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

Evolution of disturbances in stagnation point flow

The evolution of three-dimensional disturbances in an incompressible three-dimensional stagnation-point flow in an inviscid fluid is investigated. Since it is not possible to apply classical normal mode analysis to the disturbance equations for the fully three-dimensional stagnation-point flow to obtain solutions, an initial-value problem is solved instead. The evolution of the disturbances provide the necessary information to determine stability and indeed the complete transient as well. It is found that when considering the disturbance energy, the planar stagnation-point flow, which is independent of one of the transverse coordinates, represents a neutrally stable flow whereas the fully three-dimensional flow is either stable or unstable, depending on whether the flow is away from or towards the stagnation point in the transverse direction that is neglected in the planar stagnation point

Parametric perturbative study of the supercritical cross-flow boundary layer

A linear analysis of the transient evolution of small perturbations in the supercritical FSC cross-flow boundary layer is presented. We used the classical method based on the temporal evolution of individual three-dimensional travelling waves subject to near-optimal initial conditions and considered an extended portion of the parameter space. Our parametrization included the wave-number, the wave-angle, the cross-flow angle, the Hartree parameter and the Reynolds number. Special focus was given to the role played by the waveangle in inducing very steep initial transient growths in waves that proved to be stable in the long term. We found that the angular distribution of the asymptotically unstable waves and of the waves that show a transient growth depends greatly on the value of the cross flow angle and wave-angle as well as on the sign of the Hartree parameter, but depend much less on the Reynolds number. In the case of the decelerated boundary layer, at sufficiently short wavelengths, transient growths become much more rapid than the initial growth of the unstable waves. In all cases of transient growth, pressure perturbations at the wall are not synchronous with the kinetic energy of the perturbation. We present a comparison with the sub-critical results obtained by Breuer and Kuraishi (1994) for the same full range of the obliquity angle here considered