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

An exploratory analysis of the transient and long-term behavior of small three-dimensional perturbations in the circular cylinder wake

An initial-value problem (IVP) for arbitrary small three-dimensional vorticity perturbations imposed on a free shear flow is considered. The viscous perturbation equations are then combined in terms of the vorticity and velocity, and are solved by means of a combined Laplace–Fourier transform in the plane normal to the basic flow. The perturbations can be uniform or damped along the mean flow direction. This treatment allows for a simplification of the governing equations such that it is possible to observe long transients, which can last hundreds time scales. This result would not be possible over an acceptable lapse of time by carrying out a direct numerical integration of the linearized Navier–Stokes equations. The exploration is done with respect to physical inputs as the angle of obliquity, the symmetry of the perturbation, and the streamwise damping rate. The base flow is an intermediate section of the growing two-dimensional circular cylinder wake where the entrainment process is still active. Two Reynolds numbers of the order of the critical value for the onset of the first instability are considered. The early transient evolution offers very different scenarios for which we present a summary for particular cases. For example, for amplified perturbations, we have observed two kinds of transients, namely (1) a monotone amplification and (2) a sequence of growth–decrease–final growth. In the latter case, if the initial condition is an asymmetric oblique or longitudinal perturbation, the transient clearly shows an initial oscillatory time scale. That increases moving downstream, and is different from the asymptotic value. Two periodic temporal patterns are thus present in the system. Furthermore, the more a perturbation is longitudinally confined, the more it is amplified in time. The long-term behavior of two-dimensional disturbances shows excellent agreement with a recent two-dimensional spatio-temporal multiscale model analysis and with laboratory data concerning the frequency and wave length of the parallel vortex shedding in the cylinder wake