Flow around obstacles

Sammanfattning

This thesis investigates the cut finite element method (CutFEM) for numericalsimulations of incompressible fluid flow governed by the Navier–Stokes equationsin domains with time‑dependent boundaries. Unlike classical body‑fitted finiteelement methods, CutFEM employs an unfitted stationary background mesh,enabling handling of complex geometries and evolving boundaries withoutre‑meshing. We derive and implement a space–time weak formulation forthe Navier–Stokes equations, using Nitsche’s method for weak enforcementof boundary conditions and ghost‑penalty stabilization to ensure numericalrobustness on cut elements. A Newton-based linearization approach is used to treat the nonlinear convectiveterm, and temporal discretization is performed with discontinuous Galerkinmethods. The implementation relies on an open-source CutFEM code library andis validated through a set of benchmark problems. These include an experimentalorder of convergence study with a manufactured solution, time‑periodic flowaround a stationary circle, dynamic Poiseuille flow with a moving obstacle, andflow around a periodically oscillating circle. The results confirm that the CutFEM framework achieves stable and accurateapproximations for both stationary and time‑dependent obstacle problems. Theobserved convergence rates are generally consistent with theoretical expectationsand comparable to those reported in the literature, with minor deviationsattributed to differences in mesh handling and quadrature along obstacleboundaries.