Fictitious boundary and penalization methods for treatment of rigid objects in incompressible flows

Abstract

The Fictitious Boundary Method (FBM) and the Penalty Method (PM) for solving the incompressible Navier-Stokes equations modeling steady or unsteady incompressible flow around solid and rigid, non-deformable objects are presented and numerically analyzed and compared in this thesis. The proposed methods are finite element methods to simulate incompressible flows with small-scale time-(in)dependent geometrical details. The FBM, described and already validated in [1, 43, 48], is based on a finite element method background grid which covers the whole computational domain and is independent of the shape, number and size of any solid obstacle contained inside. The fluid part is computed by a multigrid finite element solver, while the behavior of the solid part is governed by the mechanics principles regarding motion and interactions of type fluid-solid, solid-solid or solid-wall collisions. A new treatment of imposing the Dirichlet boundary conditions for the case of immersed rigid boundary objects is proposed by using the penalization method as a more general framework then the FBM, but containing it as a special case. The new PM approach has a stronger mathematical background. In contrast to FBM, the PM does not imply a direct modification or artificial techniques over the matrix of the system of equations like the fictitious boundary method. A pairing of the penalty method with multigrid solvers is used, while the computational domain is fixed and needs no re-meshing during the simulations. However, the degree of geometrical details that the coarse mesh contains has an impact onto numerical results, a fact which will be investigated/ clarified in this thesis. The presented method is a finite element method, easy to be incorporated into standard CFD codes, for simulating particulate flow or, in general, flows with immersed time-(in)dependent and complicated shaped objects. The aim is to analyze and validate the penalty method and compare, qualitatively and quantitatively, with the already validated FBM regarding the aspects of accuracy of the solution, efficiency, robustness and behavior of the solvers. Different techniques to avoid the numerical difficulties that arise by using penalty method will be particularly described and analyzed.