### Perturbation dynamics in laminar and turbulent flows. Initial value problem analysis

Francesca De Santi, Tordella Daniela
2015
Stability and turbulence are often studied as separate branches of fluid dynamics, but they are actually the two faces of the same coin: the existence of equilibrium, laminar in one case and steady in the mean in the other. The link between these two faces is transition. Initial value problems are considered to analyse the dynamics of disturbances in the three phases. In the context of stability, linearised equations of motion can be used. Although this is a substantial simplification, the
more » ... ts that are obtained with this analysis are far from being trivial. The transition to turbulence through the dynamics of disturbances is discussed in the context of the zig-zag instability: a particular kind of instability that occurs in geophysical flows. Eventually, the perturbations dynamics in turbulent flows is used to analyse the mixing process between water-vapour in clouds and clear air in the surroundings, in the presence of a meteorological inversion. The evolution of the spatial-temporal perturbations is a topic of interest for most physical systems. From a physical point of view, in fact, disturbances are always present in reality and can not be eliminated or ignored, whether or not they are infinitesimal. Is therefore essential to study their spatial-temporal evolution in the two main flow regimes: laminar and turbulent. Stability, transition and turbulence are often studied as separate strands of fluid dynamics. Stability and turbulence are actually faces of the same coin. The existence of equilibrium: in one case is laminar, in the other one is steady in the mean. The link between these two faces is transition. In this thesis initial value problems are solved in order to analyses the dynamics of disturbances acting in these three different phases. In the context of stability, assuming that the velocity and pressure of the perturbations are small in comparison to those of the main flow, it is possible to use the linearised motion equations. In this way the physics is significantly simplified since the interactions between disturbances co-existing in the system (including the self-interaction) and those between the disturbances and the base flow are neglected. Although this is a substantial simplification, the results that are obtained with this analysis are far from being trivial. The transients can be very complex: for example the perturbation can initially lose kinetic energy, subsequently be amplified for a long time-interval and eventually completely decay when the asymptotic state is reached. This is just one of the possible scenarios that may occur by introducing travelling waves as perturbations in different types of shear flows: two-dimensional bounded (Poiseuille flow) or unbounded (Blasius boundary layer and wake flows) and three-dimensional (boundary layer in cross flow). When the base flow is two-dimensional, the temporal evolution of the perturbative waves is analysed in terms of the velocity at which the phase of any one frequency component of the wave travels, this is named the phase velocity. This has been poorly investigated so far. Indeed traditional studies are mostly interested in identify whether a perturbation can be stable or not. However, also the phase velocity can lead to interesting considerations. In fact, generally in laminar system more than a disturbance can coexist at the same time and therefore wave packets can form. The two main features that describe the dynamics of wave packets are precisely the phase velocity and the group velocity, the velocity with which the overall shape of the waves' amplitudes propagates through space. The relation between these velocities gives