Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions

In this work, depending on the relation between the Deborah, the Reynolds and the aspect ratio numbers, we formally derived shallow-water type systems starting from a micro-macro description for non-Newtonian fluids in a thin domain governed by an elastic dumbbell type model with a slip boundary condition at the bottom. The result has been announced by the authors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl. Springer Verlag (2010)] and in the present paper, we provide a self-contained description, complete formal derivations and various numerical computations. In particular, we extend to FENE type systems the derivation of shallowwater models for Newtonian fluids that we can find for instance in [J.-F. Gerbeau, B. Perthame, Discrete Contin. Dyn. Syst. (2001)] which assume an appropriate relation between the Reynolds number and the aspect ratio with slip boundary condition at the bottom. Under a radial hypothesis at the leading order, for small Deborah number, we find an interesting formulation where polymeric effect changes the drag term in the second order shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discuss intermediate Deborah number with a fixed Reynolds number where a strong coupling is found through a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formal level, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] including non-linear effects (shallow-water framework)

Existence of a global weak solution for a 2d viscous bi-layer Shallow Water model

We consider a non-linear viscous bi-layer shallow water model with capillarity effects and extra friction terms in a two-dimensional space. This system is issued from a derivation of three-dimensional Navier–Stokes equations with a water-depth depending on friction coefficients. We prove an existence result for a global weak solution in a periodic domain Ω = T 2

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

Formal derivation of a bilayer model coupling shallow water and Reynolds lubrication equations: evolution of a thin pollutant layer over water

In this paper a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier-Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids, thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation for the upper layer and the shallow water model for the lower one. We prove that the model is provided of a dissipative entropy inequality, up to a second order term. Moreover, we propose a correction of the model by taking into account the second order extention for the pressur that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first one we compare the numerical results with the viscous bilayer shallow water model proposed in [G. Narbona-Reina, J.D.D. Zabsonré, E.D. Fernández-Nieto, D. Bresch, CMES Comput. Model. Eng. Sci., 2009]. In the second test the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test the influence of the friction coefficient on the coastal area affected by the pollutant is studied

Formulación de tipo Petrov-Galerkin de algunos métodos distributivos: Aplicación a las ecuaciones de Navier-Stokes

En este trabajo estudiamos la resolución de las Ecuaciones de Navier-Stokes estacionarias mediante métodos distributivos no lineales. Formulamos estos métodos como métodos de tipo Petrov-Galerkin, en un contexto de discretización por el método de los elementos finitos. Utilizamos funciones tests descentradas “corriente arriba”para el tratamiento del término de convección. Esta formulación nos permite realizar el análisis de los métodos distributivos que consideramos como una extensión del análisis estándar. Presentamos resultados de existencia de solución del problema discreto, convergencia y estimaciones de error. Por último, presentamos algunos test numéricos resueltos mediante un esquema de tipo distributivo no lineal, el PSI. Estos tests muestran un comportamiento resistente a la generación de oscilaciones parásitas, y una mayor exactitud que un método de las características de primer orden

Numerical analysis of the PSI solution of advection–diffusion problems through a Petrov–Galerkin formulation

We consider a system composed by two immiscible fluids in two-dimensional space that can be modelized by a bilayer Shallow Water equations with extra friction terms and capillary effects. We give an existence theorem of global weak solutions in a periodic domain.In this paper we introduce an analysis technique for the solution of the steady advection– diffusion equation by the PSI (Positive Streamwise Implicit) method. We formulate this approximation as a nonlinear finite element Petrov–Galerkin scheme, and use tools of functional analysis to perform a convergence, error and maximum principle analysis. We prove that the scheme is first-order accurate in H1 norm, and well-balanced up to second order for convection-dominated flows. We give some numerical evidence that the scheme is only first-order accurate in L2 norm. Our analysis also holds for other nonlinear Fluctuation Splitting schemes that can be built from first-order monotone schemes by the Abgrall and Mezine’s technique

Thermal 3D CFD Simulation with Active Transparent Façade in Buildings

In recent years active façades have acquired greater importance given their capacity to improve the energy efficiency of buildings. One such type is the so-called Active Transparent Façade (ATF). A 3D numerical model based on computational fluid dynamics (CFD) and the Finite Element Method (FEM) has been generated to simulate the thermal performance of buildings equipped with this type of façade. This model is introduced for general application and allows the design parameters to be adapted for this system. The case study of Le Corbusier’s proposal for the City of Refuge in Paris, the clearest example of previous use of an ATF is examined. In addition, a proposal is presented for the energy improvement of Le Corbusier’s original solution. In order to do so, the conditions for the supply of air into the ATF cavity and in the mechanical ventilation system are assessed to guarantee comfort conditions