The influence of shear layer thickness on the stability of confined two-dimensional wakes

Abstract

The influence of confinement onto the inviscid and incompressible linear stability of the family of wakes introduced by Monkewitz [Phys. Fluids 31, 999 (1988)] is examined. The nondimensional parameters of the model, the velocity ratio Λ, defined as ratio of the velocity gap to the mean velocity, the profile shape parameter, which controls the shear layer thickness δw, and the confinement parameter, are varied and their effect onto temporal and spatiotemporal stability properties is considered. Particularly, the limit between absolute (A) and convective (C) instability is investigated as a function of the different parameters. For a given confinement, there exists an optimal value of the shear layer thickness for which the absolute instability is maximal. The absolute frequency and complex wavenumber of the mode at the C/A transition are discussed. Furthermore, the continuous profiles are approximated by means of piecewise broken-line profile, with similar spatiotemporal properties. As a typical application, a few nonparallel base flows, computed by direct numerical simulation at Re=500, are analyzed on a weakly nonparallel basis by plotting the locus of the local velocity profile in the (δw,Λ) plane. The absolute frequency at the C/A transition point is seen to predict accurately the frequency prevailing in the nonlinear direct numerical simulations. These results further help interpreting the influence of confinement on the Strouhal number measured experimentally in the wake of confined cylinders.

Counterpropagating Rossby waves in confined plane wakes

“…These diagrams are obtained by means of a Fourier inverse transform of the matrix M computed with Matlab using the command IFFT. The analyzed k-domain is 0.01 ≤ k ≤ 32 and the values of the wavenumber are equispaced with a k-step Therefore, we can conclude that the maximum of the instability at moderate confinement noticed by Juniper,4 and confirmed in his more recent analyses 5,22 and by Biancofiore & Gallaire 6 can be attributed principally to the influence of the confinement on the group velocities of the uncoupled CRWs. It remains however impossible to further attribute this phenomenon to one edge wave in particular.…”

“…In particular, we suggest that the dispersive behavior of the underlying pure counterpropagating Rossby waves, taken uncoupled, can both explain the stabilizing influence of confinement onto the temporal instability 6 and its counterintuitive spatio-temporal destabilizing influence noticed by Juniper 4 and confirmed in several further studies. 5,6,22 The outline of the paper is as follows: in Sec. II we describe the model and we obtain the dynamical system which describes the interacting counterpropagating Rossby waves, while in Sec.…”

“…This was explained as the result of an interaction of the modes of the inner and outer streams, which have a similar thickness in a moderate confinement case. This destabilizing effect was confirmed adding a finite shear layer thickness to the model, 5,6 thereby confirming the accuracy of broken-line velocity profiles for modeling the temporal and spatio-temporal instabilities of wakes. 6 The influence of the presence of confining walls on the spatio-temporal development of the varicose mode of jets and wakes was furthermore recovered by Healey,7 owing to the geometric similarity of such mode with asymmetric confined mixing layers.…”

“…The objective of the present section is to establish a connection between the CRWs perspective and the spatio-temporal analysis, particularly with the reinforcement of the absolutely unstable character of the flow at a medium confinement for the sinuous mode, noticed by Juniper 4 for the top-hat wake profile and confirmed when a finite shear layer thickness is included. 6,22 …”

“…(2), it is possible to obtain the dispersion relations 5,6 for both sinuous and varicose perturbations with the classic normal mode decomposition 23 exp [i(kx − ωt)]. These dispersion relations are algebraic equations which associate to each wavenumber k, a pair of complex conjugate angular frequencies ω.…”