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 discretization. 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 flows. A new kind of methods has been motivated from an alternative version of the monolithic 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-diffusion 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.