A "well-balanced" finite volume scheme for blood flow simulation

We are interested in simulating blood flow in arteries with a one dimensional model. Thanks to recent developments in the analysis of hyperbolic system of conservation laws (in the Saint-Venant/ shallow water equations context) we will perform a simple finite volume scheme. We focus on conservation properties of this scheme which were not previously considered. To emphasize the necessity of this scheme, we present how a too simple numerical scheme may induce spurious flows when the basic static shape of the radius changes. On contrary, the proposed scheme is "well-balanced": it preserves equilibria of Q = 0. Then examples of analytical or linearized solutions with and without viscous damping are presented to validate the calculations. The influence of abrupt change of basic radius is emphasized in the case of an aneurism.Comment: 36 page

FullSWOF: A free software package for the simulation of shallow water flows

Numerical simulations of flows are required for numerous applications, and are usually carried out using shallow water equations. We describe the FullSWOF software which is based on up-to-date finite volume methods and well-balanced schemes to solve this kind of equations. It consists of a set of open source C++ codes, freely available to the community, easy to use, and open for further development. Several features make FullSWOF particularly suitable for applications in hydrology: small water heights and wet-dry transitions are robustly handled, rainfall and infiltration are incorporated, and data from grid-based digital topographies can be used directly. A detailed mathematical description is given here, and the capabilities of FullSWOF are illustrated based on analytic solutions and datasets of real cases. The codes, available in 1D and 2D versions, have been validated on a large set of benchmark cases, which are available together with the download information and documentation at http://www.univ-orleans.fr/mapmo/soft/FullSWOF/.Comment: 38 page

SWASHES: a compilation of Shallow Water Analytic Solutions for Hydraulic and Environmental Studies

Numerous codes are being developed to solve Shallow Water equations. Because there are used in hydraulic and environmental studies, their capability to simulate properly flow dynamics is critical to guarantee infrastructure and human safety. While validating these codes is an important issue, code validations are currently restricted because analytic solutions to the Shallow Water equations are rare and have been published on an individual basis over a period of more than five decades. This article aims at making analytic solutions to the Shallow Water equations easily available to code developers and users. It compiles a significant number of analytic solutions to the Shallow Water equations that are currently scattered through the literature of various scientific disciplines. The analytic solutions are described in a unified formalism to make a consistent set of test cases. These analytic solutions encompass a wide variety of flow conditions (supercritical, subcritical, shock, etc.), in 1 or 2 space dimensions, with or without rain and soil friction, for transitory flow or steady state. The corresponding source codes are made available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow Water-based models can easily find an adaptable benchmark library to validate their numerical methods.Comment: 40 pages There are some errors in the published version. This is a corrected versio

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

We present here numerical modelling of granular flows with the $\mu(I)$ rheology in confined channels. The contribution is twofold: (i) a model to approximate the Navier-Stokes equations with the $\mu(I)$ 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 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 $\mu(I)$ but not with a constant friction coefficient

Uncertainty quantification in littoral erosion

International audienceWe aim at quantifying the impact of flow state uncertainties in lit-toral erosion to provide confidence bounds on deterministic predictions of bottom morphodynamics. Two constructions of the bathymetry standard deviation are discussed. The first construction involves directional quantile-based extreme scenarios using what is known on the flow state Probability Density Function (PDF) from on site observations. We compare this construction to a second cumulative one using the gradient by adjoint of a functional involving the energy of the system. These ingredients are illustrated for two models for the interaction between a soft bed and a flow in a shallow domain. Our aim is to keep the computational complexity comparable to the deterministic simulations taking advantage of what already available in our simulation toolbox

Beyond Shallow Water: appraisal of a numerical approach to hydraulic jumps based upon the Boundary Layer Theory

International audienceWe study the flow of a thin layer of fluid over a flat surface. Commonly, the 1-D Shallow-water or Saint-Venant set of equations are used to compute the solution of such flows. These simplified equations may be obtained through the integration of the Navier-Stokes equations over the depth of the fluid, but their solution requires the introduction of constitutive relations based on strict hypothesis on the flow régime. Here, we present an approach based on a kind of boundary layer system with hydrostatic pressure. This relaxes the need for closure relations which are instead obtained as solutions of the computation. It is then demonstrated that the corresponding closures are very dependent on the type of flow considered, for example laminar viscous slumps or hydraulic jumps. This has important practical consequences as far as the applicability of standard closures is concerned