Hydrodynamic linear stability of the two-dimensional bluff-body wake through modal analysis and initial-value problem formulation

Abstract

The stability of the two-dimensional wake behind a circular cylinder - a free flow of general interest in differing applications (from aerodynamics to environmental physics and biology) - is studied by means of two different but complementary theoretical methods. The first part of the work is focused on the asymptotic evolution of disturbances described through modal analysis, a method which allows the determination of the asymptotic stability of a flow. The stability of the intermediate and far near-parallel wake is studied by means of a multiscale approach. The disturbance is defined as the local wavenumber at order zero in the longitudinal direction and is associated to a classical spatio-temporal WKBJ analysis. The inverse of the Reynolds number is taken as the small parameter for the multiscaling. It takes into account non-parallelism effects related to the transversal dynamics of the base flow. The first order corrections find absolute instability pockets in the first part of the intermediate wake (and not in the near wake, where the recirculating eddies are, as usually seen in literature in contrast with the near-parallelism hypothesis). These regions are present for Reynolds numbers larger than $Re=35$. That is in agreement with the general notion of critical Reynolds number for the onset of the first instability of about $Re=47$. In particular, for Re=50 and Re=100, the angular frequency obtained is in agreement with global data in literature concerning numerical and experimental results. The instability is convective throughout the domain. All the stability characteristics are vanishing in the far field, a fact that is independently confirmed by the asymptotic analysis of the Orr-Sommerfeld operator. Using asymptotic Navier-Stokes expansions for the wake inner field the entrainment evolution in the intermediate and far domain is evaluated in terms of asymptotic expansion. The maximum of entrainment is reached in the region where the absolute instability pockets are found. Downstream of this region the entrainment is decreasing and eventually vanishing in the far wake. This point confirms the validity of the multiscale approach. In the second part of the thesis the stability analysis is studied as an initial-value problem to observe the transient behaviour and the asymptotic state of perturbations initially imposed. The initial-value problem allows the formulation to be extended to the near-parallel flow configuration. The initial-value method is, however, less general than the modal analysis, since many parameters, such as the polar wavenumber, the spatial damping rate, the angle of obliquity and the symmetry of the perturbation, are involved. An exploratory analysis of these parameters permits the study of different transient configurations. Before the asymptotic (stable or unstable) state is reached, maxima and minima of the perturbation energy are observed for transients lasting hundreds of time scales. In the temporal asymptotics, the initial-value problem well reproduces modal results in terms of angular frequency and temporal growth rate. Moreover, for Reynolds numbers larger than the critical one (Re_{cr} = 47), the present method gives a good prediction, in terms of wavelength and pulsation, of the vortex shedding observed in experiments. In the framework of the initial-value problem formulation, a multiscale analysis for the stability of long waves is then proposed. Even to the lowest order, the multiscaling - whose small parameter is defined as the polar wavenumber - approximates sufficiently well the full problem solution with a relevant reduction of the computational cost. The two (modal and non-modal) analyses combined together lead to a quite complete description of the bluff-body wake stability