simplified two-layer model of Savage-Hutter type to simulate tsunamis generated by landslides (see (Fern´andez et al. 2008)). A layer composed of fluidized granular material is assumed to flow within an upper layer composed of an inviscid fluid (e.g. water). The sediment layer ismodelled by a Savage-Hutter type model where buoyancy effects have been considered. The system is discretized using IFCP finite volume scheme. The first order IFCP scheme was introduced in (Fern´andez et al. 2011) and it is constructed by using a suitable decomposition of a Roe matrix by means of a parabolic viscosity matrix, that captures information of the intermediate fields (Intermediate Field Capturing Parabola). Its extension to high order and two-dimensional domains is straightforward. To conclude, some numerical examples are presente

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

[EN] 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.Ramírez-Balas, C.; Sendra, J.; Suárez, R.; Fernández-Nieto, E.; Narbona-Reina, G. (2016). The mur neutralisant as an active thermal system: Saint Gobain tests (1931) versus CFD simulation (2015). En LE CORBUSIER. 50 AÑOS DESPUÉS. Editorial Universitat Politècnica de València. 1798-1819. https://doi.org/10.4995/LC2015.2015.899OCS1798181

A semi-implicit approach for sediment transport models with gravitational effects

International audienceIn this work an efficient semi-implicit method, which is based on the theta method, for sediment bedload transport models with gravitational effects under subcritical regimes is proposed. Several families of models with gravitational effects are presented and rewritten under a general formulation that allows us to apply the semi-implicit method. In the numerical tests we focus on the application of a generalization of the Ashida-Michiue model, which includes the gradient of both the bedload and the fluid surface. Analytical steady states solutions (both lake at rest and non vanishing velocity) are deduced an approximated with the proposed scheme. In all the presented tests, the computational efforts are notably reduced thanks to the proposed method without loosing the accuracy in the results

A two-phase solid-fluid model for dense granular flows including dilatancy effects: comparison with submarine granular collapse experiments

We simulate here the collapse of granular columns immersed in a viscous fluid based on a simplified version of the model developed by [2]. The simulation quite well reproduces the dynamics and deposit of the granular mass as well as the excess pore fluid pressure measured in the laboratory experiments of [10] owing that dilatancy effects and pore pressure feedback are accounted for. In particular, the difference in the behaviour of initially loose and dense columns is reproduced numerically

Coupling superposed 1D and 2D shallow-water models: Source terms in finite volume schemes

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Non-hydrostatic layer-averaged Euler system with layerwise linear horizontal velocity

A new hierarchy of non-hydrostatic layer-averaged models for the non-stationary Euler equations is presented in this work, with improved dispersion relations. They are a generalisation of the layeraveraged models introduced in [9], named LDNH models, where the vertical profile of the horizontal velocity is layerwise constant. This assumption implies that solutions of LDNH can be seen as a first order Galerkin approximation of Euler equations. This work focuses on particular weak solutions where the horizontal velocity is layerwise linear on z and possibly discontinuous across layer interfaces. Several closure relations of the layer-averaged system with non-hydrostatic pressure are presented. The resulting models are named LIN-NH p models, with p = 0, 1, 2. Parameter p indicates the degree of the vertical velocity profile considered in the approximation of the vertical momentum equation. All the introduced models satisfy a dissipative energy balance. Finally, an analysis and a comparison of the dispersive properties of each model are carried out. We show that Models LIN-NH 1 and LIN-NH 2 provide a better dispersion relation, group velocity and shoaling than LDNH models

Non-hydrostatic layer-averaged approximation of Euler system with enhanced dispersion properties

A new family of non-hydrostatic layer-averaged models for the non-stationary Euler equations is presented in this work, with improved dispersion relations. They are a generalisation of the layer-averaged models introduced in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018), named LDNH models, where the vertical profile of the horizontal velocity is layerwise constant. This assumption implies that solutions of LDNH can be seen as a first order Galerkin approximation of Euler system. Nevertheless, it is not a fully (x, z) Galerkin discretisation of Euler system, but just in the vertical direction (z). Thus, the resulting model only depends on the horizontal space variable (x), and therefore specific and efficient numerical methods can be applied (see Escalante-Sanchez et al. in J Sci Comput 89(55):1–35, 2021). This work focuses on particular weak solutions where the horizontal velocity is layerwise linear on z and possibly discontinuous across layer interfaces. This approach allows the system to be a second-order approximation in the vertical direction of Euler system. Several closure relations of the layer-averaged system with non-hydrostatic pressure are presented. The resulting models are named LIN-NHk models, with k = 0, 1, 2. Parameter k indicates the degree of the vertical velocity profile considered in the approximation of the vertical momentum equation. All the introduced models satisfy a dissipative energy balance. Finally, an analysis and a comparison of the dispersive properties of each model are carried out. We show that Models LIN-NH1 and LIN-NH2 provide a better dispersion relation, group velocity and shoaling than LDNH models