Coupling the Stokes and Navier-Stokes equations with two scalar nonlinear parabolic equations

This work deals with a system of nonlinear parabolic equations arising in turbulence modelling. The unknowns are the N components of the velocity field u coupled with two scalar quantities θ and ϕ. The system presents nonlinear turbulent viscosity A(θ, ϕ) and nonlinear source terms of the form θ2|∇u| 2 and θϕ|∇u| 2 lying in L1. Some existence results are shown in this paper, including L∞-estimates and positivity for both θ and ϕ.Nous étudions un système non-linéaire d’équations du type parabolique provenant de la modélisation de la turbulence. Les inconnues sont les N composantes du champ des vitesses u couplées avec deux grandeurs scalaires θ et ϕ. Ce système présente un terme de diffusion non-linéaire sous forme matricielle A(θ,ϕ) et les termes sources non-linéaires θ2|∇u| 2 et θϕ|∇u| 2 appartenant à L1. On démontre alors quelques résultats d’existence de solutions, ainsi que des estimations dans L∞ et positivité pour θ et ϕ.Dirección General de Investigación Científica y Técnic

Efficient implicit solvers for models of neuronal networks

We introduce economical versions of standard implicit ODE solvers that are specifically tailored for the efficient and accurate simulation of neural networks. The specific versions of the ODE solvers proposed here, allow to achieve a significant increase in the efficiency of network simulations, by reducing the size of the algebraic system being solved at each time step, a technique inspired by very successful semi-implicit approaches in computational fluid dynamics and structural mechanics. While we focus here specifically on Explicit first step, Diagonally Implicit Runge Kutta methods (ESDIRK), similar simplifications can also be applied to any implicit ODE solver. In order to demonstrate the capabilities of the proposed methods, we consider networks based on three different single cell models with slow-fast dynamics, including the classical FitzHugh-Nagumo model, a Intracellular Calcium Concentration model and the Hindmarsh-Rose model. Numerical experiments on the simulation of networks of increasing size based on these models demonstrate the increased efficiency of the proposed methods

Computational Modeling of Gurney Flaps and Microtabs by POD Method

Gurney flaps (GFs) and microtabs (MTs) are two of the most frequently used passive flow control devices on wind turbines. They are small tabs situated close to the airfoil trailing edge and normal to the surface. A study to find the most favorable dimension and position to improve the aerodynamic performance of an airfoil is presented herein. Firstly, a parametric study of a GF on a S810 airfoil and an MT on a DU91(2)250 airfoil was carried out. To that end, 2D computational fluid dynamic simulations were performed at Re = 10(6) based on the airfoil chord length and using RANS equations. The GF and MT design parameters resulting from the computational fluid dynamics (CFD) simulations allowed the sizing of these passive flow control devices based on the airfoil's aerodynamic performance. In both types of flow control devices, the results showed an increase in the lift-to-drag ratio for all angles of attack studied in the current work. Secondly, from the data obtained by means of CFD simulations, a regular function using the proper orthogonal decomposition (POD) was used to build a reduced order method. In both flow control cases (GFs and MTs), the recursive POD method was able to accurately and very quickly reproduce the computational results with very low computational cost.The current research was partially supported by the Spanish Government with the Project: grant number: MTM2015-64577-C2-1-R

Error analysis of a subgrid eddy viscosity multi-scale discretization of the Navier-Stokes equations

We propose a finite element discretization of the Navier–Stokes equations that relies on the variational multi-scale approach together with the addition of a Smagorinsky type viscosity, in order to take into account possible subgrid turbulence. We recall that the discrete problem admits a solution and prove a priori error estimates. Next we perform the a posteriori analysis of the discretization. Some numerical experiments justify the interest of this approach

On a certified VMS-Smagorinsky Reduced Basis model with LPS pressure stabilisation

In this work we introduce a certified Reduced Basis VMS-Smagorinsky turbulence model with local projection stabilisation (LPS) on the pressure. We prove its stability for Taylor-Hood discretisations of velocity-pressure. We construct an \textit{a posteriori} error estimator for the snapshot selection through a Greedy algorithm, based on the Brezzi-Rappaz-Raviart theory of approximation of non-singular branches of non-linear PDEs. The Empirical Interpolation Method (EIM) is used for the approximation of the non-linear terms. We present some numerical tests in which we show an improved speedup on the computation of the reduced basis problem with the LPS pressure stabilisation, with respect to the method of using pressure supremizers

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

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

Computational modeling of Gurney flaps and microtabs by POD method

Gurney flaps (GFs) and microtabs (MTs) are two of the most frequently used passive flow control devices on wind turbines. They are small tabs situated close to the airfoil trailing edge and normal to the surface. A study to find the most favorable dimension and position to improve the aerodynamic performance of an airfoil is presented herein. Firstly, a parametric study of a GF on a S810 airfoil and an MT on a DU91(2)250 airfoil was carried out. To that end, 2D computational fluid dynamic simulations were performed at Re = 106 based on the airfoil chord length and using RANS equations. The GF and MT design parameters resulting from the computational fluid dynamics (CFD) simulations allowed the sizing of these passive flow control devices based on the airfoil’s aerodynamic performance. In both types of flow control devices, the results showed an increase in the lift-to-drag ratio for all angles of attack studied in the current work. Secondly, from the data obtained by means of CFD simulations, a regular function using the proper orthogonal ecomposition (POD) was used to build a reduced order method. In both flow control cases (GFs and MTs), the recursive POD method was able to accurately and very quickly reproduce the computational results with very low computational cost.Ministerio de Economía y Competitivida

Reduced basis method for the Smagorinsky model

We present a reduced basis Smagorinsky model. This model includes a non-linear eddy diffusion term that we have to treat in order to solve efficiently our reduced basis model. We approximate this non-linear term using the Empirical Interpolation Method, in order to obtain a linearised decomposition of the reduced basis Smagorinsky model. The reduced basis Smagorinsky model is decoupled in a Online/Offline procedure. First, in the Offline stage, we construct hierarchical bases in each iteration of the Greedy algorithm, by selecting the snapshots which have the maximum a posteriori error estimation value. To assure the Brezzi inf-sup condition on our reduced basis space, we have to define a supremizer operator on the pressure solution, and enrich the reduced velocity space. Then, in the Online stage, we are able to compute a speedup solution of our problem, with a good accuracy