Wakes and paths of buoyancy-driven permeable disks: a linear stability approach

In the present thesis, the modifications of the wake and path instabilities of a freely-falling or rising permeable thin disk are investigated. The flow is described via the incompressible Navier-Stokes equations coupled with the Newton equations for the disk dynamics. A membrane model based on homogenization technique is employed, valid when the flow inertia within the pores composing the micostructured membrane are negligible. The numerical implementations is validated against literature results and full-scale models. The features of the steady and vertical path of the disk are investigated performing a parametric study, varying the parameters involved. The results reveal a strong effect of the permeability on the flow morphology. The wake shows a recirculation region that detaches from the disk, becomes smaller and disappears as permeability increases. A maximum of the drag coefficient is found with increasing filtrability. An increase in the Reynolds number leads to a non-monotonous increase of the dimensions of the recirculation region and eventually a decrease, if permeability is sufficiently large. Subsequently, the bifurcations of a fixed disk are investigated via a linear stability analysis. The stability scenario is profoundly influenced by the permeability. The latter initially destabilizes the flow, but, for large values of permeability, all the instabilities are quenched and a threshold beyond which the wake is stable is identified. In a successive step, the bifurcations of the steady vertical path of a freely-falling permeable disk are investigated through a linear stability analysis, with a particular focus on the effect of permeability and disk inertia on the stability properties of the flow. Four modes of instability are found. The series of bifurcations is intriguing and the primary bifurcation is modified both by the inertia of the disk and permeability. An increase in permeability leads to the emergence of a steady mode which induces an oblique path as first bifurcation if inertia is low. For large permeability, we identify a threshold beyond which the all instabilities are quenched