Incompressible flows around a square cylinder using adaptive virtual element methods

In the thesis, a Virtual element method (VEM) is applied to a fluid dynamics problem. VEM is a numerical method that can be applied with general polygonal meshes to solve partial differential equations (PDE) problems. VEM allows the use of “hanging nodes”, nodes that divide two aligned edges of a mesh element. In this way local refinement is possible and it is useful to adapt the mesh where the physical problem requires. The fluid dynamics problem considers an incompressible flow inside a channel containing an obstacle, representing a square cylinder placed at the centre of the longitudinal axis. A parabolic velocity profile is assigned at the inflow. The range of Reynolds number is limited so that the flow remains laminar and steady. In the first part of the thesis, a diffusion equation is solved with the VEM, implementing also adaptive mesh refinement based on an a posteriori error estimate. Considering incompressible flows without convection, the discretization of the Stokes problem is implemented with the possibility of adaptive mesh refinement based on an appropriate a posteriori error estimate. In particular the fluid dynamics problem is solved analysing the resulting velocity field and meshes, with hanging nodes, coming from adaptive refinement. Finally, adding the convection terms, the Navier-Stokes problem is solved with the VEM. The model is applied to the fluid dynamics case for different Reynolds numbers in a suitable range. Results are commented and compared with analogous ones coming from computational fluid dynamics literature.