A synthetic perturbative hypothesis for the multiscale analysis of the wake instability

The paper presents a nonparallel stability analysis of the intermediate region of the two-dimensional wake behind a bluff body. In particular, it analyzes the convective instabilities using a Wentzel-Kramers-Brillouin-Jeffreys method on a basic flow previously derived from intermediate asymptotics D. Tordella and M. Belan, Phys. Fluids 15, 1897, 2003. The multiscaling is carried out to explicitly account for the effects associated to the lateral momentum dynamics at a given Reynolds number. These effects are an important feature of the base flow and are included in the perturbative equation as well as in the associated modulation equation. At the first order in the multiscaling, the disturbance is locally tuned to the property of the instability, as can be seen in the zero-order theory near-parallel parametric Orr-Sommerfeld treatment. This leads to a synthetic analysis of the nonparallel correction of the instability characteristics. The system is, in fact, considered to be locally perturbed by waves with a wave number that varies along the intermediate wake and which is equal to the wave number of the dominant saddle point of the zero order dispersion relation, taken at different Reynolds numbers. In this study, the Reynolds number is thus the only parameter. It is shown that the corrections to the frequency, and to the temporal and spatial growth rates are remarkable in the first part of the intermediate wake and lead to absolute instability in regions that extend to about ten body scales. The correction increases with the Reynolds number and agrees with data from laboratory and numerical experiments in literature. An eigenfunction and eigenvalue asymptotic analysis for the far wake is included, which is in excellent agreement with the complete problem

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

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 stabilit

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

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 results 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

Instabilities in multiphase and icing flows

The problem of the stability of water-coated ice layer is investigated for a free surface flow down an inclined plane for the cases of normal and thin (boundary- layer scale) water films. For the case of boundary-layer scale water film a Froude- based double-deck theory is developed which is then used to investigate linear two-dimensional (2D), three-dimensional (3D) and nonlinear 2D stability of the problem. The new mode of upstream-propagating instability arising because of the ice layer is found and its properties investigated. In the linear double-deck, analytic expressions for the dispersion relation and neutral curves are obtained for the case of Pr = 1. For the general case, linear stability problem is solved numerically using new 4th order finite-difference scheme developed for Orr-Sommerfeld equations. Non-linear double-deck equations are solved with a new 2nd order in space global- marching type scheme and the effects of nonlinearity are analysed. An explanation of the physical mechanism leading to the upstream propagation is derived. The effect of the intersection and branch exchange between the interfacial mode and a shear mode is discovered in a 2-fluid plane Poiseuille flow and investigated in detail using linear stability theory and the numerical approach developed for the free-surface flows. The interaction between three instability modes present in the problem is analysed. It is shown that the question of mode identity becomes complicated because of the discovered intersection and the methods of establishing mode identity are discussed. Finally, the longwave asymptotic analysis of the ice layer under a water/air plane Poiseuille flow is performed. The effect of ice on the modes present in the problem is discussed

Internal waves in fluid flows. Possible coexistence with turbulence

Waves in fluid flows represents the underlying theme of this research work. Wave interactions in fluid flows are part of multidisciplinary physics. It is known that many ideas and phenomena recur in such apparently diverse fields, as solar physics, meteorology, oceanography, aeronautical and hydraulic engineering, optics, and population dynamics. In extreme synthesis, waves in fluids include, on the one hand, surface and internal waves, their evolution, interaction and associated wave-driven mean flows; on the other hand, phenomena related to nonlinear hydrodynamic stability and, in particular, those leading to the onset of turbulence. Close similarities and key differences exist between these two classes of phenomena. In the hope to get hints on aspects of a potential overall vision, this study considers two different systems located at the opposite limits of the range of existing physical fluid flow situations: first, sheared parallel continuum flows - perfect incompressibility and charge neutrality - second, the solar wind - extreme rarefaction and electrical conductivity. Therefore, the activity carried out during the doctoral period consists of two parts. The first is focused on the propagation properties of small internal waves in parallel flows. This work was partly carried out in the framework of a MISTI-Seeds MITOR project proposed by Prof. D. Tordella (PoliTo) and Prof. G. Staffilani (MIT) on the long term interaction in fluid flows. The second part regards the analysis of solar-wind fluctuations from in situ measurements by the Voyagers spacecrafts at the edge of the heliosphere. This work was supported by a second MISTI-Seeds MITOR project, proposed by D. Tordella (PoliTo), J. D. Richardson (MIT, Kavli Institute), with the collaboration of M. Opher (BU)