Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme

A review of variational multiscale methods for the simulation of turbulent incompressible flows

Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier-Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.Ministerio de Economía y CompetitividadV Plan Propio de Investigacion (niversidad de Sevilla)Fondation Sciences Mathematiques de Pari

A review of variational multiscale methods for the simulation of turbulent incompressible flows

Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier-Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized

A variational subgrid scale model for transient incompressible flows

We introduce in this paper a variational subgrid scale model for the solution of the incompressible Navier-Stokes equations. With respect to classical multiscale-based stabilisation techniques, we retain the subgrid scale effects in the convective term and integrate the subgrid scale equation in time. The method is applied to the Navier-Stokes equations in an accelerating frame of reference and with Dirichlet (essential), Neumann (natural) and mixed boundary conditions. The concrete objective of the paper is to test a numerical algorithm for solving the non-linear subgrid scale equation and the introduction of the subgrid scale into the grid scale equation. The performance of the technique is demonstrated through the solution of two numerical examples: one to test the tracking of the subgrid scale in the convection term and the other to investigate the effects of considering the subgrid scale transient

Stabilized pressure segregation methods and their application to fluid-stucture interaction problems

In this work we design and analyze pressure segregation methods in order to approximate the Navier-Stokes equations. Pressure correction methods are widely used because they allow the decoupling of velocity and pressure computation, decreasing the computational cost. We have analyzed some of these schemes, obtaining inherent pressure stability. However, for second order accurate methods (in time) this inherent stability is too weak, requiring the introduction of a stabilized finite element methodology for the space dis- cretization. Moreover, we have carried out a complete convergence analysis of a first order pressure segregation method. We have used a stabilization technique justified from a multiscale approach that allows the use of equal velocity-pressure interpolation spaces and convection dominated °ows. A new kind of methods has been motivated from an alternative version of the mono- lithic fluid solver where the continuity equation is replaced by a discrete pressure Poisson equation. These methods belong to the family of velocity correction schemes, where it is the velocity instead of the pressure the extrapolated unknown. Some stability bounds have been proved, revealing that their inherent pressure stability is too weak. Further, predictor corrector schemes easily arise from the new monolithic system. Numerical ex- perimentation shows the good behavior of these methods. We have introduced the ALE framework in order for the fluid governing equations to be formulated on moving domains. Taking as the model equation the convection-di®usion equation, we have analyzed the blend of the ALE framework and a stabilized finite element method. We suggest a coupling procedure for the fluid-structure problem taking benefit from the ingredients previously introduced: pressure segregation methods, a stabilized finite element formulation and the ALE framework. The final algorithm, using one loop, tends to the monolithic (fluid-structure) system. This method has been applied to the simulation of bridge aerodynamics, obtaining a good convergence behavior. We end with the simulation of wind turbines. The fact that we have a rotary body surrounded by the fluid (air) has motivated the introduction of a remeshing strategy. We consider a selective remeshing procedure that only affects a tiny portion of the domain, with little impact on the overall CPU time

Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank--Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results

A posteriori error estimation and adaptivity based on VMS for the incompressible Navier–Stokes equations

In this work an explicit a posteriori error estimator for the steady incompressible Navier–Stokes equations is investigated. The error estimator is based on the variational multiscale theory, where the numerical solution is decomposed in resolved scales (FEM solution) and unresolved scales (FEM error). The error is estimated locally considering the residuals that emerge from the numerical solution and the error inverse-velocity scales, t’s, associated with each type of residual. These error scales are provided in this paper, which have been computed a-priori solving a set of local problems with unit residuals. Therefore, the computational effort to predict the error is small and its implementation in any FEM code is simple. As an application, a strategy to develop adaptive meshes with the aim of optimizing the computational effort is shown. Numerical examples are presented to test the behavior of the error estimator

On the grad-div stabilization for the steady Oseen and Navier--Stokes equations

This paper studies the parameter choice in the grad-div stabilization applied to the generalized problems of Oseen type. Stabilization parameters based on minimizing the H¹ (Ω) error of the velocity are derived which do not depend on the viscosity parameter. For the proposed parameter choices, the H¹ (Ω) error of the velocity is derived that shows a direct dependence on the viscosity parameter. Differences and common features to the situation for the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier- Stokes equations, numerical simulations were performed on a two-dimensional ow past a circular cylinder. It turns out, for the MINI element, that the best results can be obtained without grad-div stabilization