Collective behaviour of linear perturbation waves observed through the energy density spectrum

Abstract: We consider the collective behaviour of small three-dimensional transient perturbations in sheared flows. In particular, we observe their varied life history through the temporal evolution of the amplification factor. The spectrum of wave vectors considered fills the range from the size of the external flow scale to the size of the very short dissipative waves. We observe that the amplification factor distribution is scale-invariant. In the condition we analyze, the system is subject to all the physical processes included in the linearized Navier-Stokes equations. With the exception of the nonlinear interaction, these features are the same as those characterizing the turbulent state. The linearized perturbative system offers a great variety of different transient behaviours associated to the parameter combination present in the initial conditions. For the energy spectrum computed by freezing each wave at the instant where its asymptotic condition is met, we ask whether this system is able to show a power-law scaling analogous to the Kolmogorov argument. At the moment, for at least two typical shear flows, the bluff-body wake and the plane Poiseuille flow, the answer is ye

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

The role of long waves in the stability of the plane wake

This work is directed towards investigating the fate of three-dimensional long perturbation waves in a plane incompressible wake. The analysis is posed as an initial-value problem in space. More specifically, input is made at an initial location in the downstream direction and then tracing the resulting behavior further downstream subject to the restriction of finite kinetic energy. This presentation follows the outline given by Criminale and Drazin [Stud. in Applied Math. \textbf{83}, 123 (1990)] that describes the system in terms of perturbation vorticity and velocity. The analysis is based on large scale waves and expansions using multi scales and multi times for the partial differential equations. The multiscaling is based on an approach where the small parameter is linked to the perturbation property independently from the flow control parameter. Solutions of the perturbative equations are determined numerically after the introduction of a regular perturbation scheme analytically deduced up to the second order. Numerically, the complete linear system is also integrated. Since the results relevant to the complete problem are in very good agreement with the results of the first order analysis, the numerical solution at the second order was deemed not necessary. The use for an arbitrary initial-value problem will be shown to contain a wealth of information for the different transient behaviors associated to the symmetry, angle of obliquity and spatial decay of the long waves. The amplification factor of transversal perturbations never presents the trend - a growth followed by a long damping - usually seen in waves with wavenumber of order one or less. Asymptotical instability is always observed.Comment: accepted Physical Review E, March 201

Dispersive to non-dispersive transition and phase velocity transient for linear waves in plane wake and channel flows

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows, the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and non-dispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum beave in a dispersive manner. By contrast, if the wavenumber is above a specific threshold, a sharp dispersive to non-dispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocty can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.Comment: 19 pages, 11 figures. Text and discussion improved with respect to the first version. Accepted for publication on Physical Review

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