134 research outputs found
Layer-averaged Euler and Navier-Stokes equations
In this paper we propose a strategy to approximate incompressible hydrostatic
free surface Euler and Navier-Stokes models. The main advantage of the proposed
models is that the water depth is a dynamical variable of the system and hence
the model is formulated over a fixed domain.The proposed strategy extends
previous works approximating the Euler and Navier-Stokes systems using a
multilayer description. Here, the needed closure relations are obtained using
an energy-based optimality criterion instead of an asymptotic expansion.
Moreover, the layer-averaged description is successfully applied to the
Navier-Stokes system with a general form of the Cauchy stress tensor
A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation
The standard multilayer Saint-Venant system consists in introducing fluid
layers that are advected by the interfacial velocities. As a consequence there
is no mass exchanges between these layers and each layer is described by its
height and its average velocity. Here we introduce another multilayer system
with mass exchanges between the neighborhing layers where the unknowns are a
total height of water and an average velocity per layer. We derive it from
Navier-Stokes system with an hydrostatic pressure and prove energy and
hyperbolicity properties of the model. We also give a kinetic interpretation
leading to effective numerical schemes with positivity and energy properties.
Numerical tests show the versatility of the approach and its ability to compute
recirculation cases with wind forcing.Comment: Submitted to M2A
2D granular flows with the rheology and side walls friction: a well balanced multilayer discretization
We present here numerical modelling of granular flows with the
rheology in confined channels. The contribution is twofold: (i) a model to
approximate the Navier-Stokes equations with the 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 Fern\'andez-Nieto et al. (J. Fluid Mech., vol. 798, 2016, pp.
643-681). 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 balance 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
but not with a constant friction coefficient
Parallelization of a relaxation scheme modelling the bedload transport of sediments in shallow water flow
In this work we are interested in numerical simulations for bedload erosion
processes. We present a relaxation solver that we apply to moving dunes test
cases in one and two dimensions. In particular we retrieve the so-called
anti-dune process that is well described in the experiments. In order to be
able to run 2D test cases with reasonable CPU time, we also describe and apply
a parallelization procedure by using domain decomposition based on the
classical MPI library.Comment: 19 page
A dynamic multilayer shallow water model
We propose a new simple approximation of the viscous primitive equations of the ocean including Coriolis force, by a multilayer shallow water type model. Using a finite volume type discretization in the vertical direction, we show that our system is a consistent approximation of the primitive model. Existence and uniqueness of local in time strong solution is proved for the new model. Finally we design a finite volume numerical scheme, taking advantage of the shallow water type formulation and perform preliminary numerical simulations in 1D to illustrate consistency as well as a dynamic behavior (add or remove layers)
A multilayer method for the hydrostatic Navier-Stokes equations: a particular weak solution
In this work we present a mutilayer approach to the solution of non-stationnary 3D Navier-Stokes equations. We use piecewise smooth weak solutions. We approximate the velocity by a piecewise constant (in z) horizontal velocity and a linear (in z) vertical velocity in each layer, possibly discontinuous across layer interfaces. The multilayer approach is deduced by using the variational formulation and by considering a reduced family of test functions. The procedure naturally provides the mass and momentum interfaces conditions. The mass and momentum conservation across interfaces is formulated via normal flux jump conditions. The jump conditions associated to momentum conservation are formulated by means of an approximation of the vertical derivative of the velocity that appears in the stress tensor. We approximate the multilayer model for hydrostatic pressure, by using a PVM finite volume scheme and we present some numerical tests that show the main advantages of the model:
it improves the approximation of the vertical velocity, provides good predictions for viscous effects and simulates re-circulations behind solid obstacles
A multilayer shallow water system for polydisperse sedimentation
This work considers the flow of a fluid containing one disperse substance consisting of small particles that belong to different species differing in size and density. The flow is modelled by combining a multilayer shallow water approach with a polydisperse sedimentation process. This technique allows one to keep information on the vertical distribution of the solid particles in the mixture, and thereby to model the segregation of the particle species from each other, and from the fluid, taking place in the vertical direction of the gravity body force only. This polydisperse sedimentation process is described by the well-known Masliyah-Lockett-Bassoon (MLB) velocity functions. The resulting multilayer sedimentation-ow model can be written as a hyperbolic system with nonconservative products. The definitions of the nonconservative products are related to the hydrostatic pressure and to the mass and momentum hydrodynamic transfer terms between the layers. For the numerical discretization a strategy of two steps is proposed, where the first one is also divided into two parts. In the _rst step, instead of approximating the complete model, we approximate a reduced model with a smaller number of unknowns. Then, taking advantage of the fact that the concentrations are passive scalars in the system, we approximate the concentrations of the different species by an upwind scheme related to the numerical flux of the total concentration. In the second step, the effect of the transference terms defined in terms of the MLB model is introduced. These transfer terms are approximated by using a numerical ux function used to discretize the 1D vertical polydisperse model (see Bürger, GarcÃa, Karlsen and Towers, J. Eng. Math. 60 (2008), 387{425). Finally, some numerical examples are presented. Numerical results suggest that the multilayer shallow water model could be adequate in situations where the settling takes place from a suspension that undergoes horizontal movement
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.This research has been partially supported by the Spanish Government and FEDER through the research projects RTI2018-096064-B-C2(1/2) and PID2020-114688RB-I00, the Junta de AndalucÃa research project P18-RT-3163, the Junta de Andalucia-FEDER-University of Málaga research project UMA18-FEDERJA-16. Funding for open access charge: Universidad de Málaga / CBUA
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
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