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

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

Tratamiento asintótico de las condiciones de contorno para problemas de convección dominante

En este trabajo realizamos un an alisis de paso al l mite singular en una ecuaci ón evolutiva de convecci ón-difusióon, imponiendo el flujo total normal en la frontera de entrada de flujo y una condici on de tipo Newmann en el resto de la frontera. Probamos que la soluci ón de este problema converge en L2 (Q) a la de la ecuaci ón de convecci ón pura con una condicióon de contorno de tipo Dirichlet en la frontera de entrada de flujo. Adem ás convergen las derivadas convectivas en L2 (Q) y las trazas convectivas en las fronteras de entrada y salida de flujo en espacios de tipo L2. Este estudio permite justi car la forma en la que ciertos m étodos numéricos de resolucióon de modelos de convecci ón-difusi ón imponen las condiciones de contorno

A Bochev-Dohrmann-Gunzburger stabilization method for the primitive equations of the ocean

We introduce a low-order stabilized discretization of the Primitive Equations of the Ocean, with a highly reduced computational complexity. We prove stability through a specific inf-sup condition, and weak convergence to a weak solution. We also perform some numerical test for relevant flows.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo Regiona

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

A reduced discrete inf-sup condition in Lp for incompressible flows and application

In this work, we introduce a discrete specific inf-sup condition to estimate the Lp norm, 1 <p< +∞, of the pressure in a number of fluid flows. It applies to projection-based stabilized finite element discretizations of incompressible flows, typically when the velocity field has a low regularity. We derive two versions of this inf-sup condition: The first one holds on shape-regular meshes and the second one on quasi-uniform meshes. As an application, we derive reduced inf-sup conditions for the linearized Primitive equations of the Ocean that apply to the surface pressure in weighted Lp norm. This allows to prove the stability and convergence of quite general stabilized discretizations of these equations: SUPG, Least Squares, Adjoint-stabilized and OSS discretizations.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona

Numerical modeling of buoyant turbulent mixing layers

We introduce in this paper some elements for the mathematical and numerical analysis of turbulence models for oceanic surface mixing layers. In these models the turbulent diffusions are parameterized by means of the Richardson’s number, that measures the balance between stabilizing buoyancy forces and un-stabilizing shearing forces. The wellpossedness of these models is a difficult mathematical problem, due to the partial monotonic nature of the space operators involved. We analyze the existence and stability of equilibria state, and devise a conservative numerical scheme satisfying the maximum principle. We present some numerical tests for realistic flows in tropical seas that reproduce the formation of mixing layers, in agreement with the physics of the problem.Ministerio de Educación y Cienci