• direct numerical simulation
• linear stability analysis
• mixers
• rotating disk
• triple deck

This thesis aims at investigating the laminar regime of two classes of problems: the boundary layer induced by a rotating disk and the flow fields that develop inside two configurations of micro-mixers.
The first part investigates the effect of roughness elements over a disk that rotates with a constant rotational speed in still air. For a smooth disk, the steady laminar region of the flow field is described by the self-similar solution provided by von Kármán, while the perturbed flow field in the presence of roughness elements is evaluated here with an analytical model based on triple deck decomposition. The proposed model provides an analytical solution, which depends only on the roughness geometry and does not require any additional parameter. Several geometries have been tested, and the analytical results are in good agreement with DNS for both axisymmetric and three-dimensional roughness elements; in particular, for geometries that span in the radial direction, the prediction of the velocity perturbation is recovered by a local scaling that considers the development of the flow along the radius. The proposed model is particularly useful for three-dimensional roughness elements since it allows to drastically reduce the computational cost with respect to numerical simulation. In addition, the triple deck theory at the considered order predicts the effect of the pressure gradient for three-dimensional roughness elements, thus providing a realistic estimation of the velocity perturbation induced upstream.
Since the roughness is generally described by its statistics, a stochastic analysis based on the generalized polynomial chaos approach has been employed. Results show that the mean value and the standard deviation of the perturbation field are proportional to the statistics of the height distribution. Hence, it is possible to exploit the linearity of the proposed model with respect to the roughness height to rescale the mean value and the standard deviation of the perturbation for any homogenous height distribution. Moreover, in the three-dimensional case, the resulting flow field is axisymmetric, and the contribution of the azimuthal component of pressure gradient is null on average. Since the mean value of the perturbation field preserves the self-similarity of von Kármán's solution, it is possible to evaluate the mean value of the resulting flow field without averaging over one roughness wavelength.
The resulting self-similar solution has been employed to perform a convective stability analysis to quantify the effect of roughness elements onto the stability of the flow field. The theoretical analysis shows a slightly stabilizing effect for the mean value of the perturbation over a sinusoidal roughness with a maximum height equals to 0.4, even though the overall effect is lower than the one predicted by other models. Moreover, numerical tests indicate that the effect of the selected roughness is negligible on delaying the crossflow mode.
The second part of the thesis concerns two different configurations of micro-mixers, a T-mixer and a cross-shaped mixer with circular and square cross-sections, respectively. The work is focused on characterizing the flow regimes at different Reynolds numbers, the mixing level achievable and the mechanisms that originate the transition from one regime to another.
Both vortex and engulfment regime of the T-mixer with circular cross-section have been characterized numerically by varying the Reynolds number (Re) in the range 50<Re<550. Despite the seeming similarity of the vortex regime in comparison to mixers with rectangular cross-section, DNSs show that the relevant vortices are those that depart from the curved edges at the intersection of the three pipes and interact with the other vortices that originates from the upper region of the inlet pipe. This difference becomes more evident in the engulfment regime, in which the flow is practically symmetric in the first part of the outlet channel and the rotation of the vortical structures is quite limited, providing low values of the mixing level.
Since the engulfment regime of the circular cross-section does not improve substantially the mixing level, a global resolvent approach has been employed to evaluate the effect of a harmonic inlet forcing. The analysis shows that the maximum gain occurs for a steady perturbation of the inflow boundary conditions and that a slight modification of the Hagen-Poiseuille profile at inlets can lead to an unsymmetrical configuration even for Reynolds number lower than the critical one. A possible strategy to improve the mixing level has been identified by properly tilting the inlet pipes to induce a regime similar to what can be commonly found in T-mixer with rectangular cross-sections.
The second configuration investigated is a cross-shaped mixer with square cross-section. The analysis is carried out numerically by considering Reynolds numbers in the range 50<Re<330. The numerical setup has been validated by comparing the results for the steady regime with those reported in literature. By increasing the Reynolds number up to 100, the steady vortex configuration is characterized by the vortex breakdown, a phenomenon that involves the formation of a recirculating region inside the vortex itself; the evolution of the vortex breakdown has been investigated in terms of Reynolds number in the range 100<Re<270. The recirculating region lengthen by increasing the Reynolds number up to Re=230, at which vortex rings periodically detach from the downstream end of the recirculating region.
For Re>280, a further unsteady regime develops. Unlike experiments, DNSs show to be sensitive to the boundary conditions at inlets. Indeed, the dynamics that arises for a fully-developed Poiseuille is substantially different from experimental investigations; conversely, when a constant velocity profile is applied at inflows, the resulting flow regime resembles the PIV visualizations, but it exhibits an additional dynamics with respect to experimental data.